**Preface to "Mathematical Logic and Its Applications 2020"**

This Special Issue contains articles representing three directions: Descriptive set theory (DTM), exact polynomial complexity algorithms (EPA), and applications of mathematical logic and algorithm theory (Appl). We will say a few words about each of the directions.

In accordance with the classical description of Nicolas Luzin, DTM considers simple properties of simple sets of real numbers **R**. "Simple" sets are Borel sets (the smallest family containing open and closed sets in **R**<sup>n</sup> and closed with respect to the operations of countable union and countable intersection) and projective sets (the smallest family containing Borel sets and closed with respect to the operations of projecting from **R**<sup>n</sup> to **R**<sup>m</sup>, m<n, and the complement to the whole space). The question of what is a "simple" property is more complicated, but it is not important, since in fact we study a small list of individual properties, including the Lebesgue measurability, Baire property<sup>1</sup>, and the individual definability of a set, function, or real. The latest means that there is a formula that holds for a given real number and for no others. This depends on the class of formulas allowed. Such a natural class consists of formulas of the form ∀x<sup>1</sup> ∃y<sup>1</sup> ∀x<sup>2</sup> ∃y<sup>2</sup> ... ∀x<sup>n</sup> ∃y<sup>n</sup> ψ(x1, y1,...,xn, yn, x), where the variables x1, y1,...,xn, yn, x run through the whole **R**, and the elementary part ψ(x1, y1,...,xn, yn, x) is any arithmetic formula (which contains any quantifiers over the natural numbers, as well as equalities and inequalities that connect the superpositions of operations from the semiring of natural numbers). To date, the development of DTM leads to a non-trivial general cultural conclusion: every real number is *definable* (using countable ordinals<sup>2</sup>) or *random*; in the latter case it does not possess any non-trivial properties. This implies that there are absolutely undecidable statements<sup>3</sup>; as well as surprising connections between seemingly very different absolutely undecidable ones. For example, the measurability implies the Baire property for a wide class of sets. The first three articles belong to this direction. In particular, they solve the well-known problem (1948) of A. Tarski on the definability of the notion of definability itself, and prove the statement (1975) of H. Friedman.

The EPA section contains an article contributing a solution for the meaningful combinatorial and, at first glance, complicated algorithmic problem of optimization of the functional given on paths of passing from one graph to another. It is solved by an algorithm of linear complexity, being at the same time exact. The latter means that for any input data, that is for any ordered pair of graphs A and B, accompanied by costs of elementary graph transformations, the algorithm produces exactly the minimal value of the above functional (i.e., the minimum distance between A and B and the minimum path itself from A to B).

Here the complexity of the problem turned into the logical complexity of this, albeit linear, algorithm. Our goal was to draw attention to the search for, and possible discussion of, algorithmic problems that seem to require exhaustive search but are actually solved by exact algorithms of low polynomial complexity. This ensures their practical significance when working with large data (terabyte and larger sizes).

The Appl section contains two articles. First of them is devoted to the application of non-standard analysis (and other logical methods) to the problems of isomorphism in algebra and mathematical physics (the Jacobian and M. Kontsevich's conjectures, and algorithmic undecidability). The second is devoted to the application of logical and algorithmic approaches to the problem of theoretical medicine — a quantitative description of the balance and the adaptive resource of a human that determines his resistance to external influences. Applied problems in which logic and theory of algorithms have shown their usefulness could be of interest.

The Editorial Board of *Mathematics* (WoS: Q1) has announced the preparation of the issue "Mathematical Logic and Its Applications 2021"; contributions in these directions and especially in other ones of this huge mathematical area, including various applications, are invited.


**Vassily Lyubetsky, Vladimir Kanovei** *Editors*

## *Article* **Models of Set Theory in which Nonconstructible Reals First Appear at a Given Projective Level**

**Vladimir Kanovei \*,† and Vassily Lyubetsky †**

Institute for Information Transmission Problems of the Russian Academy of Sciences, 127051 Moscow, Russia; lyubetsk@iitp.ru

**\*** Correspondence: kanovei@iitp.ru

† These authors contributed equally to this work.

Received: 11 May 2020; Accepted: 27 May 2020; Published: 3 June 2020

**Abstract:** Models of set theory are defined, in which nonconstructible reals first appear on a given level of the projective hierarchy. Our main results are as follows. Suppose that *n* ≥ 2. Then: 1. If it holds in the constructible universe **<sup>L</sup>** that *<sup>a</sup>* <sup>⊆</sup> *<sup>ω</sup>* and *<sup>a</sup>* <sup>∈</sup>/ *<sup>Σ</sup>*<sup>1</sup> *<sup>n</sup>* <sup>∪</sup> *<sup>Π</sup>*<sup>1</sup> *<sup>n</sup>* , then there is a generic extension of **<sup>L</sup>** in which *<sup>a</sup>* <sup>∈</sup> <sup>Δ</sup><sup>1</sup> *<sup>n</sup>*+<sup>1</sup> but still *<sup>a</sup>* <sup>∈</sup>/ *<sup>Σ</sup>*<sup>1</sup> *<sup>n</sup>* <sup>∪</sup> *<sup>Π</sup>*<sup>1</sup> *<sup>n</sup>* , and moreover, any set *<sup>x</sup>* <sup>⊆</sup> *<sup>ω</sup>*, *<sup>x</sup>* <sup>∈</sup> *<sup>Σ</sup>*<sup>1</sup> *<sup>n</sup>* , is constructible and *Σ*<sup>1</sup> *<sup>n</sup>* in **L**. 2. There exists a generic extension **L** in which it is true that there is a nonconstructible Δ1 *<sup>n</sup>*+<sup>1</sup> set *<sup>a</sup>* <sup>⊆</sup> *<sup>ω</sup>*, but all *<sup>Σ</sup>*<sup>1</sup> *<sup>n</sup>* sets *<sup>x</sup>* <sup>⊆</sup> *<sup>ω</sup>* are constructible and even *<sup>Σ</sup>*<sup>1</sup> *<sup>n</sup>* in **L**, and in addition, **V** = **L**[*a*] in the extension. 3. There exists an generic extension of **L** in which there is a nonconstructible *Σ*1 *<sup>n</sup>*+<sup>1</sup> set *<sup>a</sup>* <sup>⊆</sup> *<sup>ω</sup>*, but all <sup>Δ</sup><sup>1</sup> *<sup>n</sup>*+<sup>1</sup> sets *<sup>x</sup>* <sup>⊆</sup> *<sup>ω</sup>* are constructible and <sup>Δ</sup><sup>1</sup> *<sup>n</sup>*+<sup>1</sup> in **L**. Thus, nonconstructible reals (here subsets of *ω*) can first appear at a given lightface projective class strictly higher than *Σ*1 <sup>2</sup> , in an appropriate generic extension of **<sup>L</sup>**. The lower limit *<sup>Σ</sup>*<sup>1</sup> <sup>2</sup> is motivated by the Shoenfield absoluteness theorem, which implies that all *Σ*<sup>1</sup> <sup>2</sup> sets *a* ⊆ *ω* are constructible. Our methods are based on almost-disjoint forcing. We add a sufficient number of generic reals to **L**, which are very similar at a given projective level *n* but discernible at the next level *n* + 1.

**Keywords:** definability; nonconstructible reals; projective hierarchy; generic models; almost disjoint forcing

**MSC:** 03E15; 03E35

#### **1. Introduction**

Problems of definability and effective construction of mathematical objects have always been in the focus of attention during the development of mathematical foundations. In particular, Hadamard, Borel, Baire, and Lebesgue, participants of the discussion published in [1], in spite of significant differences in their positions regarding problems of mathematical foundations, emphasized that a pure existence proof and a direct definition (or an effective construction) of a mathematical object required are different mathematical results, and the second one of them does not follow from the first. Problems of definability and effectivity are considered in such contemporary monographs on foundations as [2–5]. Moschovakis, one of founders of modern set theory, pointed in [6] (p. xiv), that

the central problem of descriptive set theory and definability theory in general [is] to find and study the characteristic properties of definable objects.

The general goal of the research line of this paper is to explore the existence of effectively definable structures in descriptive set theory on specific levels of the projective hierarchy. One of the directions here is the construction of set theoretic models, in which this or another problem is decided, at a predefined projective level *n*, differently than it is decided in **L**, Gödel's constructible universe, or, that is equivalent, by adding the axiom of constructibility, dubbed **V** = **L**.

Such set theoretic models are usually defined as generic extensions of **L** itself. Any such a generic extension leads to consistency and independence results in set theory, because if a sentence Φ holds in **L** or in a generic extension of **L** then Φ is consistent with the axioms of **ZFC**, the Zermelo–Fraenkel set theory (with the axiom of choice **AC**).

As a first, and perhaps most immediately interesting problem of this sort, in this paper, we consider the problem of the existence of effectively definable (that is, occurring in one of lightface classes *Σ*<sup>1</sup> *n* of the projective hierarchy) but nonconstructible reals. It follows from Shoenfield's absoluteness theorem [7] that every (lightface) *Σ*<sup>1</sup> <sup>2</sup> set *x* ⊆ *ω* belongs to **L**. Generic models, in which there exist nonconstructible reals on effective levels of the projective hierarchy higher than *Σ*<sup>1</sup> <sup>2</sup> , were defined in the early years of forcing; see a brief account in [8]. This culminated in two different generic extensions [9,10] containing a nonconstructible *Π*<sup>1</sup> <sup>2</sup> singleton, hence, a <sup>Δ</sup><sup>1</sup> <sup>3</sup> set *a* ⊆ *ω*. (We are concentrated on generic extensions of **L** in this paper, and therefore leave aside another research line, related to models with large cardinals, with many deep and fruitful results connected, in particular, with properties of *Π*<sup>1</sup> <sup>2</sup> singletons, see e.g., [11–13]).

Then it was established in [14] that for any *n* ≥ 2 there is a generic extension of **L** in which there exists a nonconstructible Δ<sup>1</sup> *<sup>n</sup>*+<sup>1</sup> real *<sup>a</sup>* <sup>⊆</sup> *<sup>ω</sup>*, but all *<sup>Σ</sup>*<sup>1</sup> *<sup>n</sup>* sets *x* ⊆ *ω* are constructible. Our motivation here is to further extend this research line. The next three theorems are the main results in this paper.

**Theorem 1.** *If* **<sup>n</sup>** <sup>≥</sup> <sup>2</sup> *and <sup>b</sup>* <sup>⊆</sup> *<sup>ω</sup>, b* <sup>∈</sup>/ *<sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>** <sup>∪</sup> *<sup>Π</sup>*<sup>1</sup> **<sup>n</sup>***, then there is a generic extension of* **<sup>L</sup>** *in which <sup>b</sup>* <sup>∈</sup> <sup>Δ</sup><sup>1</sup> **n**+1 *but still b* <sup>∈</sup>/ *<sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>** <sup>∪</sup> *<sup>Π</sup>*<sup>1</sup> **<sup>n</sup>***, and moreover, any set x* <sup>⊆</sup> *<sup>ω</sup>, x* <sup>∈</sup> *<sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>***, is constructible and Σ*<sup>1</sup> **<sup>n</sup>** *in* **L***.*

Theorem 1 shows that being at a certain lightface projective level is hardly an intrinsic property of a constructible real, unless it is already at that level in **L**. The theorem definitely fails for **n** = 1 since being Δ<sup>1</sup> <sup>2</sup> is an ablosute property of a real by the Shoenfield absoluteness theorem.

**Theorem 2.** *If* **n** ≥ 2*, then there exists a generic extension of the universe* **L** *in which it is true that*


**Theorem 3.** *If* **<sup>n</sup>** <sup>≥</sup> <sup>2</sup> *then there exists an extension of* **<sup>L</sup>** *in which there is a nonconstructible <sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> *set a* ⊆ *ω but all* Δ<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> *sets x* <sup>⊆</sup> *<sup>ω</sup> are constructible and* <sup>Δ</sup><sup>1</sup> **<sup>n</sup>**+<sup>1</sup> *in* **L***.*

The common denominator of Theorems 2 and 3 is that nonconstructible reals can first appear at a given lightface projective class strictly higher than *Σ*<sup>1</sup> <sup>2</sup> , in an appropriate generic extension of **L**. The lower limit *Σ*<sup>1</sup> <sup>2</sup> is motivated by the Shoenfield absoluteness theorem.

The generic models, which we define to prove the main theorems, make use of modifications of the almost-disjoint forcing by Jensen–Solovay [9].

Some other recent results can be mentioned here, which resemble Theorems 1–3 in that they give models in which a particular property of some kind holds at a certain pre-selected level of the projective hierarchy. Yet they are different in that they use modifications of Jensen's minimal *Π*<sup>1</sup> 2 singleton forcing [10] and its finite-support products first considered by Enayat [15], as well as its collapse-style modification by Abraham [16], rather than the almost-disjoint forcing.


#### *Organization of the Paper*

Our plan of the proofs of the main results will be to construct, in **L**, a sequence of forcing notions **P**(*ν*), *ν* < *ω*<sup>1</sup> , satisfying the following three key conditions.


Each **<sup>P</sup>**(*ν*) will be a forcing notion of almost-disjoint type, determined by a set **<sup>U</sup>**(*ν*) <sup>⊆</sup> *<sup>ω</sup><sup>ω</sup>* . To make the exposition self-contained, we review some basic details related to almost-disjoint forcing, finite-support products, and related generic extensions, taken mainly from [9], in Sections 2 and 3.

Having the construction of **P**(*ν*), *ν* < *ω*<sup>1</sup> , accomplished in Section 4, the proof of, e.g., Theorem 1 (Section 7.1) is performed as follows. Let *b* ∈ **L**, *b* ⊆ *ω* be chosen as in Theorem 1 for a given **n** ≥ 2. We consider a **P**-generic extension **L**[*G*] of **L**, where **P** = ∏*i*<*<sup>ω</sup>* **P**(*i*). Let *ai* ⊆ *ω* be the **P**(*i*)-generic real generated by the *i*th projection *Gi* of *G*; these reals are nonconstructible and **L**[*G*] = **L**[{*ai*}*i*<*ω*]. Let *<sup>z</sup>* = {0}∪{2*<sup>k</sup>* : *<sup>k</sup>* ∈ *<sup>b</sup>*}∪{2*<sup>k</sup>* + 1 : *<sup>k</sup>* ∈/ *<sup>b</sup>*} Consider the subextension **<sup>L</sup>**[{*ai*}*i*∈*z*]. Then it is true in **<sup>L</sup>**[{*ai*}*i*∈*z*] by condition 1, that

> *b* = {*k* < *ω* : there exist **P**(2*k*)-generic reals} = {*k* < *ω* : there are no **P**(2*k* + 1)-generic reals} ,

so using condition 2, we easily get *<sup>b</sup>* <sup>∈</sup> <sup>Δ</sup><sup>1</sup> **<sup>n</sup>**+<sup>1</sup> in **<sup>L</sup>**[{*ai*}*i*∈*z*]. A similar construction (but with *<sup>b</sup>* being generic over **L**) was carried out in the early years of forcing in [9] for **n** = 2, which is the least possible value. In the case **n** = 2, the fact, that all *Σ*<sup>1</sup> <sup>2</sup> sets *<sup>x</sup>* <sup>⊆</sup> *<sup>ω</sup>* in the extension belong to **<sup>L</sup>** and are *<sup>Σ</sup>*<sup>1</sup> <sup>2</sup> in **L**, is guaranteed by the Shoenfield absoluteness theorem.

If **n** ≥ 3, then the Shoenfield absoluteness argument does not work, of course. Still we can argue that any lightface *Σ*<sup>1</sup> **<sup>n</sup>** set *<sup>x</sup>* ⊆ *<sup>ω</sup>* in **<sup>L</sup>**[{*ai*}*i*∈*z*] belongs to **<sup>L</sup>** by the general forcing theory, because the product forcing **<sup>P</sup>***<sup>z</sup>* = <sup>∏</sup>*i*∈*<sup>z</sup>* **<sup>P</sup>**(*i*) ∈ **<sup>L</sup>** is homogeneous by condition 1. However this does not immediately imply the lightface definability of *b* in **L**, as **P***<sup>z</sup>* is defined via *z*, hence via *b*. To solve this difficulty, we make use of condition 3 to prove another absoluteness property: *Σ*<sup>1</sup> **<sup>n</sup>** formulas turn out to be absolute between **<sup>L</sup>**[{*ai*}*i*∈*z*] and the entire extension **<sup>L</sup>**[*G*] = **<sup>L</sup>**[{*ai*}*i*<*ω*], which is an **P**-generic extension of **L**. Here **P** = ∏*i*<*<sup>ω</sup>* **P**(*i*) is a parameter-free definable forcing in **L**, leading to the parameter-free definability of *b* in **L**. There are two issues here that need to be explained.

First, how to secure condition 3 in a sufficiently effective form. To explain the main technical device, we recall that by [9] the system of forcing notions **P**(*ν*) is the result of certain transfinite *ω*<sup>1</sup> -long construction of assembling it from countable fragments in **L**. The construction can be viewed as a maximal branch in a certain "mega-tree", say T , whose nodes are such countable fragments, and each of them is chosen to be the Gödel-least appropriate one over the previous one. The complexity of this construction is Δ<sup>1</sup> <sup>2</sup> in the codes, leading in [9] to the *<sup>Π</sup>*<sup>1</sup> <sup>2</sup> definability of the property of being generic, as in condition 2, in case **n** = 2.

To adapt this construction for the case **n** ≥ 3, our method requires us to define a maximal branch in <sup>T</sup> that intersects all dense sets in <sup>T</sup> of class *<sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>**−<sup>1</sup> . Such a construction is carried out in Section 4. This genericity-like condition of meeting all dense *Σ*<sup>1</sup> **<sup>n</sup>**−<sup>1</sup> sets, results in the *<sup>Π</sup>*<sup>1</sup> **<sup>n</sup>** definability of the property of being generic in condition 2, and also yields condition 3, since the abundance of order automorphisms of the "mega-tree" T (including those related to index permutations) allows to establish some homogeneity properties of a certain auxiliary forcing-style relation.

This auxiliary forcing-style relation, defined and studied in Sections 5 and 6. The auxiliary relation approximates the truth in **<sup>P</sup>** -generic extensions, as **<sup>L</sup>**[{*ai*}*i*∈*z*] above, up to *<sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>** formulas, but, unlike the ordinary **P** -forcing relation, is sufficiently homogeneous. In particular, it helps to obtain the mentioned absoluteness property. This will allow us to accomplish the proof of the main results, Theorem 1 together with part (i) of Theorem 2 in Section 7, part (ii) of Theorem 2 in Section 8, Theorem 3 in Section 9. The flowchart can be seen in Figure 1.

The flowchart can be seen in Figure 1. And we added the index and contents as Supplementary Materials for easy reading.

**Figure 1.** Flowchart.

*General Set-Theoretic Notation Used in This Paper*


*Mathematics* **2020**, *8*, 910


#### **2. Almost Disjoint Forcing**

In this section, we review basic definitions and results related to almost disjoint forcing, as well as some rarely used results related, for instance, to symmetries of almost disjoint forcing notions.

**Assumption 1.** *In this paper, we assume that* **L** *is the ground universe. Thus all forcing notions are defined in* **L** *while all generic extensions are those of* **L***. (In fact many intermediate results remain true w. r. t. any ground universe.)*

#### *2.1. Almost Disjoint Forcing*

We present this forcing in a form based on the fact that the set **Fun** of all functions *f* : *ω* → *ω* is almost disjoint in the sense that if *f* = *g* belong to **Fun** then the infinite sets { *f m* : *m* ∈ *ω*} and {*gm* : *m* ∈ *ω*} of finite strings have a finite intersection.

**Definition 1. Seq** <sup>=</sup> *<sup>ω</sup>*<*<sup>ω</sup>* {Λ} *= all finite non-empty strings of natural numbers. A recursive enumeration <sup>ω</sup>*<*<sup>ω</sup>* <sup>=</sup> {**s***<sup>k</sup>* : *<sup>k</sup>* <sup>∈</sup> *<sup>ω</sup>*} *is fixed, such that* **<sup>s</sup>**<sup>0</sup> <sup>=</sup> <sup>Λ</sup>*, the empty string, and* **<sup>s</sup>***<sup>k</sup>* <sup>⊆</sup> **<sup>s</sup>** <sup>=</sup><sup>⇒</sup> *<sup>k</sup> . Thus* **Seq** <sup>=</sup> *<sup>ω</sup>*<*<sup>ω</sup>* {Λ} <sup>=</sup> {**s***<sup>k</sup>* : *<sup>k</sup>* <sup>≥</sup> <sup>1</sup>}*. For any s* <sup>=</sup> **<sup>s</sup>***<sup>k</sup> , we let* num *<sup>s</sup>* <sup>=</sup> *k; in particular* num <sup>Λ</sup> <sup>=</sup> <sup>0</sup>*.*

**Fun** <sup>=</sup> *<sup>ω</sup><sup>ω</sup> = all infinite sequences of natural numbers. A set <sup>X</sup>* <sup>⊆</sup> **Fun** *is dense iff for any <sup>s</sup>* <sup>∈</sup> **Seq** *there is f* ∈ *X such that s* ⊂ *f .*

*Let S* ⊆ **Seq***, f* ∈ **Fun***. If the set S*/ *f* = {*n* : *f n* ∈ *S*} *is infinite then we say that S covers f , otherwise S does not cover f .*

We underline that Λ, the empty string, does not belong to **Seq**.

Given a set *u* ⊆ **Fun** in the ground universe, the general goal of almost disjoint forcing is to find a generic set *S* ⊆ **Seq** such that the equivalence

$$f \in \mathfrak{u} \iff \text{S} \text{ does not cover } f \tag{1}$$

holds for each *f* ∈ **Fun** in the ground universe. This goal will be achieved by a forcing *P*[*u*] introduced in Definition 4. In fact *P*[*u*] will be a part, determined by *u*, of a common reservoir *P*∗ .

**Definition 2.** *P*<sup>∗</sup> *is the set of all pairs p* = *Sp* ; *Fp of finite sets Fp* ⊆ **Fun***, Sp* ⊆ **Seq***. Elements of P*<sup>∗</sup> *will sometimes be called (forcing) conditions. If p* ∈ *P*<sup>∗</sup> *then put F*<sup>∨</sup> *<sup>p</sup>* = { *f n* : *f* ∈ *Fp* ∧ *n* ≥ 1}*. The set F*<sup>∨</sup> *<sup>p</sup> is an infinite (or else F*∨ *<sup>p</sup>* = *Fp* = ∅*) tree in* **Seq***, without terminal nodes.*

**Definition 3** (order)**.** *Let p*, *q* ∈ *P*<sup>∗</sup> *. We define q* ≤ *p (q is stronger) iff Sp* ⊆ *Sq* , *Fp* ⊆ *Fq , and the difference Sq Sp does not intersect F*∨ *<sup>p</sup> , that is, Sq* ∩ *F*<sup>∨</sup> *<sup>p</sup>* = *Sp* ∩ *F*<sup>∨</sup> *p .*

Thus any condition *p* ∈ *P*<sup>∗</sup> is a pair that consists of a "finite" component *Sp* and an "infinite" component *Fp* . Either of the components is a finite set (possibly, empty), but *Sp* consists of finite strings of integers while *Fp* consists of infinite sequences of integers that will be called functions (from *ω* to *ω*). Both components of a stronger condition *q*, naturally, increase, but strings *t* ∈ *Sq Sp* must satisfy *t* ∈/ *F*<sup>∨</sup> *<sup>p</sup>* —in other words, *t* is not a substring of any function (infinite sequence) *f* ∈ *Fp* .

If *p* ∈ *P*<sup>∗</sup> then both ∅; *Fp* and *Sp* ; ∅ belong to *P*<sup>∗</sup> and *p* ≤ *Sp* ; ∅, but *p* ≤ ∅; *Fp* may fail. In fact *p* ≤ ∅; *Fp* iff *Sp* ∩ *F*<sup>∨</sup> *<sup>p</sup>* = ∅.

**Lemma 1.** *Conditions p*, *q* ∈ *P*<sup>∗</sup> *are compatible in P*<sup>∗</sup> *iff* 1) *Sq Sp does not intersect F*<sup>∨</sup> *<sup>p</sup> , and* 2) *Sp Sq does not intersect F*∨ *<sup>q</sup> . Therefore, any p*, *q* ∈ *P*<sup>∗</sup> *are compatible in P*<sup>∗</sup> *iff p* ∧ *q* ≤ *p and p* ∧ *q* ≤ *q.*

**Proof.** The pair *p* ∧ *q* = *Sp* ∪ *Sq* ; *Fp* ∪ *Fq* is a condition in *P*<sup>∗</sup> . Moreover if 1) and 2) hold then we have *p* ∧ *q* ≤ *p* and *p* ∧ *q* ≤ *q*, thus *p*, *q* are compatible.

Now let us introduce a relativized version of *P*∗ . The parameter of relativization will be an arbitrary set *u* ⊆ **Fun** served as a reservoir of functions allowed to occur in sets *Fp* .

**Definition 4.** *If u* ⊆ **Fun** *then put P*[*u*] = {*p* ∈ *P*<sup>∗</sup> : *Fp* ⊆ *u*}*.*

Note that if *p*, *q* ∈ *P*[*u*] then *p* ∧ *q* ∈ *P*[*u*]. Thus in this case if conditions *p*, *q* are compatible in *P*<sup>∗</sup> then they are compatible in *P*[*u*], too. Therefore, we will say that conditions *p*, *q* ∈ *P*<sup>∗</sup> are compatible (or incompatible) without an indication which set *P*[*u*] containing both conditions is considered.

**Lemma 2.** *If u* ⊆ **Fun** *then P*[*u*] *is a ccc forcing.*

**Proof.** If *Sp* = *Sq* then *p* and *q* are compatible by Lemma 1. However there are only countably many sets of the form *Sp* .

#### *2.2. Almost-Disjoint Generic Extensions*

Fix, in **L**, a set *u* ⊆ **Fun** and consider a *P*[*u*]-generic extension **L**[*G*] of the ground (constructible by Assumption 1) set universe **<sup>L</sup>**, obtained by adjoining a *<sup>P</sup>*[*u*]-generic set *<sup>G</sup>* ⊆ *<sup>P</sup>*[*u*]. Put *SG* = - *<sup>p</sup>*∈*<sup>G</sup> Sp* ; thus *SG* ⊆ **Seq**. The next lemma reflects the idea of almost-disjoint forcing: elements of *u* are distinguished by the property of *SG* not covering *f* in the sense of Definition 1.

**Lemma 3.** *Suppose that u* ⊆ **Fun** *in the universe* **L***, and G* ⊆ *P*[*u*] *is a set P*[*u*]*-generic over* **L***. Then*

(i) *G belongs to* **L**[*SG*] ;

(ii) *if f* ∈ **Fun** ∩ **L** *then f* ∈ *u iff SG does not cover f* ;

(iii) *if p* ∈ *P*[*u*] *then p* ∈ *G iff sp* ⊆ *SG* ∧ (*SG sp*) ∩ (*F*<sup>∨</sup> *<sup>p</sup>* ∪ *S*<sup>∨</sup> *<sup>p</sup>* ) = ∅*.*

**Proof.** (ii) Let *f* ∈ *u*. The set *Df* = {*p* ∈ *P*[*u*] : *f* ∈ *Fp* } is dense in *P*[*u*]. (Let *q* ∈ *P*[*u*]. Define *p* ∈ *P*[*u*] so that *Sp* = *Sq* and *Fp* = *Fq* ∪ { *f* }. Then *p* ∈ *Df* and *p q*.) Therefore *Df* ∩ *G* = ∅. Pick any *p* ∈ *Df* ∩ *G*. Then *f* ∈ *Fp* . Now every *r* ∈ *G* is compatible with *p*, and hence *Sr*/ *f* ⊆ *Sp*/ *f* by Lemma 1. Thus *SG*/ *f* = *Sp*/ *f* is finite.

Let *<sup>f</sup>* <sup>∈</sup>/ *<sup>u</sup>*. The sets *Df l* <sup>=</sup> {*<sup>p</sup>* <sup>∈</sup> *<sup>P</sup>*[*u*] : sup(*Sp*/ *<sup>f</sup>*) <sup>&</sup>gt; *<sup>l</sup>*} are dense in *<sup>P</sup>*[*u*]. (If *<sup>q</sup>* <sup>∈</sup> *<sup>P</sup>*[*u*] then *Fq* is finite. As *f* ∈/ *u*, there is *m* > *l* with *f m* ∈/ *F*<sup>∨</sup> *<sup>q</sup>* . Define *p* so that *Fp* = *Fq* and *Sp* = *Sq* ∪ { *f m*}. Then *<sup>p</sup>* <sup>∈</sup> *Df l* and *<sup>p</sup> <sup>q</sup>*.) Let *<sup>p</sup>* <sup>∈</sup> *Df l* <sup>∩</sup> *<sup>G</sup>*. Then sup(*SG*/ *<sup>f</sup>*) <sup>&</sup>gt; *<sup>l</sup>*. As *<sup>l</sup>* is arbitrary, *SG*/ *<sup>f</sup>* is infinite.

(iii) Consider any *p* ∈ *P*[*u*]. Suppose that *p* ∈ *G*. Then obviously *sp* ⊆ *SG* . If there exists *s* ∈ (*SG Sp*) ∩ *F*<sup>∨</sup> *<sup>p</sup>* then by definition we have *s* ∈ *Sq* for some *q* ∈ *G*. However, then *p*, *q* are incompatible by Lemma 1, a contradiction.

Now suppose that *p* ∈/ *G*. Then there exists *q* ∈ *G* incompatible with *p*. By Lemma 1, there are two cases. First, there exists *s* ∈ (*Sq Sp*) ∩ *F*<sup>∨</sup> *<sup>p</sup>* . Then *s* ∈ *SG Sp* , so *p* is not compatible with *SG* . Second, there exists *s* ∈ (*Sp Sq*) ∩ *F*<sup>∨</sup> *<sup>q</sup>* . Then any condition *r* ≤ *q* satisfies *s* ∈/ *Sr* . Therefore *s* ∈/ *SG* , so *Sp* ⊆ *SG* , and *p* is not compatible with *SG* .

$$\text{(i)}\ G = \{ p \in P[\mathfrak{u}] : s\_p \subseteq S\_G \land (S\_G > s\_p) \cap F\_p^\vee = \mathcal{Q} \}\text{ by (iii)}.\ \ \ \Box$$

#### *2.3. Lipschitz Transformations*

Let **Lip** be the group of all ⊆-automorphisms of **Seq**; these transformations may be called Lipschitz by obvious association. Any *<sup>λ</sup>* <sup>∈</sup> **Lip** preserves the length lh of finite strings, that is, lh *<sup>s</sup>* <sup>=</sup> lh (*λ·s*) for all *<sup>s</sup>* <sup>∈</sup> **Seq**. Define the action of any transformation *<sup>λ</sup>* <sup>∈</sup> **Lip** on:


– conditions *p* ∈ *P*<sup>∗</sup> , by: *λ· p* = *λ·Sp* ; *λ· Fp*.

**Lemma 4** (routine)**.** *The action of any λ* ∈ **Lip** *is an order-preserving automorphism of P*∗. *If u* ⊆ **Fun** *and p* ∈ *P*[*u*] *then λ· p* ∈ *P*[*λ·u*]*.*

**Lemma 5.** *Suppose that u*, *v* ⊆ **Fun** *are countable sets topologically dense in* **Fun***, and p* ∈ *P*[*u*]*, q* ∈ *P*[*v*]*. Then there is λ* ∈ **Lip** *and conditions p* ∈ *P*[*u*]*, p* ≤ *p and q* ∈ *P*[*v*]*, q* ≤ *q, such that λ·u* = *v, and λ· p* = *q — therefore conditions λ· p and q are compatible in P*[*v*]*.*

**Proof.** Put bas *<sup>r</sup>* <sup>=</sup> {*s*(0) : *<sup>s</sup>* <sup>∈</sup> *Sr*}∪{ *<sup>f</sup>*(0) : *<sup>f</sup>* <sup>∈</sup> *Fr*} for any *<sup>r</sup>* <sup>∈</sup> *<sup>P</sup>*<sup>∗</sup> ; bas *<sup>r</sup>* <sup>⊆</sup> *<sup>ω</sup>* is finite. Let *<sup>M</sup>* <sup>&</sup>lt; *<sup>ω</sup>* satisfy bas *<sup>p</sup>* <sup>∪</sup> bas *<sup>q</sup>* <sup>⊆</sup> *<sup>M</sup>*. Because of density, for any *<sup>i</sup>* <sup>&</sup>lt; *<sup>M</sup>* there exist *fi* <sup>∈</sup> *<sup>u</sup>* and *gi* <sup>∈</sup> *<sup>u</sup>* such that *fi*(0) = *i* and *gi*(0) = *M* + *i*.

For any *f* = *g* ∈ **Fun**, let <sup>N</sup>(*f* , *g*) be the largest *n* with *f n* = *g n*.

We will define enumerations *u* = { *fk* : *k* < *ω*} and *u* = {*gk* : *k* < *ω*}, without repetitions, which agree with the above definition for *k* < *M* and satisfy <sup>N</sup>(*fk*, *fl*) = <sup>N</sup>(*gk*, *gl*) for all *k*, *l*, and *gk*(0) = *fk*(0) for all *k* ≥ *M*. As soon as this is accomplished, define *λ* ∈ **Lip** as follows. Consider any *<sup>s</sup>* <sup>∈</sup> **Seq** of length *<sup>m</sup>* <sup>=</sup> lh *<sup>s</sup>*. As *<sup>u</sup>* is dense, *<sup>s</sup>* <sup>=</sup> *fk <sup>m</sup>* for some *<sup>k</sup>*. Put *<sup>λ</sup>*(*s*) = *gk m*. Clearly *<sup>λ</sup>·<sup>u</sup>* <sup>=</sup> *<sup>u</sup>* , and in particular *λ· fk* = *gk* for all *k*, and hence

$$
\lambda(\*) \text{ if } k < M \text{ then } \lambda(\langle k \rangle) = \langle M+k \rangle \text{ and } \lambda(\langle M+k \rangle) = \langle k \rangle \text{, but if } k \ge 2M \text{ then } \lambda(\langle k \rangle) = \langle k \rangle \text{.}
$$

Now to define *<sup>q</sup>* put *<sup>r</sup>* <sup>=</sup> *<sup>λ</sup>· <sup>p</sup>*. Then *<sup>r</sup>* <sup>∈</sup> *<sup>P</sup>*[*v*], and bas *<sup>r</sup>* <sup>=</sup> *<sup>β</sup>·* bas *<sup>p</sup>* <sup>⊆</sup> *<sup>ω</sup> <sup>M</sup>* by (∗), since bas *<sup>p</sup>* <sup>⊆</sup> *<sup>M</sup>*. Therefore, bas *<sup>r</sup>* <sup>∩</sup> bas *<sup>q</sup>* <sup>=</sup> <sup>∅</sup> because bas *<sup>q</sup>* <sup>⊆</sup> *<sup>M</sup>* as well. It follows that conditions *<sup>r</sup>* and *q* are compatible in *P*[*v*], and hence condition *q* = *r* ∧ *q* (that is, *Sq* = *Sr* ∪ *Sq* and *Xq* = *Xr* ∪ *Xq* ) belongs to *<sup>P</sup>*[*v*], and obviously *<sup>q</sup>* <sup>≤</sup> *<sup>q</sup>*. Pretty similarly, to define *<sup>q</sup>*, we put *<sup>r</sup>* <sup>=</sup> *<sup>λ</sup>*−<sup>1</sup> *· <sup>q</sup>* <sup>∈</sup> *<sup>P</sup>*[*u*], thus bas *<sup>r</sup>* <sup>⊆</sup> *<sup>ω</sup> <sup>M</sup>*, bas *<sup>r</sup>* <sup>∩</sup> bas *<sup>p</sup>* <sup>=</sup> <sup>∅</sup>, conditions *<sup>r</sup>*, *<sup>p</sup>* are compatible, condition *<sup>p</sup>* <sup>=</sup> *<sup>p</sup>* <sup>∧</sup> *<sup>r</sup>* (that is, *Sp* = *Sp* ∪ *Sr* and *Xp* = *Xp* ∪ *Xr* ) belongs to *P*[*u*], and *p* ≤ *p*. Note that *q* = *λ·r* and *r* = *λ· p* by construction. It follows that *q* = *r* ∧ *q* = *λ·*(*p* ∧ *r*) = *λ· p* , as required.

To define *fk* and *gk* by induction, suppose that *k* ≥ *M*, *f*0, ... , *fk*−<sup>1</sup> and *g*0, ... , *gk*−<sup>1</sup> are defined, and <sup>N</sup>(*fi*, *fj*) = <sup>N</sup>(*gi*, *gj*) holds in this domain. Consider any next function *<sup>f</sup>* ∈ *<sup>u</sup>* { *<sup>f</sup>*0, ... , *fk*−<sup>1</sup> }, and let it be *fk* . There are functions *g* ∈ **Fun** satisfying <sup>N</sup>(*fj*, *fk*) = <sup>N</sup>(*gj*, *g*) for all *j* < *k*. This property of *g* is determined by a certain finite part *gm*. By the density the set *v* contains a function *g* of this type. Let *gk* be any of them. In the special case when <sup>N</sup>(*fj*, *fk*) = 0 for all *j* < *k* (then *k* ≥ 2*M*), we take any *gk* ∈ *v* satisfying <sup>N</sup>(*fj*, *fk*) = 0 for all *j* < *k* and *gk*(0) = *fk*(0).

#### *2.4. Substitution Transformations*

The next lemma (Lemma 6) will help to prove that the forcing notions considered are sufficiently homogeneous. Assume that *p*, *q* ∈ *P*<sup>∗</sup> satisfy the following requirement:

$$F\_p = F\_q \quad \text{and} \quad S\_p \cup S\_q \subseteq F\_p^\vee = F\_q^\vee. \tag{2}$$

We define a transformation *H<sup>p</sup> <sup>q</sup>* acting as follows. Let *p* ∈ *P*<sup>∗</sup> , *p* ≤ *p*. Then by definition *Sp* ⊆ *Sp* , *Fp* ⊆ *Fp* , and *Sp* ∩ *F*<sup>∨</sup> *<sup>p</sup>* <sup>=</sup> *Sp* (by (2)). We put *<sup>H</sup><sup>p</sup> <sup>q</sup>* (*p* ) = *q* := *Sq* , *Fq* , where *Fq* = *Fp* and *Sq* = (*Sp Sp*) ∪ *Sq* . Thus the difference between *Sq* and *Sp* lies entirely within the set *F*<sup>∨</sup> *<sup>p</sup>* = *F*<sup>∨</sup> *q* , and in particular *Sq* has *Sq* there while *Sp* has *Sp* there.

**Lemma 6** (routine)**.** *If p*, *q* ∈ *P*<sup>∗</sup> *, Fp* = *Fq , and Sp* ∪ *Sq* ⊆ *F*<sup>∨</sup> *<sup>p</sup>* = *F*<sup>∨</sup> *<sup>q</sup> , then*

$$H\_q^p \;:\; P = \{ p' \in P^\* : p' \le p \} \stackrel{\text{cont}}{\longrightarrow} \mathcal{Q} = \{ q' \in P^\* : q' \le q \}.$$

*is an order isomorphism, and H<sup>p</sup> <sup>q</sup>* = (*H<sup>q</sup> <sup>p</sup>*)−1. *If moreover <sup>u</sup>* <sup>⊆</sup> **Fun** *and <sup>p</sup>*, *<sup>q</sup>* <sup>∈</sup> *<sup>P</sup>*[*u*] *then <sup>H</sup><sup>p</sup> <sup>q</sup> maps the set* {*p* ∈ *P*[*u*] : *p* ≤ *p*} *onto* {*q* ∈ *P*[*u*] : *q* ≤ *q*} *order-preservingly.*

#### **3. Almost Disjoint Product Forcing**

Here we review the structure and basic properties of product almost-disjoint forcing over **L** and corresponding generic extensions of **L**. In order to support various applications, we make use of *ω*1 many independent forcing notions.

#### *3.1. Product Forcing, Systems, Restrictions*

We begin with *ω*1-products of *P*∗ after which we consider more complicated forcing notions.

**Definition 5.** *Let* I = *ω*<sup>1</sup> *. This is the index set for the forcing products considered below. Let* **P**<sup>∗</sup> *be the product of* I *copies of the set P*<sup>∗</sup> *(Definition 2), with finite support. That is,* **P**<sup>∗</sup> *consists of all functions <sup>p</sup>* : <sup>|</sup>*p*| → *<sup>P</sup>*<sup>∗</sup> *such that the set* <sup>|</sup>*p*<sup>|</sup> <sup>=</sup> dom *<sup>p</sup>* ⊆ I *is finite.*

If *p* ∈ **P**<sup>∗</sup> then put *Fp*(*ν*) = *Fp*(*ν*) and *Sp*(*ν*) = *Sp*(*ν*) for all *ν* ∈ |*p*|, so that *p*(*ν*) = *Sp*(*ν*); *Fp*(*ν*). We order **P**<sup>∗</sup> componentwise: *p* ≤ *q* iff |*q*|⊆|*p*| and *p*(*ν*) ≤ *q*(*ν*) for all *ν* ∈ |*q*|. Put

$$F\_p^\vee(\nu) = F\_{p(\nu)}^\vee = \{ f \upharpoonright m : f \in F\_p(\nu) \land m \ge 1 \}.$$

If *p*, *q* ∈ **P**<sup>∗</sup> then define a condition *r* = *p* ∧ *q* ∈ **P**<sup>∗</sup> so that |*p* ∧ *q*| = |*p*|∪|*q*|, (*p* ∧ *q*)(*ν*) = *p*(*ν*) ∧ *q*(*ν*) whenever *ν* ∈ |*p*|∩|*q*|, and if *ν* ∈ |*p*| |*q*| or *ν* ∈ |*q*| |*p*|, then (*p* ∧ *q*)(*ν*) = *p*(*ν*), resp., (*p* ∧ *q*)(*ν*) = *q*(*ν*). Then Conditions *p*, *q* are compatible iff *p* ∧ *q* ≤ *p* and *p* ∧ *q* ≤ *q*.

We consider certain subforcings of the total product almost disjoint forcing notion **P**∗ . This involves the following notion of a system.

**Definition 6.** *A system is any map U* : |*U*| → P(**Fun**) *such that* |*U*|⊆I *and each set U*(*ν*) *(ν* ∈ |*U*|*) is topologically dense in* **Fun***. A system U is:*


## **Definition 7** (restrictions)**.** *Suppose that c* ⊆ I *.*

*If p* ∈ **P**<sup>∗</sup> *then define p* = *p c* ∈ **P**<sup>∗</sup> *so that* |*p* | = *c* ∩ |*p*| *and p* (*ν*) = *p*(*ν*) *whenever ν* ∈ |*p* |*. Accordingly if U is a system then define a system U c so that* |*U c*| = *c* ∩ |*U*| *and* (*U c*)(*ν*) = *U*(*ν*) *for <sup>ν</sup>* ∈ |*U <sup>c</sup>*|*. A special case: if <sup>ν</sup>* ∈ I *then let p* =*<sup>ν</sup>* = *<sup>p</sup>* (|*p*| {*ν*}) *and U* =*<sup>ν</sup>* = *<sup>U</sup>* (|*U*| {*ν*})*.*

Note that writing *p c* or *U c*, it is not assumed that *c* ⊆ |*p*|, resp., *c* ⊆ |*U*|.

#### *3.2. Regular Forcing Notions*

Unfortunately, product forcing notions of the form **P**[*U*] (*U* being a system in **L**) do not provide us with all the definability effects we need. We will make use of certain more complicated forcing notions *K* ⊆ **P**<sup>∗</sup> in **L**. To explain the idea, let a system *U* ∈ **L** satyisfy |*U*| = *ω*. Let *G* ⊆ **P**[*U*] be generic over **L**. The sets *SG*(*ν*) = *SG*(*ν*) = - *<sup>p</sup>*∈*<sup>G</sup> Sp*(*ν*) ⊆ **Seq** then belong to **<sup>L</sup>**[*G*], and in fact **L**[*G*] = **L**[{*SG*(*ν*)}*ν*<*ω*]. As **Seq** = {**s***<sup>k</sup>* : *k* ≥ 1} (a fixed recursive enumeration, Definition 1), let *a*0[*G*] = {*k* ≥ 1 : **s***<sup>k</sup>* ∈ *S*0[*G*]} and *c* = {0} ∪ *aG*(0). Consider the model **L**[{*SG*(*ν*)}*ν*∈*c*]. The first idea is to make use of *U c*, but oops, clearly *c* ∈/ **L**, and consequently *U c* ∈/ **L** and **P**[*U c*] ∈/ **L**, so that many typical product forcing results do not apply in this case. The next definition attempts to view the problem from another angle.

**Definition 8** (in **L**)**.** *A set K* ⊆ **P**<sup>∗</sup> *is called a regular subforcing if:*


*In this case, if U is a system then define K*[*U*] = *K* ∩ **P**[*U*]*. In particular, if simply K* = **P**<sup>∗</sup> *then* **P**∗[*U*] = **P**<sup>∗</sup> ∩ **P**[*U*] = **P**[*U*]*.*

**Example 1** (trivial)**.** *If c* ⊆ I *in the ground universe* **L***, then* **P**<sup>∗</sup> *c is a regular forcing. To prove (4) of Definition 8 let p*<sup>∗</sup> = *p and d* = |*p*| ∩ *c.*

**Example 2** (less trivial)**.** *Consider the set K of all conditions p* ∈ **P**<sup>∗</sup> *such that* |*p*| ⊆ *ω and if ν* ∈ |*p*|*, ν* ≥ 1*, then* **s***<sup>ν</sup>* ∈ *Sp*(0)*. We claim that K is a regular subforcing.*

*To verify 8(2), note that if q* ∈ *K then either* 0 ∈ |*q*| *or* |*q*| = ∅*.*

*To verify 8(4), let p* ∈ **P**<sup>∗</sup> *. If* |*p*|⊆{0}*, then setting p*<sup>∗</sup> = *p and d* = |*p*| *works, so we assume that* |*p*| ⊆ {0}*. Define p*<sup>∗</sup> ∈ **P**<sup>∗</sup> *so that p*∗(*ν*) = *p*(*ν*) *for all ν* ≥ 1*, Fp*<sup>∗</sup> (0) = *Fp*(0)*, and Sp*<sup>∗</sup> (0) = *Sp*(0) ∪ {**s***<sup>ν</sup>* : *ν* ∈ *I* }*, where I consists of all ν* ∈ |*p*|*, ν* ≥ 1*, such that* **s***<sup>ν</sup>* ∈/ *Sp*(0) ∪ *F*<sup>∨</sup> *<sup>p</sup>* (0)*. Then* |*p*∗| = |*p*|∪|0|*, p*<sup>∗</sup> ≤ *p, and we have* **s***<sup>ν</sup>* ∈ *Sp*<sup>∗</sup> (0) ∪ *F*<sup>∨</sup> *<sup>p</sup>*<sup>∗</sup> (0) *(not necessarily* **s***<sup>ν</sup>* ∈ *Sp*<sup>∗</sup> (0)*) for all ν* ∈ |*p*|*, ν* ≥ 1*. Let d* ⊆ |*p*∗| *contain* 0 *and all ν* ∈ |*p*|*, ν* ≥ 1 *with* **s***<sup>ν</sup>* ∈ *Sp*<sup>∗</sup> (0)*; easily p*<sup>∗</sup> *d* ∈ *K*.

*Now let q* ∈ *K, q* ≤ *r* = *p*<sup>∗</sup> *d. Consider any index ν* ∈ |*p*∗| *d. Then* **s***<sup>ν</sup>* ∈/ *Sp*<sup>∗</sup> (0) = *Sr*(0)*, hence* **s***<sup>ν</sup>* ∈ *F*<sup>∨</sup> *<sup>p</sup>*<sup>∗</sup> (0) = *F*<sup>∨</sup> *<sup>r</sup>* (0)*. We claim that ν* ∈ | / *q*|*. Indeed otherwise* **s***<sup>ν</sup>* ∈ *Sq*(0) *as q* ∈ *K. However* **s***<sup>ν</sup>* ∈ *F*<sup>∨</sup> *<sup>r</sup>* (0) *Sr*(0) *(see above). However, this contradicts* **s***<sup>ν</sup>* ∈ *Sq*(0)*, because q* ≤ *r.*

**Theorem 4** (in **L**)**.** *The partially ordered set* **P**∗ *, and hence each* **P**[*U*]*, and generally each regular subforcing of* **P**[*U*] *(for any system U ) satisfies CCC* (countable antichain condition)*.*

**Proof.** Suppose towards the contrary that *A* ⊆ **P**<sup>∗</sup> is an uncountable antichain. We may assume that there is *m* ∈ *ω* such that |*p*| = *m* for all *p* ∈ *A*. Applying the Δ-lemma argument, we obtain an uncountable set *<sup>A</sup>* <sup>⊆</sup> *<sup>A</sup>* and a finite set *<sup>w</sup>* ⊆ I with card *<sup>w</sup>* <sup>&</sup>lt; *<sup>m</sup>* strictly, such that <sup>|</sup>*p*|∩|*q*<sup>|</sup> <sup>=</sup> *<sup>w</sup>* for all *p* = *q* in *A* . Then *A* = {*p w* : *p* ∈ *A* } is still an uncountable antichain, with |*p*| = *w* for all *p* ∈ *A* , easily leading to a contradiction (see the proof of Lemma 2).

**Lemma 7** (in **L**)**.** *If K* ⊆ **P**<sup>∗</sup> *is a regular forcing and U is a system then K*[*U*] = *K* ∩ **P**[*U*] *is a regular subforcing of* **P**[*U*]*.*

To show how (4) of Definition 8 works, we prove

**Lemma 8** (in **L**)**.** *If U is a system and K* ⊆ **P**[*U*] *is a regular subforcing of* **P**[*U*] *then any set D* ⊆ *K pre-dense in K remains pre-dense in* **P**[*U*]*.*

**Proof.** Consider any *p* ∈ **P**[*U*]. Let *p*<sup>∗</sup> ∈ **P**[*U*] and *d* ⊆ |*p*∗| satisfy (4) of Definition 8. In particular, *p*<sup>∗</sup> ≤ *p* and *p*<sup>∗</sup> *d* ∈ *K*. By the pre-density, there is a condition *q* ∈ *D* compatible with *p*<sup>∗</sup> *d*. Then by (1) of Definition 8 there is a condition *r* = *q* ∧ (*p*<sup>∗</sup> *d*) ∈ *K* such that *r* ≤ *q* and *r* ≤ *p*<sup>∗</sup> *d*. Then *r* is compatible with *p* by the choice of *p*∗ and *d*.

#### *3.3. Outline of Product and Regular Extensions*

We consider sets of the form **P**[*U*], *U* being a system in **L**, as well as regular subforcings *K* ⊆ **P**[*U*], as forcing notions over **L**. Accordingly, we will study **P**[*U*]-generic and *K*-generic extensions **L**[*G*] of the ground universe **L**. Define some elements of these extensions.

**Definition 9.** *Suppose that G* ⊆ **<sup>P</sup>**<sup>∗</sup> *. Put* |*G*| = - *<sup>p</sup>*∈*<sup>G</sup>* |*p*|*;* |*G*|⊆I *. Let*

$$S\_G(\nu) = S\_{G(\nu)} = \bigcup\_{p \in G} S\_p(\nu) \quad \text{and} \quad a\_{G(\nu)} = a\_G(\nu) = \{k \ge 1 : \mathbb{s}\_k \in S\_G(\nu)\},$$

*for any ν* ∈ I *, where G*(*ν*) = {*p*(*ν*) : *p* ∈ *G*} ⊆ *P*<sup>∗</sup> *, and* **Seq** = {**s***<sup>k</sup>* : *k* ≥ 1} *is a fixed recursive enumeration (see Definition 1).*

*Thus SG*(*ν*) ⊆ **Seq***, aG*(*ν*) ⊆ *ω* {0}*, and SG*(*ν*) = *aG*(*ν*) = ∅ *for any ν* ∈ | / *G*|*.*

*By the way, this defines a sequence SG* = {*SG*(*ν*)}*ν*∈I *of subsets of* **Seq***.*

*If c* ⊆ I *then let G c* = {*p* ∈ *G* : |*p*| ⊆ *c*}*. It will typically happen that G c* = {*p c* : *p* ∈ *G*}*. Put <sup>G</sup>* =*<sup>ν</sup>* = {*<sup>p</sup>* ∈ *<sup>G</sup>* : *<sup>ν</sup>* ∈ | / *<sup>p</sup>*|} = *<sup>G</sup>* (I {*ν*})*.*

If *U* is a system in **L**, the ground universe, then any **P**[*U*]-generic set *G* ⊆ **P**[*U*] splits into the family of sets *G*(*ν*), *ν* ∈ I , and each *G*(*ν*) is *P*[*U*(*ν*)]-generic.

**Lemma 9.** *Let U be a system and K* ⊆ **P**[*U*] *be a regular subforcing in the ground universe* **L***. Let G* ⊆ **P**[*U*] *be a set* **P**[*U*]*-generic over* **L***. Then*:


#### **Proof.** (ii) This follows from Lemma 8.

(iii) Let us show that *G c* = {*q* ∈ **P**<sup>∗</sup> : ∃ *p* ∈ *G* ∩ *K* (*p* ≤ *q*)}; this proves *G c* ∈ **L**[*G* ∩ *K*]. Suppose that *q* ∈ *G c*, so that *q* ∈ *G* and |*q*| ⊆ *c*, in other words, |*q*|⊆|*p*1|∪···∪|*pn*| for a finite set of conditions *p*1, ... , *pn* ∈ *G* ∩ *K*. Note that *p* = *p*<sup>1</sup> ∧···∧ *pn* ∈ *K* by Definition 8(1). Thus *p* ∈ *G* ∩ *K*, and |*q*|⊆|*p*|. Yet *q* ∈ *G* as well, therefore, *p* = *p* ∧ *q* ∈ *G*, and |*p* | = |*p*|. It follows that *p* ∈ *K*, by Definition 8(3), so that *p* ∈ *G* ∩ *K*. Finally *p* ≤ *q*.

Now suppose that *p* ∈ *G* ∩ *K* and *p* ≤ *q* ∈ **P**<sup>∗</sup> . Then obviously *q* belongs to **P**[*U*] (since so does *p*), hence *q* ∈ *G* (since *G* is generic). Finally |*q*|⊆|*p*| ⊆ *c*.

Let us show that *G* ∩ *K* = (*G c*) ∩ *K*; this proves *G* ∩ *K* ∈ **L**[*G c*]. Indeed if *p* ∈ *G* ∩ *K* then by definition |*p*| ⊆ *c* = |*G* ∩ *K*|, therefore *p* ∈ *G c*, as required.

(iv) This is clear since we have *<sup>G</sup>* ∩ *<sup>K</sup>* = *<sup>G</sup>* =*<sup>ν</sup>* ∩ *<sup>K</sup>* in the case considered.

(v) The set **<sup>P</sup>**[*U*] can be identified with the product **<sup>P</sup>**[*U*] =*<sup>ν</sup>* × *<sup>P</sup>*[*U*(*ν*)]. Thus *<sup>G</sup>*(*ν*) and *SG*(*ν*) are *<sup>P</sup>*[*U*(*ν*)]-generic over **<sup>L</sup>**[**P**[*U*] =*ν*].

(vi) The genericity easily follows from Definition 8(3). Then use Lemma 3.

(i) First of all, *G* = ∏*<sup>ν</sup> G*(*ν*) by the product-forcing theorem. Then, each *G*(*ν*) is recovered from the associated *SG*(*ν*) by means of a simple uniform formula, see the proof of Lemma 3(i).

*3.4. Names for Sets in Product and Regular Extensions*

For any set *X* we let **N***<sup>X</sup>* be the set of all **P**∗-names for subsets of *X*. Thus **N***<sup>X</sup>* consists of all sets *τ* ⊆ **P**<sup>∗</sup> × *X*. Let **SN***<sup>X</sup>* (small names) consist of all at most countable names *τ* ∈ **N***<sup>X</sup>* .

We define dom *<sup>τ</sup>* <sup>=</sup> {*<sup>p</sup>* : <sup>∃</sup> *<sup>x</sup>* (*<sup>p</sup>*, *<sup>x</sup>*<sup>∈</sup> *<sup>τ</sup>*)}, <sup>|</sup>*τ*<sup>|</sup> <sup>=</sup> -{|*p*<sup>|</sup> : *<sup>p</sup>* <sup>∈</sup> dom *<sup>τ</sup>*} for any name *<sup>τ</sup>*.

Say that a name *<sup>τ</sup>* is below a given *<sup>p</sup>* <sup>∈</sup> **<sup>P</sup>**<sup>∗</sup> if all *<sup>p</sup>* <sup>∈</sup> dom *<sup>τ</sup>* satisfy *<sup>p</sup>* <sup>≤</sup> *<sup>p</sup>*.

For any set *<sup>K</sup>* <sup>⊆</sup> **<sup>P</sup>**<sup>∗</sup> , we let **<sup>N</sup>***X*(*K*) be the set of all names *<sup>τ</sup>* <sup>∈</sup> **<sup>N</sup>***<sup>X</sup>* such that dom *<sup>τ</sup>* <sup>⊆</sup> *<sup>K</sup>*, and accordingly **SN***X*(*K*) = **N***X*(*K*) ∩ **SN***<sup>X</sup>* (small names). In particular, we'll consider such sets of names as **SN***X*(**P**[*U*]) and **SN***X*(**P**[*U*] *c*). Names in **N***X*(*K*) for different sets *X* will be called *K*-names. Accordingly, names in **SN***X*(*K*) for different sets *X* will be called small *K*-names.

**Definition 10** (valuations)**.** *If τ* ∈ **N***<sup>X</sup> and G* ⊆ **P**<sup>∗</sup> *then define τ*[*G*] = {*x* : ∃ *p* ∈ *G* (*p*, *x* ∈ *τ*)}*, the Gvaluation of τ; τ*[*G*] *is a subset of X.*

**Example 3** (some names)**.** *Let* ∈ **P**<sup>∗</sup> *be the empty condition, that is,* || = ∅*. This is the weakest condition in any* **<sup>P</sup>**[*U*]*. If <sup>X</sup> is a set in the ground universe then <sup>X</sup>*˘ <sup>=</sup> {, *<sup>x</sup>* : *<sup>x</sup>* <sup>∈</sup> *<sup>X</sup>*} *is a K-name for any regular forcing K* <sup>⊆</sup> **<sup>P</sup>**<sup>∗</sup> *, and <sup>X</sup>*˘ [*G*] = *X for any set G containing .*

*We will typically use breve-names like <sup>X</sup>*˘ *for sets in the ground universe, and dot-names (like* **.** *x ) for sets in generic extensions.*

*Suppose that K* ⊆ **P**<sup>∗</sup> *. Let G* = {*p*, *p* : *p* ∈ *K*}*. (In principle, G depends on K but this dependence will usually be suppressed.) Clearly G* ∈ **N***K*(*K*) *(but G* ∈/ **SN***K*(*K*) *unless K is countable), and in addition G*[*G*] = *G for any* ∅ = *G* ⊆ *K. Thus G is a name for the generic set G* ⊆ *K.*

*Similarly, G c* = {*p*, *p* : *p* ∈ *K c*} *(c* ⊆ I *) is a name for G c (see Definition 9).*

#### *3.5. Names for Functions*

For any sets *X*, *Y* let **N***<sup>X</sup> <sup>Y</sup>* be the set of all **P**∗-names for functions *X* → *Y*; it consists of all *τ* ⊆ **P**<sup>∗</sup> × (*X* × *Y*) such that the sets *τ* "*x*, *y* = {*p* : *p*,*x*, *y* ∈ *τ*} satisfy the following requirement:

> if *y* = *y* , *p* ∈ *τ* "*x*, *y*, *p* ∈ *τ* "*x*, *y* , then *p*, *p* are incompatible.

Let dom *τ* = - *<sup>x</sup>*,*<sup>y</sup> <sup>τ</sup>* "*<sup>x</sup>*, *<sup>y</sup>* and |*τ*| = -{|*p*<sup>|</sup> : *<sup>p</sup>* <sup>∈</sup> dom *<sup>τ</sup>*}.

As above, **SN***<sup>X</sup> <sup>Y</sup>* consists of all at most countable names *<sup>τ</sup>* <sup>∈</sup> **<sup>N</sup>***<sup>X</sup> Y* .

For any set *<sup>K</sup>* <sup>⊆</sup> **<sup>P</sup>**<sup>∗</sup> , we let **<sup>N</sup>***<sup>X</sup> <sup>Y</sup>* (*K*) be the set of all names *<sup>τ</sup>* <sup>∈</sup> **<sup>N</sup>***<sup>X</sup> <sup>Y</sup>* such that dom *<sup>τ</sup>* <sup>⊆</sup> *<sup>K</sup>*, and accordingly **SN***<sup>X</sup> <sup>Y</sup>* (*K*) = **<sup>N</sup>***<sup>X</sup> <sup>Y</sup>* (*K*) <sup>∩</sup> **SN***<sup>X</sup> <sup>Y</sup>* (small names).

A name *<sup>τ</sup>* <sup>∈</sup> **<sup>N</sup>***<sup>X</sup> <sup>Y</sup>* (*K*) is *K*-full iff the union *τ* "*x* = - *<sup>y</sup> τ* "*x*, *y* is pre-dense in *K* for any *x* ∈ *X*. A name *<sup>τ</sup>* <sup>∈</sup> **<sup>N</sup>***<sup>X</sup> <sup>Y</sup>* (*K*) is *K*-full below some *p*<sup>0</sup> ∈ *K*, iff all sets *τ* "*x* are pre-dense in *K* below *p*<sup>0</sup> , that is, any condition *q* ∈ *K*, *q* ≤ *p*<sup>0</sup> , is compatible with some *r* ∈ *τ<sup>x</sup>* (and this holds for all *x* ∈ *X*).

Note that **N***<sup>X</sup> <sup>Y</sup>* (*K*) <sup>⊆</sup> **<sup>N</sup>***X*×*Y*(*K*), and accordingly **SN***<sup>X</sup> <sup>Y</sup>* (*K*) <sup>⊆</sup> **SN***X*×*Y*(*K*). Thus all names in **<sup>N</sup>***<sup>X</sup> <sup>Y</sup>* (*K*) and in **SN***<sup>X</sup> <sup>Y</sup>* (*K*) are still *K*-names in the sense above.

**Corollary 1** (of Lemma 8, in **L**)**.** *If U is a system, K* ⊆ **P**[*U*] *is a regular subforcing, X*,*Y any sets, and τ is a name in* **N***<sup>X</sup> <sup>Y</sup>* (*K*)*, then τ is K-full* (*resp., K-full below p* ∈ *K* ) *iff τ is* **P**[*U*]*-full* (*resp.,* **P**[*U*]*-full below p* )*.*

Suppose that *<sup>τ</sup>* <sup>∈</sup> **<sup>N</sup>***<sup>X</sup> <sup>Y</sup>* . Call a set *G* ⊆ **P**<sup>∗</sup> minimally *τ*-generic iff it is compatible in itself (if *p*, *q* ∈ *G* then there is *r* ∈ *G* with *r* ≤ *p*, *r* ≤ *q*), and intersects each set of the form *τ* "*x*, *x* ∈ *X*. In this case put

$$\pi[G] = \{ \langle \mathbf{x}, \mathbf{y} \rangle \in \mathcal{X} \times \mathcal{Y} : (\pi'' \langle \mathbf{x}, \mathbf{y} \rangle) \cap G \neq \mathcal{Q} \},$$

so that *<sup>τ</sup>*[*G*] <sup>∈</sup> *<sup>Y</sup><sup>X</sup>* and *<sup>τ</sup>*[*G*](*x*) = *<sup>y</sup>* ⇐⇒ *<sup>τ</sup>* "*<sup>x</sup>*, *<sup>y</sup>*<sup>∩</sup> *<sup>G</sup>* <sup>=</sup> <sup>∅</sup>. If *<sup>ϕ</sup>* is a formula in which some names *<sup>τ</sup>* <sup>∈</sup> **<sup>N</sup>***<sup>X</sup> <sup>Y</sup>* occur (for various sets *X*,*Y*), and a set *G* ⊆ **P**<sup>∗</sup> is minimally *τ*-generic for any name *τ* in *ϕ*, then accordingly *ϕ*[*G*] is the result of substitution of *τ*[*G*] for each name *τ* in *ϕ*.

**Claim 1** (obvious)**.** *Suppose that, in* **<sup>L</sup>***, <sup>X</sup>*,*<sup>Y</sup> are any sets, <sup>p</sup>* <sup>∈</sup> *<sup>K</sup>* <sup>⊆</sup> **<sup>P</sup>**<sup>∗</sup> *and <sup>τ</sup>* <sup>∈</sup> **<sup>N</sup>***<sup>X</sup> <sup>Y</sup>* (*K*) *is K-full* (*resp., K-full below p* )*. Then, any set G* ⊆ *K, K-generic over* **L** (*resp., K-generic over* **L** *and containing p* )*, is minimally τ-generic.*

**Definition 11** (equivalent names)**.** *Names <sup>τ</sup>*, *<sup>μ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**P**∗) *are called equivalent iff conditions q*,*r are incompatible whenever q* ∈ *τ "m*, *j and r* ∈ *μ"m*, *k for some m and j* = *k. (Recall that τ "m*, *k* = {*p* : *p*,*m*, *k* ∈ *τ*}*.) Similarly, names τ*, *μ are equivalent below some p* ∈ **P**<sup>∗</sup> *iff the triple of conditions p*, *q*,*r is incompatible (that is, p* ∧ *q* ∧ *r is not* ≤ *than at least one of p*, *q*,*r) whenever q* ∈ *τ "m*, *j and r* ∈ *μ"m*, *k for some m and j* = *k.*

**Claim 2** (obvious)**.** *Suppose that, in* **<sup>L</sup>***, <sup>p</sup>* <sup>∈</sup> *<sup>K</sup>* <sup>⊆</sup> **<sup>P</sup>**<sup>∗</sup> *, and names <sup>μ</sup>*, *<sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(*K*) *are equivalent* (*resp., equivalent below p* )*. Then, for any G* ⊆ *K both minimally μ-generic and minimally τ-generic* (*resp., and containing p* )*, μ*[*G*] = *τ*[*G*]*.*

**Lemma 10.** *Suppose that, in* **L***, U is a system, K* ⊆ **P**[*U*] *is a regular subforcing, p*<sup>0</sup> ∈ *K, A* ⊆ *P* = {*p* ∈ *K* : *<sup>p</sup>* <sup>≤</sup> *<sup>p</sup>*<sup>0</sup> } *is a countable antichain, and, for any <sup>p</sup>* <sup>∈</sup> *A, <sup>τ</sup><sup>p</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(*K*) *is a name K-full below p*<sup>0</sup> *. Then there is a K-full name <sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(*K*)*, equivalent to τ<sup>p</sup> below p for any p* ∈ *A.*

**Proof.** Let *B* be a maximal (countable) antichain in the set of all conditions *q* ∈ *K* incompatible with *p*<sup>0</sup> . Then *A* ∪ *B* is a countable maximal antichain in *K*. We let *τ* consist of: 1) all triples *r* ∧ *q*,*k*, *m*, such that *q* ∈ *A* and *r*,*k*, *m* ∈ *τ<sup>q</sup>* , and 2) all triples *q*,*k*, 0, such that *q* ∈ *B* and *m* ∈ *ω*.

#### *3.6. Names and Sets in Generic Extensions*

For any forcing *P*, let ||−−*<sup>P</sup>* denote the *P*-forcing relation over **L** as the ground model.

**Theorem 5.** *Suppose that U is a system and K* ⊆ **P**[*U*] *a regular subforcing in* **L***. Let G* ⊆ *K be a set Kgeneric over* **L***. Then*:

(i) *if p* ∈ *K and ϕ is a closed formula with K-names as parameters, then*

*<sup>p</sup>* ||−−*<sup>K</sup> <sup>ϕ</sup> iff p* ||−−**P**[*U*] "**L**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*˘] <sup>|</sup><sup>=</sup> *<sup>ϕ</sup>*[*G*]" ;


**Proof.** (i) Suppose *<sup>p</sup>* ||−−*<sup>K</sup> <sup>ϕ</sup>*. To prove *<sup>p</sup>* ||−−**P**[*U*] "**L**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*˘] <sup>|</sup><sup>=</sup> *<sup>ϕ</sup>*[*G*]", consider a set *<sup>G</sup>* <sup>⊆</sup> **<sup>P</sup>**[*U*], **<sup>P</sup>**[*U*] generic over **L**. Then *G* ∩ *K* is *K*-generic over **L** by Lemma 8, hence *ϕ*[*G*] is true in **L**[*G* ∩ *K*], as required. Conversely assume ¬ *p* ||−−*<sup>K</sup> ϕ*. There is a condition *q* ∈ *K*, *q* ≤ *p*, *q* ||−−*<sup>k</sup>* ¬ *ϕ*. Then *<sup>q</sup>* ||−−**P**[*U*] "**L**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*˘] <sup>|</sup><sup>=</sup> <sup>¬</sup> *<sup>ϕ</sup>*[*G*]" by the above, thus *<sup>p</sup>* ||−−**P**[*U*] "**L**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*] <sup>|</sup><sup>=</sup> *<sup>ϕ</sup>*[*G*]" fails.

(ii) It follows from general forcing theory that there is a *<sup>K</sup>*-full name *<sup>σ</sup>* <sup>∈</sup> **<sup>N</sup>***<sup>X</sup> <sup>Y</sup>* (*K*), not necessarily countable, such that *f* = *σ*[*G*]. Then all sets *Qx* = *σ*"*x*, *x* ∈ *X*, are pre-dense in *K*. Put *τ* = {*p*,*x*, *y* ∈ *σ* : *x* ∈ *X* ∧ *y* ∈ *Y* ∧ *p* ∈ *Ax* } , where *Ax* ⊆ *Qx* is a maximal (countable, by Theorem 4) antichain for any *x*.

(iv) We conclude from (ii) that the set *Q* of all conditions *q* ∈ *K*, *q* ≤ *p*, such that *q* ||−−*<sup>K</sup> ϕ*(*τ*) for some name *<sup>τ</sup>* <sup>=</sup> *<sup>τ</sup><sup>q</sup>* <sup>∈</sup> **SN***<sup>X</sup> <sup>Y</sup>* (*K*), is dense in *K* below *p*. Let *A* ⊆ *Q* be a maximal antichain in *Q*; *A* is countable and pre-dense in *K* below *p*. Apply Lemma 10 to get a name *τ* as required.

**Example 4.** *Consider the regular forcing K* = **P**[*U c*]*, where U is a system and c* ⊆ I *in* **L***. If G* ⊆ **P**[*U*] *is* **P**[*U*]*-generic over* **L** *then the restricted set G c* = *G* ∩ (**P**[*U c*]) *is* **P**[*U c*]*-generic over* **L***, by Lemma 9 (with K* = **P**[*U c*]*). Furthermore, it follows from Lemma 9 and Theorem 5 that if ν* ∈ I *then SG*(*ν*) ∈ **L**[*G c*] *iff ν* ∈ *c, so that* **L**[*G c*] = **L**[{*SG*(*ν*)}*ν*∈*c*]*.*

**Example 5.** *Consider the regular forcing K defined in Example 2 in Section 3.2. Suppose that U is a system in* **L** *and G* ⊆ **P**[*U*] *is a set* **P**[*U*]*-generic over* **L***. Then K*[*U*] = *K* ∩ **P**[*U*] *is a regular subforcing of* **P**[*U*] *by Lemma 7. We conclude that G* = *G* ∩ *K is a set K*[*U*]*-generic over* **L***, by Lemma 9.*

*It follows by the definition of K that the set* |*G* | = - *<sup>p</sup>*∈*<sup>G</sup>* |*p*| *satisfies* |*G* | ⊆ *ω, contains* 0*, and if ν* ≥ 1 *then ν* ∈ |*G* | *iff* **s***<sup>ν</sup>* ∈ *SG*(0)*. Therefore, by Lemma 9 and Theorem 5, the sets G*(0) *and SG*(0) *belong to* **L**[*G* ]*, and if* 1 ≤ *ν* < *ω then SG*(*ν*) ∈ **L**[*G* ] *iff* **s***<sup>ν</sup>* ∈ *SG*(0)*. Thus*

$$\mathbf{L}[G'] = \mathbf{L}[\mathcal{S}\_G(0), \{\mathcal{S}\_G(\nu)\}\_{\mathfrak{s}\_\nu \in \mathcal{S}\_G(0)}] = \mathbf{L}[G'] = \mathbf{L}[G \upharpoonright \mathcal{c}] \text{ \(\!\! \! \! \/)}$$

*where c* = |*G* | = {0}∪{*ν* < *ω* : **s***<sup>ν</sup>* ∈ *SG*(0)} ∈/ **L***.*

#### *3.7. Transformations Related to Product Forcing*

There are three important families of transformations of the whole system of objects related to product forcing. Two of them are considered in this Subsection.

**Family 1: permutations.** If *<sup>c</sup>*, *<sup>c</sup>* ⊆ I are sets of equal cardinality then let BIJ*<sup>c</sup> <sup>c</sup>* be the set of all bijections *π* : *c* onto −→ *<sup>c</sup>* . Let <sup>|</sup>*π*<sup>|</sup> <sup>=</sup> {*<sup>ν</sup>* <sup>∈</sup> *<sup>c</sup>* : *<sup>π</sup>*(*ν*) <sup>=</sup> *<sup>ν</sup>*}∪{*<sup>ν</sup>* <sup>∈</sup> *<sup>c</sup>* : *<sup>π</sup>*−1(*ν*) <sup>=</sup> *<sup>ν</sup>*}, so that *<sup>π</sup>* is essentially a bijection *<sup>c</sup>* ∩ |*π*<sup>|</sup> onto −→ *c* ∩ |*π*|, equal to the identity on *c* |*π*| = *c* |*π*|. Define the action of any *<sup>π</sup>* <sup>∈</sup> BIJ*<sup>c</sup> <sup>c</sup>* onto:


**Lemma 11.** *If <sup>c</sup>*, *<sup>c</sup>* ⊆ I *are sets of equal cardinality and <sup>π</sup>* <sup>∈</sup> BIJ*<sup>c</sup> <sup>c</sup> then p* −→ *π· p is an order preserving bijection of* **P**<sup>∗</sup> *c onto* **P**<sup>∗</sup> *c , and if U is a system and* |*U*| ⊆ *c then* |*π·U*| ⊆ *c , and we have p* ∈ **P**[*U*] ⇐⇒ *π· p* ∈ **P**[*π·U*]*.*

**Family 2: Lipschitz transformations.** Let **Lip**<sup>I</sup> be the I-product of the group **Lip** (see Section 2.3), with countable support; this will be our second family of transformations. Thus a typical element *<sup>α</sup>* <sup>∈</sup> **Lip**<sup>I</sup> is *<sup>α</sup>* <sup>=</sup> {*αν* }*ν*∈|*α*<sup>|</sup> , where <sup>|</sup>*α*<sup>|</sup> <sup>=</sup> dom *<sup>α</sup>* ⊆ I is at most countable, and *αν* <sup>∈</sup> **Lip**, ∀ *ν*. We will routinely identify each *α* ∈ **Lip**<sup>I</sup> with its extension on I defined so that *αν* is the identity map (on **Seq**) for all *ν* ∈ I |*α*|. Keeping this identification in mind, define the action of any *α* ∈ **Lip**<sup>I</sup> on:


*Mathematics* **2020**, *8*, 910

In the first two lines, we refer to the action of *αν* ∈ **Lip** on sets *u* ⊆ **Fun** and on forcing conditions, as defined in Section 2.3.

**Lemma 12.** *If α* ∈ **Lip**<sup>I</sup> *then p* −→ *π· p is an order preserving bijection of* **P**<sup>∗</sup> *onto* **P**<sup>∗</sup> *, and if U is a system then we have p* ∈ **P**[*U*] ⇐⇒ *α· p* ∈ **P**[*α·U*]*.*

**Corollary 2** (of Lemma 5)**.** *Suppose that U*, *V are countable systems,* |*U*| = |*V*|*, and p* ∈ **P**[*U*]*, q* ∈ **P**[*V*]*. Then there is a transformation α* ∈ **Lip**<sup>I</sup> *such that*


**Proof.** Apply Lemma 5 componentwise for every *ν* ∈ |*U*| = |*U* |.

*3.8. Substitutions and Homogeneous Extensions*

Assume that conditions *p*, *q* ∈ **P**<sup>∗</sup> satisfy (2) of Section 2.4 for all *ν*, that is:

$$\left|p\right| = \left|q\right|, \text{ and } \leftS\_{\mathcal{P}}(\nu) \cup \mathcal{S}\_{\emptyset}(\nu) \subseteq F\_{\mathcal{P}}^{\vee}(\nu) = F\_{\mathcal{q}}^{\vee}(\nu) \text{ for all } \nu \in \left|p\right| = \left|q\right|. \tag{3}$$

**Definition 12.** *If* (3) *holds and <sup>p</sup>* <sup>∈</sup> **<sup>P</sup>**<sup>∗</sup> *, <sup>p</sup>* <sup>≤</sup> *p, then define <sup>q</sup>* <sup>=</sup> *<sup>H</sup><sup>p</sup> <sup>q</sup>* (*p* ) *so that* |*q* | = |*p* |*, q* (*ν*) = *p* (*ν*) *whenever ν* ∈ |*p* | |*p*|*, but q* (*ν*) = *<sup>H</sup>p*(*ν*) *<sup>q</sup>*(*ν*) (*p* (*ν*)) *for all <sup>ν</sup>* ∈ |*p*|*, where <sup>H</sup>p*(*ν*) *<sup>q</sup>*(*ν*) *is defined as in Section 2.4. This is Family 3 of transformations, called substitutions.*

**Theorem 6.** *If U is a system, and conditions p*, *q* ∈ **P**[*U*] *satisfy* (3) *above, then*

$$H\_{\emptyset}^{p}: P = \{ p' \in \mathbf{P}[\mathcal{U}] : p' \le p \} \stackrel{\text{cont}\emptyset}{\longrightarrow} \mathcal{Q} = \{ q' \in \mathbf{P}[\mathcal{U}] : q' \le q \}.$$

*is an order isomorphism.*

**Proof.** Apply Lemma 6 componentwise.

Suppose that *<sup>U</sup>*, *<sup>p</sup>*, *<sup>q</sup>* <sup>∈</sup> **<sup>P</sup>**[*U*], *<sup>H</sup><sup>p</sup> <sup>q</sup>* are as in Theorem 6. Extend the action of *<sup>H</sup><sup>p</sup> <sup>q</sup>* onto names and formulas. Recall that a name *<sup>τ</sup>* <sup>∈</sup> **<sup>N</sup>***<sup>X</sup> <sup>Y</sup>* is below *p* iff *p* ≤ *p* holds for any triple *p* ,*n*, *k* ∈ *τ*.


Forcing notions of the form **P**[*U*] are quite homogeneous by Theorem 6. The next result is a usual product forcing application of such a homogeneity.

**Theorem 7.** *Suppose that, in* **L***, U is a system, d* ⊆ *c* ⊆ I *, K is a regular subforcing of* **P**[*U d*]*, and Q* = {*p* ∈ **P**[*U c*] : *p d* ∈ *K*} = *K* × **P**[*U* (*c d*)]*. Let ϕ be a formula which contains as parameters*: (∗) *K-names, and* (†) *names of the form G e, where e* ∈ **L***, e* ⊆ *c, and G e enters ϕ only via* **L**[*G e*]*. Then*:


**Proof.** (i) Otherwise there are conditions *p*, *q* ∈ *Q* with *p d* = *q d*, *p* ||−−*<sup>Q</sup> ϕ*, but *q* ||−−*<sup>Q</sup>* ¬ *ϕ*. We can w.l. o. g. assume that *p*, *q* satisfy (3) above (otherwise extend *p*, *q* appropriately). Define *P*, *Q*, *H<sup>p</sup> <sup>q</sup>* as in Definition 12 and Theorem 6.

Let *<sup>G</sup>* <sup>⊆</sup> *<sup>Q</sup>* be a generic set containing *<sup>p</sup>*. Assuming w.l. o. g. that *<sup>G</sup>* <sup>⊆</sup> *<sup>P</sup>*, the set *<sup>H</sup>* <sup>=</sup> {*H<sup>p</sup> <sup>q</sup>* (*p* ) : *p* ∈ *G*} ⊆ *Q* will be generic as well by Theorem 6, and *q* ∈ *H*. Therefore *ϕ*[*G*] is true in **L**[*G*] but *ϕ*[*H*] is false in **L**[*H*]. Yet **L**[*G*] = **L**[*H*] since *H<sup>p</sup> <sup>q</sup>* ∈ **L**. Moreover *ϕ*[*G*] coincides with *ϕ*[*H*] since 1) *Hp <sup>q</sup>* is the identity on *d* (indeed *p d* = *q d*), and 2) if *e* ∈ **L**, *e* ⊆ *c*, then **L**[*G e*] = **L**[*H e*] since *G e*, *H e* can be obtained from each other via maps coded in **L**. This is a contradiction.

(iii) This is a particular case.

**Corollary 3.** *Under the assumptions of Theorem 7, suppose that X*, *Y are arbitrary sets in* **L***, p* ∈ *Q, and <sup>p</sup>* ||−−*<sup>Q</sup>* <sup>∃</sup> *<sup>f</sup>* <sup>∈</sup> **<sup>L</sup>**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*] (*<sup>f</sup>* <sup>∈</sup> *<sup>Y</sup><sup>X</sup>* <sup>∧</sup> *<sup>ϕ</sup>*(*f*))*. Then there is a K-full name <sup>τ</sup>* <sup>∈</sup> **SN***<sup>X</sup> <sup>Y</sup>* (*K*) *such that p d* ||−−*<sup>Q</sup> ϕ*(*τ*)*.*

**Proof.** We can assume that |*p*| ⊆ *d* by Theorem 7(iii), thus *p* = *p d* ∈ *K*. It follows from Theorems 5(ii) and 7(i) that there exist: a (countable) antichain *A* ⊆ *K* maximal below *p*, and, for any *q* ∈ *A*, a *<sup>K</sup>*-full name *<sup>τ</sup><sup>q</sup>* <sup>∈</sup> **SN***<sup>X</sup> <sup>Y</sup>* (*K*) such that *<sup>q</sup>* ||−−*<sup>Q</sup> <sup>ϕ</sup>*(*τq*). Now compose a *<sup>K</sup>*-full name *<sup>τ</sup>* <sup>∈</sup> **SN***<sup>X</sup> <sup>Y</sup>* (*K*), such that every *q* ∈ *A* forces *τ* = *τ<sup>q</sup>* , as in the proof of Theorem 5(iv).

#### **4. Basic Forcing Notion and Basic Generic Extension**

The proofs of Theorems 1–3, that follow in Sections 7–9, will have something in common. Namely the generic extensions we employ to get the results required will be parts of a basic extension, introduced and studied in this section. To define the extension, we'll define (in **L** as the ground universe) an increasing sequence {*M<sup>ξ</sup>* , *U<sup>ξ</sup>* }*ξ*<*ω*<sup>1</sup> of pairs of certain type—a Jensen–Solovay sequence, since this construction goes back to [9]—and make use of a forcing notion of the form **P**[*U*], where *U* = *<sup>ξ</sup>*<*ω*<sup>1</sup> *U<sup>ξ</sup>* . It turns out that if such a sequence is *n*-complete, in sense that it meets all sets of *n*-complexity within the whole tree of possible constructions, then the truth of analytic formulas up to level *n* in corresponding generic extensions has a remarkable connection with the forcing approximations studied in Section 5. This will allow us to convert the homogeneity of the construction of Jensen–Solovay sequences into a uniformity of the corresponding generic extensions, expressed by Theorem 13.

Recall that **V** = **L** assumed in the ground universe by Assumption 1.

#### *4.1. Jensen–Solovay Sequences*

If *U V* are systems then by definition **P**[*U*] ⊆ **P**[*V*] holds. However this is not necessarily a suitably good notion. For instance a dense set *X* ⊆ **P**[*U*] may not be pre-dense in **P**[*V*], thus if *G* ⊆ **P**[*V*] is a generic set then the "projection" *G* ∩ **P**[*U*] is not necessarily **P**[*U*]-generic. Yet there is a special type of extension of systems, introduced by Jensen and Solovay [9], which preserves the density. This method is based on the requirement that the functions in **Fun** that occur in *V* but not in *U* must be generic over a certain model that contains *U*.

Recall that **ZFC**− is **ZFC** minus the Power Set axiom, see Section 5.1 below. Let **ZFC**− <sup>1</sup> be **ZFC**<sup>−</sup> plus the axioms **V** = **L** and "every set is at most countable".

**Definition 13.** *Let U*, *U be a pair of systems. Suppose that M is any transitive model of* **ZFC**− *. Define U <sup>M</sup> U iff U U and we have:*


*Let* **JS***, Jensen–Solovay pairs, be the set of all pairs M*, *U of a transitive model M* |= **ZFC**<sup>−</sup> *and a disjoint (ν* = *ν* =⇒ *U*(*ν*) ∩ *U*(*ν* ) = ∅*) system U* ∈ *M. Let* **sJS***, small pairs, consist of all M*, *U* ∈ **JS** *such that M* |= **ZFC**<sup>−</sup> <sup>1</sup> *and M (then U as well) is countable. Define the extension relations:*

*M*, *U M* , *U iff M* ⊆ *M and U <sup>M</sup> U ;* *M*, *U*≺*M* , *U iff M*, *U M* , *U and* ∀ *ν* ∈ |*U*|(*U*(*ν*) - *U* (*ν*))*.*

It would be a vital simplification to get rid of *M* as an explicit element of the construction, e.g., by setting *U* ∗ *U* iff *U U* and there is a CTM *M* containing *U* and such that *U <sup>M</sup> U* .

**Lemma 13.** *Suppose that pairs M*, *U M* , *U M* , *U belong to* **JS***. Then M*, *U M* , *U . Thus is a partial order on* **JS***.*

**Proof.** Prove that the set *F* = - *<sup>ν</sup>*∈|*U*|(*<sup>U</sup>* (*ν*) *<sup>U</sup>*(*ν*)) is multiply Cohen generic over *<sup>M</sup>*. Consider a simple case when *f* ∈ *U* (*ν*) *U*(*ν*) and *g* ∈ *U* (*μ*) *U* (*μ*), where *ν*, *μ* ∈ |*U*|, and prove that *f* , *g* is Cohen generic over *M*. (The general case does not differ much.) By definition, *f* is Cohen generic over *M* and *g* is Cohen generic over *M* . Therefore, *g* is Cohen generic over *M*[ *f* ], which satisfies *M*[ *f* ] ⊆ *M* since *f* ∈ *M* . It remains to apply the product forcing theorem.

**Remark 1.** We routinely have *M*, *U M* , *U* (the same *U*) provided *M* ⊆ *M* . On the other hand, *M*, *U M*, *U*  (with the same *M*) is possible only in the case when Δ(*U*, *U* ) = ∅, that is, *U*(*ν*) = *U* (*ν*) for all *ν* ∈ |*U*|. In particular, if *M*, *U* ∈ **JS**, *c* ∈ *M*, *c* ⊆ |*U*|, then *M*, *U c M*, *U*.

**Lemma 14** (extension)**.** *If M*, *U* ∈ **sJS** *and z* ⊆ I *is countable, then there is a pair M* , *U*  ∈ **sJS** *such that M*, *U*≺*M* , *U and z* ⊆ |*U* |*.*

**Proof.** Let *<sup>d</sup>* <sup>=</sup> <sup>|</sup>*U*| ∪ *<sup>z</sup>*, and let *<sup>f</sup>* <sup>=</sup> { *<sup>f</sup>νk*}*ν*∈*d*,*k*<*<sup>ω</sup>* <sup>∈</sup> (**Fun**)*d*×*<sup>ω</sup>* be Cohen generic over *<sup>M</sup>*. Now define *U* (*ν*) = *U*(*ν*) ∪ { *fν<sup>k</sup>* : *k* ∈ *ω*} for each *ν* ∈ *d*, and let *M* |= **ZFC**<sup>−</sup> <sup>1</sup> be any CTM satisfying *M* ⊆ *M* and containing *U* .

**Definition 14.** *A Jensen–Solovay sequence of length λ* ≤ *ω*<sup>1</sup> *is any strictly* ≺*-increasing λ-sequence* {*M<sup>ξ</sup>* , *<sup>U</sup><sup>ξ</sup>* }*ξ*<*<sup>λ</sup> of pairs <sup>M</sup><sup>ξ</sup>* , *<sup>U</sup><sup>ξ</sup>*  ∈ **sJS***, which satisfies <sup>U</sup><sup>η</sup>* = *<sup>ξ</sup>*<*<sup>η</sup> <sup>U</sup><sup>ξ</sup> on limit steps. Let* −→**JS***<sup>λ</sup> be the set of all such sequences.*

**Lemma 15.** *Suppose that <sup>λ</sup>* <sup>≤</sup> *<sup>ω</sup>*<sup>1</sup> *is a limit ordinal, and* {*M<sup>ξ</sup>* , *<sup>U</sup><sup>ξ</sup>* }*ξ*<*<sup>λ</sup> belongs to* −→**JS***<sup>λ</sup> . Put <sup>U</sup>* <sup>=</sup> *<sup>ξ</sup>*<*<sup>λ</sup> U<sup>ξ</sup> , that is, U*(*ν*) = - *<sup>ξ</sup>*<*<sup>λ</sup> U<sup>ξ</sup>* (*ν*) *for all ν* ∈ I *.*

*Then U<sup>ξ</sup> M<sup>ξ</sup> U for every ξ .*

*If, moreover, λ* < *ω*<sup>1</sup> *and M is a CTM of* **ZFC**<sup>−</sup> <sup>1</sup> *containing* {*M<sup>ξ</sup>* , *U<sup>ξ</sup>* }*ξ*<*<sup>λ</sup> then M*, *U* ∈ **sJS** *and M<sup>ξ</sup>* , *U<sup>ξ</sup>* ≺*M*, *U for every ξ .*

**Proof.** The same idea as in the proof of Lemma 13.

#### *4.2. Stability of Dense Sets*

Assume that *M*, *U* ∈ **sJS** and *D* is a pre-dense subset of **P**[*U*] (say, a maximal antichain). If *U* is another system satisfying *U U* , then it may well happen that *D* is not maximal in **P**[*U* ]. The role of the multiple genericity requirement (a) in Definition 13, first discovered in [9], is to somehow seal the property of pre-density of sets already in *M* for any further extensions. This is the content of the following key theorem. The product forcing arguments allow us to extend the stability result to pre-dense sets not necessarily in *M*, as in items (ii), (iii) of the following theorem.

**Theorem 8.** *Assume that, in* **L***, M*, *U* ∈ **sJS***, U is a disjoint system, and U <sup>M</sup> U . If D is a pre-dense subset of* **P**[*U*] (*resp., pre-dense below some p* ∈ **P**[*U*] ) *then D remains pre-dense in* **P**[*U* ] (*resp., pre-dense in* **P**[*U* ] *below p* ) *in each of the following three cases*:


**Proof.** We consider only the case of sets *D* pre-dense in **P**[*U*] itself; the case of pre-density below some *p* ∈ **P**[*U*] is treated similarly.

(i) Suppose, towards the contrary, that a condition *p* ∈ **P**[*U* ] is incompatible with each *q* ∈ *D*. As *D* ⊆ **P**[*U*], we can w.l. o. g. assume that |*p*|⊆|*U*|.

Our plan is to define a condition *p* ∈ **P**[*U*], also incompatible with each *q* ∈ *D*, contrary to the pre-density. To maintain such a construction, consider the finite string *<sup>f</sup>* <sup>=</sup> *<sup>f</sup>*1, ... , *fm* of all elements *<sup>f</sup>* ∈ **Fun** occurring in - *<sup>ν</sup>*∈|*p*<sup>|</sup> *Fp*(*ν*) but not in *<sup>U</sup>*. It follows from *<sup>U</sup> <sup>M</sup> <sup>U</sup>* that *<sup>f</sup>* is Cohen-generic over *M*. Further analysis shows that *p* being incompatible with *D* is implied by the fact that *f* meets a certain *M*-countable family of Cohen-dense sets. Therefore, we can simulate this in *M*, getting a string *g* ∈ *M* which meets the same Cohen-dense sets, and hence yields a condition *p* ∈ **P**[*U*], also incompatible with each *q* ∈ *D*.

This argument was first carried out in [9] in full generality, where we address the reader. However, to present the key idea in sufficient detail in a somewhat simplified subcase, we assume that (1) |*p*| = {*ν*} is a singleton; *ν* ∈ |*U*|. Then *p*(*ν*) = *Sp*(*ν*); *Fp*(*ν*) ∈ *P*[*U* (*ν*)], where *Sp*(*ν*) ⊆ **Seq** and *Fp*(*ν*) ⊆ *U* (*ν*) are finite sets. The (finite) set *X* = *Fp*(*ν*) *U*(*ν*) is multiply Cohen generic over *M* since *U <sup>M</sup> U* . To make the argument even more transparent, we suppose that (2) *X* = { *f* , *g*}, where *f* = *g* and the pair *f* , *g* is Cohen generic over *M*. (The general case follows the same idea and can be found in [9]; we leave it to the reader.)

Thus *Fp*(*ν*) = *F* ∪ { *f* , *g*}, where *F* = *Fp*(*ν*) ∩ *U*(*ν*) ∈ *M* is by definition a finite set.

The plan is to replace the functions *f* , *g* by some functions *f* , *g* ∈ *U*(*ν*) so that the incompatibility of *p* with conditions in *D* will be preserved.

It holds by the choice of *p* and Lemma 1 that *D* = *D*1(*f* , *g*) ∪ *D*<sup>2</sup> , where

$$\begin{aligned} D\_1(f,\emptyset) &= \{ q \in D : A\_{\emptyset} \cap F\_p^\vee(\nu) \neq \mathcal{Q} \}, \text{ where } A\_{\emptyset} = S\_{\emptyset}(\nu) \, \lhd S\_p(\nu) \subseteq \mathbf{Seq} \, \forall \nu\\ D\_2 &= \{ q \in D : (S\_p(\nu) \, \lhd S\_q(\nu)) \cap F\_q^\vee(\nu) \neq \mathcal{Q} \} \in M; \end{aligned}$$

and *D*<sup>1</sup> depends on *f* , *g* via *Fp*(*ν*). (See Section 3.1 on notation.) The equality *D* = *D*1(*f* , *g*) ∪ *D*<sup>2</sup> ∪ *D*<sup>3</sup> can be rewritten as Δ ⊆ *D*1(*f* , *g*), where Δ = *D D*<sup>2</sup> ∈ *M*. Further, Δ ⊆ *D*1(*f* , *g*) is equivalent to

$$\mathcal{A}(\*) \quad \forall A \in \mathcal{A} \; (A \cap F\_p^\vee(\nu) \neq \mathcal{Q}), \text{ where } \mathcal{A}' = \{A\_{\emptyset} : q \in D\} \in M\_{\mathcal{A}}$$

and each *Aq* = *Sq*(*ν*) *Sp*(*ν*) ⊆ **Seq** is finite. Recall that *Fp*(*ν*) = *F* ∪ { *f* , *g*}, therefore *F*<sup>∨</sup> *<sup>p</sup>* (*ν*) = *<sup>Z</sup>* ∪ *<sup>S</sup>*(*<sup>f</sup>* , *<sup>g</sup>*), where *<sup>Z</sup>* = {*h<sup>m</sup>* : *<sup>m</sup>* ≥ <sup>1</sup> ∧ *<sup>h</sup>* ∈ *<sup>F</sup>*} ∈ *<sup>M</sup>* and *<sup>S</sup>*(*<sup>f</sup>* , *<sup>g</sup>*) = - *<sup>m</sup>*≥1{ *<sup>f</sup> m*, *<sup>g</sup>m*}. Thus (∗) is equivalent to

(†) ∀ *A* ∈ A (*A* ∩ *S*(*f* , *g*) = ∅), where A = {*Aq Z* : *q* ∈ *D*} ∈ *M*.

Note that each *A* ∈ A is a finite subset of **Seq**, so we can reenumerate A = {*A <sup>k</sup>* : *k* < *ω*} in *M* and rewrite (†) as follows:

(‡) ∀ *k* (*A <sup>k</sup>* ∩ *S*(*f* , *g*) = ∅), where each *A <sup>k</sup>* ⊆ **Seq** is finite.

As the pair *f* , *g* is Cohen-generic, there is a number *m*<sup>0</sup> such that (‡) is forced over *M* by *σ*0, *τ*<sup>0</sup>, where *σ*<sup>0</sup> = *f m*<sup>0</sup> and *τ*<sup>0</sup> = *gm*<sup>0</sup> . In other words, *A <sup>k</sup>* ∩ *S*(*f* , *g* ) = ∅ holds for all *k* whenever *f* , *g*  is Cohen-generic over *M* and *σ*<sup>0</sup> ⊂ *f* , *τ*<sup>0</sup> ⊂ *g* . It follows that for any *k* and strings *σ*, *τ* ∈ **Seq** extending resp. *σ*0, *τ*<sup>0</sup> there are strings *σ* , *τ* ∈ **Seq** extending resp. *σ*, *τ*, at least one of which extends one of *w* ∈ *A <sup>k</sup>* . This allows us to define, in *M*, a pair of *f* , *g* ∈ **Fun** such that *σ*<sup>0</sup> ⊂ *f* , *τ*<sup>0</sup> ⊂ *g* , and for any *k* at least one of *f* , *g* extends one of *w* ∈ *A <sup>k</sup>* . In other words, we have

$$
\forall k \left( A'\_k \cap \mathcal{S}(f', \emptyset') \neq \mathcal{Q} \right) \quad \text{and} \quad \forall A' \in \mathcal{A'} \left( A' \cap \mathcal{S}(f', \emptyset') \neq \mathcal{Q} \right).
$$

It follows that the condition *p* defined by |*p* | = {*ν*}, *Sp* (*ν*) = *Sp*(*ν*), *Fp* (*ν*) = *F* ∪ { *f* , *g* }, still satisfies ∀ *A* ∈ A (*A* ∩ *F*<sup>∨</sup> *<sup>p</sup>* (*ν*) = ∅) (compare with (∗)), and further *D* = *D*1(*f* , *g* ) ∪ *D*<sup>2</sup> ∪ *D*<sup>3</sup> , therefore, *p* is incompatible with each *q* ∈ *D*. Yet *p* ∈ *M* since *f* , *g* ∈ *M*, which contradicts the pre-density of *D*.

(ii) The above proof works with *M*[*G*] instead of *M* since the set *X* as in the proof is multiple Cohen generic over *M*[*G*] by the product forcing theorem.

(iii) Assuming w.l. o. g. that *H* ⊆ *U* (*ν*0) *U*(*ν*0), we conclude that *M*[*H*] is a Cohen generic extension of *M*. Following the above, let *ν* ∈ |*U*|, *ν* = *ν*<sup>0</sup> . By the definition of the set *F* = *Fp*(*ν*) *U*(*ν*) is multiply Cohen generic not only over *M* but also over *M*[*H*]. This allows to carry out the same argument as above.

**Corollary 4.** (i) *Assume that, in* **L***, M*, *U* ∈ **sJS***, and M*, *U M* , *U*  ∈ **JS***. Let a set G* ⊆ **P**[*U* ] *be* **P**[*U* ]*-generic over M . Then G* ∩ **P**[*U*] *is* **P**[*U*]*-generic over M.*

(ii) *If moreover, K* ∈ *M, K* ⊆ **P**[*U*] *is a regular subforcing, then G* ∩ *K is K-generic over M.*

**Proof.** To prove (i), note that if a set *D* ∈ *M*, *D* ⊆ **Q**(*U*), is pre-dense in **Q**(*U*), then it is pre-dense in **Q**(*U* ) by Theorem 8, and hence *G* ∩ *D* = ∅ by the genericity. To prove (ii), apply Lemma 8.

The next corollary returns us to names, the material of Sections 3.4 and 3.5.

**Corollary 5** (of Theorem 8(i))**.** *In* **L***, suppose that M*, *U* ∈ **sJS***, M*, *U M* , *U*  ∈ **JS***, and X*, *Y belong to M. Assume that <sup>τ</sup>* <sup>∈</sup> *<sup>M</sup>* <sup>∩</sup> **SN***<sup>X</sup> <sup>Y</sup>* (**P**[*U*]) *is a* **P**[*U*]*-full name. Then τ remains* **P**[*U* ]*-full. If moreover p* ∈ **P**[*U*] *and τ is* **P**[*U*]*-full below p, then τ remains* **P**[*U* ]*-full below p.*

#### *4.3. Digression: Definability in* HC

The next subsection will contain a transfinite construction of a key forcing notion in **L** relativized to HC. Recall that HC is the collection of all *hereditarily countable* sets. In particular, HC = **L***ω*<sup>1</sup> in **L**. In matters of related definability classes, we refer to e.g., Part B, Chapter 5, Section 4 in [20], or Chapter 13 in [21], on the Lévy hierarchy of <sup>∈</sup>-formulas and definability classes *<sup>Σ</sup><sup>X</sup> <sup>n</sup>* , *Π<sup>X</sup> <sup>n</sup>* , Δ*<sup>X</sup> <sup>n</sup>* for any set *X*, and especially on *Σ*HC *<sup>n</sup>* , *Π*HC *<sup>n</sup>* , ΔHC *<sup>n</sup>* for *X* = HC in Sections 8 and 9 in [22], or elsewhere. In particular,

*Σ*HC *<sup>n</sup>* = all sets *X* ⊆ HC, definable in HC by a parameter-free *Σ<sup>n</sup>* formula. **Σ**HC *<sup>n</sup>* = all sets *X* ⊆ HC definable in HC by a *Σ<sup>n</sup>* formula with sets in HC as parameters.

Something like *Σ*HC *<sup>n</sup>* (*x*), *<sup>x</sup>* <sup>∈</sup> HC, means that only *<sup>x</sup>* is admitted as a parameter, while *<sup>Σ</sup>*HC *<sup>n</sup>* (*M*), where *M* ⊆ HC is a transitive model, means that all *x* ∈ *M* are admitted as parameters. Collections like *Π*HC *<sup>n</sup>* , *Π*HC *<sup>n</sup>* (*x*), *Π*HC *<sup>n</sup>* (*M*) are defined similarly, and ΔHC *<sup>n</sup>* = *Σ*HC *<sup>n</sup>* <sup>∩</sup> *<sup>Π</sup>*HC *<sup>n</sup>* , etc.. The boldface classes are defined as follows: **Σ**HC *<sup>n</sup>* = *Σ*HC *<sup>n</sup>* (HC), **Π**HC *<sup>n</sup>* = *Π*HC *<sup>n</sup>* (HC), **Δ**HC *<sup>n</sup>* = ΔHC *<sup>n</sup>* (HC).

**Remark 2.** It is known that the classes *Σ*HC *<sup>n</sup>* , *Π*HC *<sup>n</sup>* , ΔHC *<sup>n</sup>* are equal to resp. *Σ*<sup>1</sup> *<sup>n</sup>*+<sup>1</sup> , *<sup>Π</sup>*<sup>1</sup> *<sup>n</sup>*+<sup>1</sup> , <sup>Δ</sup><sup>1</sup> *<sup>n</sup>*+<sup>1</sup> for sets of reals, and the same for parameters and boldface classes. This well-known result was explicitly mentioned in [23] (Lemma on p. 281), a detailed proof see Lemma 25.25 in [21], or Theorem 9.1 in [22].

**Remark 3.** Recall that <**<sup>L</sup>** is the Gödel wellordering of **L**, the constructible universe.

It is known that the restriction <**L** HC is a ΔHC <sup>1</sup> relation, and if *<sup>n</sup>* <sup>≥</sup> 1, *<sup>p</sup>* <sup>∈</sup> *<sup>ω</sup><sup>ω</sup>* is any parameter, and *R*(*x*, *y*, *z*, ...) is a finitary ΔHC *<sup>n</sup>* (*p*) relation on HC then the relations ∃ *x* <**<sup>L</sup>** *y R*(*x*, *y*, *z*, ...) and <sup>∀</sup> *<sup>x</sup>* <sup>&</sup>lt;**<sup>L</sup>** *y R*(*x*, *<sup>y</sup>*, *<sup>z</sup>*,...) (with arguments *<sup>y</sup>*, *<sup>z</sup>*, . . . ) are <sup>Δ</sup>HC *<sup>n</sup>* (*p*) as well.

#### *4.4. Complete Sequences and the Basic Notion of Forcing*

Say that a pair *M*, *U* ∈ **sJS** solves a set *D* ⊆ **sJS** iff either *M*, *U* ∈ *D*, or there is no pair *M* , *U* <sup>∈</sup> *<sup>D</sup>* extending *<sup>M</sup>*, *<sup>U</sup>*. Let *<sup>D</sup>*solv be the set of all pairs *<sup>M</sup>*, *<sup>U</sup>*<sup>∈</sup> **sJS** which solve *<sup>D</sup>*.

**Definition 15.** *Let <sup>n</sup>* <sup>≥</sup> <sup>3</sup>*. A sequence* {*M<sup>ξ</sup>* , *<sup>U</sup><sup>ξ</sup>* }*ξ*<*ω*<sup>1</sup> <sup>∈</sup> −→**JS***ω*<sup>1</sup> *is n- complete iff it intersects every set of the form D*solv *, where D* <sup>⊆</sup> **sJS** *is* **<sup>Σ</sup>**HC *<sup>n</sup>*−<sup>2</sup> *. (See Section 4.3 on definability classes in* HC*.)*

Let us prove the existence of complete sequences.

**Theorem 9** (in **<sup>L</sup>**)**.** *Let <sup>n</sup>* <sup>≥</sup> <sup>2</sup>*. There is a sequence* {*M<sup>ξ</sup>* , *<sup>U</sup><sup>ξ</sup>* }*ξ*<*ω*<sup>1</sup> <sup>∈</sup> −→**JS***ω*<sup>1</sup> *of class* <sup>Δ</sup>HC *<sup>n</sup>*−<sup>1</sup> *, n-complete in case n* ≥ <sup>3</sup>*, and such that <sup>ξ</sup>* ∈ |*Uξ*+1| *for all <sup>ξ</sup> — hence the limit system U* = *<sup>ξ</sup>*<*ω*<sup>1</sup> *U<sup>ξ</sup> satisfies* |*U*| = I *.*

**Proof.** Define pairs *M<sup>ξ</sup>* , *U<sup>ξ</sup>* , *ξ* < *ω*<sup>1</sup> , by induction. Let *U*<sup>0</sup> be the null system with |*U*0| = ∅, and *M*<sup>0</sup> be the least CTM of **ZFC**− <sup>1</sup> . If *<sup>λ</sup>* < *<sup>ω</sup>*<sup>1</sup> is limit then put *<sup>U</sup><sup>λ</sup>* = *<sup>ξ</sup>*<*<sup>λ</sup> U<sup>ξ</sup>* and let *M<sup>λ</sup>* be the least CTM of **ZFC**− <sup>1</sup> containing the sequence {*M<sup>ξ</sup>* , *U<sup>ξ</sup>* }*ξ*<*<sup>λ</sup>* . If *M<sup>ξ</sup>* , *U<sup>ξ</sup>*  ∈ **sJS** is defined then by Lemma 14 there is a pair *M* , *U*  ∈ **sJS** with *M<sup>ξ</sup>* , *U<sup>ξ</sup>* ≺*M* , *U*  and *ξ* ∈ |*U* |. Let Θ ⊆ *ω*<sup>1</sup> × HC be a universal *Σ*HC *<sup>n</sup>*−<sup>2</sup> set, and *<sup>D</sup><sup>ξ</sup>* <sup>=</sup> {*<sup>z</sup>* <sup>∈</sup> **sJS** : *<sup>ξ</sup>*, *<sup>z</sup>*<sup>∈</sup> <sup>Θ</sup>}. Let *<sup>M</sup>ξ*<sup>+</sup>1, *<sup>U</sup>ξ*+<sup>1</sup> be the <sup>&</sup>lt;**L**-least pair *<sup>M</sup>*, *<sup>U</sup>*<sup>∈</sup> *<sup>D</sup><sup>ξ</sup>* solv satisfying *M* , *U M*, *U*. To check the definability property use the fact mentioned by Remark 3 in Section 4.3.

Now define the basic forcing notion.

**Definition 16** (in **<sup>L</sup>**)**.** *Fix a number* **<sup>n</sup>** <sup>≥</sup> <sup>2</sup>*. Let* {**M***<sup>ξ</sup>* , **<sup>U</sup>***<sup>ξ</sup>* }*ξ*<*ω*<sup>1</sup> <sup>∈</sup> −→**JS***ω*<sup>1</sup> *be any* **<sup>n</sup>***-complete Jensen–Solovay sequence of class* ΔHC **<sup>n</sup>**−<sup>1</sup> *as in Theorem 9—in case* **<sup>n</sup>** <sup>≥</sup> <sup>3</sup>*, or just any Jensen–Solovay sequence of class* <sup>Δ</sup>HC <sup>1</sup> *—in case* **<sup>n</sup>** = <sup>2</sup>*, and in both cases <sup>ξ</sup>* ∈ |**U***ξ*+1| *for every <sup>ξ</sup>* < *<sup>ω</sup>*<sup>1</sup> *, as in Theorem 9. Put* **<sup>U</sup>** = *<sup>ξ</sup>*<*ω*<sup>1</sup> **U***<sup>ξ</sup> , so* **U** *is a system,* |**U**| = I = *<sup>ω</sup>*<sup>1</sup> *,* **<sup>U</sup>**(*ν*) = - *<sup>ξ</sup>*<*ω*1,*ν*∈|**U***<sup>ξ</sup>* <sup>|</sup> **<sup>U</sup>***<sup>ξ</sup>* (*ν*) *for all <sup>ν</sup>* ∈ I *. We finally define* **<sup>P</sup>** = **<sup>P</sup>**[**U**] *and* **P***<sup>γ</sup>* = **P**[**U***γ*] *for γ* < *ω*<sup>1</sup> *.*

Thus **P** is the product of sets **P**(*ν*) = *P*[**U**(*ν*)], *ν* ∈ I , with finite support. We proceed with a couple of simple lemmas.

**Corollary 6.** *Suppose that, in* **<sup>L</sup>***, <sup>M</sup> is a transitive model of* **ZFC**<sup>−</sup> *containing the sequence* {**M***<sup>ξ</sup>* , **<sup>U</sup>***<sup>ξ</sup>* }*ξ*<*ω*<sup>1</sup> <sup>∈</sup> −→**JS***ω*<sup>1</sup> *of Definition 16. Then, for every <sup>ξ</sup>* <sup>&</sup>lt; *<sup>ω</sup>*<sup>1</sup> :


**Proof.** To prove (i) use Lemma 15. Both claims of (ii) hold by Definition 13. To prove (iii) and (iv) use Corollary 5. Finally, (v) follows from (iii).

Now let us address definability issues.

**Lemma 16** (in **<sup>L</sup>**)**.** *The binary relation f* <sup>∈</sup> **<sup>U</sup>**(*ν*) *is* <sup>Δ</sup>HC **<sup>n</sup>**−<sup>1</sup> *.*

*The sets* **P** *and* **SN***<sup>ω</sup> <sup>ω</sup>*(**P**) (**P**-names for functions in **Fun**) *are* ΔHC **<sup>n</sup>**−<sup>1</sup> *. The set of all* **P***-full names in* **SN***<sup>ω</sup> <sup>ω</sup>*(**P**) *is* ΔHC **<sup>n</sup>**−<sup>1</sup> *.*

**Proof.** The sequence {**M***<sup>ξ</sup>* , **<sup>U</sup>***<sup>ξ</sup>* }*ξ*<*ω*<sup>1</sup> is <sup>Δ</sup>HC **<sup>n</sup>**−<sup>1</sup> by definition, hence the relation *<sup>f</sup>* <sup>∈</sup> **<sup>U</sup>**(*ν*) is *<sup>Σ</sup>*HC **<sup>n</sup>**−<sup>1</sup> . On the other hand, if *f* ∈ **Fun** belongs to some **M***<sup>ξ</sup>* then *f* ∈ **U**(*ν*) obviously implies *f* ∈ **U***<sup>ξ</sup>* (*ν*), leading to a *Π*HC **<sup>n</sup>**−<sup>1</sup> definition of the relation *<sup>f</sup>* <sup>∈</sup> **<sup>U</sup>**(*ν*). To prove the last claim, note that by Corollary <sup>5</sup> if a name *<sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**P***<sup>ξ</sup>* ) ∩ **M***<sup>ξ</sup>* is **P***<sup>ξ</sup>* -full then it remains **P**-full.

#### *4.5. Basic Generic Extension and Regular Subextensions*

Recall that an integer **n** ≥ 2 and sets **U***<sup>ξ</sup>* , **M***<sup>ξ</sup>* , **U**, **P***<sup>ξ</sup>* , **P** are fixed in **L** by Definition 16. These sets are fixed for the remainder.

Suppose that, in **L**, *K* ⊆ **P** is a regular subforcing. If *G* ⊆ **P** is a set **P**-generic over **L** then *G* ∩ *K* is *K*-generic over **L** by Lemma 9(vi), and hence **L**[*G* ∩ *K*] is a *K*-generic extension of **L**. The following formulas **Γ***<sup>i</sup>* (*i* ∈ I ) will give us a useful coding tool in extensions of this form:

**<sup>Γ</sup>***ν*(*S*) :=def *<sup>ν</sup>* ∈I∧ *<sup>S</sup>* <sup>⊆</sup> **Seq** ∧ ∀ *<sup>f</sup>* <sup>∈</sup> **Fun** <sup>∩</sup> **<sup>L</sup>** (*<sup>f</sup>* <sup>∈</sup> **<sup>U</sup>**(*ν*) ⇐⇒ max (*S*/ *<sup>f</sup>*) <sup>&</sup>lt; *<sup>ω</sup>*).

This is based on the next two results. Recall that |*<sup>G</sup>* ∩ *<sup>K</sup>*| = - *<sup>p</sup>*∈*G*∩*<sup>K</sup>* |*p*|.

**Lemma 17. Γ***ν*(*S*) *as a binary relation belongs to Π*HC **<sup>n</sup>**−<sup>1</sup> *in any cardinal-preserving generic extension of* **<sup>L</sup>***.*

**Proof.** The set *<sup>W</sup>* <sup>=</sup> {*ν*, *<sup>f</sup>* : *<sup>ν</sup>* ∈ I∧ *<sup>f</sup>* <sup>∈</sup> **<sup>U</sup>**(*ν*)} is <sup>Δ</sup>HC **<sup>n</sup>**−<sup>1</sup> in **<sup>L</sup>**, by Lemma 16, and hence so is *W* = {*ν*, *f* : *ν* ∈ I∧ *f* ∈ **Fun U**(*ν*)}. Let *ϕ*(*ν*, *f*) and *ϕ* (*ν*, *<sup>f</sup>*) be *<sup>Σ</sup>***n**−<sup>1</sup> formulas that define resp. *W*, *W* in HC, in **L**. Then, in any generic extension of **L**, **Γ***ν*(*S*) is equivalent to *ν* ∈I∧ *S* ⊆ **Seq** ∧ ∀ *<sup>f</sup>* <sup>∈</sup> **Fun** <sup>∩</sup> **<sup>L</sup>** <sup>Ψ</sup>(*ν*, *<sup>f</sup>*), where <sup>Ψ</sup>(*ν*, *<sup>f</sup>*) is the *<sup>Π</sup>***n**−<sup>1</sup> formula

$$\left( (\mathbf{L} \mid = q(\mathbf{v}, f)) \implies \max \left( \mathbf{S} / f \right) < \omega \right) \land \left( (\mathbf{L} \mid = q'(\mathbf{v}, f)) \implies \max \left( \mathbf{S} / f \right) = \omega \right). \tag{7}$$

**Theorem 10.** *Suppose that, in* **L***, K* ⊆ **P** *is a regular subforcing. Let G* ⊆ **P** *be* **P***-generic over* **L***. Then*:


**Proof.** (i) This is a corollary of Lemma 9(vi).

(ii) Suppose towards the contrary that some *<sup>S</sup>* ∈ **<sup>L</sup>**[*<sup>G</sup>* ∩ *<sup>K</sup>*] satisfies **<sup>Γ</sup>***ν*(*S*). Note that *<sup>S</sup>* ∈ **<sup>L</sup>**[*G* =*ν*] by Lemma 9(iv). Now we can forget about the given set *K*. It follows from Theorem 5(iii) (with *<sup>K</sup>* = **<sup>P</sup>** =*<sup>ν</sup>* ), that there is a name *<sup>τ</sup>* ∈ **SNSeq**(**P** =*ν*) such that *<sup>S</sup>* = *<sup>τ</sup>*[*G* =*ν*]. There is an ordinal *<sup>ξ</sup>* < *<sup>ω</sup>*<sup>1</sup> satisfying *<sup>τ</sup>* <sup>∈</sup> **<sup>M</sup>***<sup>ξ</sup>* and *<sup>τ</sup>* <sup>∈</sup> **SNSeq**(**P***<sup>ξ</sup>* =*ν*). Then *<sup>S</sup>* <sup>=</sup> *<sup>τ</sup>*[*G<sup>ξ</sup>* =*ν*], where *<sup>G</sup><sup>ξ</sup>* <sup>=</sup> *<sup>G</sup>* <sup>∩</sup> **<sup>P</sup>***<sup>ξ</sup>* is **<sup>P</sup>***<sup>ξ</sup>* -generic over **<sup>M</sup>***<sup>ξ</sup>* by Corollary 6, and hence *<sup>S</sup>* belongs to **<sup>M</sup>***<sup>ξ</sup>* [*G<sup>ξ</sup>* =*ν*].

Note that **U**(*ν*) is uncountable by Corollary 6(ii), and hence *F* = **U**(*ν*) **U***<sup>ξ</sup>* (*ν*) is uncountable. Let *<sup>f</sup>* <sup>∈</sup> *<sup>F</sup>*. Then *<sup>f</sup>* is Cohen generic over the model **<sup>M</sup>***<sup>ξ</sup>* by Corollary 6. On the other hand *<sup>G</sup><sup>ξ</sup>* =*<sup>ν</sup>* is **<sup>P</sup>***<sup>ξ</sup>* =*ν*-generic over **<sup>M</sup>***<sup>ξ</sup>* [ *<sup>f</sup>* ] by Theorem 8(iii). Therefore *<sup>f</sup>* is Cohen generic over **<sup>M</sup>***<sup>ξ</sup>* [*G<sup>ξ</sup>* =*ν*] as well.

Recall that *<sup>S</sup>* <sup>∈</sup> **<sup>M</sup>***<sup>ξ</sup>* [*G<sup>ξ</sup>* =*ν*] and **<sup>Γ</sup>***ν*(*S*) holds, hence max (*S*/ *<sup>f</sup>*) <sup>&</sup>lt; *<sup>ω</sup>*. As *<sup>f</sup>* is Cohen generic over **<sup>M</sup>***<sup>ξ</sup>* [*G<sup>ξ</sup>* =*ν*], it follows that there is a string *<sup>s</sup>* <sup>∈</sup> **Seq**, *<sup>s</sup>* <sup>⊂</sup> *<sup>f</sup>* , such that *<sup>S</sup>* contains no strings extending *s*. Take any *μ* ∈ I , *j* = *ν*. By Corollary 6(ii), there exists a function *g* ∈ **U**(*μ*) **U***<sup>ξ</sup>* (*μ*), *g* ∈/ **U**(*ν*), satisfying *<sup>s</sup>* <sup>⊂</sup> *<sup>g</sup>*. Then, max (*S*/*g*) = *<sup>ω</sup>* by **<sup>Γ</sup>***ν*(*S*). However, this is absurd by the choice of *<sup>s</sup>*.

**Corollary 7.** *Suppose that, in* **L***, K* ⊆ **P** *is a regular forcing. Let G* ⊆ **P** *be a set* **P***-generic over* **L***. Then*


**Proof.** Claim (i) follows from the theorem, because by the regularity we have *<sup>G</sup>* ∩ *<sup>K</sup>* ∈ **<sup>L</sup>**[*G* =*ν*] for all *ν* ∈ | / *G* ∩ *K*|. Claim (ii) immediately follows from Lemma 17. To prove (iii) note that, by (i) and (ii), it holds in **<sup>L</sup>**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*] that the set <sup>|</sup>*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*<sup>|</sup> is defined by a *<sup>Σ</sup>*HC **<sup>n</sup>** formula ∃ *S* **Γ***ν*(*S*) in HC.

#### **5. Forcing Approximations**

Here we define and study here an important forcing-like relation **forc**. It will give us control over various phenomena of analytic definability in the generic extensions considered.

We continue to assume **V** = **L** in the ground universe by Assumption 1.

#### *5.1. Models and Absolute Sets*

To consider transitive models of weaker theories, we let **ZFC**− be **ZFC** minus the Power Set axiom, with the schema of Collection instead of replacement, and **AC** in the form of well-orderability of every set. See [24] on **ZFC**− in detail.

Let **ZFC**− <sup>1</sup> be **ZFC**<sup>−</sup> plus the axioms **V** = **L** and "every set is at most countable".

Let *<sup>W</sup>* <sup>⊆</sup> HC. By definition, a set *<sup>X</sup>* <sup>⊆</sup> HC is <sup>Δ</sup>HC <sup>1</sup> (*W*) iff there exist a *Σ*<sup>1</sup> formula *σ*(*x*) and a *Π*<sup>1</sup> formula *π*(*x*), with sets in *W* as parameters, such that

$$\mathbf{X} = \{ \mathbf{x} \in \mathbf{HC} : \sigma^{\mathrm{HC}}(\mathbf{x}) \} = \{ \mathbf{x} \in \mathbf{HC} : \pi^{\mathrm{HC}}(\mathbf{x}) \}, \tag{4}$$

in particular, we have *<sup>σ</sup>*HC(*x*) ⇐⇒ *<sup>π</sup>*HC(*x*) for all *<sup>x</sup>*. However, generally speaking, this does not imply that *<sup>X</sup>* <sup>∩</sup> *<sup>M</sup>* <sup>∈</sup> <sup>Δ</sup>*<sup>M</sup>* <sup>1</sup> (*W*), where *M* ∈ HC is a countable transitive model (CTM). The goal of the next two definitions is to distinguish and formalize this kind of absoluteness.

**Definition 17.** *If for a given* ΔHC <sup>1</sup> (*W*) *set X, there exists such a pair of formulas, containing only parameters in <sup>W</sup> and satisfying* (4) *and* <sup>∀</sup> *<sup>x</sup>* <sup>∈</sup> *<sup>M</sup>* (*σM*(*x*) ⇐⇒ *<sup>π</sup>M*(*x*)) *for all countable transitive models <sup>M</sup>* <sup>|</sup><sup>=</sup> **ZFC**<sup>−</sup> *containing all parameters that occur in σ and/or in π, then we say that X is absolute* ΔHC <sup>1</sup> (*W*)*. In this case, if <sup>M</sup> is as indicated then the set <sup>X</sup>* <sup>∩</sup> *<sup>M</sup> is* <sup>Δ</sup>*<sup>M</sup>* <sup>1</sup> (*W*) *via the same pair of formulas. In particular, any* <sup>Δ</sup>HC <sup>0</sup> (*W*) *set is absolute* ΔHC <sup>1</sup> (*W*) *by obvious reasons.*

**Definition 18.** *In continuation of the last definition, a function f* : *D* → HC*, defined on a set D* ⊆ HC*, is absolute* ΔHC <sup>1</sup> (*W*) *function, if <sup>f</sup> is absolute* <sup>Δ</sup>HC <sup>1</sup> (*W*) *as a set of pairs in the sense of Definition 17, and in addition, if M* |= **ZFC**<sup>−</sup> *is a CTM and x* ∈ *D* ∩ *M then f*(*x*) ∈ *M.*

#### *5.2. Formulas*

Here we introduce a language that will help us study analytic definability in **P**[*U*]-generic extensions, for different systems *U*, and their submodels.

**Definition 19.** *Let* <sup>L</sup> *be the 2nd order Peano language, with variables of type 1 over <sup>ω</sup><sup>ω</sup> . If <sup>K</sup>* <sup>⊆</sup> **<sup>P</sup>**<sup>∗</sup> *then an* L(*K*) *formula is any formula of* L*, with some free variables of types* 0, 1 *replaced by resp. numbers in ω and names in* **SN***<sup>ω</sup> <sup>ω</sup>*(*K*)*, and some type 1 quantifiers are allowed to have bounding indices <sup>B</sup> (i.e.,* <sup>∃</sup>*<sup>B</sup> ,* <sup>∀</sup>*<sup>B</sup> ) such that B* ⊆ I *is finite or countable.*

Typically *K* will be a regular forcing in Definition 19, in the sense of Definition 8, or a regular subforcing of the form *K*[*U*], *U* being a system.

If *ϕ* is a L(**P**∗) formula then let

NAM *ϕ* = the set of all names *τ* that occur in *ϕ*; |*ϕ*| = - *<sup>τ</sup>*∈NAM *<sup>ϕ</sup>* |*τ*| (at most countable); IND *ϕ* = the set of all quantifier indices *B* which occur in *ϕ*; ||*ϕ*|| <sup>=</sup> <sup>|</sup>*ϕ*| ∪ - IND *ϕ* (at most countable).

Note that |*ϕ*| ⊆ ||*ϕ*|| ⊆ I provided *ϕ* is an L(**P**∗) formula.

If a set *G* ⊆ **P**<sup>∗</sup> is minimally *ϕ*-generic (i.e., minimally *τ*-generic w. r. t. every name *τ* ∈ NAM *ϕ*, in the sense of Section 3.5), then let the valuation *ϕ*[*G*] be the result of substitution of *τ*[*G*] for any name *<sup>τ</sup>* <sup>∈</sup> NAM *<sup>ϕ</sup>*, and changing each quantifier <sup>∃</sup>*Bx*, <sup>∀</sup>*Bx* to <sup>∃</sup> (<sup>∀</sup> ) *<sup>x</sup>* <sup>∈</sup> *<sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>**[*G <sup>B</sup>*] respectively, while index-free type 1 quantifiers are relativized to *<sup>ω</sup><sup>ω</sup>* ; *<sup>ϕ</sup>*[*G*] is a formula of <sup>L</sup> with real parameters, and with some quantifiers of type 1 explicitly relativized to certain submodels of **L**[*G*].

An arithmetic formula in <sup>L</sup>(*K*) is a formula with no quantifiers of type 1 (names in **SN***<sup>ω</sup> <sup>ω</sup>*(*K*) as in Definition <sup>19</sup> are allowed). If *<sup>n</sup>* <sup>&</sup>lt; *<sup>ω</sup>* then let a <sup>L</sup>*Σ*<sup>1</sup> *<sup>n</sup>*(*K*), resp., <sup>L</sup>*Π*<sup>1</sup> *<sup>n</sup>*(*K*) formula be a formula of the form

$$\exists^{\diamond} \exists^{\diamond} \mathbf{x}\_1 \forall^{\diamond} \mathbf{x}\_2 \dots \forall^{\diamond} (\exists^{\diamond}) \mathbf{x}\_{n-1} \exists \left(\forall\right) \mathbf{x}\_n \,\forall\,\,\,\forall^{\diamond} \mathbf{x}\_1 \exists^{\diamond} \mathbf{x}\_2 \dots \exists^{\diamond} (\forall^{\diamond}) \mathbf{x}\_{n-1} \,\forall\,(\exists) \,\mathbf{x}\_n \,\forall^{\diamond}$$

respectively, where *<sup>ψ</sup>* is an arithmetic formula in <sup>L</sup>(*K*), all variables *xi* are of type 1 (over *<sup>ω</sup><sup>ω</sup>* ), the sign ◦ means that this quantifier can have a bounding index as in Definition 19, and it is required that the rightmost (closest to the kernel *ψ*) quantifier doesn't have a bounding index.

If in addition *M* |= **ZFC**<sup>−</sup> is a transitive model and *U* ∈ *M* a system then define

<sup>L</sup>*Σ*<sup>1</sup> *<sup>n</sup>*(*K*[*U*], *<sup>M</sup>*) = all <sup>L</sup>*Σ*<sup>1</sup> *<sup>n</sup>*(*K*) formulas *<sup>ϕ</sup>* such that NAM *<sup>ϕ</sup>* <sup>⊆</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(*K*[*U*]) ∩ *M* and all indices *B* ∈ IND *ϕ* belong to *M* and satisfy *B* ⊆ |*U*|.

Define <sup>L</sup>*Π*<sup>1</sup> *<sup>n</sup>*(*K*[*U*], *<sup>M</sup>*) similarly. All formulas in <sup>L</sup>*Σ*<sup>1</sup> *<sup>n</sup>*(*K*[*U*], *<sup>M</sup>*)∪ L*Π*<sup>1</sup> *<sup>n</sup>*(*K*[*U*], *M*) are by definition (finite) strings in *M*.

#### *5.3. Forcing Approximation*

The next definition invents a convenient forcing-type relation **forc** for pairs *M*, *U* in **sJS** and formulas *ϕ* in L(*K*[*U*]), associated with the truth in *K*[*U*]-generic extensions of **L**, where *K* ⊆ **P**<sup>∗</sup> is a regular forcing. Recall that *K*[*U*] = *K* ∩ **P**[*U*] whenever *K* ⊆ **P**<sup>∗</sup> is a regular forcing and *U* is a system.

**Definition 20** (in **L**)**.** *We introduce a relation p <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> ϕ. First of all,*

	- *(a) M*, *U* ∈ **sJS***,*
	- *(b) K* <sup>⊆</sup> **<sup>P</sup>**<sup>∗</sup> *is a regular forcing and an absolute* <sup>Δ</sup>HC <sup>1</sup> (*M*) *set,*
	- *(c) p belongs to K*[*U*] *(a regular subforcing of* **P**[*U*] *by Lemma 7),*
	- *(d) <sup>ϕ</sup> is a closed formula in* <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* (*K*[*U*], *<sup>M</sup>*) ∪ L*Σ*<sup>1</sup> *<sup>k</sup>*+1(*K*[*U*], *M*) *for some k* ≥ 1*, and each name τ* ∈ NAM *ϕ is K*[*U*]*-full below p.*

*Under these assumptions, the sets U*, *K*[*U*], *p*, NAM *ϕ*, IND *ϕ belong to M. The property of K*[*U*] *fullness in* (F1)*d is equivalent to just* **P**[*U*]*-fullness, by Corollary 1, since K*[*U*] *is a regular subforcing of* **P**[*U*] *by Lemma 7.*

*The definition of* **forc** *goes on by induction on the complexity of formulas.*

	- *(a) p <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup>* <sup>∃</sup>*Bx <sup>ϕ</sup>*(*x*) *iff there is a name <sup>τ</sup>* <sup>∈</sup> *<sup>M</sup>* <sup>∩</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(*K*[*U*] *B*)*, K*[*U*]*-full below p (by* (F1)*d) and such that p <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> ϕ*(*τ*)*.*
	- *(b) p <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup>* <sup>∃</sup> *<sup>x</sup> <sup>ϕ</sup>*(*x*) *iff there is a name <sup>τ</sup>* <sup>∈</sup> *<sup>M</sup>* <sup>∩</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(*K*[*U*])*, K*[*U*]*-full below p (by* (F1)*d) and such that p <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> ϕ*(*τ*)*.*

**Lemma 18** (in **L**)**.** *Let K*, *M*, *U*, *p*, *ϕ satisfy* (F1) *of Definition 20. Then*:


Thus by the first claim of the lemma **forc** is monotone w. r. t. both the extension of pairs in **sJS** and the strengthening of forcing conditions.

**Proof.** (i) Let *<sup>ϕ</sup>* <sup>=</sup> *<sup>ϕ</sup>*(*τ*1, ... , *<sup>τ</sup>m*) be a closed formula in <sup>L</sup>*Π*<sup>1</sup> <sup>1</sup> (*K*[*U*], *M*), where all names *τ<sup>j</sup>* ∈ **SN***<sup>ω</sup> <sup>ω</sup>*(*K*[*U*]) ∩ *M* are *K*[*U*]-full below the condition *p* ∈ *K*[*U*] considered. Then all names *τ<sup>j</sup>* remain *K*[*U* ]-full below *p*, and below *q* as well since *q* ≤ *p*, by Corollary 5. Consider a set *G* ⊆ *K*[*U* ], *K*[*U* ]-generic over *M* and containing *q*. We have to prove that *ϕ*[*G* ] is true in *M* [*G* ]. Note that the set *G* = *G* ∩ *K*[*U*] is *K*[*U*]-generic over *M* by Corollary 4, and we have *p* ∈ *G*. Moreover the valuation *ϕ*[*G* ] coincides with *ϕ*[*G*] since all names in *ϕ* belong to **SN***<sup>ω</sup> <sup>ω</sup>*(*K*[*U*]). *ϕ*[*G*] is true in *M*[*G*] as *p <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> ϕ*. It remains to apply Mostowski's absoluteness between the models *M*[*G*] ⊆ *M* [*G* ].

The inductive steps related to (F3), (F4) of Definition 20 are easy.

Claim (ii) immediately follows from (F4) of Definition 20.

The next theorem classifies the complexity of **forc** in terms of projective hierarchy. Recall that all formulas in - *n* <sup>L</sup>*Σ*<sup>1</sup> *<sup>n</sup>*(*K*, *<sup>M</sup>*) ∪ L*Π*<sup>1</sup> *<sup>n</sup>*(*K*, *M*) are by definition (finite) strings in *M*. This allows us to consider and analyze sets

**Forc***<sup>K</sup> w*(*Σ*<sup>1</sup> *<sup>n</sup>*) = *<sup>M</sup>*, *<sup>U</sup>*, *<sup>p</sup>*, *<sup>ϕ</sup>* : *<sup>M</sup>*, *<sup>U</sup>*<sup>∈</sup> **sJS** <sup>∧</sup> *<sup>w</sup>* <sup>∈</sup> *<sup>M</sup>* <sup>∧</sup> *<sup>p</sup>* <sup>∈</sup> *<sup>K</sup>*[*U*] <sup>∧</sup> *<sup>ϕ</sup>* is a closed <sup>L</sup>*Σ*<sup>1</sup> *<sup>n</sup>*(*K*[*U*], *<sup>M</sup>*) formula <sup>∧</sup> *<sup>p</sup> <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> <sup>ϕ</sup>* ;

and similarly defined **Forc***<sup>K</sup> w*(*Π*<sup>1</sup> *<sup>n</sup>*), where it is assumed that *<sup>w</sup>* <sup>∈</sup> *<sup>ω</sup><sup>ω</sup>* and *<sup>K</sup>* <sup>⊆</sup> **<sup>P</sup>**<sup>∗</sup> is a regular forcing and an absolute ΔHC <sup>1</sup> (*w*) set.

**Theorem 11** (in **<sup>L</sup>**)**.** *Let w* <sup>∈</sup> *<sup>ω</sup><sup>ω</sup> and K* <sup>⊆</sup> **<sup>P</sup>**<sup>∗</sup> *be a regular forcing and an absolute* <sup>Δ</sup>HC <sup>1</sup> (*w*) *set. Then*:


**Proof** (sketch)**.** Suppose that *<sup>ϕ</sup>* is <sup>L</sup>*Π*<sup>1</sup> <sup>1</sup> . Under the assumptions of the theorem, items (F1)a, (F1)c, (F1)d of Definition 20(F1) are ΔHC <sup>1</sup> (*w*) relations, (F1)b is automatic, while (F2) is reducible to a forcing relation over *M* that we can relativize to *M*. The inductive step goes on straightforwardly using (F3), (F4) of Definition 20. Note that the quantifier over names in (F3) is a bounded quantifier (bounded by *M*), hence it does not add any extra complexity.

#### *5.4. Advanced Properties of Forcing Approximations*

The following lemma works whenever the domain *K* ⊆ **P**<sup>∗</sup> (a regular forcing) of conditions *p* related to the definition of *p <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> ϕ* is bounded by a set *c* ⊆ I . (Compare with Theorem 7.)

**Lemma 19** (restriction lemma, in **L**)**.** *Suppose that K*, *M*, *U*, *p*, *ϕ satisfy* (F1) *of Definition 20, a set <sup>c</sup>* ⊆ I *is absolute* <sup>Δ</sup>HC <sup>1</sup> (*M*)*, K* <sup>⊆</sup> **<sup>P</sup>**<sup>∗</sup> *c, and p <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> <sup>ϕ</sup>. Then p <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> <sup>c</sup> ϕ.*

Note that |*U*| ⊆ *c* is not assumed in the lemma. On the other hand, we have |*p*| ⊆ *c* by Definition 20(F1)c, because *p* ∈ *K*[*U*] and *K* ⊆ **P**<sup>∗</sup> *c*, and |*ϕ*| ⊆ *c* holds because *ϕ* is an L(*K*[*U*]) formula. In addition, *U c* ∈ *M* by the choice of *c*.

**Proof.** The direction ⇐= immediately follows from Lemma 18(i) since we have *M*, *U c M*, *U* by Remark 1 in Section 4.1. Prove the opposite implication by induction.

**Case of** <sup>L</sup>*Π*<sup>1</sup> <sup>1</sup> **formulas**: *K*[*U*] = *K*[*U c*] under the assumptions of the lemma.

**Step** <sup>L</sup>*Π*<sup>1</sup> *<sup>n</sup>* → L*Σ*<sup>1</sup> *<sup>n</sup>*+<sup>1</sup> **.** Let *<sup>ψ</sup>*(*x*) be a <sup>L</sup>*Π*<sup>1</sup> *<sup>n</sup>*(*K*[*U*], *<sup>M</sup>*) formula, and *<sup>ϕ</sup>* be <sup>∃</sup>*Bx <sup>ψ</sup>*(*x*), *<sup>B</sup>* ⊆ I , *<sup>B</sup>* <sup>∈</sup> *<sup>M</sup>*. If *p <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> <sup>ϕ</sup>* then there is a name *<sup>τ</sup>* <sup>∈</sup> *<sup>M</sup>* <sup>∩</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(*K*[*U*] *B*) such that *p <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> ψ*(*τ*). We conclude that *p <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> <sup>c</sup> <sup>ψ</sup>*(*τ*) by the inductive hypothesis. However we have **SN***<sup>ω</sup> <sup>ω</sup>*(*K*[*U*] *B*) = **SN***<sup>ω</sup> <sup>ω</sup>*(*K*[*U c*] *B*) since <sup>|</sup>*K*| ⊆ *<sup>c</sup>*. Thus *<sup>p</sup> <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> <sup>c</sup> ϕ*. The case *ϕ* being ∃ *x ψ*(*x*) is similar.

**Step** <sup>L</sup>*Σ*<sup>1</sup> *<sup>n</sup>* → L*Π*<sup>1</sup> *<sup>n</sup>* **,** *<sup>n</sup>* <sup>≥</sup> <sup>2</sup>**.** Let *<sup>ϕ</sup>* be a <sup>L</sup>*Π*<sup>1</sup> *<sup>n</sup>*(*K*[*U*], *M*) formula. Suppose towards the contrary that *p <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> <sup>ϕ</sup>* holds, but *<sup>p</sup> <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> <sup>c</sup> ϕ* fails, so that there exist a pair *M* , *V* ∈ **sJS** and a condition *q* ∈ *K*[*V*], such that *M*, *U c M* , *<sup>V</sup>*, *<sup>q</sup>* <sup>≤</sup> *<sup>p</sup>*, and *<sup>q</sup> <sup>K</sup>***forc***<sup>M</sup> <sup>V</sup> <sup>ϕ</sup>*<sup>¬</sup> . Then *<sup>q</sup> <sup>K</sup>***forc***<sup>M</sup> <sup>V</sup> <sup>c</sup> <sup>ϕ</sup>*<sup>¬</sup> by the inductive hypothesis. Note that |*q*| ⊆ *c* by the choice of *K*, but not necessarily |*V*| ⊆ *c*.

Define a system *W* ∈ *M* such that |*W*| = (|*V*| ∩ *c*) ∪ (|*U*| *c*), *W* (|*V*| ∩ *c*) = *V* (|*V*| ∩ *c*), and *W* (|*U*| *c*) = *U* (|*U*| *c*). Then *M* , *V c M* , *<sup>W</sup>*, therefore still *<sup>q</sup> <sup>K</sup>***forc***<sup>M</sup> <sup>W</sup> ϕ*<sup>¬</sup> by Lemma 18(i).

Now we claim that *M*, *U M* , *W*. Indeed, suppose that *ν* ∈ |*U*|. If *ν* ∈/ *c* then *W*(*ν*) = *U*(*ν*). If *ν* ∈ *c* then *U*(*ν*) ⊆ *V*(*ν*) = *W*(*ν*) by construction. It follows that |*U*|⊆|*W*|, *U W*, and Δ(*U*, *W*) ⊆ Δ(*U c*, *V*)—which implies *U <sup>M</sup> W*, since *M*, *U c M* , *V*. Thus *M*, *U M* , *W*.

We have *<sup>q</sup>* <sup>≤</sup> *<sup>p</sup>* as well. This contradicts the assumption *<sup>p</sup> <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> ϕ* by Lemma 18(ii).

**Lemma 20** (in **L**)**.** *Let K*, *M*, *U*, *p*, *ϕ*, *k satisfy* (F1) *of Definition 20,* NAM *ϕ* = {*τ*1, ... , *τ<sup>m</sup>* }*, μ*1, ... , *μ<sup>m</sup> be another list of names in* **SN***<sup>ω</sup> <sup>ω</sup>*(*K*[*U*]) ∩ *M, K*[*U*]*-full below p and such that τ and μ are equivalent below p for each* = 1, . . . , *m. Then p <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> <sup>ϕ</sup>*(*τ*1,..., *<sup>τ</sup>m*) *iff p <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> ϕ*(*μ*1,..., *μm*)*.*

**Proof.** It suffices to consider the case of *Π*<sup>1</sup> <sup>1</sup> formulas; the induction steps <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* → L*Σ*<sup>1</sup> *<sup>k</sup>*+<sup>1</sup> and <sup>L</sup>*Σ*<sup>1</sup> *<sup>k</sup>* → L*Π*<sup>1</sup> *<sup>k</sup>* are rather easy.

Suppose that *<sup>ϕ</sup>* is <sup>L</sup>*Π*<sup>1</sup> <sup>1</sup> and *<sup>p</sup> <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> ϕ*(*τ*1, ... , *τm*). Suppose that *G* ⊆ *K*[*U*] is a set *K*[*U*]-generic over *M*, and *p* ∈ *G*. We claim that *τ*[*G*] = *μ*[*G*] for all ; this obviously implies the result required. Suppose that this is not the case. Then, by definition, there exist numbers *m* and *j* = *k* and conditions *q* ∈ *G* ∩ (*τ* "*m*, *j*) and *r* ∈ *G* ∩ (*μ*"*m*, *k*). Then *p*, *q*,*r* must be compatible (as elements of the same generic set), which is a contradiction.

**Lemma 21** (in **<sup>L</sup>**)**.** *Suppose that <sup>K</sup>*, *<sup>M</sup>*, *<sup>U</sup>*, *<sup>p</sup>*, *<sup>ϕ</sup>*, *<sup>k</sup> satisfy* (F1) *of Definition 20, <sup>ϕ</sup> is* <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* (*K*[*U*], *M*)*, <sup>P</sup>* <sup>=</sup> {*<sup>q</sup>* <sup>∈</sup> *<sup>K</sup>*[*U*] : *<sup>q</sup>* <sup>≤</sup> *<sup>p</sup>*}*, a set <sup>A</sup>* <sup>∈</sup> *M, <sup>A</sup>* <sup>⊆</sup> *<sup>P</sup> is a maximal antichain in P, and <sup>q</sup> <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> ϕ for all q* ∈ *A. Then p <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> ϕ.*

**Proof.** If *<sup>ϕ</sup>* is a <sup>L</sup>*Π*<sup>1</sup> <sup>1</sup> formula then the result follows from (F2) of Definition 20 and known properties of the ordinary forcing over *M*. Now let *ϕ* be *Π*<sup>1</sup> *<sup>k</sup>* , *<sup>k</sup>* <sup>≥</sup> 2. Suppose towards the contrary that *<sup>p</sup> <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> ϕ* fails. Then there exist: a pair *M* , *U*  ∈ **sJS** extending *M*, *U*, and a condition *r* ∈ *K*[*U* ], *r* ≤ *p*, such that *r <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> ϕ*<sup>¬</sup> . Note that *A* remains a maximal antichain in the set *Q* = {*q* ∈ **P**[*U*] : *q* ≤ *p*} (bigger than *P* above), by Lemma 8. Therefore, *A* is still a maximal antichain in *Q* = {*q* ∈ **P**[*U* ] : *q* ≤ *p*}, by Theorem 8(i), hence a maximal antichain in *P* = {*q* ∈ *K*[*U* ] : *q* ≤ *p*}. It follows that *r* is compatible in *K*[*U* ] with at least one condition *<sup>q</sup>* <sup>∈</sup> *<sup>A</sup>*. However, *<sup>r</sup> <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> <sup>ϕ</sup>*<sup>¬</sup> while *<sup>q</sup> <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> ϕ*, easily leading to a contradiction with Lemma 19.

#### *5.5. Transformations and Invariance*

Here we show that, under certain assumptions, the transformations of the first two groups defined in Section 3.7 preserve forcing approximations **forc**. This is not an absolutely elementary thing: there is no way to reasonably apply transformations to transitive models *M* involved in the definition of **forc** . What we can do is to require that the transformations involved belong to the models involved. This leads to certain complications of different sort.

**Family 1: permutations.** First of all we have to extend the definition of the action of *π* in Section 3.7 to include formulas. Suppose that *<sup>c</sup>*, *<sup>c</sup>* ⊆ I . Define the action of any *<sup>π</sup>* <sup>∈</sup> BIJ*<sup>c</sup> <sup>c</sup>* onto formulas *ϕ* of L(**P**∗) such that ||*ϕ*|| ⊆ *c* :

– to get *πϕ* substitute *π·τ* for any *τ* ∈ NAM *ϕ* and *π·B* for any *B* ∈ IND *ϕ*.

**Lemma 22.** *Suppose that M*, *U*, *K*, *p*, *ϕ satisfy* (F1) *of Definition 20, sets c*, *c* ⊆ I *have equal cardinality and are absolute* ΔHC <sup>1</sup> (*M*)*, <sup>π</sup>* <sup>∈</sup> BIJ*<sup>c</sup> <sup>c</sup> is an absolute* <sup>Δ</sup>HC <sup>1</sup> (*M*) *function, and* ||*ϕ*|| ⊆ *c,* |*U*| ⊆ *c, K* ⊆ **P**<sup>∗</sup> *c.*

*Then, p <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> <sup>ϕ</sup> iff* (*π· <sup>p</sup>*) *<sup>π</sup> ·K***forc***<sup>M</sup> <sup>π</sup> ·<sup>U</sup> πϕ.*

**Proof.** Under the assumptions of the lemma, in particular, the requirement of *c*, *c* , *π* being absolute ΔHC <sup>1</sup> (*M*), *π* acts as an isomorphism on all relevant domains and preserves all relevant relations between the objects involved. Thus *M*, *π·U*, *π·K*, *π· p*, *πϕ* still satisfy Definition 20(F1), and ||*πϕ*|| ⊆ *c* , |*π·U*| ⊆ *c* , *π·K* ⊆ **P**<sup>∗</sup> *c* . (For instance, to show that *π·U* still belongs to *M*, note that the set <sup>|</sup>*U*| ⊆ *<sup>c</sup>* belongs to *<sup>M</sup>*, thus *<sup>π</sup>* <sup>|</sup>*U*| ∈ *<sup>M</sup>*, too, since *<sup>π</sup>* is an absolute <sup>Δ</sup>HC <sup>1</sup> (*M*) function.) This allows to prove the lemma by induction on the complexity of *ϕ*.

Suppose that *<sup>ϕ</sup>* is a closed formula in <sup>L</sup>*Π*<sup>1</sup> <sup>1</sup> (*K*[*U*], *M*). Then *πϕ* is a closed formula in <sup>L</sup>*Π*<sup>1</sup> <sup>1</sup> ((*π·K*)[*π·U*], *M*). Then easily *P* = (*π·K*)[*π·U*] = *π·*(*K*[*U*]) ⊆ **P**<sup>∗</sup> is a set in *M* order isomorphic to *P* = *K*[*U*] itself by means of the map *p* −→ *π· p*. Moreover a set *G* ⊆ *P* is *P*generic over *M* iff *π·G* is, accordingly, *P* -generic over *M* and the valuated formulas *ϕ*[*G*] and (*πϕ*)[*π·G*] coincide. Now the result for *<sup>Π</sup>*<sup>1</sup> <sup>1</sup> formulas follows from (F2) of Definition 20.

**Step** *Π*<sup>1</sup> *<sup>n</sup>* <sup>→</sup> *<sup>Σ</sup>*<sup>1</sup> *<sup>n</sup>*+<sup>1</sup> **,** *<sup>n</sup>* <sup>≥</sup> 1. Let *<sup>ψ</sup>*(*x*) be a <sup>L</sup>*Π*<sup>1</sup> *<sup>n</sup>*(*K*[*U*], *M*) formula, and *ϕ* be ∃ *x ψ*(*x*). Assume *p <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> <sup>ϕ</sup>*. By definition there is a name *<sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(*K*[*U*]) <sup>∩</sup> *<sup>M</sup>* such that *<sup>p</sup> <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> ψ*(*τ*). Then, by the inductive hypothesis, *<sup>π</sup>· <sup>p</sup> <sup>π</sup> ·K***forc***<sup>M</sup> <sup>π</sup> ·<sup>U</sup>* (*πψ*)(*π·τ*), and hence by definition *<sup>π</sup>· <sup>p</sup> <sup>π</sup> ·K***forc***<sup>M</sup> <sup>π</sup> ·<sup>U</sup> πϕ*. The case of *<sup>ϕ</sup>* being <sup>∃</sup>*Bx <sup>ψ</sup>*(*x*) is similar.

**Step** *Σ*<sup>1</sup> *<sup>n</sup>* <sup>→</sup> *<sup>Π</sup>*<sup>1</sup> *<sup>n</sup>* **,** *<sup>n</sup>* <sup>≥</sup> 2. This is somewhat less trivial. Assume that *<sup>ϕ</sup>* is a closed <sup>L</sup>*Π*<sup>1</sup> *<sup>n</sup>*(*K*[*U*], *M*) formula; all names in *ϕ* belong to **SN***<sup>ω</sup> <sup>ω</sup>*(*K*[*U*]) ∩ *M* and are *K*[*U*]-full below a given *p* ∈ *K*[*U*]. Then, by rather obvious reasons, *πϕ* is a closed <sup>L</sup>*Π*<sup>1</sup> *<sup>n</sup>*((*π·K*)[*π·U*], *M*) formula, whose all names belong to **SN***<sup>ω</sup> <sup>ω</sup>*((*π·K*)[*π·U*]) <sup>∩</sup> *<sup>M</sup>* and are (*π·K*)[*π·U*]-full below *<sup>π</sup>· <sup>p</sup>*. Suppose that *<sup>p</sup> <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> ϕ* **fails**. By definition there exist a pair *M*1, *U*1 ∈ **sJS** with *M*, *U M*1, *U*<sup>1</sup>, and a condition *q* ∈ *K*[*U*1], *q* ≤ *p*, such that *q <sup>K</sup>***forc***M*<sup>1</sup> *<sup>U</sup>*<sup>1</sup> *<sup>ϕ</sup>*<sup>¬</sup> . We can also assume by Lemma 19, that <sup>|</sup>*U*1| ⊆ *<sup>c</sup>*. Then (*π· <sup>q</sup>*) *<sup>π</sup> ·K***forc***M*<sup>1</sup> *<sup>π</sup> ·U*<sup>1</sup> *πϕ* by the inductive hypothesis. Yet the pair *M*1, *π·U*<sup>1</sup> belongs to **sJS** and extends *M*, *π·U*. (As *π* is absolute ΔHC <sup>1</sup> (*M*) and *U* ∈ *M*, the restriction *π* |*U*| belongs to *M*.) In addition, *π· q* ∈ (*π·K*)[*π·U*1], and *<sup>π</sup>· <sup>q</sup>* <sup>≤</sup> *<sup>π</sup>· <sup>p</sup>*. Therefore the statement (*π· <sup>p</sup>*) *<sup>K</sup>***forc***<sup>M</sup> <sup>π</sup> ·<sup>U</sup> πϕ* fails, as required.

**Family 2: Lipschitz transformations.** We extend the action of *α* ∈ **Lip**<sup>I</sup> to formulas of L(**P**∗):

– to get *πϕ* substitute *π·τ* for any *τ* ∈ NAM *ϕ* but do not change the quantifier indices *B*.

Note that the action of any *α* ∈ **Lip**<sup>I</sup> ∩ *M* on systems, conditions, names, and formulas, as defined there, is absolute ΔHC <sup>1</sup> (*M*). This allows to prove the next invariance lemma similarly to Lemma 22, which we leave for the reader.

**Lemma 23.** *Suppose that <sup>M</sup>*, *<sup>U</sup>*, *<sup>K</sup>*, *<sup>p</sup>*, *<sup>ϕ</sup> satisfy* (F1) *of Definition 20, and <sup>α</sup>* <sup>∈</sup> **Lip**<sup>I</sup> <sup>∩</sup> *M. Then <sup>p</sup> <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> ϕ iff* (*α· <sup>p</sup>*) *<sup>α</sup> ·K***forc***<sup>M</sup> <sup>α</sup> ·<sup>U</sup> <sup>α</sup>ϕ.*

#### **6. Elementary Equivalence Theorem**

This section presents further properties of **P**-generic extensions of **L** and their subextensions, including Theorem 13 and its corollaties on the elementary equivalence of different subextensions.

**Assumption 2.** *We continue to assume* **V** = **L** *in the ground universe. Below in this section, a number* **<sup>n</sup>** ≥ <sup>2</sup> *is fixed, and pairs* **<sup>M</sup>***<sup>ξ</sup>* , **<sup>U</sup>***<sup>ξ</sup> , the system* **<sup>U</sup>** = *<sup>ξ</sup>*<*ω*<sup>1</sup> **U***<sup>ξ</sup> , the forcing notions* **P***<sup>ξ</sup>* = **P**[**U***<sup>ξ</sup>* ] *and* **P** = **P**[**U**] = - *<sup>ξ</sup>*<*ω*<sup>1</sup> **P***<sup>ξ</sup> are as in Definition 16 for this* **n***.*

#### *6.1. Further Properties of Forcing Approximations*

Coming back to the complete sequence of pairs **M***<sup>ξ</sup>* , **U***<sup>ξ</sup>*  introduced by Definition 16, we consider the auxiliary forcing relation **forc** with respect to those pairs. We begin with the following definition. **Definition 21** (in **L**)**.** *Let K* ⊆ **P**<sup>∗</sup> *be a regular forcing. Recall that*

$$K[\mathbb{U}] = K \cap \mathbb{P} \quad \text{and} \quad K[\mathbb{U}\_{\xi}] = K \cap \mathbb{P}[\mathbb{U}\_{\xi}] = K \cap \mathbb{P}\_{\xi}$$

*for any <sup>ξ</sup>* <sup>&</sup>lt; *<sup>ω</sup>*<sup>1</sup> *. Let <sup>p</sup> <sup>K</sup>***forc***<sup>ξ</sup> <sup>ϕ</sup> mean <sup>p</sup> <sup>K</sup>***forcM***<sup>ξ</sup>* **<sup>U</sup>***<sup>ξ</sup> <sup>ϕ</sup>—then by definition <sup>K</sup> has to be an absolute* <sup>Δ</sup>HC <sup>1</sup> (**M***<sup>ξ</sup>* ) *set, by the way. We let p <sup>K</sup>***forc**<sup>∞</sup> *<sup>ϕ</sup> mean:* <sup>∃</sup> *<sup>ξ</sup>* <sup>&</sup>lt; *<sup>ω</sup>*<sup>1</sup> (*<sup>p</sup> <sup>K</sup>***forc***<sup>ξ</sup> <sup>ϕ</sup>*)*.*

Thus, if *<sup>p</sup> <sup>K</sup>***forc***<sup>ξ</sup> <sup>ϕ</sup>* then definitely *<sup>K</sup>* is an absolute <sup>Δ</sup>HC <sup>1</sup> (**M***<sup>ξ</sup>* ) set, *p* ∈ *K*[**U***<sup>ξ</sup>* ], *ϕ* is a formula with names in **<sup>M</sup>***<sup>ξ</sup>* <sup>∩</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(*K*[**U***<sup>ξ</sup>* ]) as parameters, all names *τ* ∈ NAM *ϕ* are *K*[**U***<sup>ξ</sup>* ]-full below *p*, all indices *B* ∈ IND *ϕ* belong to **M***<sup>ξ</sup>* . The following is an easy consequence of Lemma 18.

**Lemma 24** (in **<sup>L</sup>**)**.** *Let <sup>K</sup>* <sup>⊆</sup> **<sup>P</sup>**<sup>∗</sup> *be a regular forcing. Assume that <sup>ϕ</sup> is a closed formula in* <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* (*K*[**U**]) ∪ <sup>L</sup>*Σ*<sup>1</sup> *<sup>k</sup>*+1(*K*[**U**])*,* 1 ≤ *k, p* ∈ *K*[**U**]*. Then:*


**Proof.** (iii) As *<sup>A</sup>* is a countable set, there exists an ordinal *<sup>ξ</sup>* <sup>&</sup>lt; *<sup>ω</sup>*<sup>1</sup> such that *<sup>q</sup> <sup>K</sup>***forc***<sup>ξ</sup> <sup>ϕ</sup>* for all *<sup>q</sup>* <sup>∈</sup> *<sup>A</sup>*. Apply Lemma 21.

**Lemma 25** (in **<sup>L</sup>**)**.** *Assume that <sup>K</sup>* <sup>⊆</sup> **<sup>P</sup>**<sup>∗</sup> *is a regular forcing, <sup>ϕ</sup> is a closed formula in* <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* (*K*[**U**]) ∪ <sup>L</sup>*Σ*<sup>1</sup> *<sup>k</sup>* (*K*[**U**])*,* <sup>2</sup> <sup>≤</sup> *<sup>k</sup>* <sup>&</sup>lt; **<sup>n</sup>***, <sup>p</sup>* <sup>∈</sup> *<sup>K</sup>*[**U**]*, all names in <sup>ϕ</sup> are <sup>K</sup>*[**U**]*-full below p, and finally <sup>w</sup>* <sup>∈</sup> *<sup>ω</sup><sup>ω</sup> and <sup>K</sup> is absolute* ΔHC <sup>1</sup> (*w*)*. Then*:


**Proof.** (i) As any name is a countable object, there is an ordinal *η* < *ω*<sup>1</sup> such that *p* ∈ *K*[**U***η*], *w* ∈ **M***<sup>η</sup>* , and all names in *<sup>ϕ</sup>* belong to **<sup>M</sup>***<sup>η</sup>* <sup>∩</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(*K*[**U***η*]); then all names in *ϕ* are *K*[**U***η*]-full below *p*, of course. As *k* < **n**, the set *D* of all pairs *M*, *U* ∈ **sJS** that extend **M***η*, **U***<sup>η</sup>* and there is a condition *q* ∈ *K*[*U*], *q p*, satisfying *q <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> <sup>ϕ</sup>*<sup>¬</sup> , belongs to **<sup>Σ</sup>**HC *<sup>n</sup>*−<sup>2</sup> by Theorem 11. Therefore, by the *<sup>n</sup>*-completeness of the sequence {**M***<sup>ξ</sup>* , **<sup>U</sup>***<sup>ξ</sup>* }*ξ*<*ω*<sup>1</sup> , there is an ordinal *<sup>ζ</sup>* , *<sup>η</sup> <sup>ζ</sup>* <sup>&</sup>lt; *<sup>ω</sup>*<sup>1</sup> , such that **<sup>M</sup>***<sup>ζ</sup>* , **<sup>U</sup>***<sup>ζ</sup>* <sup>∈</sup> *<sup>D</sup>*solv . (By the way, this is the only use of the **n**-completeness!)

We have two cases.

Case 1: **<sup>M</sup>***<sup>ζ</sup>* , **<sup>U</sup>***<sup>ζ</sup>* <sup>∈</sup> *<sup>D</sup>*. Then there is a condition *<sup>q</sup>* <sup>∈</sup> *<sup>K</sup>*[**U***<sup>ζ</sup>* ], *<sup>q</sup>* <sup>≤</sup> *<sup>p</sup>*, satisfying *<sup>q</sup> <sup>K</sup>***forc***<sup>ζ</sup> <sup>ϕ</sup>*<sup>¬</sup> . However, obviously *q* ∈ *K*[**U**].

Case 2: there is no pair *<sup>M</sup>*, *<sup>U</sup>*<sup>∈</sup> *<sup>D</sup>* extending **<sup>M</sup>***<sup>ζ</sup>* , **<sup>U</sup>***<sup>ζ</sup>* . Prove *<sup>p</sup> <sup>K</sup>***forc***<sup>ζ</sup> <sup>ϕ</sup>*. Suppose otherwise. Then by the choice of *η* and (F4) of Definition 20 there exist a pair *M*, *U* ∈ **sJS** extending **M***<sup>ζ</sup>* , **U***<sup>ζ</sup>* , and a condition *<sup>q</sup>* <sup>∈</sup> *<sup>K</sup>*[*U*], *<sup>q</sup>* <sup>≤</sup> *<sup>p</sup>*, such that *<sup>q</sup> <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> ϕ*<sup>¬</sup> . Then *M*, *U* ∈ *D*, a contradiction.

(ii) Suppose that there is no condition *<sup>q</sup>* <sup>∈</sup> *<sup>K</sup>*[**U**], *<sup>q</sup>* <sup>≤</sup> *<sup>p</sup>*, with *<sup>q</sup> <sup>K</sup>***forc**<sup>∞</sup> *<sup>ϕ</sup>*<sup>¬</sup> . Then by (i) the set *<sup>Q</sup>* <sup>=</sup> {*<sup>q</sup>* <sup>∈</sup> *<sup>K</sup>*[**U**] : *<sup>q</sup>* <sup>≤</sup> *<sup>p</sup>* <sup>∧</sup> *<sup>q</sup> <sup>K</sup>***forc**<sup>∞</sup> *<sup>ϕ</sup>*} is dense in *<sup>K</sup>*[**U**] below *<sup>p</sup>*. Let *<sup>A</sup>* <sup>⊆</sup> *<sup>Q</sup>* be a maximal antichain. It remains to apply Lemma 24(iii).

#### *6.2. Relations to the Truth in Generic Extensions*

According to the next theorem, the truth in the generic extensions considered is connected in the usual way with the relation forc<sup>∞</sup> up to the **n**-th level of analytic hierarchy. Recall that **V** = **L** is assumed in the ground universe.

**Theorem 12.** *Assume that, in* **<sup>L</sup>***, <sup>K</sup>* <sup>⊆</sup> **<sup>P</sup>**<sup>∗</sup> *is a regular forcing, <sup>ϕ</sup> is a closed formula in* <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* (*K*[**U**]) ∪ <sup>L</sup>*Σ*<sup>1</sup> *<sup>k</sup>*+1(*K*[**U**])*,* <sup>1</sup> <sup>≤</sup> *<sup>k</sup>* <sup>≤</sup> **<sup>n</sup>***, all names in* NAM *<sup>ϕ</sup> are K*[**U**]*-full, w* <sup>∈</sup> *<sup>ω</sup>ω*, *and K is an absolute* <sup>Δ</sup>HC <sup>1</sup> (*w*) *set. Let G* ⊆ **P** *be a* **P***-generic set over* **L***. Then*:


The formulas *ϕ*[*G*], *ϕ*[*G* ∩ *K*] coincide under the assumptions of the theorem.

**Proof.** (ii) We argue by induction on the complexity of *ϕ*.

**The case of** <sup>L</sup>*Π*<sup>1</sup> <sup>1</sup> **formulas**. Let *<sup>ϕ</sup>* be a closed formula in <sup>L</sup>*Π*<sup>1</sup> <sup>1</sup> (*K*[**U**]). As names in the formulas considered are countable objects, there is an ordinal *ξ* < *ω*<sup>1</sup> such that *w* ∈ **M***<sup>ξ</sup>* and *ϕ* is a <sup>L</sup>*Π*<sup>1</sup> <sup>1</sup> (*K*[**U***<sup>ξ</sup>* ], **M***<sup>ξ</sup>* ) formula. As *G* ⊆ **P** is **P**-generic over **L**, the smaller set *G<sup>ξ</sup>* = *G* ∩ *K*[**U***<sup>ξ</sup>* ] is *K*[**U***<sup>ξ</sup>* ]-generic over **M***<sup>ξ</sup>* by Corollary 4, and the formulas *ϕ*[*G*], *ϕ*[*G<sup>ξ</sup>* ] coincide by the choice of *ξ* . Therefore if *ϕ*[*G*] holds in **L**[*G* ∩ *K*[**U**]] then *ϕ*[*G<sup>ξ</sup>* ] holds in **M***<sup>ξ</sup>* [*G<sup>ξ</sup>* ], by Shoenfield's absoluteness theorem, and hence there is a condition *<sup>p</sup>* <sup>∈</sup> *<sup>G</sup><sup>ξ</sup>* which *<sup>K</sup>*[**U***<sup>ξ</sup>* ]-forces *<sup>ϕ</sup>* over **<sup>M</sup>***<sup>ξ</sup>* , that is, *<sup>p</sup> <sup>K</sup>***forc***<sup>ξ</sup> <sup>ϕ</sup>* by (F2) of Definition 20, and finally *<sup>p</sup> <sup>K</sup>***forc**<sup>∞</sup> *<sup>ϕ</sup>*, as required. If conversely, *<sup>p</sup>* <sup>∈</sup> *<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*[**U**], *<sup>ζ</sup>* , *<sup>ξ</sup>* <sup>≤</sup> *<sup>ζ</sup>* <sup>&</sup>lt; *<sup>ω</sup>*<sup>1</sup> , and *<sup>p</sup> <sup>K</sup>***forc***<sup>ζ</sup> <sup>ϕ</sup>*, then by definition *p K*[**U***<sup>ζ</sup>* ]-forces *<sup>ϕ</sup>* over **<sup>M</sup>***<sup>ξ</sup>* . It follows that *<sup>ϕ</sup>*[*G<sup>ξ</sup>* ] holds in **<sup>M</sup>***<sup>ξ</sup>* [*G<sup>ξ</sup>* ], and hence *ϕ*[*G*] holds in **L**[*G* ∩ *K*[**U**]] as well by the Shoenfield absoluteness.

**Step** <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* → L*Σ*<sup>1</sup> *<sup>k</sup>*+<sup>1</sup> **,** *<sup>k</sup>* <sup>&</sup>lt; **<sup>n</sup>.** Let *<sup>ϕ</sup>*(*x*) be a <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* (*K*[**U**]) formula; let us prove the result for <sup>∃</sup> *<sup>x</sup> <sup>ϕ</sup>*(*x*). If *<sup>p</sup>* <sup>∈</sup> *<sup>G</sup>* and *<sup>p</sup> <sup>K</sup>***forc***<sup>ξ</sup>* <sup>∃</sup> *<sup>x</sup> <sup>ϕ</sup>*(*x*) then by definition there is a name *<sup>τ</sup>* <sup>∈</sup> **<sup>M</sup>***<sup>ξ</sup>* <sup>∩</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(*K*[**U***<sup>ξ</sup>* ]), *<sup>K</sup>*[**U***<sup>ξ</sup>* ]-full below *<sup>p</sup>*, and such that *<sup>p</sup> <sup>K</sup>***forc***<sup>ξ</sup> <sup>ϕ</sup>*(*τ*). By Lemma 10, there is a *<sup>K</sup>*[**U***<sup>ξ</sup>* ]-full name *<sup>τ</sup>* <sup>∈</sup> **<sup>M</sup>***<sup>ξ</sup>* <sup>∩</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(*K*[**U***<sup>ξ</sup>* ]), equivalent to *<sup>τ</sup>* below *<sup>p</sup>*. Then *<sup>p</sup> <sup>K</sup>***forc***<sup>ξ</sup> <sup>ϕ</sup>*(*<sup>τ</sup>* ) by Lemma 20. Note that *τ* is **P***<sup>ξ</sup>* -full by Corollary 1, hence **P**-full by Corollary 6(iv), and *K*[**U**]-full, too. It follows that *ϕ*(*τ* )[*G*] holds in **L**[*G* ∩ *K*[**U**]] by the inductive hypothesis, thus (∃ *x ϕ*(*x*))[*G*] holds in **L**[*G* ∩ *K*[**U**]] because *τ* [*G*] = *τ*[*G*] ∈ **L**[*G* ∩ *K*[**U**]] by the choice of *τ*.

If conversely (∃ *x ϕ*(*x*))[*G*] is true in **L**[*G* ∩ *K*[**U**]] then by definition there is an element *x* ∈ **L**[*G* ∩ *K*] = **L**[*G* ∩ *K*[**U**]] such that *ϕ*[*G*](*x*) is true in **L**[*G* ∩ *K*[**U**]]. By Theorem 5(ii), there is a *K*[**U**] full name *<sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(*K*[**U**]) such that *x* = *τ*[*G*]. Thus *ϕ*(*τ*)[*G*] is true in **L**[*G* ∩ *K*[**U**]]. Note that *τ* is **P**full as well, by Corollary 1, and hence *K*[**U**]-full, too. By the inductive hypothesis, there is a condition *<sup>p</sup>* <sup>∈</sup> *<sup>G</sup>* such that *<sup>p</sup> <sup>K</sup>***forc**<sup>∞</sup> *<sup>ϕ</sup>*(*τ*). It follows that *<sup>p</sup> <sup>K</sup>***forc**<sup>∞</sup> <sup>∃</sup> *<sup>x</sup> <sup>ϕ</sup>*(*x*).

**Step** <sup>L</sup>*Σ*<sup>1</sup> *<sup>k</sup>* → L*Π*<sup>1</sup> *<sup>k</sup>* **,** <sup>2</sup> <sup>≤</sup> *<sup>k</sup>* <sup>&</sup>lt; **<sup>n</sup>.** Prove the theorem for a <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* (*K*[**U**]) formula *ϕ*, assuming that the result holds for *ϕ*¬ . If *ϕ*[*G*] is false in **L**[*G*] then *ϕ*¬[*G*] is true. Thus by the inductive hypothesis, there is a condition *<sup>p</sup>* <sup>∈</sup> *<sup>G</sup>* such that *<sup>p</sup> <sup>K</sup>***forc**<sup>∞</sup> *<sup>ϕ</sup>*<sup>¬</sup> . Then *<sup>q</sup> <sup>K</sup>***forc**<sup>∞</sup> *<sup>ϕ</sup>* for any *<sup>q</sup>* <sup>∈</sup> *<sup>G</sup>* is impossible by Lemma 24(ii). Conversely, suppose that *<sup>p</sup> <sup>K</sup>***forc**<sup>∞</sup> *<sup>ϕ</sup>* fails for all *<sup>p</sup>* <sup>∈</sup> *<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*. Then by Lemma 25(i) there is *<sup>q</sup>* <sup>∈</sup> *<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*[**U**] such that *<sup>q</sup> <sup>K</sup>***forc**<sup>∞</sup> *<sup>ϕ</sup>*<sup>¬</sup> . It follows that *<sup>ϕ</sup>*¬[*G*] is true in **<sup>L</sup>**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*[**U**]] by the inductive hypothesis, therefore *ϕ*[*G*] is false.

(i) Let *<sup>ϕ</sup>* be a <sup>L</sup>*Π*<sup>1</sup> **<sup>n</sup>**(*K*[**U**]) formula, *<sup>p</sup>* <sup>∈</sup> *<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*[**U**], *<sup>p</sup> <sup>K</sup>***forc**<sup>∞</sup> *<sup>ϕ</sup>*. By Lemma 24(ii), there is no *<sup>q</sup>* <sup>∈</sup> *<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*[**U**] such that *<sup>q</sup> <sup>K</sup>***forc**<sup>∞</sup> *<sup>ϕ</sup>*<sup>¬</sup> . However, *<sup>ϕ</sup>*<sup>¬</sup> is <sup>L</sup>*Σ*<sup>1</sup> **<sup>n</sup>**(*K*[**U**]), thus ¬ *ϕ*[*G*] in **L**[*G* ∩ *K*] holds by (ii).

Finally prove (i) for a formula *<sup>ϕ</sup>* :<sup>=</sup> <sup>∃</sup> *<sup>x</sup> <sup>ψ</sup>*(*x*), *<sup>ψ</sup>* being <sup>L</sup>*Π*<sup>1</sup> **<sup>n</sup>**(*K*[**U**]). Suppose that *p* ∈ *G* ∩ *K*[**U**] and *<sup>p</sup> <sup>K</sup>***forc**<sup>∞</sup> *<sup>ϕ</sup>*. Then there is a name *<sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(*K*[**U**]), *K*[**U**]-full below *p* and such that *p <sup>K</sup>***forc**<sup>∞</sup> *ψ*(*τ*). We can w.l. o. g. assume that *τ* is totally *K*[**U**]-full, by Lemmas 10 and 20. We have to prove that the formula *ψ*(*τ*)[*G*], that is, *ψ*[*G*](*τ*[*G*]), holds in **L**[*G* ∩ *K*]—then *ϕ*[*G*] holds in **L**[*G* ∩ *K*] as well. Suppose to the contrary that *ψ*(*τ*)[*G*] fails in **L**[*G* ∩ *K*]. However, *ψ*(*τ*) <sup>¬</sup> is a *Σ*<sup>1</sup> *<sup>n</sup>* formula. Therefore, by the first claim of the lemma, there is a condition *<sup>q</sup>* <sup>∈</sup> *<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>* such that *<sup>q</sup> <sup>K</sup>***forc**<sup>∞</sup> *<sup>ψ</sup>*(*τ*) ¬ . However, *p <sup>K</sup>***forc**<sup>∞</sup> *ψ*(*τ*) and *p*, *q* are compatible (as they belong to the same generic set). This contradicts Lemma 24(ii).

#### *6.3. Consequences for the Ordinary Forcing Relation*

For any forcing *P* ∈ **L**, we let ||−−*<sup>P</sup>* be the ordinary *P*-forcing relation over **L** as the ground universe. In particular ||−−**<sup>P</sup>** is the **P**-forcing relation over **L**.

**Corollary 8** (in **L**)**.** *Under the assumptions of Theorem 12, let p* ∈ *K*[**U**]*. Then*:

(i) *if <sup>ϕ</sup> is* <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* (*K*[**U**]) *or* <sup>L</sup>*Σ*<sup>1</sup> *<sup>k</sup>*+1(*K*[**U**]) *and p <sup>K</sup>***forc**<sup>∞</sup> *<sup>ϕ</sup>, then p* ||−−*K*[**U**] *<sup>ϕ</sup>* ;


$$\exists \; q \stackrel{\cdots \atop \cdots \atop q}{\in} \tilde{K}[\mathbb{U}] \; (q \le p \land q \, \, ^\mathbb{K} \mathbf{forc}\_{\infty} \, \, \! \! \! / \,\! \mathbf{forc}\_{\infty} \, \! \! \! \! \mathbf{for} \, \! \! \! \! \ \! \! \mathbf{for} \, \! \! \! \! \! \ \! \! \mathbf{for} \, \! \! \! \! \! \! \mathbf{for} \, \! \! \! \! \! \! \! \! \mathbf{for} \, \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!$$

(iv) *if k* <sup>&</sup>lt; **<sup>n</sup>** *strictly, <sup>ϕ</sup> is* <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* (*K*[**U**])*, and p* ||−−*K*[**U**] *<sup>ϕ</sup> then p <sup>K</sup>***forc**<sup>∞</sup> *<sup>ϕ</sup>* .

**Proof.** (i) follows from Theorem 12(i).

(iii) Let *G* ⊆ **P** be **P**-generic over **L**, and *p* ∈ *G*. If *p* ||−−*K*[**U**] *ϕ* then *ϕ*[*G*] is true in **L**[*G* ∩ *K*[**U**]], and hence there is *<sup>r</sup>* <sup>∈</sup> *<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>* such that *<sup>r</sup> <sup>K</sup>***forc**<sup>∞</sup> *<sup>ϕ</sup>*, by Theorem 12. However, then *<sup>p</sup>*,*<sup>r</sup>* are compatible (as members of *G*), hence *q* = *p* ∧ *r* still is a condition, and *q* ∈ *K*[**U**].

(iv) If *<sup>p</sup> <sup>K</sup>***forc**<sup>∞</sup> *<sup>ϕ</sup>* fails, then by Lemma 25(ii) there is a condition *<sup>q</sup>* <sup>∈</sup> *<sup>K</sup>*[**U**], *<sup>q</sup>* <sup>≤</sup> *<sup>p</sup>*, such that *<sup>q</sup> <sup>K</sup>***forc**<sup>∞</sup> *<sup>ϕ</sup>*<sup>¬</sup> . Then *<sup>q</sup>* ||−−*K*[**U**] *<sup>ϕ</sup>*<sup>¬</sup> by (i), thus *<sup>p</sup>* ||−−*K*[**U**] *<sup>ϕ</sup>* fails.

(ii) Suppose that *<sup>q</sup>* <sup>∈</sup> *<sup>K</sup>*[**U**], *<sup>q</sup>* <sup>≤</sup> *<sup>p</sup>*, *<sup>q</sup> <sup>K</sup>***forc**<sup>∞</sup> *<sup>ϕ</sup>*<sup>¬</sup> . Then *<sup>q</sup>* ||−−*K*[**U**] <sup>¬</sup> *<sup>ϕ</sup>* by (i), and hence *<sup>p</sup>* ||−−*K*[**U**] *<sup>ϕ</sup>* fails. Now suppose that *p* ||−−*K*[**U**] *ϕ* fails. Then there is a condition *r* ∈ *K*[**U**], *r* ≤ *p*, *r* ||−−*K*[**U**] *ϕ*<sup>¬</sup> . However, then, by (iii), there is a condition *<sup>q</sup>* <sup>∈</sup> *<sup>K</sup>*[**U**], *<sup>q</sup>* <sup>≤</sup> *<sup>r</sup>*, *<sup>q</sup> <sup>K</sup>***forc**<sup>∞</sup> *<sup>ϕ</sup>*<sup>¬</sup> , as required.

The next corollary evaluates the complexity of the ordinary forcing relations ||−−*K*[**U**] . The result is related to formulas in classes <sup>L</sup>*Π*<sup>1</sup> **<sup>n</sup>** and higher.

**Corollary 9** (in **L**)**.** *Let ϕ*(*x*1, ... , *xm*) *be an* L(∅) *formula (that is, no names), and K* ⊆ **P**<sup>∗</sup> *be a regular forcing. Suppose that w* <sup>∈</sup> *<sup>ω</sup><sup>ω</sup> , and K is an absolute* <sup>Δ</sup>HC <sup>1</sup> (*w*) *set. Then*:

(i) *if <sup>ϕ</sup> belongs to* <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup> , k* <sup>≥</sup> **<sup>n</sup>***, then the following set is <sup>Π</sup>*HC *<sup>k</sup>*−1(*w*) :

$$\begin{array}{rcll} \mathsf{FORC}\_{K}(\mathsf{q}) &=& \left\{ & \langle p, \tau\_{1}, \dots, \tau\_{m} \rangle : p \in K[\mathbb{U}] \right\} \wedge \\ & & \tau\_{1}, \dots, \tau\_{m} \in \mathsf{SN}\_{\omega}^{\omega}(K[\mathbb{U}]) \text{ are } K[\mathbb{U}]\text{-full names} \wedge \\ & & p \mid \vdash\_{K[\mathbb{U}]} \, q(\tau\_{1}, \dots, \tau\_{m}) \; \Big|\; \, ; \end{array}$$

(ii) *similarly, if <sup>ϕ</sup> is* <sup>L</sup>*Σ*<sup>1</sup> *<sup>k</sup> , k* > **<sup>n</sup>***, then* **FORC***K*(*ϕ*) *is <sup>Σ</sup>*HC *<sup>k</sup>*−1(*w*)*.*

**Proof.** We argue by induction on *<sup>k</sup>* <sup>≥</sup> **<sup>n</sup>**. Suppose that *<sup>ϕ</sup>* is <sup>L</sup>*Π*<sup>1</sup> **<sup>n</sup>** and *<sup>τ</sup>*1, ... , *<sup>τ</sup><sup>m</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(*K*[**U**]) are *K*[**U**]-full names. It follows from Corollary 8(ii) that *p*, *τ*1,..., *τm* ∈ **FORC***K*(*ϕ*) iff

$$\neg \exists \; \emptyset \; \mathsf{x} \; \mathsf{q} \; \neg \mathsf{a} \; \mathsf{q} \in \mathsf{K}[\mathbb{U}\_{\mathsf{f}}] \; (\mathsf{q} \leq p \land q \; \mathsf{?} \; \mathsf{forc}\_{\mathsf{U}\_{\mathsf{f}}}^{\mathsf{P}\_{\mathsf{f}}} \; \mathsf{q} \; \neg (\mathsf{r}\_{1}, \ldots, \mathsf{r}\_{\mathsf{M}})) \; \mathsf{x}$$

The formula *<sup>q</sup> <sup>K</sup>***forcM***<sup>ξ</sup>* **<sup>U</sup>***<sup>ξ</sup> <sup>ϕ</sup>*¬(*τ*1,..., *<sup>τ</sup>m*) can be replaced by

$$\langle \mathbb{M}\_{\vec{\mathbb{S}}'} \mathbb{U}\_{\vec{\mathbb{S}}'} q, q(\tau\_1, \dots, \tau\_m) \rangle \in \mathbf{Forc}\_w^K(\Sigma\_\mathbf{r}^1)$$

(see a definition in Theorem 11). However, **Forc***<sup>K</sup> w*(*Σ*<sup>1</sup> **<sup>n</sup>**) is ΔHC **<sup>n</sup>**−1(*w*) by Theorem <sup>11</sup> (even *<sup>Π</sup>*HC **<sup>n</sup>**−2(*w*) provided **<sup>n</sup>** <sup>≥</sup> 3 ). On the other hand, the maps *<sup>ξ</sup>* −→ **<sup>M</sup>***<sup>ξ</sup>* and *<sup>ξ</sup>* −→ **<sup>U</sup>***<sup>ξ</sup>* are <sup>Δ</sup>HC **<sup>n</sup>**−<sup>1</sup> by construction (Definition 16). As *K* is ΔHC <sup>1</sup> (*w*), it easily follows that *<sup>ξ</sup>* −→ *<sup>K</sup>*[**U***<sup>ξ</sup>* ] is <sup>Δ</sup>HC **<sup>n</sup>**−1(*w*). We conclude that **FORC***K*(*ϕ*) is *Π*HC **<sup>n</sup>**−1(*w*).

**Step** <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* → L*Σ*<sup>1</sup> *<sup>k</sup>*+<sup>1</sup> . Suppose that *<sup>ϕ</sup>*(*τ*) is a <sup>L</sup>*Σ*<sup>1</sup> *<sup>k</sup>*+<sup>1</sup> formula of the form ∃ *y ψ*(*y*,*τ*), where accordingly *<sup>ψ</sup>* is <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* . Let us show that simply

$$\left(\left\langle p, \vec{\tau} \right\rangle \in \mathsf{FORC}\_{k}(\varphi) \iff \exists \, \sigma \in \mathsf{SN}\_{\omega}^{\omega}(\mathsf{K}[\mathbb{U}]) \left(\left\langle p, \sigma, \vec{\tau} \right\rangle \in \mathsf{FORC}\_{k}(\psi)\right),\tag{5}$$

which obviously suffices to carry out the step <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* → L*Σ*<sup>1</sup> *<sup>k</sup>*+<sup>1</sup> .

If *σ* is a name as in the right-hand side then obviously any *p* forces *σ*[*G*] ∈ **L**[*G* ∩ *K*[**U**]], and on the other hand by definition *p* ||−−*K*[**U**] *ψ*(*σ*,*τ*). Thus *p* ||−−*K*[**U**] *ϕ*(*τ*), hence, *p*,*τ* ∈ **FORC***K*(*ϕ*), as required. Now suppose that *p* ||−−*K*[**U**] *ϕ*(*τ*). This means, by definition, that *p* ||−−*K*[**U**] ∃ *y ψ*(*y*,*τ*). By Theorem 5(iv), there is a *<sup>K</sup>*[**U**]-full name *<sup>σ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(*K*[**U**]) such that *p* ||−−*K*[**U**] *ψ*(*σ*,*τ*), thus *p*, *σ*,*τ* ∈ **FORC***K*(*ψ*).

**Step** <sup>L</sup>*Σ*<sup>1</sup> *<sup>k</sup>* → L*Π*<sup>1</sup> *<sup>k</sup>* **,** *<sup>k</sup>* <sup>&</sup>gt; **<sup>n</sup>**. Suppose that *<sup>ϕ</sup>* is a <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* formula; accordingly, *<sup>ϕ</sup>*<sup>¬</sup> is <sup>L</sup>*Σ*<sup>1</sup> *<sup>k</sup>* . It is clear that, under the assumptions that *<sup>p</sup>* <sup>∈</sup> *<sup>K</sup>*[**U**] and *<sup>τ</sup>*1, ... , *<sup>τ</sup><sup>m</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(*K*[**U**]) are *K*[**U**]-full names, the following holds:

$$\langle p, \vec{\tau} \rangle \in \mathbf{FORC}\_{K}(\varphi) \iff \neg \exists \, q \in K[\mathbb{U}] \left( q \le p \land \langle p, \vec{\tau} \rangle \in \mathbf{FORC}\_{K}(\varphi^{-}) \right),\tag{6}$$

which is sufficient to accomplish the step <sup>L</sup>*Σ*<sup>1</sup> *<sup>k</sup>* → L*Π*<sup>1</sup> *k* .

#### *6.4. Elementary Equivalence Theorem*

According to Theorem 10, sets *S* satisfying **Γ***i*(*S*) are different for different indices *i* ∈ I , and the difference can be determined, in the extensions of the form **L**[*G z*], at the level *Π*HC **<sup>n</sup>**−<sup>1</sup> by Corollary 7, that is, *Π*<sup>1</sup> **<sup>n</sup>** (see Remark 2 in Section 4.3). On the other hand, the extensions considered remain rather amorphous w. r. t. lower levels of definability, as witnessed by the following key theorem.

**Theorem 13.** *Suppose that, in* **<sup>L</sup>** : *<sup>d</sup>* ⊆ I *, <sup>w</sup>* <sup>∈</sup> *<sup>ω</sup>ω*, *sets <sup>b</sup>*, *<sup>c</sup>* <sup>⊆</sup> *<sup>d</sup>* <sup>=</sup> <sup>I</sup> *<sup>d</sup> have equal cardinality, <sup>d</sup> is uncountable, <sup>K</sup>* <sup>⊆</sup> **<sup>P</sup>**<sup>∗</sup> *<sup>d</sup> is a regular forcing,* <sup>Ψ</sup>(*y*) *is a <sup>Π</sup>*<sup>1</sup> **<sup>n</sup>**−<sup>1</sup> *formula with parameters in <sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*]*, and K*, *b*, *c*, *d are absolute* ΔHC <sup>1</sup> (*w*) *sets. Let G* ⊆ **P** *be* **P***-generic over* **L***.*

*Then, if there is a real <sup>y</sup>* <sup>∈</sup> *<sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*, *<sup>G</sup> <sup>b</sup>*] *such that* <sup>Ψ</sup>(*y*) *holds in* **<sup>L</sup>**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*, *<sup>G</sup> <sup>d</sup>*]*, then there exists y* ∈ **L**[*G* ∩ *K*, *G c*] *such that* Ψ(*y* ) *holds in* **<sup>L</sup>**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*, *<sup>G</sup> <sup>d</sup>*] .

Recall that ΔHC <sup>1</sup> (*w*) means that *<sup>w</sup>* is admitted as the only parameter. The assumption that *<sup>d</sup>* is uncountable, can be avoided at the cost of extra complications, but the case of *d* countable is not considered below. The proof makes use of the transformations introduced in Section 3.7.

**Proof.** As all cardinals are preserved in **L**[*G*], we w.l. o. g. assume that *b*, *c* are countably infinite (or finite of equal cardinality) in **L**. Suppose towards the contrary that


By Theorem 5(ii), for every real parameter *<sup>z</sup>* in <sup>Ψ</sup> there is a *<sup>K</sup>*[**U**]-full name *<sup>τ</sup><sup>z</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(*K*[**U**]) such that *z* = *τz*[*G*]. Replace each parameter *z* in Ψ(*x*) by such a name *τ<sup>z</sup>* in **L**, and let *ψ*(*x*) be the <sup>L</sup>*Π*<sup>1</sup> **<sup>n</sup>**−1(*K*[**U**]) formula obtained. Then <sup>|</sup>*ψ*| ⊆ *<sup>d</sup>*. Further, the set

$$K' = \{ p \in \mathbf{P}^\* \restriction (d \cup b) : p \upharpoonright d \in K \} = K \times (\mathbf{P}^\* \restriction b) \subseteq \mathbf{P}^\* \restriction (d \cup b)$$

is a regular forcing, and **L**[*G* ∩ *K*, *G b*] = **L**[*G* ∩ *K* ]. Choose *y* by (A). Once again, Theorem 5(ii), yields a *K* [**U**]-full name *<sup>τ</sup><sup>y</sup>* <sup>∈</sup> **SN***<sup>ω</sup> ω*(*K* [**U**]) such that *y* = *τy*[*G*]. The name *τ<sup>y</sup>* is small, hence the set |*τy*| ⊆ *d* ∪ *b* is countable (in **L**). We let *d*<sup>0</sup> = |*τy*| ∩ *d*; the set *B* = *d*<sup>0</sup> ∪ *b* is still countable and |*τy*| ⊆ *B*. Thus the formula <sup>∃</sup>*By <sup>ψ</sup>*(*y*)[*G*] is true in **<sup>L</sup>**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*, *<sup>G</sup> <sup>d</sup>*].

Now let *<sup>Q</sup>* <sup>=</sup> {*<sup>p</sup>* <sup>∈</sup> **<sup>P</sup>**<sup>∗</sup> : *<sup>p</sup> <sup>d</sup>* <sup>∈</sup> *<sup>K</sup>*} <sup>=</sup> *<sup>K</sup>* <sup>×</sup> (**P**<sup>∗</sup> *<sup>d</sup>*). Thus *<sup>Q</sup>* is a regular forcing, and **<sup>L</sup>**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*, *<sup>G</sup> <sup>d</sup>*] = **<sup>L</sup>**[*<sup>G</sup>* <sup>∩</sup> *<sup>Q</sup>*] = **<sup>L</sup>**[*<sup>G</sup>* <sup>∩</sup> *<sup>Q</sup>*[**U**]]. Therefore <sup>∃</sup>*By <sup>ψ</sup>*(*y*)[*G*] is true in **<sup>L</sup>**[*<sup>G</sup>* <sup>∩</sup> *<sup>Q</sup>*[**U**]] by the above. It follows by Theorem 12(ii) that there is a condition *<sup>p</sup>* <sup>∈</sup> *<sup>G</sup>* <sup>∩</sup> *<sup>Q</sup>* such that *<sup>p</sup> <sup>Q</sup>***forc**<sup>∞</sup> <sup>∃</sup>*By <sup>ψ</sup>*(*y*), and, by (B), we can also assume that *p Q*[**U**]-forces ¬ ∃*Cy <sup>ψ</sup>*(*y*) over **<sup>L</sup>** where *<sup>C</sup>* <sup>=</sup> *<sup>d</sup>*<sup>0</sup> <sup>∪</sup> *<sup>c</sup>*. Further, in **<sup>L</sup>**, there exists an ordinal *ξ* < *ω*<sup>1</sup> such that

$$p^{\mathsf{Q}} \mathsf{f} \mathsf{or} \mathsf{c}\_{ll}^{M} \exists^{B} y \,\psi(y),$$

where *M* = **M***<sup>ξ</sup>* and *U* = **U***<sup>ξ</sup>* , and in addition the countable sets *d*0, *b*, *c* belong to *M*, *w* ∈ *M*, *<sup>p</sup>* <sup>∈</sup> *<sup>Q</sup>*[*U*], *<sup>d</sup>*<sup>0</sup> <sup>∪</sup> *<sup>b</sup>* <sup>∪</sup> *<sup>c</sup>* <sup>⊆</sup> *<sup>A</sup>* <sup>=</sup> <sup>|</sup>*U*|, and all names in *<sup>ψ</sup>* belong to *<sup>M</sup>* <sup>∩</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(*K*[*U*]), so that *ψ*(*x*) is a <sup>L</sup>*Π*<sup>1</sup> **<sup>n</sup>**−1(*K*[*U*], *<sup>M</sup>*) formula.

Now we can assume that both sets |*U*| (*d* ∪ *b*) and |*U*| (*d* ∪ *c*) are infinite. (Otherwise take a suitably bigger *<sup>ξ</sup>* .) Then there is a bijection *<sup>f</sup>* <sup>∈</sup> *<sup>M</sup>*, *<sup>f</sup>* : <sup>|</sup>*U*<sup>|</sup> onto −→ |*U*|, such that *<sup>f</sup> <sup>d</sup>* is the identity and *f* [*b*] = *c*. Define a bijection *π* ∈ BIJ<sup>I</sup> <sup>I</sup> such that *<sup>π</sup>* <sup>|</sup>*U*<sup>|</sup> coincides with *<sup>f</sup>* and *<sup>π</sup>* (<sup>I</sup> <sup>|</sup>*U*|) is the identity. Let *q* = *π· p* and *V* = *π·U*. Acting by *π* on (7), we obtain, by Lemma 22,

$$\neq q^{\text{Q}} \mathbf{forc}\_V^M \exists^C y \,\psi(y),\tag{8}$$

Comments: 1) *π·Q* = *Q* since *π d* is the identity by construction and *K* ⊆ **P**<sup>∗</sup> *d*; 2) *π·B* = *π*[*B*] = *f* [*B*] = *C* by construction; 3) *π·ψ*(*x*) is *ψ*(*x*) because |*ψ*| ⊆ *d* and *π d* is the identity.

Note that *V* ∈ *M* is a system with |*V*| = *π·*|*U*| = |*U*|, and *p* ∈ *U*, *q* ∈ *V*, *U d* = *V d* and *q d* = *p d* by the choice of *π* and *f* . In addition, *U*, *V* are countable systems in *M* |= **ZFC**<sup>−</sup> 1 . Corollary 2 yields a transformation *α* ∈ **Lip**<sup>I</sup> in *M* such that |*α*| = |*U*| = |*V*|, *α·V* = *U*, conditions *q* = *α· q* ∈ *Q*[*U*] and *p* are compatible, and *α d* is the identity (as *U d* = *V d* and *p d* = *q d*). However, then *<sup>α</sup>·<sup>Q</sup>* <sup>=</sup> *<sup>Q</sup>*, and *<sup>α</sup>*(∃*Cx <sup>ψ</sup>*(*x*)) coincides with <sup>∃</sup>*Cx <sup>ψ</sup>*(*x*) since <sup>|</sup>*ψ*| ⊆ *<sup>d</sup>*. Therefore *q <sup>Q</sup>***forc***<sup>M</sup> <sup>U</sup>* <sup>∃</sup>*Cy <sup>ψ</sup>*(*y*) by (8) and Lemma 23. This implies *<sup>q</sup> <sup>Q</sup>***forc**<sup>∞</sup> <sup>∃</sup>*Cy <sup>ψ</sup>*(*y*). We conclude that *<sup>q</sup> <sup>Q</sup>*[**U**]-forces <sup>∃</sup>*Cy <sup>ψ</sup>*(*y*) over **<sup>L</sup>**, by Corollary 8(i). However, *<sup>q</sup>* is compatible with *<sup>p</sup>* and *<sup>p</sup>* forces the negation of this sentence. The contradiction completes the proof.

**Corollary 10.** *Under the assumptions of Theorem 13, if c is uncountable in* **L***, then* **L**[*G* ∩ *K*, *G c*] *is an elementary submodel of* **<sup>L</sup>**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*, *<sup>G</sup> <sup>d</sup>*] *w. r. t. all <sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>** *formulas with parameters in <sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*, *<sup>G</sup> <sup>c</sup>*]*.*

**Proof.** Prove by induction that if *k* ≤ **n** then **L**[*G* ∩ *K*, *G c*] is an elementary submodel of **L**[*G* ∩ *K*, *G d*] w. r. t. all *Σ*<sup>1</sup> *<sup>k</sup>* formulas with parameters in **L**[*G* ∩ *K*, *G c*]. If *k* = 2 then the result holds by the Shoenfield absoluteness theorem. It remains to carry out the step *k* → *k* + 1 (*k* < **n**). Let *ϕ*(*x*) be a *Π*<sup>1</sup> *<sup>k</sup>* formula with parameters in **<sup>L</sup>**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*, *<sup>G</sup> <sup>c</sup>*]; we have to prove the result for the *<sup>Σ</sup>*<sup>1</sup> *k* formula ∃ *x ϕ*(*x*), assuming *k* < **n**. First of all, as the cardinals are preserved, there is a set *δ* ∈ **L**, *<sup>δ</sup>* <sup>⊆</sup> *<sup>d</sup>* , countable in **<sup>L</sup>** and such that all parameters of *<sup>ϕ</sup>* belong to **<sup>L</sup>**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*, *<sup>G</sup> <sup>δ</sup>*]. Let *<sup>d</sup>* <sup>=</sup> *<sup>d</sup>* <sup>∪</sup> *<sup>δ</sup>* and *K* = {*p* ∈ **P**<sup>∗</sup> *d* : *p d* ∈ *K*}; we can identify *K* with *K* × (**P**<sup>∗</sup> *δ*), of course. Then, in **L**, *K* is a regular forcing, *K* ⊆ **P**<sup>∗</sup> *d* , and all parameters of *ϕ* belong to **L**[*G* ∩ *K* ].

Now suppose that <sup>∃</sup> *<sup>x</sup> <sup>ϕ</sup>*(*x*) holds in **<sup>L</sup>**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*, *<sup>G</sup> <sup>d</sup>*], the bigger of the two models of the lemma. Let this be witnessed by a real *<sup>x</sup>*<sup>0</sup> <sup>∈</sup> **<sup>L</sup>**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*, *<sup>G</sup> <sup>d</sup>*] = **<sup>L</sup>**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>* , *G* (*d* )], where (*d* ) <sup>=</sup> <sup>I</sup> *<sup>d</sup>* <sup>=</sup> *<sup>d</sup> <sup>δ</sup>*, so that *<sup>ϕ</sup>*(*x*0) holds in the model **<sup>L</sup>**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*, *<sup>G</sup> <sup>d</sup>*] = **<sup>L</sup>**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>* , *G* (*d* )]. As the cardinals are preserved, there is a set *b* ∈ **L**, *b* ⊆ (*d* ) , countably infinite in **L** and such that *x*<sup>0</sup> belongs to **L**[*G* ∩ *K* , *G b* ]. Since *c* is uncountable, there exists a set *c* ∈ **L**, *c* ⊆ (*d* ) <sup>∩</sup> *<sup>c</sup>*, countably infinite in **<sup>L</sup>**. By the choice of *<sup>δ</sup>*, there is a real *<sup>w</sup>* <sup>∈</sup> *<sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>** such that the sets *<sup>K</sup>* , *<sup>d</sup>* , *<sup>c</sup>* , *<sup>b</sup>* are absolute ΔHC <sup>1</sup> (*w* ) in **L**. By Theorem 13, there is a real *y*<sup>0</sup> ∈ **L**[*G* ∩ *K* , *G c* ] such that *ϕ*(*y*0) holds in **L**[*G* ∩ *K* , *G* (*d* )] = **<sup>L</sup>**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*, *<sup>G</sup> <sup>d</sup>*], and then in **<sup>L</sup>**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*, *<sup>G</sup> <sup>c</sup>*] by the inductive assumption.

Note that if say *c* is uncountable but *b* countable, and *d* is countable, then Theorem 13 fails by means of the formula "there is a real *x* such that all reals belong to **L**[*x*, *G* ∩ *K*]", and *G* ∩ *K* is equiconstructible with a real in this case.

**Question 1.** It would be very interesting to figure out whether Theorem 13 and Corollary 10 hold also for sets *b*, *c* not necessarily constructible.

The following corollary presents a partial positive result.

A set *<sup>z</sup>* ⊆ I <sup>=</sup> *<sup>ω</sup>***<sup>L</sup>** <sup>1</sup> is bounded iff there is *<sup>α</sup>* < *<sup>ω</sup>***<sup>L</sup>** <sup>1</sup> such that *z* ⊆ *α*.

**Corollary 11.** *Suppose that G* ⊆ **P** *is* **P***-generic over* **L***, and z* ⊆ I *is a set unbounded in* I *, locally constructible in the sense that z* ∩ *α* ∈ **L** *for all α* ∈ I *, and all* **L***-cardinals are preserved in* **L**[*G z*]*. Then* **L**[*G z*] *is elementarily equivalent to* **L**[*G*] *w. r. t. all Σ*<sup>1</sup> **<sup>n</sup>** *formulas with parameters in* **L**[*G z*]*.*

Remark: under the assumptions of the corollary, it is not necessary that **L**[*G z*] ⊆ **L**[*G*], since the set *<sup>z</sup>* is not assumed to belong to **<sup>L</sup>**[*G*], but we necessarily have **<sup>L</sup>**[*G <sup>z</sup>*] <sup>∩</sup> *<sup>ω</sup><sup>ω</sup>* <sup>⊆</sup> **<sup>L</sup>**[*G*] <sup>∩</sup> *<sup>ω</sup><sup>ω</sup>* by rather obvious reasons.

**Proof.** Prove by induction that for any *k* ≤ **n**, **L**[*G z*] is elementarily equivalent to **L**[*G*] w. r. t. all *Σ*1 *<sup>k</sup>* formulas with parameters in **L**[*G z*]. For *k* = 2 use Shoenfield's absoluteness. To carry out the step *<sup>k</sup>* <sup>→</sup> *<sup>k</sup>* <sup>+</sup> 1 (*<sup>k</sup>* <sup>&</sup>lt; **<sup>n</sup>**), let *<sup>ϕ</sup>* :<sup>=</sup> <sup>∃</sup> *<sup>y</sup> <sup>ψ</sup>*(*y*) be a *<sup>Σ</sup>*<sup>1</sup> *<sup>k</sup>*+<sup>1</sup> formula with parameters in **L**[*G z*]. Then, by the choice of *z*, 1) there is a set *d* ∈ **L**, *d* ⊆ *z*, countable in **L** and such that all parameters in *ϕ* belong to **L**[*G d*], and 2) there is a set *e* ∈ **L**, *e* ⊆ *z d*, countably infinite in **L**.

Now suppose that ∃ *y ψ*(*y*) is true in **L**[*G*]. This is witnessed by a real *y* ∈ **L**[*G* (*d* ∪ *e* )] for a set *e* ∈ **L**, *e* ⊆ I *d*, countably infinite in **L**. Then, by Theorem 13 with *K* = **P**<sup>∗</sup> *d*, there is a real *y* ∈ **L**[*G* (*d* ∪ *e*)], hence, *y* ∈ **L**[*G z*], such that *ψ*(*y*) is true in **L**[*G*]. However, then *ψ*(*y*) is true in **L**[*G z*] by the inductive hypothesis. Hence *ϕ* is true in **L**[*G z*] as well, as required.

#### **7. Application 1: Nonconstructible** Δ<sup>1</sup> *<sup>n</sup>* **Reals**

In this section, we proveTheorems 1 and 2(i). Theorem 1 provides change of definability of reals situated in the ground set universe **<sup>L</sup>**, in generic extensions of **<sup>L</sup>**. Thus, any real *<sup>a</sup>* <sup>∈</sup>/ *<sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>** <sup>∪</sup> *<sup>Π</sup>*<sup>1</sup> **<sup>n</sup>** in **L** can be placed exactly at Δ<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> in an appropriate (almost disjoint) extension of **L**. Theorem 2 contains several results for nonconstructible reals. The proofs of these results will make use of various results in Sections 5 and 6, in particular, a result (Theorem 11) related to definability of relevant forcing relations.

**Assumption 3.** *We continue to assume* **V** = **L** *in the ground universe. We fix an integer* **n** ≥ 2*, for which Theorems 1 and 2 will be proved, and make use of a system* **U** *and the forcing notion* **P** = **P**[**U**] *as in Definition 16; both* **U** *and* **P** *belong to* **L***.*

#### *7.1. Changing Definability of an Old Real*

**Proof** (Theorem 1)**.** Fix a set *<sup>b</sup>* <sup>⊆</sup> *<sup>ω</sup>*, *<sup>b</sup>* <sup>∈</sup>/ *<sup>Σ</sup>*<sup>1</sup> *<sup>n</sup>* <sup>∪</sup> *<sup>Π</sup>*<sup>1</sup> *<sup>n</sup>* , in **L**, and define

$$\mathcal{L} = \{2k : k \in b\} \cup \{2k + 1 : k \notin b\} \text{ and } K = \mathbf{P}^\* \mid \mathcal{c} = \{p \in \mathbf{P}^\* : |p| \subseteq \mathcal{c}\} \dots$$

Thus *c* ⊆ *ω* ⊆ I = *ω*<sup>1</sup> , *c* ∈ **L**, *K* ⊆ **P**<sup>∗</sup> is a regular forcing. Let *G* ⊆ **P** be a **P**-generic set over **L**. Then the set *G* ∩ *K* = *G c* is *K*[**U**]-generic over **L** by Lemma 9(ii), where *K*[**U**] = *K* ∩ **P**[**U**], as usual.

Define *S*(*ν*) = *SG*(*ν*) ⊆ **Seq** and *a<sup>ν</sup>* = *aG*(*ν*) = {*k* ≥ 1 : **s***<sup>k</sup>* ∈ *SG*(*ν*)} for every *ν*, as in Definition 9. We assert that the submodel **L**[*G c*] = **L**[*G* ∩ *K*] = **L**[{*am* }*m*∈*c*] of the whole generic extension **L**[*G*] witnesses Theorem 1. This amounts to the two following claims:

**Claim 3.** *It is true in* **L**[*G c*] *that c is Σ*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> *, therefore b is* <sup>Δ</sup><sup>1</sup> **<sup>n</sup>**+<sup>1</sup> *.*

**Proof.** By definition we have *<sup>c</sup>* <sup>=</sup> <sup>|</sup>*K*<sup>|</sup> <sup>=</sup> <sup>|</sup>*<sup>K</sup>* <sup>∩</sup> *<sup>G</sup>*|. Therefore *<sup>c</sup>* is *<sup>Σ</sup>*HC **<sup>n</sup>** in **L**[*G c*] by Corollary 7(iii), hence *Σ*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> (see Remark <sup>2</sup> in Section 4.3), and *<sup>b</sup>* <sup>=</sup> {*<sup>k</sup>* : <sup>2</sup>*<sup>k</sup>* <sup>∈</sup> *<sup>c</sup>*} <sup>=</sup> {*<sup>k</sup>* : <sup>2</sup>*<sup>k</sup>* <sup>+</sup> <sup>1</sup> <sup>∈</sup>/ *<sup>c</sup>*} ∈ <sup>Δ</sup><sup>1</sup> **<sup>n</sup>**+<sup>1</sup> , as required. In more detail,

$$\begin{aligned} \mathcal{C} &=& \{ m : \mathbb{S}\_{\mathcal{G}}(m) \in \mathbf{L}[\mathbf{G} \restriction \mathbf{c}] \} &=& \{ m : \mathbf{L}[\mathbf{G} \restriction \mathbf{c}] \mathrel{\mathop{=}} \exists \mathcal{S} \sqcap\_{\mathcal{W}}(\mathbf{S}) \}, \text{ hence} \\\ a &=& \{ k : \mathbb{S}\_{\mathcal{G}}(2k) \in \mathbf{L}[\mathbf{G} \restriction \mathbf{c}] \} &=& \{ k : \mathbf{L}[\mathbf{G} \restriction \mathbf{c}] \mathrel{\mathop{=}} \exists \mathcal{S} \sqcap\_{2k}(\mathbf{S}) \} \\\ &=& \{ k : \mathbb{S}\_{\mathcal{G}}(2k+1) \notin \mathbf{L}[\mathbf{G} \restriction \mathbf{c}] \} &=& \{ k : \mathbf{L}[\mathbf{G} \restriction \mathbf{c}] \mathrel{\mathop{=}} \sqcap\_{2k+1}(\mathbf{S}) \} \end{aligned}$$

by Theorem 10, and it remains to apply Lemma 17.

**Claim 4.** *In* **<sup>L</sup>**[*G <sup>c</sup>*] : *if x* <sup>⊆</sup> *<sup>ω</sup> is <sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>** *then x* <sup>∈</sup> **<sup>L</sup>** *and x is <sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>** *in* **L***.*

**Proof** (Claim 4)**.** Let *<sup>x</sup>* <sup>=</sup> {*<sup>m</sup>* : *<sup>ϕ</sup>*(*m*)} in **<sup>L</sup>**[*G <sup>c</sup>*], where *<sup>ϕ</sup>*(*m*) is a *<sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>** formula. Define *c* = *ω*, *K* = **P**∗ *ω*, and *K* [*U*] = *K* ∩ **P**[*U*], as usual. Prove that

$$m \in \mathbf{x} \iff \exists \left< M, lI \right> \in \mathbf{s} \mathbf{f} \mathbf{S} \; \exists \; p \in \mathbf{K}' [lI] \; (p \; \, ^K \mathbf{f} \mathbf{orc} \mathbf{c}\_{ll}^M \; \; \! \; \mathbf{y}(m)). \tag{9}$$

The right-hand side of (9) is relativized to **L** and is *Σ*<sup>1</sup> *<sup>n</sup>* in **L** by Theorem 11. Thus (9) implies Claim 4.

To verify =⇒ in (9), suppose that *m* ∈ *x*, that is, *ϕ*(*m*) holds in **L**[*G c*] = **L**[*G* ∩ *K*]. Then by Theorem 12(ii) there is a condition *<sup>p</sup>* <sup>∈</sup> *<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>* such that *<sup>p</sup> <sup>K</sup>***forc**<sup>∞</sup> *<sup>ϕ</sup>*(*m*), that is, *<sup>p</sup> <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> ϕ*(*m*), where *M* = **M***<sup>ξ</sup>* , *U* = **U***<sup>ξ</sup>* for some *ξ* < *ω*<sup>1</sup> . As **M***<sup>ξ</sup>* = *M* |= **ZFC**<sup>−</sup> <sup>1</sup> , *M* contains *c*, *c* , and the increasing bijection *<sup>π</sup>* <sup>∈</sup> BIJ*<sup>c</sup> <sup>c</sup>* . It follows that *<sup>q</sup> <sup>K</sup>* **forc***<sup>M</sup> <sup>U</sup> ϕ*(*m*), by Lemma 22, where *U* = *π·U* and *q* = *π· p*, as obviously *π·K* = *K* . This implies the right-hand side of (9).

To verify ⇐= , let *M* , *U*  ∈ **sJS**, *p* ∈ *K* [*U* ], and *p <sup>K</sup>* **forc***<sup>M</sup> <sup>U</sup> ϕ*(*m*). Suppose towards the contrary that *ϕ*(*m*) fails in **L**[*G* ∩ *K*], so that there is a condition *q* ∈ *G* ∩ *K* such that *q* ||−−*K*[**U**] ¬ *ϕ*(*m*). Then *q* ∈ *K*[**U**] (since *G* ⊆ **P**), and hence there is an ordinal *ξ* < *ω*<sup>1</sup> such that *q* ∈ *K*[**U***<sup>ξ</sup>* ], *ω* ∪ |*U* | ⊆ <sup>|</sup>**U***<sup>ξ</sup>* <sup>|</sup> and *<sup>M</sup>* <sup>⊆</sup> **<sup>M</sup>***<sup>ξ</sup>* . Then still *<sup>p</sup> <sup>K</sup>* **forcM***<sup>ξ</sup> <sup>U</sup> <sup>ϕ</sup>*(*m*) by Lemma 18, and Lemma <sup>22</sup> implies *<sup>p</sup> <sup>K</sup>***forcM***<sup>ξ</sup> <sup>U</sup> ϕ*(*m*), where *<sup>p</sup>* <sup>=</sup> *<sup>π</sup>*−<sup>1</sup> *· <sup>p</sup>* and *<sup>U</sup>* <sup>=</sup> *<sup>π</sup>*−<sup>1</sup> *·<sup>U</sup>* . (By obvious reasons, *<sup>K</sup>* <sup>=</sup> *<sup>π</sup>*−<sup>1</sup> *·<sup>K</sup>* .) Note that <sup>|</sup>*U*|⊆|**U***<sup>ξ</sup>* <sup>|</sup> by the choice of *ξ* . Therefore, we can define a system *V* ∈ **M***<sup>ξ</sup>* such that *V* |*U*| = *U* and *V*(*ν*) = **U***<sup>ξ</sup>* (*ν*) for all *<sup>ν</sup>* ∈ | / *<sup>U</sup>*|. Then obviously **<sup>M</sup>***<sup>ξ</sup>* , *<sup>U</sup>* **<sup>M</sup>***<sup>ξ</sup>* , *<sup>V</sup>*, therefore *<sup>p</sup> <sup>K</sup>***forcM***<sup>ξ</sup> <sup>V</sup> ϕ*(*m*).

Now, *V* and **U***<sup>ξ</sup>* are countable systems in **M***<sup>ξ</sup>* with |*V*| = |**U***<sup>ξ</sup>* | and *p* ∈ *K*[*V*] but *q* ∈ *K*[**U***<sup>ξ</sup>* ]. Corollary 2 yields a transformation *α* ∈ **Lip**<sup>I</sup> in *M* such that |*α*| ⊆ *c*, *α·V* = **U***<sup>ξ</sup>* , and conditions *<sup>r</sup>* <sup>=</sup> *<sup>α</sup>· <sup>p</sup>* <sup>∈</sup> *<sup>K</sup>*[**U***<sup>ξ</sup>* ] and *<sup>q</sup>* are compatible. Then *<sup>r</sup> <sup>K</sup>***forcM***<sup>ξ</sup>* **<sup>U</sup>***<sup>ξ</sup> <sup>ϕ</sup>*(*m*) by Lemma 23. (Comment: *<sup>α</sup><sup>ϕ</sup>* is *<sup>ϕ</sup>*, and *α·K* = *K* because regular forcings of the form *K* = **P**<sup>∗</sup> *c* are invariant w. r. t. the transformations in **Lip**<sup>I</sup> .) Thus *<sup>r</sup> <sup>K</sup>***forc**<sup>∞</sup> *<sup>ϕ</sup>*(*m*), and hence *<sup>r</sup>* ||−−*K*[**U**] *<sup>ϕ</sup>*(*m*) by Corollary 8(i). However, *<sup>r</sup>* is compatible with *q*, and *q* forces the opposite, a contradiction. This ends the proof of (9). (Claim 4)

(Theorem 1)

#### *7.2. Nonconstructible* Δ<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> *Real, Part 1*

Here we begin the proof of Theorem 2(i). Suppose that a set *G* ⊆ **P** is **P**-generic over **L**. Define *S*(*ν*) = *SG*(*ν*) ⊆ **Seq** and *a<sup>ν</sup>* = *aG*(*ν*) = {*k* ≥ 1 : **s***<sup>k</sup>* ∈ *SG*(*ν*)} for every *ν* as in Definition 9. Emulating the construction in Section 7.1, put

$$z = z\_G = \{0\} \cup \{2k + 2 : k \in a\_0\} \cup \{2k + 1 : k \notin a\_0\}.\tag{10}$$

The sets *SG*(*ν*) and *a<sup>ν</sup>* do not belong to **L**, accordingly, *z* = *zG* ∈ **L**[*a*0] **L**—unlike *c* in Section 7.1. Nevertheless, we are going to prove that the extension **L**[*G z*] = **L**[{*am* }*m*∈*z*] witnesses Theorem 2(i) with *a* = *a*<sup>0</sup> .

Note that the setup here is not exactly the same as in the proof of Theorem 1 in Section 7.1 since the set *z* does not belong to **L**, the ground universe. Therefore we cannot treat **P**∗ *z* as a forcing in **L**. Instead of **P**<sup>∗</sup> *z*, we make use of the set *K* of all conditions *p* ∈ **P**<sup>∗</sup> *ω* such that for any *k* ≥ 1:


as well as the related set *K*[**U**] = *K* ∩ **P** = *K* ∩ **P**[**U**].

**Lemma 26.** *K is a regular forcing in* **L***. If G* ⊆ **P** *is* **P***-generic over* **L** *then G* ∩ *K* = *G* ∩ *K*[**U**] *is a set K*[**U**] *generic over* **L** *and* **L**[*G* ∩ *K*] = **L**[*G zG*].

**Proof.** The nontrivial item of the regularity property here is (4) of Definition 8. If *p* ∈ **P**<sup>∗</sup> then define *p*<sup>∗</sup> ∈ **P**<sup>∗</sup> to be equal to *p* everywhere except for *Sp*<sup>∗</sup> (0) = *Sp*(0) ∪ *S*, where *S* consists of all strings *s* = **s***<sup>k</sup>* such that 1) 2*k* ∈ |*p*| or 2*k* − 1 ∈ |*p*|, and 2) *s* ∈/ *F*<sup>∨</sup> *<sup>p</sup>* (0) (to make sure that *p*<sup>∗</sup> ≤ *p*). Now we let *d* contain 0, all numbers 2*k* ∈ |*p*∗| such that **s***<sup>k</sup>* ∈ *Sp*<sup>∗</sup> (0), and all numbers 2*k* − 1 ∈ |*p*∗| such that **s***<sup>k</sup>* ∈ *F*<sup>∨</sup> *<sup>p</sup>* (0) *Sp*<sup>∗</sup> (0). (Compare to Example 2 in Section 3.2!)

The rest of the lemma follows from Lemma 9.

Thus extensions of the form **L**[*G zG*] considered here are exactly *K*[**U**]-generic extensions of **L**. To check that those extensions satisfy Theorem 2(i), we prove the following Claims 5 and 6. The first of them is entirely similar to Claim 3, so the proof is omitted (left to the reader).

**Claim 5.** *It is true in* **L**[*G z*] *that z is Σ*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> *, therefore a*<sup>0</sup> *is* <sup>Δ</sup><sup>1</sup> **<sup>n</sup>**+<sup>1</sup> *.*

**Claim 6.** *In* **<sup>L</sup>**[*G <sup>z</sup>*]*, if x* <sup>⊆</sup> *<sup>ω</sup> is <sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>***, then x* <sup>∈</sup> **<sup>L</sup>** *and x is <sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>** *in* **L***.*

The proof of this claim involves the following lemma.

**Lemma 27** (proved below in Section 7.3)**.** *Suppose that M*, *U* ∈ **sJS***, p* ∈ *K*[*U*]*, q* ∈ *K*[**U**]*. Let* Φ *be any closed parameter-free Σ*<sup>1</sup> **<sup>n</sup>** *formula. Then it is impossible that simultaneously q* ||−−*K*[**U**] <sup>¬</sup> <sup>Φ</sup> *and p <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup>* Φ*.*

**Proof** (Claim 6 from the lemma)**.** Assume that *x* = {*m* : *ϕ*(*m*)} in **L**[*G c*] = **L**[*G* ∩ *K*], where *ϕ*(*m*) is a *Σ*<sup>1</sup> **<sup>n</sup>** formula. We claim that then

$$m \in \mathbf{x} \iff \exists \left( \mathcal{M}, \mathcal{U} \right) \in \mathbf{s} \mathbf{JS} \exists \left p \in \mathbb{K}[\mathcal{U}] \left( p^{\nkern-1.1em} \mathbf{forrc}^{\mathcal{M}}\_{ll} \; \middle|\; \mathcal{q}\left(m\right)\right). \tag{11}$$

This proves Claim 6, of course, by Theorem 11. Now let us check (11) itself; this will be similar to the proof of (9) in Section 7.1.

Assume that *m* ∈ *x*, that is, *ϕ*(*m*) holds in **L**[*G* ∩ *K*]. By Theorem 12(ii) there is a condition *<sup>p</sup>* <sup>∈</sup> *<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>* such that *<sup>p</sup> <sup>K</sup>***forc**<sup>∞</sup> *<sup>ϕ</sup>*(*m*), that is, *<sup>p</sup> <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> ϕ*(*m*), where *M* = **M***<sup>ξ</sup>* , *U* = **U***<sup>ξ</sup>* , *ξ* < *ω*<sup>1</sup> . However, this implies the right-hand side of (9).

Now assume that *<sup>M</sup>*, *<sup>U</sup>*<sup>∈</sup> **sJS**, *<sup>p</sup>* <sup>∈</sup> *<sup>K</sup>*[*U*], and *<sup>p</sup> <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup> ϕ*(*m*). Suppose towards the contrary that *ϕ*(*m*) is false in **L**[*G* ∩ *K*], so that there is a condition *q* ∈ *G* ∩ *K* such that *q* ||−−*K*[**U**] ¬ *ϕ*(*m*). However, this contradicts Lemma 27. (Claim 6 and Theorem 2(i) modulo Lemma 27)

#### *7.3. Nonconstructible* Δ<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> *Real, Part 2*

We continue the proof of Theorem 2(i).

The proof of Lemma 27 that follows below makes use of transformations in BIJ*<sup>ω</sup> <sup>ω</sup>* (bijections of *<sup>ω</sup>*) and those in the set **Lip***<sup>ω</sup>* <sup>=</sup> {*<sup>α</sup>* <sup>∈</sup> **Lip**<sup>I</sup> : <sup>|</sup>*α*| ⊆ *<sup>ω</sup>*}, essentially the *<sup>ω</sup>*-product of **Lip**. Yet this will be somewhat more complicated than the proof of Theorem 1 above, because in this case *K* is not preserved under the action of arbitrary transformations in BIJ*<sup>ω</sup> <sup>ω</sup>* and **Lip***<sup>ω</sup>* . This is why we have to consider certain combinations of those transformations.

Namely consider superpositions of the form *<sup>σ</sup>* <sup>=</sup> *<sup>π</sup>* ◦ *<sup>α</sup>*, where *<sup>π</sup>* <sup>∈</sup> BIJ*<sup>ω</sup> <sup>ω</sup>* and *<sup>α</sup>* <sup>∈</sup> **Lip***<sup>ω</sup>* . (Such *<sup>σ</sup>* acts so that *σ· x* = *π·*(*α· x*) for any applicable object *x*.)

**Remark 4.** The set Σ of all *σ* of this form is a group under the superposition, because the transformations of the two families considered commute so that *α* ◦ *π* = *π* ◦ *α* , where *α* = *π·α*, that is, *α <sup>k</sup>* = *απ*(*k*) for all *k*.

**Definition 22.** *A transformation σ* = *π* ◦ *α* ∈ Σ *is K-preserving, if p* ∈ *K* ⇐⇒ *σ· p* ∈ *K for all p* ∈ **P**<sup>∗</sup> *ω.*

Not all *<sup>π</sup>* <sup>∈</sup> BIJ*<sup>ω</sup> <sup>ω</sup>* are *<sup>K</sup>*-preserving, and neither is any *<sup>α</sup>* <sup>∈</sup> **Lip***<sup>ω</sup>* with *<sup>α</sup>*<sup>0</sup> <sup>=</sup> the identity. Yet there are plenty of *K*-preserving transformations in Σ.

**Lemma 28.** *Let U*, *V be countable systems with* |*U*| = |*V*| = *ω, and p* ∈ *K*[*U*], *q* ∈ *K*[*V*]*. There is a Kpreserving transformation σ* = *π* ◦ *α* ∈ Σ *such that σ·U* = *V, and the conditions σ· p and q are compatible.*

**Proof.** First of all, Lemma 5 yields a transformation *α*<sup>0</sup> ∈ **Lip** such that *α*<sup>0</sup> *·U*(0) = *V*(0) and the conditions *<sup>α</sup>*<sup>0</sup> *· <sup>p</sup>*(0) and *<sup>q</sup>*(0) (in *<sup>P</sup>*<sup>∗</sup> ) are compatible. Define *<sup>α</sup>* <sup>=</sup> {*αi*}*i*∈*<sup>ω</sup>* <sup>∈</sup> **Lip***<sup>ω</sup>* so that *<sup>α</sup>*<sup>0</sup> has just been defined, and *α<sup>k</sup>* = the identity for all *k* > 0. Note that *α*<sup>0</sup> is a ⊆-preserving bijection of the set **Seq** of all non-empty strings of integers. Let *f* : *ω* onto −→ *ω* be the associated permutation of integers, so that *<sup>f</sup>*(*k*) = *<sup>n</sup>* iff *<sup>α</sup>*0(**s***k*) = **<sup>s</sup>***<sup>n</sup>* (and *<sup>f</sup>*(0) = 0 ). Define *<sup>π</sup>* <sup>∈</sup> BIJ*<sup>ω</sup> <sup>ω</sup>* so that *π*(0) = 0 and then *π*(2*k* + 2) = 2 *f*(*k*) + 2 and *π*(2*k* + 1) = 2 *f*(*k*) + 1. It is quite obvious that *ρ* = *π* ◦ *α* is *K*-preserving. Let *U* = *ρ·U* and *p* = *ρ· p*. Thus *U* is a countable system with |*U* | = *ω*, *p* ∈ *K*[*U* ], and in addition *U* (0) = *V*(0) and the conditions *p* (0) = *α*<sup>0</sup> *· p*(0) and *q*(0) are compatible.

It follows from Lemma <sup>5</sup> that there is a transformation *<sup>γ</sup>* <sup>=</sup> {*γν* }*ν*<*<sup>ω</sup>* <sup>∈</sup> **Lip***<sup>ω</sup>* such that *<sup>γ</sup>*<sup>0</sup> is the identity (and hence *γ* is *K*-preserving) and for any *k* ≥ 1 we have *γ<sup>k</sup> ·U* (*k*) = *V*(*k*) and *γ· p* (*k*) is compatible with *q*(*k*). We conclude that the transformation *σ* = *γ* ◦ *ρ* = *γ* ◦ *π* ◦ *α* is *K*-preserving, *V* = *γ·U* = *σ·U*, and the condition *γ· p* = (*γ* ◦ *π* ◦ *α*)*· p* is compatible with *q*. Then, we have *σ* ∈ Σ by Remark 4 in Section 7.3.

**Proof** (Lemma 27)**.** Suppose towards the contrary that both *<sup>q</sup>* ||−−*K*[**U**] <sup>¬</sup> <sup>Φ</sup> and *<sup>p</sup> <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup>* Φ. By the way we can w.l. o. g. assume that |*U*| ⊆ *ω*, by Lemma 19, and moreover, that |*U*| = *ω* exactly. (Otherwise extend *U* by *U*(*ν*) = *Q* for all *ν* ∈ *ω* |*U*|, where *Q* = all eventually-0 functions *f* ∈ **Fun**.)

There is an ordinal *ξ* < *ω*<sup>1</sup> such that *q* ∈ *K*[**U***<sup>ξ</sup>* ], *ω* ⊆ |**U***<sup>ξ</sup>* |, and *M* ⊆ **M***<sup>ξ</sup>* . Let *V* = **U***<sup>ξ</sup> ω*. Note that |*q*| ⊆ *ω* since *K* ⊆ **P**<sup>∗</sup> *ω*. Thus *q* ∈ *K*[*V*]. Apply Lemma 28 in **M***<sup>ξ</sup>* . It gives a *K*-preserving transformation *σ* = *α* ◦ *π* ∈ **M***<sup>ξ</sup>* such that *σ·U* = *V* and the conditions *r* = *σ· p* and *q* (both in *K*[*V*]) are compatible. On the other hand, we have *<sup>r</sup> <sup>K</sup>***forcM***<sup>ξ</sup> <sup>V</sup>* <sup>Φ</sup> by Lemmas <sup>22</sup> and 23, and hence *<sup>r</sup> <sup>K</sup>***forcM***<sup>ξ</sup>* **U***ξ* Φ by Lemma 18, that is, *<sup>r</sup> <sup>K</sup>***forc**<sup>∞</sup> <sup>Φ</sup>. Thus *<sup>r</sup>* ||−−*K*[**U**] <sup>Φ</sup> by Corollary 8(i). However, *<sup>r</sup>* is compatible with *<sup>q</sup>*, and *q* forces the opposite, a contradiction. (Lemma 27) (Claim 6) (Theorem 2(i))

#### **8. Application 2: Nonconstructible Self-Definable** Δ<sup>1</sup> *<sup>n</sup>* **Reals**

Note that the set *a* as in Theorem 2(i) is definable in the generic extension of **L**, considered in Section 7.2, by means of other reals in that extension, including those which do not necessarily belong to **L**[*a*]. Claim (ii) of Theorem 2 achieves the same effect with the advantage that *a* is definable inside **L**[*a*].

The key idea (originally from [9] Section 4) can be explained as follows. Recall that a set of the form *a*<sup>0</sup> = *aG*(0) was made definable in a generic extension of the form **L**[*G zG*] by means of the presence/absense of other sets of the form *SG*(*ν*), *ν* < *ω*, in **L**[*G z*], see Sections 7.2 and 7.3. Our plan will now be to make each of the according sets *aG*(*ν*) ∈ **L**[*G z*] (note that *aG*(*ν*) ⊆ *ω* {0}, see Definition 9), as well as the whole sequence of them, Δ<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> -definable in **L**[*G z*]. In order to do this, we need to develop a suitable coding construction.

**Assumption 4.** *We continue to assume* **V** = **L** *in the ground universe. We fix an integer* **n** ≥ 2*, for which Theorem 1*(ii) *will be proved, and make use of a system* **U** *and the forcing notion* **P** = **P**[**U**] *as in Definition 16; both* **U** *and* **P** *belong to* **L***.*

*8.1. Nonconstructible Self-Definable* Δ<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> *Reals: The Model*

Here we begin **the proof of Theorem <sup>2</sup>**(ii)**.** Recall that *<sup>ω</sup><sup>ω</sup>* <sup>=</sup> {**s***<sup>k</sup>* : *<sup>k</sup>* <sup>&</sup>lt; *<sup>ω</sup>*} is a fixed recursive enumeration of strings of natural numbers, such that **s**<sup>0</sup> = Λ, the empty string, and **s***<sup>k</sup>* ⊆ **s***<sup>k</sup>* =⇒ *k* ≤ *k* . Let *k <sup>i</sup>* <sup>=</sup> num (**s***<sup>k</sup> i*), thus **s** *k i* = **s***<sup>k</sup> i*. Then we have:

• Each set *L*(*k*) = {*k <sup>i</sup>* : *<sup>i</sup>* <sup>&</sup>lt; *<sup>ω</sup>*} ⊆ *<sup>ω</sup>* is countably infinite, *<sup>k</sup>* <sup>&</sup>lt; min*<sup>i</sup> k i* , *k* = *k* =⇒ *L*(*k*) ∩ *L*(*k* ) = ∅ and *i* = *j* =⇒ *k <sup>i</sup>* = *k <sup>j</sup>* , and finally each *m* ≥ 1 is equal to *k <sup>i</sup>* for exactly one pair of indices of *i*, *k* < *ω*.

Define a partial order on *ω* so that *i k* iff **s***<sup>i</sup>* ⊂ **s***<sup>k</sup>* . Obviously *k k <sup>i</sup>* for all *i*, *k* ∈ *ω*, and 0 is the -least element.

For any sequence *a* = {*ak*}*k*<*<sup>ω</sup>* of sets *ak* ⊆ *ω*, we define a set *ζ<sup>a</sup>* ⊆ *ω* so that:

1) 0 ∈ *ζ<sup>a</sup>* ;


The next theorem obviously implies Theorem 2(ii).

**Theorem 14.** *Let G* ⊆ **P** *be* **P***-generic over* **L***. Define a*[*G*] = {*aG*(*i*)}*i*<*<sup>ω</sup> and ζ* = *ζa*[*G*] ⊆ *ω. Then* **L**[*ζ*] = **L**[*G ζ*]*, and it holds in* **L**[*ζ*] *that*:


**Proof** (will continue towards the end of Section 7)**.** Our arguments will be a more elaborate version of arguments in Sections 7.2, 7.3. We'll make use of the set *K* of all conditions *p* ∈ **P**<sup>∗</sup> *ω* such that for all *i* and *k*:


(compare to (A), (B) in Section 7.2), and the related set *K*[**U**] = *K* ∩ **P**.

**Lemma 29.** *K is a regular forcing in* **L***. If G* ⊆ **P** *is a set* **P***-generic over* **L** *then G* ∩ *K* = *G* ∩ *K*[**U**] *is K*[**U**] *generic over* **L***,* |*G* ∩ *K*| = *ζa*[*G*] *, and accordingly* **L**[*G* ∩ *K*] = **L**[*G ζa*[*G*]] = **L**[*ζa*[*G*]]*.*

**Proof.** As above, the nontrivial item of the regularity property is (4) of Definition 8. Suppose that *p* ∈ **P**<sup>∗</sup> . Then |*p*| ⊆ *ω* is finite. Let *δ* be the least -initial segment of *ω* satisfying |*p*| ⊆ *δ*; *δ* is finite as well. Define *p*<sup>∗</sup> ∈ **P**<sup>∗</sup> so that |*p*∗| = *δ* and *Fp*<sup>∗</sup> (*k*) = *Fp*(*k*) for all *k*, but the sets *Sp*<sup>∗</sup> (*k*) may be strictly bigger than the corresponding sets *Sp*(*k*). The definition of *Sp*<sup>∗</sup> (*k*) goes on by -inverse induction on *k* ∈ *δ*. If *k* ∈ *δ* is -maximal in *δ* then obviously *k* ∈ |*p*|, and we put *Sp*<sup>∗</sup> (*k*) = *Sp*(*k*). Assume that *k* ∈ *δ* is not -maximal in *δ*, and the value of *p*∗(*k <sup>m</sup>*) = *Sp*<sup>∗</sup> (*k <sup>m</sup>*); *Fp*(*k <sup>m</sup>*) is defined for all *m* such that *k <sup>m</sup>* ∈ *δ*. Put *Sp*<sup>∗</sup> (*k*) = *Sp*(*k*) ∪ *S*, where *S* consists of all strings *s* = **s***<sup>i</sup>* such that


By definition, |*p*∗| = *δ*, and if *i*, *k* ∈ *ω* and at least one of the numbers *k* <sup>2</sup>*i*+<sup>1</sup> , *k* <sup>2</sup>*<sup>i</sup>* belongs to *δ*, then the string **s***<sup>i</sup>* belongs to *F*∨ *<sup>p</sup>*<sup>∗</sup> (*k*) ∪ *Sp*<sup>∗</sup> (*k*).

Now we define a set *d* ⊆ *δ* so that the decision whether a number *k* ∈ *δ* belongs to *d* is made by direct -induction. We put 0 ∈ *d*. Suppose that some *k* ∈ *δ* already belongs to *d*. We define:


A simple verification that *p*∗ and *d* satisfy Definition 8(4) is left to the reader.

Further, the set *G* ∩ *K* = *G* ∩ *K*[**U**] is *K*[**U**]-generic by Lemma 9(ii).

By definition if *k* ∈ *ζa*[*G*] then *aG*(*k*) = {*i* : *k* <sup>2</sup>*<sup>i</sup>* ∈ *ζa*[*G*]} = {*i* : *k* <sup>2</sup>*i*+<sup>1</sup> ∈/ *ζa*[*G*]} ∈ **L**[*ζa*[*G*]], therefore *G ζa*[*G*] ∈ **L**[*ζa*[*G*]] and **L**[*G ζa*[*G*]] = **L**[*ζa*[*G*]].

Now to prove **L**[*G* ∩ *K*[**U**]] = **L**[*G ζa*[*G*]] it remains to show that |*G* ∩ *K*| = *ζa*[*G*]—then use Lemma 9(iii). Note that both |*p*| for any *p* ∈ *K* and *ζa*[*G*] are -initial segments. Thus it suffices to check that if *k* ∈ |*G* ∩ *K*| ∩ *ζa*[*G*] then

$$\mathcal{C}\_{2i+1}^{k} \in |G \cap K| \iff \mathcal{C}\_{2i+1}^{k} \in \mathbb{Z}\_{\vec{\sigma}[G]} \quad \text{and} \quad \mathcal{C}\_{2i}^{k} \in |G \cap K| \iff \mathcal{C}\_{2i}^{k} \in \mathbb{Z}\_{\vec{\sigma}[G]}.$$

Prove, e.g., the first equivalence. Suppose that *k* <sup>2</sup>*i*+<sup>1</sup> ∈ |*G* ∩ *K*|. Then *k* <sup>2</sup>*i*+<sup>1</sup> ∈ |*p*| for some *p* ∈ *K* in *G*, and we have **s***<sup>i</sup>* ∈ *F*<sup>∨</sup> *<sup>p</sup>* (*k*) *Sp*(*k*) by (B), so that **s***<sup>i</sup>* ∈/ *SG*(*k*) and accordingly *i* ∈/ *aG*(*k*), thus by definition *k* <sup>2</sup>*i*+<sup>1</sup> ∈ *ζa*[*G*] . Suppose conversely that *k* <sup>2</sup>*i*+<sup>1</sup> ∈ *ζa*[*G*] . Then by definition *i* ∈/ *aG*(*k*), hence **s***<sup>i</sup>* ∈/ *GG*(*k*). This must be forced by some *p* ∈ *K* ∩ *G*, and, as *k* ∈ |*G* ∩ *K*|, we can assume that *k* ∈ |*p*|. However, in this case forcing **s***<sup>i</sup>* ∈/ *GG*(*k*) means by necessity that just **s***<sup>i</sup>* ∈ *F*<sup>∨</sup> *<sup>p</sup>* (*k*) *Sp*(*k*), so there exists a stronger condition *p* ∈ *K* ∩ *G* with *k* <sup>2</sup>*i*+<sup>1</sup> ∈ |*p* |. We conclude that *k* <sup>2</sup>*i*+<sup>1</sup> ∈ |*G* ∩ *K*|. (Lemma)

It follows that *ζa*[*G*] is *Σ*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> in **L**[*G ζ*] by Corollary 7. On the other hand, by definition, if *k* ∈ *ζa*[*G*] , then, for any *k*, we have *k* <sup>2</sup>*<sup>i</sup>* ∈ *ζa*[*G*] iff *k* <sup>2</sup>*i*+<sup>1</sup> <sup>∈</sup>/ *<sup>ζ</sup>a*[*G*] . This easily leads to a *<sup>Π</sup>*<sup>1</sup> *<sup>n</sup>*+<sup>1</sup> definition of *ζa*[*G*] . Thus *ζa*[*G*] is Δ<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> in **L**[*G ζ*], and hence we have claim (i) of Theorem 14. The proof of claim (ii) follows in the next two subsections.

**Remark 5.** A slightly more elaborate argument, like in the end of Section 4 in [9], shows that even more {*ζa*[*G*]} is a *<sup>Π</sup>*<sup>1</sup> **<sup>n</sup>** singleton in **L**[*ζa*[*G*]] since *ζa*[*G*] is equalto the only set *ζ* ⊆ *ω* in **L**[*ζa*[*G*]] satisfyings the following requirements:

(a) 0 ∈ *ζ* , and if *k* ∈/ *ζ* then *k* <sup>2</sup>*<sup>i</sup>* ∈/ *ζ* and *k* <sup>2</sup>*<sup>i</sup>* ∈/ *ζ* for all *i*;

(b) if *k* ∈ *ζ* then we have *k* <sup>2</sup>*<sup>i</sup>* ∈ *ζ* iff *k* <sup>2</sup>*i*+<sup>1</sup> ∈/ *ζ* for every *i*, and

(c) if *k* ∈ *ζ* then the set *Sζ<sup>k</sup>* = {**s***<sup>i</sup>* : *k* <sup>2</sup>*<sup>i</sup>* ∈ *ζ*} satisfies **Γ***k*(*Sζk*).

The conjunction of them amounts to a *Π*<sup>1</sup> **<sup>n</sup>** definition of {*ζ*} in **L**[*ζ*].

#### *8.2. Key Lemma*

As in Section 7.2, Claim (ii) of Theorem 14 is a consequence of the following lemma (the key lemma from the title), the proof of which will end the proof of theorems 14 and 2(ii).

**Lemma 30** (in **L**)**.** *Suppose that M*, *U* ∈ **sJS***, p* ∈ *K*[*U*]*, q* ∈ *K*[**U**]*. Let* Φ *be any closed parameter-free Σ*1 **<sup>n</sup>** *formula. Then it is impossible that simultaneously q* ||−−*K*[**U**] <sup>¬</sup> <sup>Φ</sup> *and p <sup>K</sup>***forc***<sup>M</sup> <sup>U</sup>* Φ*.*

Following Definition 22, a transformation *σ* ∈ Σ (see Remark 4 in Section 7.3 on Σ) is called *K*preserving, if *p* ∈ *K* ⇐⇒ *σ· p* ∈ *K* for all *p* ∈ **P**<sup>∗</sup> *ω*. Clearly the regular forcing *K* here is different (and way more complex in some aspects) than *K* in Section 7.3. The following lemma is analogous to Lemma 28.

**Lemma 31** (in **L**)**.** *Suppose that U*, *V are countable systems with* |*U*| = |*V*| = *ω, and p* ∈ *K*[*U*], *q* ∈ *K*[*V*]*. Then there is a K-preserving transformation σ* ∈ Σ *such that σ·U* = *V, and the conditions σ· p and q are compatible.*

**Proof.** The proof resembles the proof of Lemma 28, but is somewhat more complicated. Essentially, we'll have a ramified *ω*-long iteration in which the construction employed in Lemma 28 will be just one step. We define -cones *Ck* = {*i* ∈ *ω* : *k i*} and *C <sup>k</sup>* = *Ck* ∪ {*k*} for any *k* ∈ *ω*.

**Claim 7.** *If <sup>α</sup>* <sup>=</sup> {*αk*}*k*<*<sup>ω</sup>* <sup>∈</sup> **Lip***<sup>ω</sup> , <sup>k</sup>*<sup>0</sup> <sup>∈</sup> *<sup>ω</sup>, and <sup>α</sup><sup>k</sup> is the identity for each <sup>k</sup>* <sup>=</sup> *<sup>k</sup>*<sup>0</sup> *then there is a bijection <sup>π</sup>* <sup>=</sup> *<sup>π</sup>*[*αk*<sup>0</sup> ] <sup>∈</sup> BIJ*<sup>ω</sup> <sup>ω</sup> , recursive in α, -preserving, and such that π*(*k*) = *k for all k* ∈/ *Ck*<sup>0</sup> *and π* ◦ *α is Kpreserving.*

**Proof.** Note that *αk*<sup>0</sup> is a ⊆-preserving bijection of the set **Seq** of all finite non-empty strings of integers. Let *<sup>f</sup>* <sup>=</sup> *<sup>f</sup>αk*<sup>0</sup> : *<sup>ω</sup>* onto −→ *ω* be the associated permutation of integers, so that *f*(*i*) = *j* iff *α*0(**s***i*) = **s***<sup>j</sup>* . Let the transformation *π* = *π*[*αk*<sup>0</sup> ] be the identity outside of the strict -cone *Ck*<sup>0</sup> ; in particular, *π*(*k*0) = *k*<sup>0</sup> . Beyond this, put *π*(*k*0 <sup>2</sup>*i*) = *k*0 <sup>2</sup> *<sup>f</sup>*(*i*) and *π*(*k*0 <sup>2</sup>*i*+1) = *k*0 <sup>2</sup> *<sup>f</sup>*(*i*)+<sup>1</sup> for all *i*. Now, if *k* ∈ *Ck*<sup>0</sup> and *π*(*k*) = *k* is defined then put *π*(*k* <sup>2</sup>*m*) = *k* <sup>2</sup>*<sup>m</sup>* for all *m*. (Claim)

#### *8.3. Matching Permutation*

Now, in continuation of the proof of Lemma 31, given any *<sup>α</sup>* <sup>∈</sup> **Lip***<sup>ω</sup>* we outline a construction of a permutation <sup>Π</sup> <sup>∈</sup> BIJ*<sup>ω</sup> <sup>ω</sup>* such that the superposition *α* ◦ Π is *K*-preserving. Suppose that *α* = {*αk*}*k*<*<sup>ω</sup>* <sup>∈</sup> **Lip***<sup>ω</sup>* . We define


The whole sequence of transformations is thereby specified by the choice of the components *αm km* ∈ **Lip**, *m* ∈ *ω*; we address this issue below. Now put

$$T\_{\mathfrak{m}} = \mathfrak{p}\_{\mathfrak{m}} \circ \cdots \circ \mathfrak{p}\_2 \circ \mathfrak{p}\_1 \circ \mathfrak{p}\_0 \in \Sigma, \quad \Pi\_{\mathfrak{m}} = \pi\_{\mathfrak{m}} \circ \cdots \circ \pi\_2 \circ \pi\_1 \circ \pi\_0 \in \text{Bil}^{\omega}\_{\omega}.\tag{12}$$


**Proof.** (i) Suppose that *k* < *ω* belongs to some *Dm* . Prove that any number *j* = *k* <sup>2</sup>*<sup>i</sup>* or *j* = *k* <sup>2</sup>*i*+<sup>1</sup> , *i* < *ω*, also belongs to some *Dm* . By definition *k* = Π*m*(*k*) ∈ *dm* . The number *j* = Π*m*(*j*) either belongs to *dm* , QED, or is -minimal in *ω dm* . In the latter case, we have ¬ *km j* for all *m* > *m*, and hence Π*<sup>m</sup>* (*j*) is equal to *j* for every *m* > *m*. Take *m* > *m* big enough for *j* ∈ *dm* ; then *j* ∈ *Dm* . To prove (ii) apply assumption (II) above. Finally (iii) easily follows from items (i), (ii).

The transformation Π as in item (iii) of the claim can be understood as the infinite superposition ···◦ *π<sup>m</sup>* ◦···◦ *π*<sup>2</sup> ◦ *π*<sup>1</sup> ◦ *π*<sup>0</sup> .

**Claim 9.** *Suppose that m* ≤ *i, U is a system,* |*U*| = *ω, and p* ∈ **P**<sup>∗</sup> *,* |*p*| ⊆ *ω. Then* (*Ti ·U*)(*km*) = ((*α* ◦ Π)*·U*)(*km*) *and* (*Ti · p*)(*km*) = ((*α* ◦ Π)*· p*)(*km*)*.*

**Proof.** By Claim 8(ii), there is an index *j* ∈ *Dm* such that *km* = Π(*j*) = Π*i*(*j*) for all *i* ≥ *m*. Thus (*Ti ·U*)(*km*) is equal to *<sup>α</sup><sup>m</sup> km ·U*(*j*) = *<sup>α</sup><sup>m</sup> km ·*((*Ti ·U*)(*km*)).

The argument for *p* is similar. (Claim)

It follows that the superposition *α* ◦ Π ∈ Σ is *K*-preserving. Indeed, since sets |*p*| are finite, if *p* ∈ *K* then there is *m* such that |*p*| ⊆ *dm* ∩ *Dm* . However, then (*α* ◦ Π)*· p* = *Ti · p* by Claim 9, and on the other hand *Ti* is *K*-preserving as a finite superposition of *K*-preserving transformations *ρ<sup>m</sup>* .

#### *8.4. Final Argument*

Now let *U*, *V*, *p*, *q* be as in Lemma 31. To accomplish the proof of Lemma 31, we note that the construction of *<sup>α</sup><sup>m</sup>* , *<sup>π</sup><sup>m</sup>* , *<sup>ρ</sup><sup>m</sup>* depends on *<sup>α</sup>km* rather than on *<sup>α</sup>* <sup>=</sup> {*αk*}*k*<*<sup>ω</sup>* <sup>∈</sup> **Lip***<sup>ω</sup>* as a whole. This enables us to carry out the following definition of *αkm* ∈ **Lip** (*m* ∈ *ω*) by induction on *m*.

**Definition 23.** *Choose, using Lemma 5, a transformation αk*<sup>0</sup> ∈ **Lip** *such that αk*<sup>0</sup> *·U*(*k*0) = *V*(*k*0) *and the conditions αk*<sup>0</sup> *· p*(*k*0) *and q*(*k*0) *(in P*<sup>∗</sup> *) are compatible.*

*Now suppose that transformations αk*<sup>0</sup> , ... , *αkm* ∈ **Lip** *have been defined, and define αkm*+<sup>1</sup> ∈ **Lip***. Note that km*+<sup>1</sup> *is a -minimal element in ω dm , where dm* = {*k*0, ... , *km* }*, as above. First of all if μ* ≤ *m then define:*


*Define* <sup>Π</sup>*<sup>m</sup> and Tm by* (12) *above. Put <sup>U</sup><sup>m</sup>* = *Tm ·<sup>U</sup> and <sup>p</sup><sup>m</sup>* = *Tm · p. By Lemma 5, there is a transformation <sup>α</sup>km*+<sup>1</sup> <sup>∈</sup> **Lip** *such that <sup>α</sup>km*+<sup>1</sup> *·Um*(*km*+1) = *<sup>V</sup>*(*km*+1) *and the conditions <sup>α</sup>km*+<sup>1</sup> *· <sup>p</sup>m*(*km*+1) *and q*(*km*+1) *are compatible.*

After we have defined *αkm* ∈ **Lip** by induction on *m*, let's take the transformation *α* = {*αk*}*k*<*<sup>ω</sup>* ∈ **Lip***<sup>ω</sup>* as the input of the construction in Section 8.3. The latter gives us a permutation <sup>Π</sup> <sup>∈</sup> BIJ*<sup>ω</sup> ω* of Claim 8, such that the superposition *σ* = *α* ◦ Π ∈ Σ is *K*-preserving. It remains to check that 1) *σ·U* = *V* and that 2) *σ· p* and *q* are compatible conditions.

To prove 1), consider any *k* = *km*+<sup>1</sup> ∈ *ω*. (The argument will also work for the case *m* = −1, that is, *k* = 0.) By definition, we have

$$V(k\_{m+1}) = \mathfrak{a}\_{k\_{m+1}} \cdot \mathcal{U}^{\mathfrak{m}}(k\_{m+1}) = (\mathfrak{a}^{m+1} \cdot \mathcal{U}^{\mathfrak{m}})(k\_{m+1}) \dots$$

and hence, as obviously *πm*+1(*km*+1) = *km*+<sup>1</sup> ,

$$V(k\_{m+1}) = ( (\pi^{m+1} \circ \mathfrak{a}^{m+1} \circ T\_m) \cdot lI) / (k\_{m+1}) = (T\_{m+1} \cdot lI) / (k\_{m+1}) \text{ .}$$

therefore *V*(*km*+1) = ((*α* ◦ Π)*·U*)(*km*+1)=(*σ·U*)(*km*+1) by Claim 9, as required. (Lemma 31)

**Proof** (Lemma 30)**.** Similar to the proof of Lemma 27, but using Lemma 31 just proved.

(Theorem 14) (Theorem 2(ii))

#### **9. Application 3: Nonconstructible** *Σ*<sup>1</sup> *<sup>n</sup>* **Reals**

Here we prove Theorem 3.

**Assumption 5.** *We continue to assume* **V** = **L** *in the ground universe. We fix an integer* **n** ≥ 2*, for which Theorem 3 will be proved, and make use of a system* **U** *and the forcing notion* **P** = **P**[**U**] *as in Definition 16; both* **U** *and* **P** *belong to* **L***.*

#### *9.1. Nonconstructible Σ*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> *Reals: The Model*

The most obvious idea as of getting an extension required is to slightly modify the proof of Theorem 2(ii) in the following direction. Suppose that *G* ⊆ **P** be **P**-generic over **L**, and let *SG*(*ν*) and *a<sup>ν</sup>* = *aG*(*ν*) = {*k* ≥ 1 : **s***<sup>k</sup>* ∈ *S*(*i*)} be defined as in Definition 9. We proved (see the proof of Theorem 2(i) above) that if

$$z = \{0\} \cup \{2k + 2 : k \in a\_0\} \cup \{2k + 1 : k \notin a\_0\}$$

by (10) of Section 7.2 then the set *a*<sup>0</sup> is Δ<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> in **L**[*G z*], and the part {2*k* + 2 : *k* ∈ *a*<sup>0</sup> } of *z* is responsible for *a*<sup>0</sup> being *Σ*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> in **L**[*G z*] (by means of the equality *a*<sup>0</sup> = {*k* : ∃ *S* **Γ**2*k*+2(*S*)}) while the part {2*<sup>k</sup>* <sup>+</sup> 1 : *<sup>k</sup>* <sup>∈</sup>/ *<sup>a</sup>*<sup>0</sup> } is responsible for *<sup>a</sup>*<sup>0</sup> being *<sup>Π</sup>*<sup>1</sup> *<sup>n</sup>*+<sup>1</sup> in **L**[*G z*] (by means of the equality *a*<sup>0</sup> = {*k* : ¬ ∃ *S* **Γ**2*k*+1(*S*)}). As now the second part is not needed, one might hope that if *y* is defined by

$$y = y\_G := \{0\} \cup a\_0 = \{0\} \cup a\_G(0) \tag{13}$$

then **L**[*G y*] will be a model for Theorem 3. At least *a*<sup>0</sup> will be *Σ*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> in **L**[*G y*] by exactly the same reasons. However we have not been able to prove the second part of the theorem, i.e., that all reals Δ1 **<sup>n</sup>**+<sup>1</sup> in **L**[*G y*] belong to **L**. The point of difficulty is the following hypothesis:

**Conjecture 1.** Under the assumptions above, if *<sup>m</sup>* <sup>∈</sup>/ *<sup>y</sup>* <sup>=</sup> *yG* then any parameter-free *<sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> formula true in **L**[*G y*] is true in **L**[*G y*, *am*] as well.

We definitely cannot expect the conjecture to be true for formulas with parameters in **L**[*G y*] (the smaller model) since if *<sup>p</sup>* ∈ **<sup>L</sup>**[*G <sup>y</sup>*], *<sup>p</sup>* ⊆ *<sup>ω</sup>* codes the sequence {*ai*}*i*∈*<sup>y</sup>* then **Fun** ⊆ **<sup>L</sup>**[*p*] is true in **L**[*G y*] but false in **L**[*G y*, *am*].

We have a near-counterexample to Conjecture 1: the formula ∃ *x* (**Γ**0(*x*) ∧ **Fun** ⊆ **L**[*x*]) of class *Σ*1 **<sup>n</sup>**+<sup>1</sup> (assuming **n** ≥ 3 ) holds in **L**[*a*0] and fails in **L**[*a*0, *a*1]. The set *y* = {*a*<sup>0</sup> } is definitely not of the form (13), so this is not literally a counterexample, yet it casts doubts on the approach based on (13).

Now we describe the extension involved in the proof of Theorem 3.

The model we define will be a submodel of the whole extension **L**[*G*], where *G* is **P**-generic over **L**, and a set *y* of (13) is involved in the definition. We let

$$\mathcal{Y} = \mathcal{Y}\_{\mathbb{G}} = \mathcal{Y}\_{\mathbb{G}} \cup \left(\mathcal{Z} \smile \omega\right) = \{0\} \cup \mathfrak{a}\_0 \cup \left(\mathcal{Z} \smile \omega\right), \tag{14}$$

where *a*<sup>0</sup> = *aG*(0) (then *Y* ∈ **L**[*a*0] **L**) and *yG* is defined by (13). The goal is to prove that **L**[*GY*] witnesses Theorem 3 with *a* = *a*<sup>0</sup> . The task splits in two claims:

**Claim 10.** *In* **L**[*GY*]*, y is Σ*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> *, therefore a*<sup>0</sup> *is <sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> *as well.*

**Claim 11.** *In* **<sup>L</sup>**[*GY*]*, if x* <sup>⊆</sup> *<sup>ω</sup> is* <sup>Δ</sup><sup>1</sup> **<sup>n</sup>**+<sup>1</sup> *then x* <sup>∈</sup> **<sup>L</sup>** *and x is* <sup>Δ</sup><sup>1</sup> **<sup>n</sup>**+<sup>1</sup> *in* **L***.*

Claim 10 is established just as similar claims above, so we leave it for the reader.

Let us concentrate on Claim 11. We make use of the set *K*<sup>0</sup> of all conditions *p* ∈ **P**<sup>∗</sup> *ω* such that

$$\text{if } k \ge 1 \text{ and } k \in |p|, \text{ then } \circ\_k \in S\_p(0) \text{ (= Example 2 in Section 3.2);}\tag{15}$$

as well as the related sets: *K* = *K*<sup>0</sup> × (**P**<sup>∗</sup> (I *ω*)) = {*p* ∈ **P**<sup>∗</sup> : *p ω* ∈ *K*<sup>0</sup> }, *K*0[**U**] = *K*<sup>0</sup> ∩ **P**, and accordingly *K*[**U**] = *K* ∩ **P**.

**Lemma 32.** *It is true in* **L** *that*: *K*<sup>0</sup> *and K are regular forcings and absolute* ΔHC <sup>1</sup> *sets, and if z* ⊆ I *contains* 0 *then the restrictions K z, K*<sup>0</sup> *z are regular forcings, too.*

*If G* ⊆ **P** *is a set* **P***-generic over* **L** *then G* ∩ *K* = *G* ∩ *K*[**U**] *is a set K*[**U**]*-generic over* **L***, G* ∩ *K*<sup>0</sup> = *G* ∩ *K*0[**U**] *is a set K*0[**U**]*-generic over* **L***, and*

$$\mathbf{L}[G \cap K\_{\mathbf{0}}] = \mathbf{L}[G \restriction y\_{G}] \,, \quad \mathbf{L}[G \cap K] = \mathbf{L}[G \restriction Y\_{G}] = \mathbf{L}[G \restriction y\_{G}, G \restriction (\mathcal{T} \ltimes \omega)] \,,$$

**Proof.** To check (4) of Definition 8 for *K*<sup>0</sup> see Example 2 in Section 3.2. To prove, that the set *K*<sup>0</sup> *z* = {*p* ∈ *K*<sup>0</sup> : |*p*| ⊆ *z*} (*z* ∈ **L**, *z* ⊆ *ω*) is regular, argue as in Example 2 in Section 3.2. The rest of the lemma is easy: apply Lemma 9.

#### *9.2. Key Lemma*

Here we establish the following key lemma. Recall that sets *yG* , *YG* are defined by (13) and (14).

**Lemma 33.** *Suppose that G* ⊆ **P** *is* **P***-generic over* **L***, and yG* = {0} ∪ *aG*(0)*, y* ⊆ *ω, the symmetric difference δ* = *y* Δ *yG is finite, and* 0 ∈/ *δ. Then the models* **L**[*GYG*] = **L**[*G yG*, *G* (I *ω*)] *and* **<sup>L</sup>**[*G <sup>y</sup>*, *<sup>G</sup>* (<sup>I</sup> *<sup>ω</sup>*)] *are <sup>K</sup>*[**U**]*-generic extensions of* **<sup>L</sup>***, elementarily equivalent w. r. t. all <sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>** *formulas with parameters in the common part* **L**[*G* (*yG* ∩ *y*), *G* (I *ω*)] *of the two models.*

**Proof.** That **L**[*G yG*, *G* (I *ω*)] = **L**[*G* ∩ *K*[**U**]] is a *K*[**U**]-generic extension of **L** follows from Lemma 32. Consider **L**[*G y*, *G* (I *ω*)], the other model.

Let *u* = *y yG* and *v* = *yG y*; thus *δ* = *u* ∪ *v*. Then *v* ⊆ *aG*(0) but *u* ∩ *aG*(0) = ∅ by the definition of *yG* . In other words, the finite disjoint sets *<sup>S</sup><sup>u</sup>* <sup>=</sup> {**s***<sup>k</sup>* : *<sup>k</sup>* <sup>∈</sup> *<sup>u</sup>*} and *<sup>S</sup><sup>v</sup>* <sup>=</sup> {**s***<sup>k</sup>* : *<sup>k</sup>* <sup>∈</sup> *<sup>v</sup>*} satisfy *<sup>S</sup><sup>v</sup>* <sup>⊆</sup> *SG*(0) but *<sup>S</sup><sup>u</sup>* <sup>∩</sup> *SG*(0) = <sup>∅</sup>. It follows that there is a condition *<sup>p</sup>* <sup>∈</sup> *<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>*[**U**] such that <sup>|</sup>*p*<sup>|</sup> <sup>=</sup> {0}, *<sup>S</sup><sup>v</sup>* <sup>⊆</sup> *Sp*(0), and *<sup>S</sup><sup>u</sup>* <sup>⊆</sup> *<sup>F</sup>*<sup>∨</sup> *<sup>p</sup>* (0) *Sp*(0). We can increase *Fp*(0) if necessary for *Sp*(0) ⊆ *F*<sup>∨</sup> *<sup>p</sup>* (0) (a technical requirement) to hold.

Now let *q* be a condition obtained by the following modification of *p*: still |*q*| = {0} and *Fq*(0) = *Fp*(0) (therefore, *<sup>q</sup>* belongs to *<sup>K</sup>*[**U**] together with *<sup>p</sup>*), and *Sq*(0)=(*Sp*(0) <sup>∪</sup> *<sup>S</sup>u*) *<sup>S</sup><sup>v</sup>* . It is clear that *Sq*(0) ⊆ *F*<sup>∨</sup> *<sup>q</sup>* (0) = *F*<sup>∨</sup> *<sup>p</sup>* (0), so *p*, *q* satisfy (3) in Section 3.7. Therefore the map (Definition 12)

$$H\_q^p: P = \{ p' \in \mathbf{P}^\* : p' \le p \} \xrightarrow{\text{cont}} \mathbb{Q} = \{ q' \in \mathbf{P}^\* : q' \le q \}.$$

is an order isomorphism of *P* onto *Q* by Theorem 6, acting so that:

(∗) if *<sup>p</sup>* <sup>∈</sup> *<sup>P</sup>* then *<sup>q</sup>* <sup>=</sup> *<sup>H</sup><sup>p</sup> <sup>q</sup>* (*p* ) satisfies |*p* | = |*q* |, *p* (*i*) = *q* (*i*) for all *i* = 0, and even *Fq* (0) = *Fp* (0), but *Sq* (0)=(*Sp* (0) <sup>∪</sup> *<sup>S</sup>u*) *<sup>S</sup><sup>v</sup>* .

We conclude that *H<sup>p</sup> <sup>q</sup>* also is an order isomorphism of *P* ∩ **P** onto *Q* ∩ **P** by (∗), and hence the set *<sup>H</sup>* <sup>=</sup> {*H<sup>p</sup> <sup>q</sup>* (*p* ) : *p* ∈ *G*} ⊆ *Q* is **P**-generic over **L**. Moreover it follows from (∗) that *SH*(*i*) = *SG*(*i*) and *aH*(*i*) = *aG*(*i*) for all *<sup>i</sup>* <sup>&</sup>gt; 0, but *SH*(0)=(*SG*(0) <sup>∪</sup> *<sup>S</sup>u*) *<sup>S</sup><sup>v</sup>* and *aH*(0)=(*aG*(0) <sup>∪</sup> *<sup>u</sup>*) *<sup>v</sup>*. Therefore *yH* = (*yG* ∪ *u*) *v* = *y*, thus **L**[*G y*, *G* (I *ω*)] is a *K*[**U**]-generic extension of **L**.

As for the elementary equivalence claim, note first of all that the common part **L**[*G* (*yG* ∩ *y*), *G* (I *ω*)] of the two models also is a *K*[**U**]-generic extension of **L** by the above. (Take *yG* ∩ *y* as a new *y*.) Thus in fact it suffices to prove that under the assumptions of the theorem if *j* ∈ *ω yG* then **<sup>L</sup>**[*G yG*, *aG*(*j*), *<sup>G</sup>* (<sup>I</sup> *<sup>ω</sup>*)] is an elementary extension of **<sup>L</sup>**[*G yG*, *<sup>G</sup>* (<sup>I</sup> *<sup>ω</sup>*)] w. r. t. all *<sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>** formulas.

Let Φ be a closed *Σ*<sup>1</sup> **<sup>n</sup>** formula with parameters in **L**[*G yG*, *G* (I *ω*)]. It can be deduced, using either Theorem 5(ii) or directly the CCC property of **P** (Theorem 4) that there is an ordinal *γ*, *ω* ≤ *γ* < *ω*<sup>1</sup> , such that all parameters of Φ belong to **L**[*G yG*, *G h*], where *h* = *γ ω*.

Put *d* = *γ* {*j*}; the sets *b* = I *γ*, *c* = *b* ∪ {*j*} have cardinality *ω*<sup>1</sup> , and *Y* = *h* ∪ *b* while *Y* ∪ {*j*} = *h* ∪ *c*. It follows from Lemma 32 that *K* = *K d* is a regular forcing, and in fact *G γ* ⊆ *K* since *<sup>j</sup>* <sup>∈</sup>/ *yG* . Moreover, by definition all of *<sup>K</sup>*<sup>0</sup> , *<sup>K</sup>*, *<sup>K</sup>* , *<sup>d</sup>*, *<sup>b</sup>*, *<sup>c</sup>* are absolute <sup>Δ</sup>HC <sup>1</sup> (*w*) sets in **L** for some *<sup>w</sup>* <sup>∈</sup> *<sup>ω</sup><sup>ω</sup>* . Therefore by Corollary <sup>10</sup> <sup>Φ</sup> is simultaneously true in **<sup>L</sup>**[*<sup>G</sup>* <sup>∩</sup> *<sup>K</sup>* , *G b*] and in **L**[*G* ∩ *K* , *G c*]. However,

$$\mathbf{L}[G \cap K', G \upharpoonright b] = \mathbf{L}[G \cap K\_0, G \upharpoonright (\gamma > \omega), G \upharpoonright (\mathcal{L} \ltimes \gamma)] = \mathbf{L}[y\_{G\prime}, G \upharpoonright (\mathcal{L} \ltimes \omega)],$$

and similarly **L**[*G* ∩ *K* , *G c*] = **L**[*yG*, *aG*(*j*), *G* (I *ω*)], as required.

#### *9.3. Second Key Lemma*

In continuation of the proof of Claim 11, we establish another key lemma (Lemma 35). Suppose that

(I) *<sup>G</sup>* <sup>⊆</sup> **<sup>P</sup>** is **<sup>P</sup>**-generic over **<sup>L</sup>**, *<sup>x</sup>* <sup>⊆</sup> *<sup>ω</sup>*, *<sup>x</sup>* <sup>∈</sup> **<sup>L</sup>**[*GYG*], and *<sup>ϕ</sup>*(*m*), *<sup>ψ</sup>*(*m*) are parameter-free *<sup>Σ</sup>*<sup>1</sup> **n**+1 formulas that give a Δ<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> definition for *x* = {*m* ∈ *ω* : *ϕ*(*m*)} = {*m* : ¬ *ψ*(*m*)} in **L**[*GYG*].

Thus it is true in **L**[*G*] that "the equivalence ∀ *m* (*ϕ*(*m*) ⇐⇒ ¬ *ψ*(*m*)) holds in the model **L**[*GYG*]". It follows that there is a condition *p*<sup>0</sup> ∈ *G* with

(II) *p*<sup>0</sup> ||−−**<sup>P</sup>** "**L**[*GYG*] |= ∀ *m* (*ϕ*(*m*) ⇐⇒ ¬ *ψ*(*m*))".

**Lemma 34.** *If p*<sup>0</sup> ∈ *G satisfies* (II) *then so does p*<sup>0</sup> {0}*.*

**Proof.** We assume w.l. o. g. that 0 ∈ |*p*0|. Let *u* = |*p*0| {0}. In the context of Theorem 7, put *d* = I , *c* = *ω u*, and *K* = *K*<sup>0</sup> *c* (a regular forcing by Lemma 32). Then *YG* = (I *ω*) ∪ *yG* = (I *ω*) ∪ (*yG* ∩ *c*) ∪ (*yG* ∩ *u*), hence

$$\mathbf{L}[G\upharpoonright\mathbf{Y}\_G] = \mathbf{L}[G\upharpoonright(\mathcal{Z}\lnot\mathcal{L}\omega)] \cup \mathbf{L}[G\upharpoonright(\mathcal{Y}\_G\lnot\mathcal{L})] \cup \mathbf{L}[G\upharpoonright(\mathcal{Y}\_G\lnot\mathcal{U})] \cup$$

Here I *ω* ⊆ *d c* is constructible while *yG* ∩ *u* ⊆ *d c* is finite and hence constructible as well. We conclude by Theorem 7(i) that *p*<sup>0</sup> *c* **P**-forces "**L**[*GYG*] |= ∀ *m* (*ϕ*(*m*) ⇐⇒ ¬ *ψ*(*m*))". However, *c* ∩ |*p*0| = {0}, so we are done.

Following the lemma, fix a condition *p*<sup>0</sup> ∈ *G* satisfying |*p*0| = {0} and (II).

**Lemma 35.** *Assume* (I) *and* (II) *above. Let m* < *ω. Then the sentence ϕ*(*m*) *is K*[**U**]*-decided by p*<sup>0</sup> : *either p*<sup>0</sup> ||−−*K*[**U**] *ϕ*(*m*) *or p*<sup>0</sup> ||−−*K*[**U**] ¬ *ϕ*(*m*)*.*

**Proof.** It will be technically easier to establish the result in the following form equivalent to the original form by Theorem 5(i):

1◦: *the sentence* "**L**[*GYG*] |= *ϕ*(*m*)" *is* **P***-decided by p*<sup>0</sup> *.*

Assume that this fails; then there exist two conditions *p*, *q* ∈ **P** stronger than *p*<sup>0</sup> and satisfying:

2◦: *q* ||−−**<sup>P</sup>** "**L**[*GYG*] |= *ϕ*(*m*)" and *p* ||−−**<sup>P</sup>** "**L**[*GYG*] |= ¬ *ϕ*(*m*)".

We can assume that |*p*| = |*q*| = {0}; otherwise apply Lemma 34 to formulas *ϕ* and ¬ *ϕ*. Strengthening *p*, *q*, if necessary, we can w.l. o. g. assume that

(a) *Fq*(0) = *Fp*(0) and *Sp*(0) ∪ *Sq*(0) ⊆ *F*<sup>∨</sup> *<sup>p</sup>* (0) = *F*<sup>∨</sup> *<sup>q</sup>* (0). (= (3) in Section 3.8.)

Working towards a contradiction, we w.l. o. g. assume that, in addition to (a), the following holds:

(b) the symmetric difference *Sp*(0) Δ *Sq*(0) contains a single element *s* ∈ **Seq**.

(Any pair of conditions *p*, *q* ≤ *p*<sup>0</sup> satisfying (a) can be connected by a finite chain of conditions in which any two neighbours satisfy (b) and are ≤ *p*<sup>0</sup> .)

Thus suppose that *p*, *q* ≤ *p*<sup>0</sup> , |*p*| = |*q*| = {0}, (a), (b), 2◦ hold; the goal is to infer a contradiction. The associated transformation *H<sup>p</sup> <sup>q</sup>* (Definition 12) maps *P* = {*p* ∈ **P** : *p* ≤ *p*} onto *Q* = {*q* ∈ **P** : *q* ≤ *q*} order-preservingly by Theorem 6. Let *G* ⊆ *P* be a set **P**-generic over **L** and containing *p*. Then *<sup>H</sup>* <sup>=</sup> {*H<sup>p</sup> <sup>q</sup>* (*p* ) : *p* ∈ *G*} ⊆ *Q* is **P**-generic as well, *q* ∈ *H*, and hence **L**[*HYH*] |= *ϕ*(*m*), while **L**[*HYG*] |= ¬ *ϕ*(*m*) by 2◦.

*Case 1*: *Sp*(0) = *Sq*(0) ∪ {*s*}, where *<sup>s</sup>* <sup>=</sup> **<sup>s</sup>** <sup>∈</sup> **Seq** *Sq*(0). Then the map *<sup>H</sup><sup>p</sup> <sup>q</sup>* acts so that *q* = *H<sup>p</sup> <sup>q</sup>* (*p* ) is defined by |*p* | = |*q* |⊇|*p*| = |*q*|, *p* (*ν*) = *q* (*ν*) for all *ν* ∈ I , *ν* = 0, *Fq* (0) = *Fp* (0), but *Sp* (0) = *Sq* (0) ∪ {*s*}. It follows that *SH*(*ν*) = *SG*(*ν*) for all *ν* = 0 but *SG*(0) = *SH*(0) ∪ {*s*}. Thus *aG*(*ν*) = *aH*(*ν*) for *ν* = 0 but *aG*(0) = *aH*(0) ∪ {} since *s* = **s** . In other words, *aG*(0) = *aH*(0) ∪ {}, therefore *yG* <sup>=</sup> *yH* ∪ {} and **<sup>L</sup>**[*GYG*] = **<sup>L</sup>**[*HYH*, *aH*()]. It follows from Lemma <sup>33</sup> that any *<sup>Σ</sup>*<sup>1</sup> **n**+1 formula true in **L**[*HYH*] remains true in **L**[*GYG*]. In particular, **L**[*GYG*] |= *ϕ*(*m*), a contradiction.

*Case 2*: *Sq*(0) = *Sp*(0) ∪ {*s*}, where *s* = **s** ∈ **Seq** *Sp*(0). Then, similarly to the above, *aG*(*ν*) = *aH*(*ν*) for *ν* = 0, but *aH*(0) = *aG*(0) ∪ {}. Therefore, *yH* = *yG* ∪ {} and **L**[*HYH*] = **L**[*GYG*, *aG*()]. Thus any *Π*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> formula true in **L**[*HYH*] remains true in **L**[*GYG*] by Lemma 33. Apply this to the formula ¬ *ψ*(*m*), equivalent to *ϕ*(*m*) in both models by (II) above. (Note that *p*, *q* ≤ *p*<sup>0</sup> , hence *p*<sup>0</sup> ∈ *G* ∩ *H*.) We have **L**[*GYG*] |= *ϕ*(*m*), a contradiction. (Lemma 35)

#### *9.4. Final Argument*

Here we finish the proof of both Claim 11 in Section 9.1 and Theorem 3. Suppose that *G* ⊆ **P** is a set **P**-generic over **L**, *Y* = *YG* , and a set *x* ⊆ *ω* in **L**[*GY*], formulas *ϕ*, *ψ*, and a condition *p*<sup>0</sup> satisfy assumptions (I), (II) above. Then, by Lemma 35,

$$\mathbf{x} = \{m < \omega : p\_0 \parallel \!\!\!- \_{K[\mathbb{U}]} \boldsymbol{\varrho}(m)\} = \{m : A(p\_0, m)\},\tag{16}$$

where, in **<sup>L</sup>**, *<sup>A</sup>* <sup>⊆</sup> *<sup>K</sup>*[**U**] <sup>×</sup> *<sup>ω</sup>* is a *<sup>Σ</sup>*HC **<sup>n</sup>** set such that *A*(*p*, *m*) ⇐⇒ *p* ||−−*K*[**U**] *ϕ*(*m*) for all *p* ∈ *K*[**U**] and *m* (Corollary 9). It follows that, in **L**, *x* is *Σ*HC **<sup>n</sup>** (*p*0), hence *Σ*<sup>1</sup> **<sup>n</sup>**+1(*w*) (see Remark 2 in Section 4.3), where *<sup>w</sup>* <sup>∈</sup> **<sup>L</sup>** <sup>∩</sup> *<sup>ω</sup><sup>ω</sup>* is a suitable code of *<sup>p</sup>*<sup>0</sup> .

To eliminate *p*<sup>0</sup> , consider the set *Q* of all conditions *p* ∈ *K*[**U**] such that |*p*| = |*p*0| and *Sp*(*ν*) = *Sp*<sup>0</sup> (*ν*) for all *ν* ∈ |*p*| = |*p*0|. Note that *K*[**U**] = *K* ∩ **P** is a set of the same complexity as **P**, that is, ΔHC **<sup>n</sup>**−<sup>1</sup> , and hence so is *<sup>Q</sup>* because <sup>|</sup>*p*0<sup>|</sup> and all *Sp*<sup>0</sup> (*ν*), *<sup>ν</sup>* ∈ |*p*0<sup>|</sup> are finite sets. It follows that *<sup>Q</sup>* is <sup>Δ</sup>HC **<sup>n</sup>**−<sup>1</sup> .

We now claim that, in **L**, *x* = {*m* ∈ *ω* : ∃ *p* ∈ *Q A*(*p*, *m*)} ; this obviously yields *x* being lightface *Σ*1 *<sup>n</sup>*+<sup>1</sup> in **L**. Indeed ⊆ follows by taking *p* = *p*<sup>0</sup> ∈ *Q* and applying (16). Now suppose that *p* ∈ *Q* and *A*(*p*, *m*), that is, *p* ||−−*K*[**U**] *ϕ*(*m*). Recall that *p*<sup>0</sup> decides *ϕ*(*m*) by Lemma 35. However, *p*<sup>0</sup> ||−−*K*[**U**] ¬ *ϕ*(*m*) is impossible since any condition in *Q* is compatible with *p*<sup>0</sup> . Therefore *p*<sup>0</sup> ||−−*K*[**U**] *ϕ*(*m*) as required. Thus *<sup>x</sup>* <sup>∈</sup> *<sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> in **L** is established.

That the complementary set *ω x* is *Σ*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> as well is verified the same way, using the formula *ψ* instead of *ϕ*. (Theorem 3)

#### **10. Conclusions and Some Further Results**

With proofs of the main theorems accomplished, in this final section some further results are briefly discussed, which we plan to achieve and publish elsewhere.

#### *10.1. Separation*

This is another application of submodels of the same basic model. Recall that given a class **K** of pointsets, the separation principle **K**-**Sep** claims that any two disjoint **K**-sets in the same space can be separated by a set in **<sup>K</sup>** <sup>∩</sup> **<sup>K</sup>** , where **<sup>K</sup>** consists of all complements of **<sup>K</sup>**-sets. The separation principle was introduced by N. Luzin. Luzin proved (see [25]) that **Σ**<sup>1</sup> <sup>1</sup>-**Sep** holds, and then P. Novikov [26,27] demonstrated that **Π**<sup>1</sup> <sup>1</sup>-**Sep** fails, while at the second projective level, the other way around, **<sup>Π</sup>**<sup>1</sup> <sup>2</sup>-**Sep** holds but **Σ**<sup>1</sup> <sup>2</sup>-**Sep** fails.

As for higher projective levels, the separation problem belongs to a considerable list of problems related to the projective hieharchy in Luzin's book [25], Chapter V. Further development of set theory showed that Luzin's problems are very hard to solve. Some of them are now known to be independent of the Zermelo–Fraenkel set theory **ZFC**, while some others are still open in different aspects, but it is known that adding Gödel's axiom of constructibility **V** = **L** solves most of them. In particular, **V** = **L** implies [28,29] that **Π**<sup>1</sup> *<sup>n</sup>*-**Sep** holds but **Σ**<sup>1</sup> *<sup>n</sup>*-**Sep** fails for all *n* ≥ 3—similarly to the classical case *n* = 2. It follows that the statement <sup>∀</sup> *<sup>n</sup>* <sup>≥</sup> <sup>3</sup> (**Π**<sup>1</sup> *<sup>n</sup>*-**Sep** ∧ ¬ **<sup>Σ</sup>**<sup>1</sup> *<sup>n</sup>*-**Sep**) is consistent with **ZFC**, and the problem is then to find a model in which we have **Σ**<sup>1</sup> *<sup>n</sup>*-**Sep** and/or <sup>¬</sup> **<sup>Π</sup>**<sup>1</sup> *<sup>n</sup>*-**Sep** (opposite to the state of affairs in **L**) for one or several or all indices *n* ≥ 3. This was the content of problems P 3029 and 3030 in the survey [8] of early years of forcing.

This turns out a very difficult question, and still open in its general forms, especially w. r. t. **Σ**<sup>1</sup> *n*-**Sep**. (Compare to Problem 9 in [30], Section 9.) As for the <sup>¬</sup> **<sup>Π</sup>**<sup>1</sup> *<sup>n</sup>*-**Sep** side, there are indications in the set-theoretic literature, that generic extensions, where both **Σ**<sup>1</sup> *<sup>n</sup>*-**Sep** and **Π**<sup>1</sup> *<sup>n</sup>*-**Sep** fail, are constructed by L. Harrington for *n* = 3 (see 5B.3 in [6]) and for arbitrary *n* ≥ 3 (see [8] and [31], p. 230). These results were indeed announced in Harrington's handwritten notes (Addendum A in [32]), with brief outline of some key arguments related mainly to case *n* = 3 and based on almost-disjoint forcing. There are no such results in Harrington's published works, assumed methods in their principal part (arbitrary *n*) are not used even for any other results, and separability theorems in this context are not considered. An article by Harrington, entitled "Consistency and independence results in descriptive set theory", which apparently might have contained these results, was announced in the References list in [31], to appear in *Ann. of Math.*, 1978, but in fact it has never been published.

The following conjecture concludes Addendum A of Harrington's note [32]:

*In fact (we believe) there is a model of* **ZFC** *in which Separation fails for all of the following at once*: **Σ**1 *<sup>n</sup> ,* **Π**<sup>1</sup> *<sup>n</sup> ,* <sup>3</sup> <sup>≤</sup> *<sup>n</sup>* <sup>&</sup>lt; *<sup>ω</sup>,* **<sup>Σ</sup>***<sup>m</sup> <sup>n</sup> ,* **Π***<sup>m</sup> <sup>n</sup> ,* <sup>1</sup> <sup>≤</sup> *<sup>n</sup>* <sup>&</sup>lt; *<sup>ω</sup>,* <sup>2</sup> <sup>≤</sup> *<sup>m</sup>* <sup>&</sup>lt; *<sup>ω</sup>.* ( **<sup>Σ</sup>***<sup>m</sup> <sup>n</sup>* , **Π***<sup>m</sup> <sup>n</sup>* are classes arising in the type-theoretic hierarchy)*.*

The hypothesis is partially confirmed by the following our theorem (to appear elsewhere).

**Theorem 15** (originally Harrington [32])**.** *If* **<sup>n</sup>** <sup>≥</sup> <sup>2</sup> *then there is a generic extension of* **<sup>L</sup>** *in which* **<sup>Π</sup>**<sup>1</sup> **<sup>n</sup>**+1*-* **Sep** *and* **Σ**<sup>1</sup> **<sup>n</sup>**+1*-***Sep** *fail, and moreover*


*Moreover there is a generic extension of* **L** *in which* (i) *and* (i) *simultaneously hold for all* **n** ≥ 2*.*

Note that generic models are defined in [33] in which both **Σ**<sup>1</sup> <sup>3</sup>-**Sep** and **<sup>Π</sup>**<sup>1</sup> <sup>3</sup>-**Sep** fail. We used different technique in [33], mostly related to Jensen's minimal *Π*<sup>1</sup> <sup>2</sup> singleton forcing [10] and its iterated forms (see [34–36]) rather than the almost-disjoint forcing as in this paper.

#### *10.2. Projections of Uniform Sets*

In his monograph [25] (pp. 276–291) Nikolas Luzin formulated a number of problems about the structure of the projective classes **Σ**<sup>1</sup> *<sup>n</sup>* , **Π**<sup>1</sup> *<sup>n</sup>* , **Δ**<sup>1</sup> *<sup>n</sup>* (or **A***n*, **CA***n*, **B***<sup>n</sup>* in the old notational system). Their general meaning was to extend the results obtained by Luzin himself and P. S. Novikov for classes **Σ**<sup>1</sup> 1 , **Π**<sup>1</sup> <sup>1</sup> , **<sup>Δ</sup>**<sup>1</sup> <sup>1</sup> (level *n* = 1 of the projective hierarchy) to higher levels. Among these problems, the following stands out, along with the separation problem discussed above:

**Projection problem:** given *<sup>n</sup>* <sup>≥</sup> 2, find out the nature of projections of uniform (planar) **<sup>Π</sup>**<sup>1</sup> *<sup>n</sup>* sets in comparison with the class **Σ**<sup>1</sup> *<sup>n</sup>*+<sup>1</sup> of arbitrary projections of **<sup>Π</sup>**<sup>1</sup> *<sup>n</sup>* sets and with the narrower class **Δ**1 *<sup>n</sup>* . (A planar set is *uniform*, if it intersects every vertical line at no more than one point.)

Further research has shown the key importance of structural theorems on projective classes for the development of descriptive set theory. For example, separation principles play essential role in research on subsystems of second-order arithmetic, in particular, in the context of reverse mathematics [5].

If *n* = 1 then every **Σ**<sup>1</sup> <sup>2</sup> set is equal to the projection of a uniform **<sup>Π</sup>**<sup>1</sup> <sup>1</sup> set by the Novikov–Kondo–Addison uniformization theorem [6, 4E.4]. Under **V** = **L**, the uniformization theorem fails for classes **Π**<sup>1</sup> *<sup>n</sup>* , *<sup>n</sup>* <sup>≥</sup> 2, but nevertheless it is known that if *<sup>n</sup>* <sup>≥</sup> 2 then every **<sup>Σ</sup>**<sup>1</sup> *<sup>n</sup>*+<sup>1</sup> set is equal to the projection of a uniform **Π**<sup>1</sup> *<sup>n</sup>* set [37]. The next theorem (to appear elsewhere) demonstrates that this property is violated in suitable generic models.

**Theorem 16.** *If* **n** ≥ 2 *then there is a generic extension of* **L** *in which*:


Problem 87 in [38] requires to prove that for each *n* > 2 there is a model of

$$\textbf{ZFC} + \text{\textquotedblleft the constructible reals are precisely the } \Lambda\_n^1 \text{ reals\textquotedblright.} \tag{17}$$

It is noted in the very end of [38] that Harrington had solved this problem affirmatively. Indeed, a sketch is given in the same handwritten notes [32], of a generic extension of **L**, in which it is true that *<sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>** <sup>=</sup> <sup>Δ</sup><sup>1</sup> <sup>3</sup> , as well as a few sentences added as how Harrington planned to get a model in which *<sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>** <sup>=</sup> <sup>Δ</sup><sup>1</sup> *<sup>n</sup>* holds for a given (arbitrary) *<sup>n</sup>* <sup>≥</sup> 3, and a model in which *<sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>** <sup>=</sup> <sup>Δ</sup><sup>1</sup> <sup>∞</sup> , where Δ1 <sup>∞</sup> = - *<sup>n</sup>* Δ<sup>1</sup> *<sup>n</sup>* (all analytically definable reals). This positively solves Problem 87, including the case *n* = ∞. Full proofs have never been published except for an independent proof of the consistency of *<sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>** <sup>=</sup> <sup>Δ</sup><sup>1</sup> <sup>∞</sup> in [39]. Our plan will be to restore Harrington's proof of the next theorem elsewhere.

	- (ii) *There is a generic extension of* **<sup>L</sup>** *in which it is true that <sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>** <sup>=</sup> <sup>Δ</sup><sup>1</sup> ∞ *.*

Friedman concludes [38] with a modified version of the above problem, given as Problem 87 : find a model of

$$\text{ZFC + ''for any reals x, y, we have: 'if x \in \mathbf{L}[y] \text{ then x is } \Delta\_3^1 \text{ in } y''. \tag{18}$$

This was solved in the positive by David [40], yet so far it is unknown whether this result generalizes to higher classes Δ<sup>1</sup> *<sup>n</sup>* , *<sup>n</sup>* <sup>≥</sup> 4, or <sup>Δ</sup><sup>1</sup> <sup>∞</sup> . We also note that problems (17) and (18) were known in the early years of forcing, see, e.g., problems P 3110, 3111, 3112 in [8].

#### *10.4. Axiom Schemata in 2nd Order Arithmetic*

Different axiomatic systems in second-order arithmetic **Z**<sup>2</sup> is widely represented in modern research, in particular, in the context of reverse mathematics and other sections of proof theory. See e.g., Simpson [5] (Part B), and numerous articles, and from older sources—for example, Kreisel [41], where the choice of subsystems is called the central problem. These systems are obtained by joining a particular combination of comprehension schema **CA**, countable choice **AC**, dependent choice **DC**, transfinite induction **TI** and recursion **TR** , etc., to the basic theory, say **ACA**<sup>0</sup> . The schemata can be specifieded by the complexity of the core formula in the Kleene hierarchy, as well as by allowing or prohibiting parameters. (For the importance of parameters, see [41], section III.)

The relationships between the subsystems have been actively studied. In particular, it is known that *Σ*<sup>1</sup> *<sup>n</sup>*+<sup>1</sup> -**CA** is strictly stronger, than *<sup>Σ</sup>*<sup>1</sup> *<sup>n</sup>* -**CA**, and the same for **AC** and **DC**. Proofs of these results in e.g., [5, Chapter VII] use the fact that the schema at a higher quantifier level allows to get strictly more countable ordinals, than the schema at a lower level, but in essence, it is utilized that the (*n* + 1)th level schema proves the consistency of the *n*th level schema.

A few more complex results are known, where the compared systems are equiconsistent, despite the increase in quantifier complexity in the schemata, so the consistency argument doesn't work. It such a case one has to resort to set theoretic methods. This is the old result of A. Levy [42] that *Σ*<sup>1</sup> <sup>3</sup> -**AC** does not follow from **CA**, as well as a recent theorem in [43] saying that *Σ*<sup>1</sup> <sup>3</sup> -**DC** does not follow from **AC**; both are obtained using complex generic models of **ZF** without the full axiom of choice. The task of our further research in this direction will be to prove consistency theorems that demonstrate the importance of both the quantifier complexity and the presence of parameters in the **Z**<sup>2</sup> schemata.

**Theorem 18** (to appear elsewhere)**.** *If* **<sup>n</sup>** <sup>≥</sup> <sup>2</sup>*, then the theory* **ACA**<sup>0</sup> <sup>+</sup> **CA**<sup>∗</sup> <sup>+</sup> **<sup>Σ</sup>**<sup>1</sup> **<sup>n</sup>**-**CA** *does not imply Σ*1 **<sup>n</sup>**+<sup>1</sup> *-***CA** (*unless inconsistent, of course*)*.*

Here **CA**\* is the parameter-free part of the comprehension schema **CA**. Thus, both the quantifier complexity and the presence of parameters are essential for the deductive power of the comprehension schema in second-order arithmetic.

**Theorem 19** (to appear elsewhere)**.** *If* **<sup>n</sup>** <sup>≥</sup> <sup>2</sup>*, then the theory* **ACA**<sup>0</sup> <sup>+</sup> **CA** <sup>+</sup> **AC** <sup>+</sup> **<sup>Σ</sup>**<sup>1</sup> **<sup>n</sup>**-**DC** *does not imply Σ*1 **<sup>n</sup>**+<sup>1</sup> *-***DC** (*unless inconsistent*)*.*

**Remark 6.** We are grateful to one of the reviewers for pointing out possible connections of our research with some questions of fuzzy set theory [44,45], yet this issue cannot be considered for a short time.

**Supplementary Materials:** The following are available online at http://www.mdpi.com/2227-7390/8/6/910/s1.

**Author Contributions:** Conceptualization, V.K. and V.L.; methodology, V.K. and V.L.; validation, V.K. and V.L.; formal analysis, V.K. and V.L.; investigation, V.K. and V.L.; writing original draft preparation, V.K.; writing review and editing, V.K.; project administration, V.L.; funding acquisition, V.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Russian Foundation for Basic Research RFBR grant number 18-29-13037.

**Acknowledgments:** We thank the anonymous reviewers for their thorough review and highly appreciate the comments and suggestions, which significantly contributed to improving the quality of the publication.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

#### *Article* **On the Δ<sup>1</sup>** *<sup>n</sup>* **Problem of Harvey Friedman**

## **Vladimir Kanovei \*,† and Vassily Lyubetsky †**

Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, Russia; lyubetsk@iitp.ru


Received: 2 August 2020; Accepted: 23 August 2020; Published: 1 September 2020

**Abstract:** In this paper, we prove the following. If *n* ≥ 3, then there is a generic extension of **L**, the constructible universe, in which it is true that the set P(*ω*) ∩ **L** of all constructible reals (here—subsets of *<sup>ω</sup>*) is equal to the set <sup>P</sup>(*ω*) <sup>∩</sup> *<sup>Δ</sup>*<sup>1</sup> *<sup>n</sup>* of all (lightface) *Δ*<sup>1</sup> *<sup>n</sup>* reals. The result was announced long ago by Leo Harrington, but its proof has never been published. Our methods are based on almost-disjoint forcing. To obtain a generic extension as required, we make use of a forcing notion of the form **Q** = **C** × ∏*<sup>ν</sup>* **Q***<sup>ν</sup>* in **L**, where **C** adds a generic collapse surjection *b* from *ω* onto <sup>P</sup>(*ω*) <sup>∩</sup> **<sup>L</sup>**, whereas each **<sup>Q</sup>***<sup>ν</sup>* , *<sup>ν</sup>* <sup>&</sup>lt; *<sup>ω</sup>***<sup>L</sup>** <sup>2</sup> , is an almost-disjoint forcing notion in the *ω*1-version, that adjoins a subset *S<sup>ν</sup>* of *ω***<sup>L</sup>** <sup>1</sup> . The forcing notions involved are independent in the sense that no **Q***<sup>ν</sup>* -generic object can be added by the product of **C** and all **Q***<sup>ξ</sup>* , *ξ* = *ν*. This allows the definition of each constructible real by a *Σ*<sup>1</sup> *<sup>n</sup>* formula in a suitably constructed subextension of the **Q**-generic extension. The subextension is generated by the surjection *<sup>b</sup>*, sets *<sup>S</sup>ω*·*k*+*<sup>j</sup>* with *<sup>j</sup>* ∈ *<sup>b</sup>*(*k*), and sets *<sup>S</sup><sup>ξ</sup>* with *ξ* ≥ *ω* · *ω*. A special character of the construction of forcing notions **Q***<sup>ν</sup>* is **L**, which depends on a given *n* ≥ 3, obscures things with definability in the subextension enough for vice versa any *Δ*1 *<sup>n</sup>* real to be constructible; here the method of *hidden invariance* is applied. A discussion of possible further applications is added in the conclusive section.

**Keywords:** Harvey Friedman's problem; definability; nonconstructible reals; projective hierarchy; generic models; almost-disjoint forcing

**MSC:** 03E15; 03E35

Dedicated to the 70-th Anniversary of A. L. Semenov.

#### **1. Introduction**

Problem 87 in Harvey Friedman's treatise *One hundred and two problems in mathematical logic* [1] requires proof that for each *n* in the domain 2 < *n* ≤ *ω* there is a model of

$$\text{ZFC + \text{\textquotedblleft}the constructible reals are precisely the } \boldsymbol{\Delta}\_n^{\dagger} \text{ reals\text{\textquotedblright}.}\tag{1}$$

(For *n* ≤ 2 this is definitely impossible e.g., by the Shoenfield's absoluteness theorem.) This problem was generally known in the early years of forcing, see, e.g., problems 3110, 3111, 3112 in an early survey [2] (the original preprint of 1968) by Mathias. At the very end of [1], it is noted that Leo Harrington had solved this problem affirmatively. For a similar remark, see [2] (p. 166), a comment to P 3110. And indeed, Harrington's handwritten notes [3] (pp. 1–4) contain a sketch of a generic extension of **<sup>L</sup>**, based on the almost-disjoint forcing of Jensen and Solovay [4], in which it is true that *<sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>** <sup>=</sup> *<sup>Δ</sup>*<sup>1</sup> 3. Then a few sentences are added on page 5 of [3], which explain, as how Harrington planned to get a model in which *<sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>** <sup>=</sup> *<sup>Δ</sup>*<sup>1</sup> *<sup>n</sup>* holds for a given (arbitrary) natural index *n* ≥ 3, and a model in which

*<sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>** <sup>=</sup> *<sup>Δ</sup>*<sup>1</sup> <sup>∞</sup>, where *Δ*<sup>1</sup> <sup>∞</sup> = - *<sup>n</sup> Δ*<sup>1</sup> *<sup>n</sup>* (all analytically definable reals). This positively solves Problem 87, including the case *n* = ∞. Different cases of higher order definability are observed in [3] as well.

Yet no detailed proofs have ever emerged in Harrington's published works. An article by Harrington, entitled "Consistency and independence results in descriptive set theory", which apparently might have contained these results among others, was announced in the References list in Peter Hinman's book [5] (p. 462) to appear in *Ann. of Math.*, 1978, but in fact, this or similar article has never been published by Harrington.

One may note that finding a model for (1) belongs to the "definability of definable" type of mathematical problems, introduced by Alfred Tarski in [6], where the definability properties of the set *D*1*<sup>M</sup>* , of all sets *x* ⊆ *ω* definable by a parameter-free type-theoretic formula with quantifiers bounded by type *M*, are discussed for different values of *M* < *ω*. In this context, case *n* = ∞ in (1) is equivalent to case *M* = 1 in the Tarski problem, whereas cases *n* < ∞ in (1) can be seen as refinements of case *m* = 1 in the Tarski problem, because classes *Δ*<sup>1</sup> *<sup>n</sup>* are well-defined subclasses of *D*<sup>11</sup> = - *<sup>n</sup>*<*<sup>ω</sup> Δ*<sup>1</sup> *n*.

The goal of this paper is to present a complete proof of the following part of Harrington's statement that solves the mentioned Friedman's problem. No such proof has been known so far in mathematical publications, and this is the **motivation** for our research.

**Theorem 1** (Harrington)**.** *If* 2 ≤ **n** < ∞ *then there is a generic extension of* **L** *in which it is true that the constructible reals are precisely the Δ*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> *reals.*

The *Δ*<sup>1</sup> <sup>∞</sup> case of Harrington's result, as well as different results related to Tarski's problems in [6], will be subject of a forthcoming publication.

This paper is dedicated to the proof of Theorem 1. This will be another application of the technique introduced in our previous paper [7] in this Journal, and in that sense this paper is a continuation and development of the research started in [7]. However, the problem considered here, i.e., getting a model for (1), is different from and irreducible to the problems considered in [7] and related to definability and constructability of individual reals. Subsequently the technique applied in [7] is considerably modified and developed here for the purposes of this new application. In particular, as the models involved here by necessity satisfy *ω*<sup>1</sup> **<sup>L</sup>** < *ω*<sup>1</sup> (unlike the models considered in [7], which satisfy the equality *ω*<sup>1</sup> **<sup>L</sup>** = *ω*1), the almost-disjoint forcing is combined with a cardinal collapse forcing in this paper. And hence we will have to substantially deviate from the layout in [7], towards a modification that shifts the whole almost-disjoint machinery from *ω* to *ω*1.

Section 2: we set up the almost-disjoint forcing in the *ω*1-version. That is, we consider the sets **SEQ** = (*ω*1)<*ω*<sup>1</sup> and **FUN** = (*ω*1)*ω*<sup>1</sup> in **<sup>L</sup>**, the constructible universe, and, given *<sup>u</sup>* <sup>⊆</sup> **FUN**, we define a forcing notion *Q*[*u*] which adds a set *G* ⊆ **SEQ** such that if *f* ∈ **FUN** in **L** then *G* covers *f* iff *f* ∈/ *u*, where covering means that *f ξ* ∈ *G* for unbounded-many *ξ* < *ω*<sup>1</sup> **<sup>L</sup>**. We also consider two types of transformations related to forcing notions of the form *Q*[*u*].

Section 3. We let I = *ω*<sup>2</sup> **<sup>L</sup>** be the index set. Arguing in **L**, we consider *systems <sup>U</sup>* = {*U*(*ν*)}*ν*∈I , where each *<sup>U</sup>*(*ν*) ⊆ **FUN** is dense. Given such *<sup>U</sup>*, the *product almost-disjoint forcing* **<sup>Q</sup>**[*U*] = **<sup>C</sup>** <sup>×</sup> <sup>∏</sup>*ν*∈I<sup>+</sup> *<sup>Q</sup>*[*U*(*ν*)] (with finite support) is defined in **<sup>L</sup>**, where **<sup>C</sup>** = (P(*ω*))<*<sup>ω</sup>* is a version of Cohen's collapse forcing. Such a forcing notion adjoins a generic map *<sup>b</sup><sup>G</sup>* : *<sup>ω</sup>* onto −→ P(*ω*) ∩ **L** to **L**, and adds an array of sets *G*(*ν*) ⊆ **SEQ** (where *ν* ∈ I ) as well, so that each *G*(*ν*) is a *Q*[*U*(*ν*)]-generic set over **L**. We also investigate the structure of related product-generic extensions and their subextensions, and transformations of forcing notions of the form **Q**[*U*].

Section 4. Given **n** ≥ 2, we define a system **U** ∈ **L** as above, which has some remarkable properties, in particular, (1) being *Q*[**U**(*ν*)]-generic is essentially a *Π*<sup>1</sup> **<sup>n</sup>** property in all suitable generic extensions, (2) if *ν* ∈ I and *G* ⊆ **Q**[**U**] is generic over **L**, then the extension **L**[*b<sup>G</sup>* , {*G*(*ν* )}*ν* =*ν*] contains no *Q*[**U**(*ν*)]-generic reals, and (3) all submodels of **L**[*G*] of certain kind are elementarily equivalent w. r. t. *Σ*<sup>1</sup> **<sup>n</sup>** formulas. The latter property is summarized in the key technical instrument, Theorem 4 (the elementary equivalence theorem), whose proof is placed in a separate Section 6. To prove Theorem 1, we make use of a related generic extension **<sup>L</sup>**[*b<sup>G</sup>* , {*G*(*ν*)}*ν*∈*W*[*G*]], where

$$\mathcal{W}[\mathcal{G}] = \mathcal{w}[\mathcal{G}] \cup \mathcal{W} = \left\{ \omega \cdot k + 2^j : j \in \mathsf{b}\_{\mathcal{G}}(k) \right\} \cup \left\{ \omega \cdot k + 3^j : j, k < \omega \right\} \cup \left\{ \nu \in \mathcal{T} : \nu \ge \omega^2 \right\}$$

(see Lemma 23), and · is the ordinal multiplication. The first term in *W*[*G*] provides a suitable definition of each set *<sup>x</sup>* = *<sup>b</sup>G*(*k*) ∈ **<sup>L</sup>** in the model **<sup>L</sup>**[*b<sup>G</sup>* , {*G*(*ν*)}*ν*∈*W*[*G*]], namely

$$\mathbf{b}\_G(k) = \{ j \text{ : there exists a } Q[\!\!\!\!\! \!\! (\nu)\!]\text{-generic set over } \mathbf{L} \},$$

while the second and third terms in *W*[*G*] are added for technical reasons. The proof itself goes on in Section 4.5, modulo Theorem 4.

We introduce *forcing approximations* in Section 5, a forcing-like relation used to prove the elementary equivalence theorem. Its key advantage is the invariance under some transformations, including the permutations of the index set I , see Section 5.4. The actual forcing notion **Q** = **Q**[**U**] is absolutely not invariant under permutations, but the **n**-completeness property, maintained through the inductive construction of **U** in **L**, allows us to prove that the auxiliary forcing relation is connected to the truth in **Q**-generic extensions exactly as the true **Q**-forcing relation does—up to the level *Σ*1 **<sup>n</sup>** of the projective hierarchy (Lemma 33). We call this construction *the hidden invariance technique* (see Section 6.1).

Finally, Section 6 presents the proof of the elementary equivalence theorem, with the help of forcing approximations, and hence completes the proof of Theorem 1.

The flowchart can be seen in Figure 1 on page 3. And we added the index and contents as Supplementary Materials for easy reading.

**Figure 1.** Flowchart.

#### **2. Almost-Disjoint Forcing**

Almost-disjoint forcing as a set theoretic tool was invented by Jensen and Solovay [4]. It has been applied in many research directions in modern set theory, in particular, in our paper [7] in this Journal. Here we make use of a considerably different version of the almost-disjoint forcing technique, which, comparably to [7], (1) considers countable cardinality instead of finite cardinalities in some key positions, (2) accordingly considers cardinality *ω*<sup>1</sup> instead of countable cardinality. In particular, sequences of finite length change to those of length < *ω*1. And so on.

**Assumption 1.** *During arguments in this section, we assume that the ground set universe is* **L***, the constructible universe. Recall that in* **L***,* HC = **L***ω*<sup>1</sup> *and* H*ω*<sup>2</sup> = **L***ω*<sup>2</sup> *.*

For the sake of brevity, we call *<sup>ω</sup>*1*-size sets* those *<sup>X</sup>* satisfying card *<sup>X</sup>* <sup>≤</sup> *<sup>ω</sup>*1.

#### *2.1. Almost-Disjoint Forcing: ω*1*-Version*

This subsection contains a review the basic notation related to almost-disjoint forcing in the *ω*1- version. Arguing in **L**, we put **FUN** = *ω*<sup>1</sup> *<sup>ω</sup>*<sup>1</sup> = all *ω*1-sequences of ordinals < *ω*1.


The following or very similar version of the almost-disjoint forcing was defined by Jensen and Solovay in [4] ([§ 5]). Its goal can be formulated as follows: given a set *u* ⊆ **FUN** in the ground universe, find a generic set *S* ⊆ **SEQ** such that the equivalence

$$f \in \mathfrak{u} \iff \text{S} \text{ does not cover } f \tag{2}$$

holds for each *f* ∈ **FUN** in the ground universe.

**Definition 1** (in **L**)**.** *Q*<sup>∗</sup> *is the set of all pairs p* = *Sp* ; *Fp of finite sets Fp* ⊆ **FUN***, Sp* ⊆ **SEQ***. Elements of Q*<sup>∗</sup> *will be called (forcing) conditions. If p* ∈ *Q*<sup>∗</sup> *then put*

$$F\_p^\vee = \{ f \upharpoonright \xi : f \in F\_p \land 1 \le \xi < \omega\_1 \} \dots$$

*a tree in* **SEQ***. If p*, *q* ∈ *Q*<sup>∗</sup> *then let p* ∧ *q* = *Sp* ∪ *Sq* ; *Fp* ∪ *Fq ; a condition in Q*<sup>∗</sup> *.*

*Let p*, *q* ∈ *Q*<sup>∗</sup> *. Define q p (that is, q is stronger as a forcing condition) iff Sp* ⊆ *Sq* , *Fp* ⊆ *Fq , and the difference Sq Sp does not intersect F*∨ *<sup>p</sup> , i.e., Sq* ∩ *F*<sup>∨</sup> *<sup>p</sup>* = *Sp* ∩ *F*<sup>∨</sup> *<sup>p</sup> . Clearly, we have q p iff Sp* ⊆ *Sq* , *Fp* ⊆ *Fq , and Sq* ∩ *F*<sup>∨</sup> *<sup>p</sup>* = *Sp* ∩ *F*<sup>∨</sup> *p .*

**Lemma 1** (in **L**)**.** *Conditions p*, *q* ∈ *Q*<sup>∗</sup> *are compatible in Q*<sup>∗</sup> *iff* (1) *Sq Sp does not intersect F*<sup>∨</sup> *<sup>p</sup> , and* (2) *Sp Sq does not intersect F*∨ *<sup>q</sup> . Therefore any p*, *q* ∈ *P*<sup>∗</sup> *are compatible in P*<sup>∗</sup> *iff p* ∧ *q p and p* ∧ *q q.*

**Proof.** If (1), (2) hold then *p* ∧ *q p* and *p* ∧ *q q*, thus *p*, *q* are compatible.

If *u* ⊆ **FUN** then put *Q*[*u*] = {*p* ∈ *Q*<sup>∗</sup> : *Fp* ⊆ *u*}.

Any conditions *p*, *q* ∈ *Q*[*u*] are compatible in *Q*[*u*] iff they are compatible in *Q*<sup>∗</sup> iff the condition *p* ∧ *q* = *Sp* ∪ *Sq* ; *Fp* ∪ *Fq* ∈ *Q*[*u*] satisfies both *p* ∧ *q p* and *p* ∧ *q q*. Therefore, we can say that conditions *p*, *q* ∈ *Q*<sup>∗</sup> are compatible (or incompatible) without an explicit indication which forcing notion *Q*[*u*] containing *p*, *q* is considered.

**Lemma 2** (in **<sup>L</sup>**)**.** *If u* <sup>⊆</sup> **FUN** *and A* <sup>⊆</sup> *<sup>Q</sup>*[*u*] *is an antichain then* card *<sup>A</sup>* <sup>≤</sup> *<sup>ω</sup>*1*.*

**Proof.** Suppose towards the contrary that card *<sup>A</sup>* <sup>&</sup>gt; *<sup>ω</sup>*1. If *<sup>p</sup>* <sup>=</sup> *<sup>q</sup>* in *<sup>A</sup>* are incompatible then obviously *Sp* = *Sq* . Yet {*Sp* : *p* ∈ *Q*<sup>∗</sup> } = all finite subsets of **SEQ**, is a set of cardinality *ω*1, a contradiction.

#### *2.2. Almost-Disjoint Generic Extensions*

To work with **L**-sets **FUN** and **SEQ** in generic extensions of **L**, possibly in those obtained by means of cardinal collapse, we let

$$\mathbf{F}\mathbf{U}\mathbf{N}^{\mathbf{L}} = (\omega\_1^{\mathbf{L}})^{\omega\_1^{\mathbf{L}}} \cap \mathbf{L} \quad \text{and} \quad \mathbf{S}\mathbf{e}\mathbf{Q}^{\mathbf{L}} = ((\omega\_1^{\mathbf{L}})^{<\omega\_1^{\mathbf{L}}} \cap \mathbf{L}) \tag{3}$$

—in other words, **FUN<sup>L</sup>** and **SEQ<sup>L</sup>** are just **FUN** and **SEQ** defined in **L**.

**Lemma 3.** *Suppose that in* **L***, u* ⊆ **FUN** *is dense. Let G* ⊆ *Q*[*u*] *be a set Q*[*u*]*-generic over* **L***. We define SG* = - *<sup>p</sup>*∈*<sup>G</sup> Sp* ; *thus SG* <sup>⊆</sup> **SEQL***. Then*


$$(\text{iii})\quad \mathbf{L}[G] = \mathbf{L}[S\_G] \; ;$$

(iv) *if f* <sup>∈</sup> **FUNL** *u then Xf* <sup>=</sup> {*<sup>ξ</sup>* <sup>&</sup>lt; *<sup>ω</sup>***<sup>L</sup>** <sup>1</sup> : *<sup>f</sup> <sup>ξ</sup>* <sup>∈</sup> *SG* } *is a cofinal subset of <sup>ω</sup>***<sup>L</sup>** <sup>1</sup> *of order type ω* ;

$$\text{(v)}\quad\omega\_1^{\mathsf{L}[G]} = \omega\_2^{\mathsf{L}}.$$

**Proof.** (i) Consider any *f* ∈ *u*. We claim that *Df* = {*p* ∈ *P*[*u*] : *f* ∈ *Fp* } is dense in *P*[*u*]. (Indeed if *q* ∈ *P*[*u*] then define *p* ∈ *P*[*u*] by *Sp* = *Sq* and *Fp* = *Fq* ∪ { *f* }; we have *p* ∈ *Df* and *p q*.) It follows that *Df* ∩ *G* = ∅. Choose any *p* ∈ *Df* ∩ *G*; we have *f* ∈ *Fp* . Each condition *r* ∈ *G* is compatible with *p*, therefore, by Lemma 1, *Sr*/ *f* ⊆ *Sp*/ *f* . We conclude that *SG*/ *f* = *Sp*/ *f* .

Now assume that *<sup>f</sup>* <sup>∈</sup>/ *<sup>u</sup>*. The set *Df l* <sup>=</sup> {*<sup>p</sup>* <sup>∈</sup> *<sup>P</sup>*[*u*] : sup(*Sp*/ *<sup>f</sup>*) <sup>&</sup>gt; *<sup>l</sup>*} is dense in *<sup>P</sup>*[*u*] for any *l* < *ω*. (Let *q* ∈ *P*[*u*]. Then *Fq* is finite. There exists *m* > *l* with *f m* ∈/ *F*<sup>∨</sup> *<sup>q</sup>* , since *f* ∈/ *u*. Define a condition *p* by *Fp* = *Fq* and *Sp* = *Sq* ∪ { *f m*}; we have *p* ∈ *Df l* and *p q*.) Pick, by the density, any *<sup>p</sup>* <sup>∈</sup> *Df l* <sup>∩</sup> *<sup>G</sup>*. Then sup(*SG*/ *<sup>f</sup>*) <sup>&</sup>gt; *<sup>l</sup>*. We conclude that *SG*/ *<sup>f</sup>* is infinite because *<sup>l</sup>* is arbitrary.

(ii) Let *p* ∈ *G*. Then obviously *sp* ⊆ *SG* . If there exists *s* ∈ (*SG Sp*) ∩ *F*<sup>∨</sup> *<sup>p</sup>* then *s* ∈ *Sq* for some *q* ∈ *G*. Then conditions *p*, *q* are incompatible by Lemma 1, which is a contradiction.

Now assume that *p* ∈ *P*[*u*] *G*. There is a condition *q* ∈ *G* incompatible with *p*. We have two cases by Lemma 1. First, there is some *s* ∈ (*Sq Sp*) ∩ *F*<sup>∨</sup> *<sup>p</sup>* . Then *s* ∈ *SG Sp* , so *p* is not compatible with *SG* . Second, there is some *s* ∈ (*Sp Sq*) ∩ *F*<sup>∨</sup> *<sup>q</sup>* . In this case, *s* ∈/ *Sr* holds for any condition *r* ≤ *q*. It follows that *s* ∈/ *SG* , hence *Sp* ⊆ *SG* , and *p* cannot be compatible with *SG* .

Further it follows from (ii) that *G* = {*p* ∈ *P*[*u*] : *sp* ⊆ *SG* ∧ (*SG sp*) ∩ *F*<sup>∨</sup> *<sup>p</sup>* = ∅}, hence, we have (iii). Claim (v) is an immediate corollary of (iv) since *ω***<sup>L</sup>** <sup>2</sup> remains a cardinal in **L**[*G*] by Lemma 2.

Finally, to prove (iv) let *<sup>f</sup>* <sup>∈</sup> **FUN<sup>L</sup>** *<sup>u</sup>* and *<sup>λ</sup>* <sup>&</sup>lt; *<sup>ω</sup>***<sup>L</sup>** <sup>1</sup> . The set *Df <sup>λ</sup>* of all conditions *p* ∈ *Q*[*u*], such that *f λ* ⊂ *g* for some *g* ∈ *Sp* , is dense in **Q**[*u*]. Therefore *G* contains some *p* ∈ *Df <sup>λ</sup>* . Let this be witnessed by some *g* ∈ *Sp* . Now, if *ξ* < *λ* belongs to *Xf* , so that *s* = *f ξ* ∈ *SG* , then *s* must belong to *Sp* by (ii), therefore *<sup>ξ</sup>* belongs to the finite set {lh *<sup>s</sup>* : *<sup>s</sup>* <sup>∈</sup> *Sp* }. Thus, *Xf* <sup>∩</sup> *<sup>λ</sup>* is finite. That *Xf* <sup>∩</sup> *<sup>ω</sup>***<sup>L</sup>** <sup>1</sup> is infinite follows from (i) (recall that *f* ∈/ *u*).

Now we consider two types of transformations related to the forcing notion *Q*∗ .

#### *2.3. Lipschitz Transformations*

We argue in **L**. Let **LIP** be the group of all ⊆-automorphisms of **SEQ**, called *Lipschitz transformations*. Any *<sup>λ</sup>* <sup>∈</sup> **LIP** preserves the *length* lh of sequences, i.e., lh *<sup>s</sup>* <sup>=</sup> lh (*λ·s*) for all *s* ∈ **SEQ** . Any transformation *λ* ∈ **LIP** acts on:


**Lemma 4** (routine)**.** *The action of any λ* ∈ **LIP** *is an order-preserving automorphism of Q*∗. *If u* ⊆ **FUN** *and p* ∈ *Q*[*u*] *then λ· p* ∈ *Q*[*λ·u*]*.*

We proceed with an important existence lemma. If *f* = *g* belongs to **FUN** then let *β*(*f* , *g*) be equal to the least ordinal *β* < *ω*<sup>1</sup> such that *f*(*β*) = *g*(*β*) (or, similarly, the largest ordinal *β* with *f β* = *g β*). Say that sets *X*,*Y* ⊆ **FUN** are *intersection-similar*, or *i-similar* for brevity, if there is a bijection *b* : *X* onto −→ *Y* such that *β*(*f* , *g*) = *β*(*b*(*f*), *b*(*g*)) for all *f* = *g* in *X*—such a bijection *b* will be called *an i-similarity bijection.*

**Lemma 5.** *Suppose that u*, *v* ⊆ **FUN** *are ω*1*-sizesets, dense in* **FUN***. Then u*, *v are i-similar. Moreover, if X* ⊆ *u, Y* ⊆ *v are finite and i-similar then*


**Proof.** The key argument is that if *<sup>A</sup>* <sup>⊆</sup> *<sup>u</sup>*, *<sup>B</sup>* <sup>⊆</sup> *<sup>v</sup>* are at most countable, *<sup>b</sup>* : *<sup>A</sup>* onto −→ *B* is an i-similarity bijection, and *f* ∈ *u A*, then by the density of *v* there is *g* ∈ *v B* such that the extended map *<sup>b</sup>* ∪ {*<sup>f</sup>* , *<sup>g</sup>*} : *<sup>A</sup>* ∪ { *<sup>f</sup>* } onto −→ *B* ∪ {*g*} is still an i-similarity bijection. This allows proof of (i), iteratively extending an initial i-similarity bijection *<sup>b</sup>*<sup>0</sup> : *<sup>X</sup>* onto −→ *Y* by a *ω*1-step back-and-forth argument involving eventually all elements *<sup>f</sup>* <sup>∈</sup> *<sup>u</sup>* and *<sup>g</sup>* <sup>∈</sup> *<sup>v</sup>*, to an i-similarity bijection *<sup>u</sup>* onto −→ *v* required. See the proof of Lemma 5 in [7] for more detail.

To get (ii) from (i), consider any sequence *<sup>s</sup>* <sup>∈</sup> **SEQ**. Let *<sup>β</sup>* <sup>=</sup> lh *<sup>s</sup>*. As *<sup>u</sup>* is dense, there exist *f* , *f* ∈ *u* such that *β*(*f* , *f* ) = *β* and *s* ⊂ *f* , *s* ⊂ *f* . Put *g* = *b*(*f*), *g* = *b*(*f* ). Then still *β*(*g*, *g* ) = *β*, hence *g β* = *g β*. Therefore, we can define *λ*(*s*) = *g β* = *g β*.

#### *2.4. Substitution Transformations*

We continue to argue in **L**. Assume that conditions *p*, *q* ∈ *Q*<sup>∗</sup> satisfy

$$F\_p = F\_q \quad \text{and} \quad S\_p \cup S\_q \subseteq F\_p^\vee = F\_q^\vee. \tag{4}$$

We define a transformation *hpq* acting as follows.

If *p* = *q* then define *hpq*(*r*) = *r* for all *r* ∈ *Q*<sup>∗</sup> , the identity.

Suppose that *p* = *q*. Then *p*, *q* are incompatible by (4) and Lemma 1. Define *dpq* = {*r* ∈ *Q*<sup>∗</sup> : *r p* ∨ *r q*}, the *domain* of *hpq* . Let *r* ∈ *dpq* . We put *hpq*(*r*) = *r* := *Sr* , *Fr* , where *Fr* = *Fr* and

$$S\_{r'} = \begin{cases} \begin{array}{ll} (\mathcal{S}\_r \smile \mathcal{S}\_p) \cup \mathcal{S}\_q & \text{in case} \quad r \lessapprox p \end{array}, \\\ (\mathcal{S}\_r \smile \mathcal{S}\_q) \cup \mathcal{S}\_p & \text{in case} \quad r \lessapprox q \end{cases},\tag{5}$$

Thus, assuming (4), the difference between *Sr* and *Sr* lies entirely within the set *X* = *F*<sup>∨</sup> *<sup>p</sup>* = *F*<sup>∨</sup> *<sup>q</sup>* , so that if *r p* then *Sr* ∩ *X* = *Sp* but *Sr* ∩ *X* = *Sq* , while if *r q* then *Sr* ∩ *X* = *Sq* but *Sr* ∩ *X* = *Sp* .

**Lemma 6.** (i) *If u* ⊆ **FUN** *is dense and p*0, *q*<sup>0</sup> ∈ *Q*[*u*] *then there exist conditions p*, *q* ∈ *Q*[*u*] *with p p*<sup>0</sup> *, q q*<sup>0</sup> *, satisfying* (4)*.*


**Proof.** (i) By the density of *u* there is a finite set *F* ⊆ **FUN** satisfying *Fp* ∪ *Fq* ⊆ *F* and *Sp* ∪ *Sq* ⊆ *F*<sup>∨</sup> = { *f ξ* : *f* ∈ *F* ∧ 1 ≤ *ξ* < *ω*1}. Put *p* = *Sp*, *F* and *q* = *Sq*, *F*. Claims (ii), (iii) are routine.

Please note that unlike the Lipschitz transformations above, transformations of the form *hpq* , called *substitutions* in this paper, act within any given forcing notion of the form *Q*[*u*] by claim (iii) of the lemma, and hence the forcing notions of the form *Q*[*u*] considered are sufficiently homogeneous.

#### **3. Almost-Disjoint Product Forcing**

Here we review the structure and basic properties of product almost-disjoint forcing and corresponding generic extensions in the *ω*1-version. There is an important issue here: a forcing **C**, which collapses *ω*<sup>1</sup> to *ω*, enters as a factor in the product forcing notions considered.

#### *3.1. Product Forcing*

In **L**, we define **C** = P(*ω*) <sup>&</sup>lt;*<sup>ω</sup>* , the set of all finite sequences of subsets of *ω*, an ordinary forcing to collapse P(*ω*) ∩ **L** down to *ω*. We will make use of an *ω*2-product of *Q*<sup>∗</sup> with **C** as an extra factor. (In fact, **C** can be eliminated since *Q*<sup>∗</sup> collapses *ω***<sup>L</sup>** <sup>1</sup> anyway by Lemma 3 (v). Yet the presence of **C** somehow facilitates the arguments since **C** has a more transparent forcing structure.)

Technically, we put <sup>I</sup> <sup>=</sup> *<sup>ω</sup>*<sup>2</sup> (in **<sup>L</sup>**) and consider the index set <sup>I</sup><sup>+</sup> <sup>=</sup> I ∪ {−1}. Let **<sup>Q</sup>**<sup>∗</sup> be the finite-support product of **C** and I copies of *Q*<sup>∗</sup> (Definition 1 in Section 2.1), ordered componentwise. That is, **<sup>Q</sup>**<sup>∗</sup> consists of all maps *<sup>p</sup>* defined on a finite set dom *<sup>p</sup>* <sup>=</sup> <sup>|</sup>*p*<sup>|</sup> <sup>+</sup> ⊆ I<sup>+</sup> so that *<sup>p</sup>*(*ν*) <sup>∈</sup> *<sup>Q</sup>*<sup>∗</sup> for all *ν* ∈ |*p*| := |*p*| <sup>+</sup> {−1}, and if <sup>−</sup><sup>1</sup> ∈ |*p*<sup>|</sup> <sup>+</sup> then *<sup>b</sup><sup>p</sup>* :<sup>=</sup> *<sup>p</sup>*(−1) <sup>∈</sup> **<sup>C</sup>**. If *<sup>p</sup>* <sup>∈</sup> **<sup>Q</sup>**<sup>∗</sup> then put *Fp*(*ν*) = *Fp*(*ν*) and *Sp*(*ν*) = *Sp*(*ν*) for all *ν* ∈ |*p*|, so that *p*(*ν*) = *Sp*(*ν*); *Fp*(*ν*).

We order **Q**<sup>∗</sup> componentwise: *p q* ( *p* is stronger as a forcing condition) iff |*q*| <sup>+</sup> ⊆ |*p*<sup>|</sup> <sup>+</sup> , *<sup>b</sup><sup>q</sup>* <sup>⊆</sup> *<sup>b</sup><sup>p</sup>* in case −1 ∈ |*q*| <sup>+</sup> , and *<sup>p</sup>*(*ν*) *<sup>q</sup>*(*ν*) in *<sup>Q</sup>*<sup>∗</sup> for all *<sup>ν</sup>* ∈ |*q*|. Put

$$F\_p^\vee(\nu) = F\_{p(\nu)}^\vee = \{ f \upharpoonright \xi : f \in F\_p(\nu) \land 1 \le \xi < \omega\_1 \}.$$

In particular, **<sup>Q</sup>**<sup>∗</sup> contains *the empty condition* ∈ **<sup>Q</sup>**<sup>∗</sup> satisfying ||<sup>+</sup> <sup>=</sup> <sup>∅</sup>; obviously is the -least (and weakest as a forcing condition) element of **Q**∗ .

Because of the factor **C**, it takes some effort to define *p* ∧ *q* for *p*, *q* ∈ **Q**<sup>∗</sup> , and only assuming that *bp*, *b<sup>q</sup>* are compatible, i.e., *b<sup>p</sup>* ⊆ *b<sup>q</sup>* or *b<sup>q</sup>* ⊆ *b<sup>p</sup>* . In such a case define *p* ∧ *q* ∈ **Q**<sup>∗</sup> as follows. First, |*p* ∧ *q*| <sup>+</sup> <sup>=</sup> <sup>|</sup>*p*<sup>|</sup> <sup>+</sup> ∪ |*q*<sup>|</sup> <sup>+</sup> . If *<sup>ν</sup>* ∈ |*p*<sup>|</sup> <sup>+</sup> <sup>|</sup>*q*<sup>|</sup> <sup>+</sup> then put (*<sup>p</sup>* <sup>∧</sup> *<sup>q</sup>*)(*ν*) = *<sup>p</sup>*(*ν*), and similarly if *ν* ∈ |*q*| <sup>+</sup> <sup>|</sup>*p*<sup>|</sup> <sup>+</sup> then (*<sup>p</sup>* <sup>∧</sup> *<sup>q</sup>*)(*ν*) = *<sup>q</sup>*(*ν*). Now suppose that *<sup>ν</sup>* ∈ |*p*<sup>|</sup> <sup>+</sup> ∩ |*q*<sup>|</sup> + .

If *ν* = −1 then (*p* ∧ *q*)(*ν*) = *p*(*ν*) ∧ *q*(*ν*) in the sense of Definition 1 in Section 2.1.

If *ν* = −1 ∈ |*p*| <sup>+</sup> ∩ |*q*<sup>|</sup> <sup>+</sup> , then, by the compatibility, either *<sup>b</sup><sup>p</sup>* <sup>⊆</sup> *<sup>b</sup>q*—and then define *<sup>b</sup>p*∧*<sup>q</sup>* <sup>=</sup> *<sup>b</sup><sup>q</sup>* , or *b<sup>q</sup>* ⊆ *bp*—and then accordingly *bp*∧*<sup>q</sup>* = *b<sup>p</sup>* .

**Lemma 7.** *Let p*, *q* ∈ **Q**<sup>∗</sup> *be compatible. Then* (*p* ∧ *q*) ∈ **Q**<sup>∗</sup> *,* (*p* ∧ *q*) *p,* (*p* ∧ *q*) *q, and if r* ∈ **Q**<sup>∗</sup> *, r p, r q, then r* (*p* ∧ *q*)*.*

#### *3.2. Systems*

Arguing in **L**, we consider certain subforcings of the total product forcing notion **Q**∗ .

*Mathematics* **2020**, *8*, 1477

Let a *system* be any map *U* : |*U*| → P(**FUN**) such that |*U*|⊆I , each set *U*(*ν*) (*ν* ∈ |*U*|) is dense in **FUN**, and the *components U*(*ν*) ⊆ **FUN** (*ν* ∈ |*U*|) are pairwise disjoint.


$$\mathbf{Q}[\mathcal{U}] = \left\{ p \in \mathbf{Q}^\* : |p| \subseteq |\mathcal{U}| \land \forall \, \nu \in |p| \left( F\_p(\nu) \subseteq \mathcal{U}(\nu) \right) \right\}.$$

Suppose that *<sup>c</sup>* ⊆ I<sup>+</sup> . If *<sup>p</sup>* <sup>∈</sup> **<sup>Q</sup>**<sup>∗</sup> then define *<sup>p</sup>* <sup>=</sup> *<sup>p</sup> <sup>c</sup>* <sup>∈</sup> **<sup>Q</sup>**<sup>∗</sup> so that <sup>|</sup>*<sup>p</sup>* | <sup>+</sup> <sup>=</sup> *<sup>c</sup>* ∩ |*p*<sup>|</sup> <sup>+</sup> and *p* (*ν*) = *p*(*ν*) whenever *ν* ∈ |*p* | <sup>+</sup> . A special case: if *<sup>ν</sup>* ∈ I<sup>+</sup> then let *<sup>p</sup>* =*<sup>ν</sup>* <sup>=</sup> *<sup>p</sup>* (|*p*<sup>|</sup> <sup>+</sup> {*ν*}). Similarly, if *U* is a system then define a system *U* = *U c* so that |*U* | = *c* ∩ |*U*| and *U* (*ν*) = *U*(*ν*) whenever *ν* ∈ |*U* <sup>|</sup>. A special case: if *<sup>ν</sup>* ∈ I<sup>+</sup> then let *<sup>U</sup>* =*<sup>ν</sup>* <sup>=</sup> *<sup>U</sup>* (|*p*<sup>|</sup> {*ν*}). And if *<sup>Q</sup>* <sup>⊆</sup> **<sup>Q</sup>**<sup>∗</sup> then let *Q c* = {*p* ∈ *Q* : |*p*| <sup>+</sup> <sup>⊆</sup> *<sup>c</sup>*} (will usually coincide with {*p <sup>c</sup>* : *<sup>p</sup>* <sup>∈</sup> *<sup>Q</sup>*}.

Writing *p c*, *U c* etc., it is not assumed that *c* ⊆ |*p*| + .

**Lemma 8** (in **<sup>L</sup>**)**.** *If U is a system and A* <sup>⊆</sup> **<sup>Q</sup>**[*U*] *is an antichain then* card *<sup>A</sup>* <sup>≤</sup> *<sup>ω</sup>*1*.*

**Proof.** Suppose that card *A* > *ω*1. As card**C** = *ω*1, we can w.l. o. g. assume that *b<sup>p</sup>* = *b<sup>q</sup>* for all *p*, *q* ∈ *A*. It follows by the Δ-system lemma that there is a set *A* ⊆ *A* of the same cardinality card *<sup>A</sup>* <sup>=</sup> card *<sup>A</sup>* <sup>&</sup>gt; *<sup>ω</sup>*1, and a finite set *<sup>d</sup>* ⊆ I<sup>+</sup> , such that <sup>|</sup>*p*<sup>|</sup> <sup>+</sup> <sup>=</sup> *<sup>d</sup>* for all *<sup>p</sup>* <sup>∈</sup> *<sup>A</sup>* . Then we have *Sp* = *Sq* for all *p* = *q* in *A* , easily leading to a contradiction, as in the proof of Lemma 2.

#### *3.3. Outline of Product Extensions*

We consider sets of the form **Q**[*U*], *U* being a system in **L**, as forcing notions over **L**. Accordingly, we'll study **Q**[*U*]-generic extensions **L**[*G*] of the ground universe **L**. Define some elements of these extensions. Suppose that *<sup>G</sup>* ⊆ **<sup>Q</sup>**<sup>∗</sup> . Put |*G*| = - *<sup>p</sup>*∈*<sup>G</sup>* |*p*|; |*G*|⊆I . Let

$$\mathfrak{b}\_{\mathbb{G}} = \bigcup\_{p \in \mathbb{G}} \mathfrak{b}\_{p \cdot \prime} \quad \text{and} \quad \mathfrak{S}\_{\mathbb{G}}(\nu) = \mathfrak{S}\_{\mathbb{G}(\nu)} = \bigcup\_{p \in \mathbb{G}} \mathfrak{S}\_{p}(\nu).$$

for any *ν* ∈ |*G*|, where *G*(*ν*) = {*p*(*ν*) : *p* ∈ *G*} ⊆ *Q*<sup>∗</sup> .

Thus, *SG*(*ν*) <sup>⊆</sup> **SEQL**, and *SG*(*ν*) = <sup>∅</sup> for any *<sup>ν</sup>* ∈ | / *<sup>G</sup>*|.

By the way, this defines a sequence *SG* = {*SG*(*ν*)}*ν*∈I of subsets of **SEQ**.

If *<sup>c</sup>* ⊆ I<sup>+</sup> then let *<sup>G</sup> <sup>c</sup>* <sup>=</sup> {*<sup>p</sup>* <sup>∈</sup> *<sup>G</sup>* : <sup>|</sup>*p*<sup>|</sup> <sup>+</sup> <sup>⊆</sup> *<sup>c</sup>*}. It will typically happen that *<sup>G</sup> <sup>c</sup>* <sup>=</sup> {*p <sup>c</sup>* : *<sup>p</sup>* <sup>∈</sup> *<sup>G</sup>*}. Put *<sup>G</sup>* =*<sup>ν</sup>* = {*<sup>p</sup>* ∈ *<sup>G</sup>* : *<sup>ν</sup>* ∈ | / *<sup>p</sup>*| <sup>+</sup>} <sup>=</sup> *<sup>G</sup>* (I<sup>+</sup> {*ν*}).

If *U* is a system in **L**, then any **Q**[*U*]-generic set *G* ⊆ **Q**[*U*] splits into the family of sets *G*(*ν*), *<sup>ν</sup>* ∈ I , and a separate map *<sup>b</sup><sup>G</sup>* : *<sup>ω</sup>* onto −→ P(*ω*) ∩ **L**. It will follow from (ii) of the next lemma that **Q**[*U*] generic extensions of **L** satisfy *ω*<sup>1</sup> = *ω***<sup>L</sup>** 2 .

**Lemma 9.** *Let U be a system in* **L***, and G* ⊆ **Q**[*U*] *be a set* **Q**[*U*]*-generic over* **L***. Then*:

(i) *b<sup>G</sup> is a* **C***-generic map from ω onto* P(*ω*) ∩ **L** ;

$$\text{(ii)}\quad \text{if } \boldsymbol{\nu} \in \mathcal{T} \text{ then } \mathbf{L}[\mathbf{G}(\boldsymbol{\nu})] = \mathbf{L}[\mathbf{S}\_{\mathbf{G}}(\boldsymbol{\nu})] \text{ and } \boldsymbol{\omega}\_{1}^{\mathsf{L}[\boldsymbol{b}\_{\mathsf{G}}]} = \boldsymbol{\omega}\_{1}^{\mathsf{L}[\mathbf{G}(\boldsymbol{\nu})]} = \boldsymbol{\omega}\_{2}^{\mathsf{L}} = \boldsymbol{\omega}\_{1}^{\mathsf{L}[\mathbf{G}]}, \text{ ;}$$

$$\text{(iii)}\quad \mathbf{L}[G] = \mathbf{L}[\mathcal{S}\_G] \text{ and } |G|^+ = \mathcal{T};$$


(vi) *if <sup>ν</sup>* ∈ I *then the set <sup>G</sup>*(*ν*) = {*p*(*ν*) : *<sup>p</sup>* <sup>∈</sup> *<sup>G</sup>*} ∈ **<sup>L</sup>**[*G*] *is <sup>P</sup>*[*U*(*ν*)]*-generic over* **<sup>L</sup>***, hence if <sup>f</sup>* <sup>∈</sup> **FUN<sup>L</sup>** *then f* ∈ *U*(*ν*) *iff SG*(*ν*) *does not cover f* .

**Proof.** Proofs of (i) and (iii)–(vi) are similar to ([7] (Lemma 9)). To prove *ω*<sup>1</sup> **<sup>L</sup>**[*G*(*ν*)] = *ω***<sup>L</sup>** <sup>2</sup> in (ii) apply Lemma 3 (v). Finally, to see that *ω***<sup>L</sup>** <sup>2</sup> remains a cardinal in **L**[*G*] apply Lemma 8.

#### *3.4. Names for Sets in Product Extensions*

The next definition introduces *names* for elements of product-generic extensions of **L** considered. Assume that in **L**, *K* ⊆ **Q**<sup>∗</sup> , e.g., *K* = **Q**[*U*], where *U* is a system, and *X* is any set. By **N***X*(*K*) (*K-names* for subsets of *X*) we denote the set of all sets *τ* ⊆ *K* × *X* in **L**. Furthermore, **SN***X*(*K*) (small names) consist of all *<sup>ω</sup>*1-size names *<sup>τ</sup>* <sup>∈</sup> **SN***X*(*K*); in other words, it is required that card *<sup>τ</sup>* <sup>≤</sup> *<sup>ω</sup>*1.

Suppose that *τ* ∈ **N***X*(**Q**∗). We put

$$\mathsf{dom}\,\mathsf{\tau} = \left\{ p: \exists \,\mathsf{x} \left( \langle p, \mathsf{x} \rangle \in \mathsf{\tau} \right), \quad |\mathsf{\tau}|^{+} = \bigcup\_{p \in \mathsf{dom}\,\mathsf{\tau}} \left| p \right|^{+}, \quad |\mathsf{\tau}| = \bigcup\_{p \in \mathsf{dom}\,\mathsf{\tau}} \left| p \right|. \right\}$$

If *G* ⊆ **Q**<sup>∗</sup> then define

$$\pi[G] = \{ \mathbf{x} \in X : (\pi''\mathbf{x}) \cap G \neq \mathcal{Q} \}, \quad \text{where} \quad \pi''\mathbf{x} = \{ p : \langle p, \mathbf{x} \rangle \in \mathsf{r} \},$$

so that *<sup>τ</sup>*[*G*] <sup>⊆</sup> *<sup>X</sup>*. If *<sup>ϕ</sup>* is a formula in which some names *<sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**∗) occur, and *G* ⊆ **Q**<sup>∗</sup> , then accordingly *ϕ*[*G*] is the result of substitution of *τ*[*G*] for each name *τ* in *ϕ*.

**Lemma 10.** *Suppose that <sup>X</sup>* <sup>∈</sup> **<sup>L</sup>***,* card *<sup>X</sup>* <sup>≤</sup> *<sup>ω</sup>*<sup>1</sup> *in* **<sup>L</sup>***, <sup>U</sup> is a system in* **<sup>L</sup>***, and <sup>G</sup>* <sup>⊆</sup> **<sup>Q</sup>**[*U*] *is a set* **<sup>Q</sup>**[*U*] *generic over* **L***. Then for any set Y* ∈ **L**[*G*]*, Y* ⊆ *X*, *there is a name τ* ∈ **SN***X*(**Q**[*U*]) *in* **L** *such that Y* = *τ*[*G*]*. If in addition <sup>c</sup>* <sup>∈</sup> **<sup>L</sup>***, c* ⊆ I<sup>+</sup> *, and <sup>Y</sup>* <sup>∈</sup> **<sup>L</sup>**[*G <sup>c</sup>*]*, then there is a name <sup>τ</sup>* <sup>∈</sup> **SN***X*(**Q**[*U*] *<sup>c</sup>*) *in* **<sup>L</sup>** *such that Y* = *τ*[*G*]*.*

**Proof.** It follows from general forcing theory that there is a name *σ* ∈ **N***X*(**Q**[*U*]), not necessarily an *ω*1-size name, such that *X* = *σ*[*G*]. Let *Qx* = *σ*"*x* for all *x* ∈ *X*. Arguing in **L**, put

$$\tau = \{ \langle p, \mathfrak{x} \rangle \in \sigma : \mathfrak{x} \in X \land p \in A\_{\mathfrak{x}} \} \ \mathsf{A}$$

where *Ax* <sup>⊆</sup> *Qx* is a maximal antichain for any *<sup>x</sup>*. We observe that card *Ax* <sup>≤</sup> *<sup>ω</sup>*<sup>1</sup> in **<sup>L</sup>** for all *<sup>x</sup>* by Lemma 8, hence *τ* ∈ **SN***X*(**Q**[*U*]). And on the other hand, we have *τ*[*G*] = *σ*[*G*] = *Y*.

To prove the additional claim, note that by the product forcing theorem if *Y* ∈ **L**[*G c*] then the original name *σ* can be chosen in **N***X*(**Q**[*U*] *c*), and repeat the argument.

#### *3.5. Names for Reals in Product Extensions*

Now we introduce *names for reals* (elements of *ω<sup>ω</sup>* ) in generic extensions of **L** considered. This is an important particular case of the content of Section 3.4.

Assume that in **<sup>L</sup>**, *<sup>K</sup>* <sup>⊆</sup> **<sup>Q</sup>**<sup>∗</sup> , e.g., *<sup>K</sup>* <sup>=</sup> **<sup>Q</sup>**[*U*], where *<sup>U</sup>* is a system. By **<sup>N</sup>***<sup>ω</sup> <sup>ω</sup>*(*K*) (*K-names* for reals in *<sup>ω</sup><sup>ω</sup>* ) we denote the set of all *<sup>τ</sup>* <sup>⊆</sup> *<sup>K</sup>* <sup>×</sup> (*<sup>ω</sup>* <sup>×</sup> *<sup>ω</sup>*) such that the sets *<sup>τ</sup>* "*<sup>j</sup>*, *<sup>k</sup>* <sup>=</sup> {*<sup>p</sup>* : *<sup>p</sup>*,*<sup>j</sup>*, *<sup>k</sup>*<sup>∈</sup> *<sup>τ</sup>*} satisfy the following requirement:

> if *k* = *k* , *p* ∈ *τ* "*j*, *k*, *p* ∈ *τ* "*j*, *k* , then conditions *p*, *p* are incompatible.

We let *τ* "*j* = - *<sup>k</sup> <sup>τ</sup>* "*<sup>j</sup>*, *<sup>k</sup>*, dom *<sup>τ</sup>* <sup>=</sup> - *<sup>j</sup>*,*k*<*<sup>ω</sup> τ* "*j*, *k*, |*τ*| <sup>+</sup> = -{|*p*| <sup>+</sup> : *<sup>p</sup>* <sup>∈</sup> dom *<sup>τ</sup>*}.

Let **SN***<sup>ω</sup> <sup>ω</sup>*(*K*) (small names) consist of all *<sup>ω</sup>*1-size names *<sup>τ</sup>* <sup>∈</sup> **<sup>N</sup>***<sup>ω</sup> <sup>ω</sup>*(*K*); in other words, it is required that card (*<sup>τ</sup>* "*<sup>j</sup>*, *<sup>k</sup>*) <sup>≤</sup> *<sup>ω</sup>*<sup>1</sup> for all *<sup>j</sup>*, *<sup>k</sup>* <sup>&</sup>lt; *<sup>ω</sup>*.

Define the restrictions **SN***<sup>ω</sup> <sup>ω</sup>*(*K*) *<sup>c</sup>* <sup>=</sup> {*<sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(*K*) : |*τ*| <sup>+</sup> <sup>⊆</sup> *<sup>c</sup>*}.

A name *<sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(*K*) is *K-full* iff the set *<sup>τ</sup>* "*<sup>j</sup>* is pre-dense in *<sup>K</sup>* for any *<sup>j</sup>* <sup>&</sup>lt; *<sup>ω</sup>*. A name *<sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(*K*) is *K*-full *below* some *p*<sup>0</sup> ∈ *K*, iff all sets *τ* "*j* are pre-dense in *K* below *p*<sup>0</sup> , i.e., any condition *q* ∈ *K*, *q p*<sup>0</sup> , is compatible with some *r* ∈ *τ<sup>j</sup>* (and this holds for all *j* < *ω*).

Suppose that *<sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**∗). A set *G* ⊆ *K* is *minimally τ-generic* iff it is compatible in itself (if *p*, *q* ∈ *G* then there is *r* ∈ *G* with *r p*, *r q*), and intersects each set *τ* "*x*, *x* ∈ *X*. In this case, put

$$\pi[G] = \{ \langle j, k \rangle \in \omega^{\omega} \times \omega^{\omega} : (\pi'' \langle j, k \rangle) \cap G \neq \boxtimes \},$$

so that *<sup>τ</sup>*[*G*] <sup>∈</sup> *<sup>ω</sup><sup>ω</sup>* and *<sup>τ</sup>*[*G*](*j*) = *<sup>k</sup>* ⇐⇒ *<sup>τ</sup>* "*<sup>j</sup>*, *<sup>k</sup>*<sup>∩</sup> *<sup>G</sup>* <sup>=</sup> <sup>∅</sup>. If *<sup>ϕ</sup>* is a formula in which some names *<sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**∗) occur, and a set *G* ⊆ **Q**<sup>∗</sup> is minimally *τ*-generic for any name *τ* in *ϕ*, then accordingly *ϕ*[*G*] is the result of substitution of *τ*[*G*] for each name *τ* in *ϕ*.

**Lemma 11.** *Suppose that U is a system in* **L***, and G* ⊆ **Q**[*U*] *is* **Q**[*U*]*-generic over* **L***. Then for any real <sup>x</sup>* <sup>∈</sup> **<sup>L</sup>**[*G*] <sup>∩</sup> *<sup>ω</sup><sup>ω</sup> there is a* **<sup>Q</sup>**[*U*]*-full name <sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**[*U*]) *in* **L** *such that x* = *τ*[*G*]*. If in addition c* ∈ **L***, <sup>c</sup>* ⊆ I<sup>+</sup> *, and x* <sup>∈</sup> **<sup>L</sup>**[*G <sup>c</sup>*]*, then there is a* **<sup>Q</sup>**[*U*]*-full name <sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**[*U*] *c*) *in* **L** *such that x* = *τ*[*G*]*.*

**Proof.** It follows from general forcing theory that there is a **<sup>Q</sup>**[*U*]-full name *<sup>σ</sup>* <sup>∈</sup> **<sup>N</sup>***<sup>ω</sup> <sup>ω</sup>*(**Q**[*U*]), not necessarily an *ω*1-size name, such that *f* = *σ*[*G*]. Then all sets *Qj* = *σ*"*j*, *j* < *ω*, are pre-dense in **Q**[*U*]. Arguing in **L**, put *τ* = {*p*,*j*, *k* ∈ *σ* : *j*, *k* < *ω* ∧ *p* ∈ *Aj*}, where *Aj* ⊆ *Qj* is a maximal antichain for any *<sup>j</sup>* <sup>&</sup>lt; *<sup>ω</sup>*. We conclude by Lemma <sup>8</sup> that card *Aj* <sup>≤</sup> *<sup>ω</sup>*<sup>1</sup> in **<sup>L</sup>** for all *<sup>j</sup>*, hence in fact *<sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**[*U*]). And on the other hand, we have *τ*[*G*] = *σ*[*G*] = *f* .

**Equivalent names**. Names *<sup>τ</sup>*, *<sup>μ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**∗) are *equivalent* iff conditions *q*,*r* are incompatible whenever *q* ∈ *τ* "*j*, *k* and *r* ∈ *μ*"*j*, *k*  for some *j* and *k* = *k*. Names *τ*, *μ* are equivalent *below* some *p* ∈ **Q**<sup>∗</sup> iff the triple of conditions *p*, *q*,*r* is incompatible (that is, no common strengthening) whenever *q* ∈ *τ* "*j*, *k* and *r* ∈ *μ*"*j*, *k*  for some *j* and *k* = *k*.

**Lemma 12.** *Suppose that in* **<sup>L</sup>***, <sup>p</sup>* <sup>∈</sup> **<sup>Q</sup>**<sup>∗</sup> *, and names <sup>μ</sup>*, *<sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**∗) *are equivalent* (*resp., equivalent below p* )*. If G* ⊆ **Q**<sup>∗</sup> *is minimally μ-generic and minimally τ-generic* (*resp., and containing p* )*, then μ*[*G*] = *τ*[*G*]*.*

**Proof.** Suppose that this is not the case. Then by definition there exist numbers *j* and *k* = *k* and conditions *q* ∈ *G* ∩ (*τ* "*j*, *k*) and *r* ∈ *G* ∩ (*μ*"*j*, *k* ). Then *p*, *q*,*r* are compatible (as elements of the same generic set), contradiction.

The next lemma provides a useful transformation of names. Recall that *p* ∧ *p* is defined in Section 3.1.

**Lemma 13** (in **<sup>L</sup>**)**.** *If p* <sup>∈</sup> **<sup>Q</sup>**<sup>∗</sup> *and <sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**∗)*, then*

> *τ<sup>p</sup>* = {*p* ∧ *p*,*j*, *k* : *p* ,*j*, *k* ∈ *τ* and *p* is compatible with *p*}

*is still a name in* **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**∗)*, equivalent to τ below p, and* |*τp*| <sup>+</sup> ⊆ |*τ*<sup>|</sup> <sup>+</sup> ∪ |*p*<sup>|</sup> + *. If U is a system and p* <sup>∈</sup> **<sup>Q</sup>**[*U*]*, <sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**[*U*])*, then <sup>τ</sup><sup>p</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**[*U*])*. Moreover, if τ is* **Q**[*U*]*-full below p then τ<sup>p</sup> is* **Q**[*U*]*-full below p, too.*

**Proof.** Routine.

## *3.6. Permutations*

We continue to argue in **L**. There are three important families of transformations of the whole system of objects related to product forcing, considered in this Subsection and the two following ones.

We begin with *permutations*, the first family. Let **BIJ** be the set of all bijections *<sup>π</sup>* : <sup>I</sup> onto −→ I , i.e., permutations of the set I , such that the set |*π*| = {*ν* ∈ I : *π*(*ν*) = *ν*} (the *essential domain*) satisfies card <sup>|</sup>*π*| ≤ *<sup>ω</sup>*1. Please note that *<sup>π</sup>* is the identity outside of <sup>|</sup>*π*|. Any permutation *<sup>π</sup>* <sup>∈</sup> **BIJ** acts onto:


**Lemma 14** (routine)**.** *If π* ∈ **BIJ** *then p* −→ *π· p is an order-preserving bijection of* **Q**<sup>∗</sup> *onto* **Q**<sup>∗</sup> *, and if U is a system then p* ∈ **Q**[*U*] ⇐⇒ *π· p* ∈ **Q**[*π·U*]*.*

#### *3.7. Multi-Lipschitz Transformations*

Still arguing in **L**, we let **LIP**<sup>I</sup> be the I-product of the group **LIP** (see Section 2.3), this will be our *second family* of transformations, called *multi-Lipschitz*. Thus, a typical element *λ* ∈ **LIP**<sup>I</sup> is *<sup>λ</sup>* <sup>=</sup> {*λν* }*ν*∈|*λ*<sup>|</sup> , where <sup>|</sup>*λ*<sup>|</sup> <sup>=</sup> dom *<sup>λ</sup>* ⊆ I<sup>+</sup> has *<sup>ω</sup>*1-size, *λν* <sup>∈</sup> **LIP**, <sup>∀</sup> *<sup>ν</sup>*. Define the action of any *λ* ∈ **LIP**<sup>I</sup> on:


In the first two items, we refer to the action of *λν* ∈ **LIP** on sets *u* ⊆ **FUN** and on forcing conditions, as defined in Section 2.3.

**Lemma 15** (routine)**.** *If λ* ∈ **LIP**<sup>I</sup> *then p* −→ *π· p is an order-preserving bijection of* **Q**<sup>∗</sup> *onto* **Q**<sup>∗</sup> *, and if U is a system then p* ∈ **Q**[*U*] ⇐⇒ *λ· p* ∈ **Q**[*λ·U*]*.*

**Lemma 16.** *Suppose that U*, *V are systems,* |*U*| = |*V*|*, p* ∈ **Q**[*U*]*, q* ∈ **Q**[*V*]*,* |*p*| = |*q*|*, and sets F*<sup>∨</sup> *<sup>p</sup>* (*ν*)*, F*∨ *<sup>q</sup>* (*ν*) *are i-similar for all ν* ∈ |*p*| = |*q*|*. Then there is λ* ∈ **LIP**<sup>I</sup> *such that* |*λ*| = |*U*| = |*V*|*, λ·U* = *V, and F*∨ *<sup>q</sup>* (*ν*) = *F*<sup>∨</sup> *<sup>λ</sup>· <sup>p</sup>*(*ν*) *for all <sup>ν</sup>* ∈ |*p*<sup>|</sup> <sup>=</sup> <sup>|</sup>*q*|*.*

**Proof.** Apply Lemma 5 componentwise for every *ν* ∈ I .

*3.8. Multi-Substitutions*

Assume that conditions *p*, *q* ∈ **Q**<sup>∗</sup> satisfy the following:

$$\begin{aligned} \text{(6i)} \quad -1 \in |p|^+ = |q|^+ \quad \text{and} \quad \text{1h} \, b\_p = 1 \text{h} \, b\_{q'} \quad \text{and} \\\text{(6ii)} \quad \text{if } \nu \in |p| \text{ then } F\_{\overline{p}}(\nu) = F\_{\overline{q}}(\nu) \text{ and } S\_{\overline{p}}(\nu) \cup S\_{\overline{q}}(\nu) \subseteq F\_{\overline{p}}^\vee(\nu) = F\_{\overline{q}}^\vee(\nu) \,. \end{aligned} \tag{6b}$$

In particular, (4) of Section 2.4 holds for all *ν*. We define a transformation *Hpq* acting as follows. First, we let **D***pq* , the domain of *Hpq* , contain all conditions *r* ∈ **Q**<sup>∗</sup> such that


Please note that all conditions *r p* and all *r p* belong to **D***pq* . On the other hand, if *r* ∈ **Q**<sup>∗</sup> satisfies |*r*|∩|*p*| = ∅ and (a), then *r* belongs to **D***pq* as well. In particular, ∈ **D***pq* .

If *r* ∈ **D***pq* , then define *r* = *Hpq*(*r*) ∈ **Q**<sup>∗</sup> so that |*r* | <sup>+</sup> <sup>=</sup> <sup>|</sup>*r*<sup>|</sup> <sup>+</sup> and:


Transformations of the form *Hpq* will be called *multi-substitutions*.


**Proof.** Apply Lemma 6 componentwise.

**Corollary 1** (of Lemma 17)**.** *If U is a system then* **Q**[*U*] *is* **homogeneous** *in the following sense*: *if p*0, *q*<sup>0</sup> ∈ **Q**[*U*] *then there exist stronger conditions p p*<sup>0</sup> *and q q*<sup>0</sup> *in* **Q**[*U*]*, such that the according lower cones* {*p* ∈ **Q**[*U*] : *p p*} *and* {*q* ∈ **Q**[*U*] : *q q*} *are order-isomorphic.*

**Action of** *Hpq* **on names.** Assume that conditions *<sup>p</sup>*, *<sup>q</sup>* <sup>∈</sup> **<sup>Q</sup>**<sup>∗</sup> satisfy (6). Let **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**∗)*pq* contain all names *<sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**∗) such that dom *<sup>τ</sup>* <sup>⊆</sup> **<sup>D</sup>***pq* . If *<sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**∗)*pq* then put

$$H\_{\mathbb{pq}} \cdot \pi = \{ \langle H\_{\mathbb{pq}}(p'), \langle n, k \rangle \rangle : \langle p', \langle n, k \rangle \rangle \in \pi \} \ . $$

Then obviously *Hpq ·<sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**∗)*qp* .

#### **4. The Basic Forcing Notion and the Model**

In this paper, we let **ZFC**− be **ZFC** minus the Power Set axiom, with the schema of Collection instead of Replacement, with **AC** is assumed in the form of well-orderability of every set, and with the axiom: "*ω*<sup>1</sup> exists". See [8] on versions of **ZFC** sans the Power Set axiom in detail.

Let **ZFC**− <sup>2</sup> be **ZFC**<sup>−</sup> plus the axioms: **<sup>V</sup>** <sup>=</sup> **<sup>L</sup>**, and the axiom "every set *<sup>x</sup>* satisfies card *<sup>x</sup>* <sup>≤</sup> *<sup>ω</sup>*1".

#### *4.1. Jensen—Solovay Sequences*

Arguing in **L**, let *U*, *V* be systems. Suppose that *M* is any transitive model of **ZFC**− 2 . Define *U <sup>M</sup> U* iff *U U* and the following holds:


Let **JS**, *Jensen—Solovay pairs*, be the set of all pairs *M*, *U* of:

− a transitive model *M* |= **ZFC**<sup>−</sup> <sup>2</sup> , and a system *U*, − such that the sets *ω*<sup>1</sup> and *U* belong to *M*—then sets **SEQ** , **Q**[*U*] also belong to *M*.

Let **sJS**, *small Jensen—Solovay pairs*, be the set of all pairs *M*, *U* ∈ **JS** such that both *U* and *M* have cardinality ≤ *ω*1. We define:

*M*, *U M* , *U*  (*M* , *U*  extends *M*, *U*) iff *M* ⊆ *M* and *U <sup>M</sup> U* ; *M*, *U*≺*M* , *U*  (strict extension) iff *M*, *U M* , *U*  and ∀ *ν* ∈ I (*U*(*ν*) - *U* (*ν*)).

**Lemma 18** (in **<sup>L</sup>**)**.** *If <sup>M</sup>*, *<sup>U</sup>*<sup>∈</sup> **sJS** *and <sup>z</sup>* ⊆ I *,* card *<sup>z</sup>* <sup>≤</sup> *<sup>ω</sup>*1*, then there is a pair <sup>M</sup>* , *U*  ∈ **sJS***, such that M*, *U*≺*M* , *U and z* ⊆ |*U* |*.*

**Proof.** Let *d* = |*U*| ∪ *z*. By definition **SEQ** is *ω-closed* as a forcing: any ⊆-increasing sequence {*sn* }*n*<*<sup>ω</sup>* of *sn* ∈ **SEQ** has the least upper bound in **SEQ**, equal to the union of all *sn* . It follows that the countable-support product **SEQ**(*d*×*ω*1) is *<sup>ω</sup>*-closed, too. Therefore, as card *<sup>M</sup>* <sup>≤</sup> *<sup>ω</sup>*1, there exists a system *<sup>f</sup>* <sup>=</sup> { *<sup>f</sup>νξ* }*ν*∈*d*, *<sup>ξ</sup>*<*ω*<sup>1</sup> <sup>∈</sup> (**Fun**)*d*×*ω*<sup>1</sup> , **SEQ**(*d*×*ω*1)-generic over *<sup>M</sup>*. Now define *U* (*ν*) = *U*(*ν*) ∪ { *fνξ* : *ξ* < *ω*1} for each *ν* ∈ *d* (assuming that *U*(*ν*) = ∅ in case *ν* ∈ | / *U*|), and let *M* |= **ZFC**<sup>−</sup> <sup>1</sup> be any transitive model of cardinality *ω*1, satisfying *M* ⊆ *M* and containing *U* .

**Lemma 19** (in **L**)**.** *Suppose that pairs M*, *U M* , *U M* , *U belong to* **JS***. Then M*, *U M* , *U . Thus is a partial order on* **JS***.*

**Proof.** We claim that *F* = - *<sup>ν</sup>*∈|*U*|(*<sup>U</sup>* (*ν*) *<sup>U</sup>*(*ν*)) is multiply **SEQ** -generic over *<sup>M</sup>*. Suppose, for the sake of brevity, that *F* = { *f* , *g*}, where *f* ∈ *U* (*ν*) *U*(*ν*)—then *f* ∈ *M* , *g* ∈ *U* (*μ*) *U* (*μ*), and *ν*, *μ* ∈ |*U*|. (The general case does not differ much.) By definition, *f* is Cohen generic over *M* and *g* is Cohen generic over *M* . Therefore, *g* is Cohen generic over *M*[ *f* ], because *M*[ *f* ] ⊆ *M* (as *f* ∈ *M* ). It remains to apply the product forcing theorem.

Now, still in **L**, a *Jensen—Solovay sequence* of length *λ* ≤ *ω*<sup>2</sup> is any strictly ≺-increasing *λ*-sequence {*M<sup>ξ</sup>* , *<sup>U</sup><sup>ξ</sup>* }*ξ*<*<sup>λ</sup>* of pairs *<sup>M</sup><sup>ξ</sup>* , *<sup>U</sup><sup>ξ</sup>*  ∈ **sJS**, satisfying *<sup>U</sup><sup>η</sup>* = *<sup>ξ</sup>*<*<sup>η</sup> <sup>U</sup><sup>ξ</sup>* on limit steps. Let −→**JS***<sup>λ</sup>* be the set of all such sequences.

**Lemma 20** (in **<sup>L</sup>**)**.** *Let <sup>λ</sup> be a limit ordinal, and* {*M<sup>ξ</sup>* , *<sup>U</sup><sup>ξ</sup>* }*ξ*<*<sup>λ</sup>* <sup>∈</sup> −→**JS***<sup>λ</sup> . Put U* <sup>=</sup> *<sup>ξ</sup>*<*<sup>λ</sup> U<sup>ξ</sup> . Then*


**Proof.** The same arguments work as in the proof of Lemma 19.

#### *4.2. Stability of Dense Sets*

If *U* is a system, *D* is a pre-dense subset of **P**[*U*], and *U* is another system extending *U*, then in principle *D* does not necessarily remain maximal in **P**[*U* ], a bigger set. This is where the genericity requirement (a) in Section 4.1 plays its role to *seal* the pre-density of sets in *M* w. r. t. further extensions. This is the content of the following key theorem. Moreover, the product forcing arguments will allow us to extend the stability result in pre-dense sets not necessarily in *M*, as in items (ii), (iii) of the theorem.

**Theorem 2** (stability of dense sets)**.** *Assume that, in* **L***, M*, *U* ∈ **sJS***, U is a system, and U <sup>M</sup> U . If D is a pre-dense subset of* **Q**[*U*] (*resp., pre-dense below some p* ∈ **Q**[*U*] ) *then D remains pre-dense in* **Q**[*U* ] (*resp., pre-dense below p* ) *in each of the following three cases*:


**Proof.** Arguing in **L**, we consider only the case of sets *D* pre-dense in **Q**[*U*] itself; the case of pre-density below some *p* ∈ **Q**[*U*] is treated similarly.

(i) Suppose, towards the contrary, that a condition *p* ∈ **Q**[*U* ] is incompatible with each *q* ∈ *D*. As *D* ⊆ **P**[*U*], we can w.l. o. g. assume that |*p*|⊆|*U*|.

We are going to define a condition *p* ∈ **Q**[*U*], also incompatible with each *q* ∈ *D*, contrary to the pre-density. To maintain the construction, consider the finite sequence *<sup>f</sup>* <sup>=</sup> *<sup>f</sup>*1, ... , *fm* of all elements *<sup>f</sup>* ∈ **FUN** occurring in - *<sup>ν</sup>*∈|*p*<sup>|</sup> *Fp*(*ν*) but not in *<sup>U</sup>*. It follows from *<sup>U</sup> <sup>M</sup> <sup>U</sup>* that *<sup>f</sup>* is **SEQ***<sup>m</sup>* -generic over *M*. Moreover, *p* being incompatible with *D* is implied by the fact that *f* meets a certain family of dense sets in **SEQ***<sup>m</sup>* , of cardinality <sup>≤</sup> *<sup>ω</sup>*<sup>1</sup> in *<sup>M</sup>*. Therefore, we will be able to simulate this in *<sup>M</sup>*, getting a sequence *g* ∈ *M* which meets the same dense sets, and hence yields a condition *p* ∈ **Q**[*U*], also incompatible with each *q* ∈ *D*.

To present the key idea in sufficient detail in a rather simplified subcase, we assume that

$$|p| = \{\nu\_0\} \text{ is a singleton}; \nu\_0 \in |\mathcal{U}|.\tag{7}$$

Then *p*(*ν*0) = *Sp*(*ν*0); *Fp*(*ν*0) ∈ *Q*[*U* (*ν*0)], where *Sp*(*ν*0) ⊆ **SEQ** and *Fp*(*ν*0) ⊆ *U* (*ν*0) are finite sets. The (finite) set *X* = *Fp*(*ν*0) *U*(*ν*0) is multiply **SEQ**-generic over *M* since *U <sup>M</sup> U* . To make the argument even more transparent, we suppose that

$$X = \{f, \emptyset\}, \text{ where } f \neq \emptyset \text{ and the pair } \langle f, \emptyset \rangle \text{ is } \mathsf{SEQ}^2\text{-generic over } M. \tag{8}$$

(The general case follows the same idea and can be found in [4]; we leave it to the reader.)

Thus, *Fp*(*ν*0) = *F* ∪ { *f* , *g*}, where *F* = *Fp*(*ν*0) ∩ *U*(*ν*0) ∈ *M* is by definition a finite set.

The plan is to replace the functions *f* , *g* by some functions *f* , *g* ∈ *U*(*ν*0) so that the incompatibility of *p* with conditions in *D* will be preserved.

It holds by the choice of *p* and Lemma 1 that *D* = *D*1(*f* , *g*) ∪ *D*<sup>2</sup> , where

$$\begin{aligned} D\_1(f,\emptyset) &= \{ q \in D : A\_q \cap F\_p^\vee(\mathbb{v}\_0) \neq \mathcal{Q} \}, \text{ where } A\_q = S\_q(\mathbb{v}\_0) \; \sim S\_p(\mathbb{v}\_0) \subseteq \mathbf{Set} \; ; \\ D\_2 &= \{ q \in D : (\mathcal{S}\_p(\mathbb{v}\_0) \, \sim S\_q(\mathbb{v})) \cap F\_q^\vee(\mathbb{v}\_0) \neq \mathcal{Q} \} \in M; \end{aligned}$$

and *D*<sup>1</sup> depends on *f* , *g* via *Fp*(*ν*0). The equality *D* = *D*1(*f* , *g*) ∪ *D*<sup>2</sup> can be rewritten as Δ ⊆ *D*1(*f* , *g*), where Δ = *D D*<sup>2</sup> ∈ *M*. Furthermore, Δ ⊆ *D*1(*f* , *g*) is equivalent to

$$\forall A \in \omega' \left( A \cap F\_p^\vee(\nu) \neq \mathcal{Q} \right), \text{ where } \omega' = \left\{ A\_{\emptyset} : q \in D \right\} \in M,\tag{9}$$

and each *Aq* = *Sq*(*ν*0) *Sp*(*ν*0) ⊆ **SEQ** is finite. Recall that *Fp*(*ν*0) = *F* ∪ { *f* , *g*}, therefore *F*∨ *<sup>p</sup>* (*ν*0) = *Z* ∪ *S*(*f* , *g*), where *Z* = {*h μ* : 1 ≤ *μ* < *ω*<sup>1</sup> ∧ *h* ∈ *F*} ∈ *M* and *S*(*f* , *g*) = - 1≤*μ*<*ω*<sup>1</sup> { *f μ*, *g μ*}. Thus, (9) is equivalent to

$$\forall A' \in \omega'^\prime \left( A' \cap \mathcal{S}(f, \emptyset) \neq \mathcal{Q} \right), \text{ where } \omega'^\prime = \{ A\_\emptyset \, : \, \mathcal{Q} \in D \} \in M. \tag{10}$$

Please note that each *A* ∈ A is a finite subset of **SEQ** , so we can re-enumerate A = {*A <sup>κ</sup>* : *κ* < *ω*1} in *M* and rewrite (10) as follows:

$$\forall \mathbf{x} < \omega\_1 (A'\_{\mathbf{x}} \cap \mathbf{S}(f, \mathbf{g}) \neq \mathcal{Q}), \text{ where each } A'\_{\mathbf{x}} \subseteq \mathbf{S} \mathbf{z} \mathbf{Q} \text{ is finite.} \tag{11}$$

As the pair *f* , *g* is **SEQ** -generic, there is an index *μ*<sup>0</sup> < *ω*<sup>1</sup> such that (11) is forced over *M* by *σ*0, *τ*<sup>0</sup>, where *σ*<sup>0</sup> = *f μ*<sup>0</sup> and *τ*<sup>0</sup> = *g μ*<sup>0</sup> . In other words, *A <sup>κ</sup>* ∩ *S*(*f* , *g* ) = ∅ holds for all *κ* < *ω*<sup>1</sup> whenever *f* , *g*  is **SEQ**-generic over *M* and *σ*<sup>0</sup> ⊂ *f* , *τ*<sup>0</sup> ⊂ *g* . It follows that for any *κ* < *ω*<sup>1</sup> and sequences *σ*, *τ* ∈ **SEQ** extending resp. *σ*0, *τ*<sup>0</sup> there are sequences *σ* , *τ* ∈ **SEQ** extending resp. *σ*, *τ*, at least one of which extends one of sequences *w* ∈ *A <sup>κ</sup>* . This allows us to define, in *M*, a pair of sequences *f* , *g* ∈ **FUN**, such that *σ*<sup>0</sup> ⊂ *f* , *τ*<sup>0</sup> ⊂ *g* , and for any *κ* < *ω*<sup>1</sup> at least one of *f* , *g* extends one of *w* ∈ *A <sup>k</sup>* . In other words, we have

$$
\forall \mathfrak{x} < \omega\_1 (A'\_{\mathfrak{x}} \cap \mathcal{S}(f', \mathfrak{g'}) \neq \mathcal{Q}) \quad \text{and} \quad \forall A' \in \mathfrak{x}' (A' \cap \mathcal{S}(f', \mathfrak{g'}) \neq \mathcal{Q}).
$$

It follows that the condition *p* defined by |*p* | = {*ν*<sup>0</sup> }, *Sp* (*ν*0) = *Sp*(*ν*0), *Fp* (*ν*0) = *F* ∪ { *f* , *g* }, still satisfies ∀ *A* ∈ A (*A* ∩ *F*<sup>∨</sup> *<sup>p</sup>* (*ν*0) = ∅) (compare with (9)), and further *D* = *D*1(*f* , *g* ) ∪ *D*<sup>2</sup> , thus *p* is incompatible with each *q* ∈ *D*. Yet *p* ∈ *M* since *f* , *g* ∈ *M*, which contradicts the pre-density of *D*.

(ii) The above proof works with *M*[*G*] instead of *M* since the set *X* as in the proof is multiple **SEQ**-generic over *M*[*G*] by the product forcing theorem.

(iii) Assuming w.l. o. g. that *H* ⊆ *U* (*ν*1) *U*(*ν*1), we conclude that *M*[*H*] is a **SEQ** -generic extension of *M*. Now, if *p* ∈ **Q**[*U* ] =*ν*<sup>1</sup> , then, following the above argument, let *<sup>ν</sup>*<sup>0</sup> ∈ |*p*|, *<sup>ν</sup>*<sup>0</sup> = *<sup>ν</sup>*<sup>1</sup> . By the definition of the set *F* = *Fp*(*ν*0) *U*(*ν*0) is multiply **SEQ**-generic not only over *M* but also over *M*[*H*]. This allows the carrying out of the same argument as above.

**Corollary 2.** *Under the assumptions of Theorem 2, if a set G* ⊆ **Q**[*U* ] *is* **Q**[*U* ]*-generic over a transitive model M* |= **ZFC**<sup>−</sup> <sup>2</sup> *containing M and U* (*including the case M* = **L**)*, then the intersection G* ∩ **Q**[*U*] *is* **Q**[*U*]*-generic over M.*

**Proof.** If a set *D* ∈ *M*, *D* ⊆ **Q**[*U*], is pre-dense in **Q**[*U*], then it is pre-dense in **Q**[*U* ] by Theorem 2, and hence *G* ∩ *D* = ∅ by the genericity.

**Corollary 3** (in **<sup>L</sup>**)**.** *Under the assumptions of Theorem 2, if <sup>τ</sup>* <sup>∈</sup> *<sup>M</sup>* <sup>∩</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**[*U*]) *is a* **Q**[*U*]*-full name then τ remains* **Q**[*U* ]*-full, and if p* ∈ **Q**[*U*] *and τ is* **Q**[*U*]*-full below p, then τ remains* **Q**[*U* ]*-full below p.*

#### *4.3. Complete Sequences and the Basic Forcing Notion*

In **L**, we say that a pair *M*, *U* ∈ **sJS** *solves* a set *D* ⊆ **sJS** iff either *M*, *U* ∈ *D* or there is no pair *M* , *U* <sup>∈</sup> *<sup>D</sup>* that extends *<sup>M</sup>*, *<sup>U</sup>*. Let *<sup>D</sup>*solv be the set of all pairs *<sup>M</sup>*, *<sup>U</sup>*<sup>∈</sup> **sJS** which solve a given set *<sup>D</sup>* <sup>⊆</sup> **sJS**. A sequence {*M<sup>ξ</sup>* , *<sup>U</sup><sup>ξ</sup>* }*ξ*<*ω*<sup>2</sup> <sup>∈</sup> −→**JS***ω*<sup>2</sup> is called *<sup>n</sup>*-*complete* (*<sup>n</sup>* <sup>≥</sup> 3 ) iff it intersects every set of the form *<sup>D</sup>*solv , where *<sup>D</sup>* <sup>⊆</sup> **sJS** is a *<sup>Σ</sup>*H*ω*<sup>2</sup> *<sup>n</sup>*−2(H*ω*2) set.

Recall that H*ω*<sup>2</sup> is the collection of all sets *x* whose transitive closure TC(*x*) has cardinality card (TC(*x*)) < *<sup>ω</sup>*2. Furthermore, *<sup>Σ</sup>*H*ω*<sup>2</sup> *<sup>n</sup>*−2(H*ω*2) means definability by a *<sup>Σ</sup>n*−<sup>2</sup> formula of the <sup>∈</sup> language, in which any definability parameters in <sup>H</sup>*ω*<sup>2</sup> are allowed, while *<sup>Σ</sup>*H*ω*<sup>2</sup> *<sup>n</sup>*−<sup>2</sup> means parameter-free definability. Similarly, *Δ*H*ω*<sup>2</sup> *n*−1 ({*ω*1}) in the next theorem means that *ω*<sup>1</sup> is allowed as a sole parameter. It is a simple exercise that sets {**SEQ**} and **SEQ** are *<sup>Δ</sup>*H*ω*<sup>2</sup> <sup>1</sup> ({*ω*1}) under **V** = **L**.

Generally, we refer to e.g., ([9] (Part B, 5.4)), or ([10] (Chapter 13)) on the Lévy hierarchy of <sup>∈</sup>-formulas and definability classes *<sup>Σ</sup><sup>H</sup> <sup>n</sup>* , *Π<sup>H</sup> <sup>n</sup>* , *Δ<sup>H</sup> <sup>n</sup>* for any transitive set *H*.

**Theorem 3** (in **<sup>L</sup>**)**.** *Let <sup>n</sup>* <sup>≥</sup> <sup>2</sup>*. There is a sequence* {*M<sup>ξ</sup>* , *<sup>U</sup><sup>ξ</sup>* }*ξ*<*ω*<sup>2</sup> <sup>∈</sup> −→**JS***ω*<sup>2</sup> *of class <sup>Δ</sup>*H*ω*<sup>2</sup> *n*−1 ({*ω*1})*, hence, Δ*H*ω*<sup>2</sup> *<sup>n</sup>*−<sup>1</sup> *in case n* <sup>≥</sup> <sup>3</sup>*, n-complete in case n* <sup>≥</sup> <sup>3</sup>*, and such that <sup>ξ</sup>* ∈ |*Uξ*+1<sup>|</sup> *for all <sup>ξ</sup>* <sup>&</sup>lt; *<sup>ω</sup>*2*.*

**Proof.** To account for*ω*<sup>1</sup> as a parameter, note that the set *<sup>ω</sup>*<sup>1</sup> is *<sup>Σ</sup>*H*ω*<sup>2</sup> <sup>1</sup> , and hence the singleton {*ω*1} is *Δ*H*ω*<sup>2</sup> <sup>2</sup> . Indeed "being *<sup>ω</sup>*1" is equivalent to the conjunction of "being uncountable"—which is *<sup>Π</sup>*H*ω*<sup>2</sup> <sup>1</sup> , and "every smaller ordinal is countable"—which is *Σ*H*ω*<sup>2</sup> <sup>1</sup> since the quantifier "for all smaller ordinals" is bounded, hence, it does not increase the complexity.

It follows that *Δ*H*ω*<sup>2</sup> *n*−1 ({*ω*1}) = *<sup>Δ</sup>*H*ω*<sup>2</sup> *<sup>n</sup>*−<sup>1</sup> in case *<sup>n</sup>* <sup>≥</sup> 3, supporting the "hence" claim of the theorem.

Then, it can be verified that the sets *Q*<sup>∗</sup> , **Q**<sup>∗</sup> , **sJS** are *Δ*H*ω*<sup>2</sup> <sup>1</sup> ({*ω*1}). (Indeed "being finite" and "being countable" are *Δ*H*ω*<sup>2</sup> <sup>1</sup> relations, while "being of cardinality *<sup>ω</sup>*1" is *<sup>Δ</sup>*H*ω*<sup>2</sup> <sup>1</sup> ({*ω*1}); the *Π*<sup>1</sup> definition says that there is no injection from *ω*<sup>1</sup> into a given set.)

Define pairs *M<sup>ξ</sup>* , *U<sup>ξ</sup>* , *ξ* < *ω*2, by induction. Let *U*<sup>0</sup> be the null system with |*U*0| = ∅, and *M*<sup>0</sup> be the least CTM of **ZFC**− <sup>2</sup> . If *<sup>λ</sup>* < *<sup>ω</sup>*<sup>1</sup> is a limit, then put *<sup>U</sup><sup>λ</sup>* = *<sup>ξ</sup>*<*<sup>λ</sup> U<sup>ξ</sup>* and let *M<sup>λ</sup>* be the least CTM of **ZFC**− <sup>2</sup> containing the sequence {*M<sup>ξ</sup>* , *U<sup>ξ</sup>* }*ξ*<*<sup>λ</sup>* . If *M<sup>ξ</sup>* , *U<sup>ξ</sup>*  ∈ **sJS** is defined, then by Lemma 18 there is a pair *M* , *U*  ∈ **sJS** with *M<sup>ξ</sup>* , *U<sup>ξ</sup>* ≺*M* , *U*  and *ξ* ∈ |*U* |. Further let Θ ⊆ *ω*1× H*ω*<sup>2</sup> be a universal *Σ*H*ω*<sup>2</sup> *<sup>n</sup>*−<sup>2</sup> set, and if *<sup>ξ</sup>* <sup>&</sup>lt; *<sup>ω</sup>*<sup>2</sup> then *<sup>D</sup><sup>ξ</sup>* <sup>=</sup> {*<sup>z</sup>* <sup>∈</sup> **sJS** : *<sup>ξ</sup>*, *<sup>z</sup>*<sup>∈</sup> <sup>Θ</sup>}. Let *<sup>M</sup>ξ*<sup>+</sup>1, *<sup>U</sup>ξ*+<sup>1</sup> be the <sup>&</sup>lt;**L**least pair *M*, *U* ∈ *D<sup>ξ</sup>* solv satisfying *<sup>M</sup>* , *U M*, *U*, where <**<sup>L</sup>** is the Gödel wellordering of **L**, the constructible universe. This completes the inductive construction of *M<sup>ξ</sup>* , *U<sup>ξ</sup>*  ∈ **sJS**, *ξ* < *ω*2.

To check the definability property, make use of the well-known fact that the restriction <**L** H*ω*<sup>2</sup> is a *Δ*H*ω*<sup>2</sup> <sup>1</sup> relation, and if *<sup>n</sup>* <sup>≥</sup> 1, *<sup>p</sup>* <sup>∈</sup> *<sup>ω</sup><sup>ω</sup>* is any parameter, and *<sup>R</sup>*(*x*, *<sup>y</sup>*, *<sup>z</sup>*, ...) is a finitary *<sup>Δ</sup>*H*ω*<sup>2</sup> *<sup>n</sup>* (*p*) relation on HC then the relations ∃ *x* <**<sup>L</sup>** *y R*(*x*, *y*, *z*, ...) and ∀ *x* <**<sup>L</sup>** *y R*(*x*, *y*, *z*, ...) (with arguments *<sup>y</sup>*, *<sup>z</sup>*, . . . ) are *<sup>Δ</sup>*H*ω*<sup>2</sup> *<sup>n</sup>* (*p*) as well.

## **Definition 2** (in **L**)**.** *Fix a number* **n** ≥ 2 *during the proof of Theorem 1.*


*We define* **Q** = **Q**[**U**] *(the basic forcing notion), and* **Q***<sup>ξ</sup>* = **Q**[**U***<sup>ξ</sup>* ] *for ξ* < *ω*2*. Thus,* **Q** *is the finite-support product of the set* **C** *and sets* **Q**(*ν*) = *Q*[**U**(*ν*)]*, i* ∈ I *; so that* **Q** ∈ **L***.*

**Corollary 4.** *Suppose that in* **L***, ξ* < *ω*<sup>2</sup> *and M is a TM of* **ZFC**<sup>−</sup> <sup>2</sup> *containing the sequence* **js***. Then*

(i) *<sup>M</sup>*, **<sup>U</sup>**<sup>∈</sup> **JS***,* **<sup>M</sup>***<sup>ξ</sup>* , **<sup>U</sup>***<sup>ξ</sup>* <sup>≺</sup>*M*, **<sup>U</sup>** *, and if <sup>ν</sup>* ∈ I *then* card(**U***<sup>ξ</sup>* (*ν*)) = *<sup>ω</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>ω</sup>*<sup>2</sup> <sup>=</sup> card(**U**(*ν*)) *in* **<sup>L</sup>***.*

(ii) *If G* <sup>⊆</sup> **<sup>Q</sup>** *is a set* **<sup>Q</sup>***-generic over* **<sup>L</sup>** *then the set G<sup>ξ</sup>* <sup>=</sup> *<sup>G</sup>* <sup>∩</sup> **<sup>Q</sup>***<sup>ξ</sup> is* **<sup>Q</sup>***<sup>ξ</sup> -generic over* **<sup>M</sup>***<sup>ξ</sup> .*

**Proof.** Make use of Lemma 20 and Corollary 2 in Section 4.2.

**Lemma 21** (in **<sup>L</sup>**)**.** *The binary relation <sup>f</sup>* <sup>∈</sup> **<sup>U</sup>**(*ν*)*, the sets* **<sup>Q</sup>** *and* **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**) (**Q**-names for reals in *ω<sup>ω</sup>* )*, and the set of all* **Q***-full names in* **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**) *are <sup>Δ</sup>*H*ω*<sup>2</sup> **n**−1 ({*ω*1})*, and even <sup>Δ</sup>*H*ω*<sup>2</sup> **<sup>n</sup>**−<sup>1</sup> *in case* **<sup>n</sup>** <sup>≥</sup> <sup>3</sup>*.*

**Proof.** The sequence {**M***<sup>ξ</sup>* , **<sup>U</sup>***<sup>ξ</sup>* }*ξ*<*ω*<sup>1</sup> is *<sup>Δ</sup>*H*ω*<sup>2</sup> **<sup>n</sup>**−<sup>1</sup> by definition, hence the relation *<sup>f</sup>* <sup>∈</sup> **<sup>U</sup>**(*ν*) is *<sup>Σ</sup>*H*ω*<sup>2</sup> **<sup>n</sup>**−<sup>1</sup> . On the other hand, if *f* ∈ **Fun** belongs to some **M***<sup>ξ</sup>* then *f* ∈ **U**(*ν*) obviously implies *f* ∈ **U***<sup>ξ</sup>* (*ν*), leading to a *Π*HC **<sup>n</sup>**−<sup>1</sup> definition of the relation *<sup>f</sup>* <sup>∈</sup> **<sup>U</sup>**(*ν*). To prove the last claim, note that by Corollary <sup>3</sup> if a name *<sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**P***<sup>ξ</sup>* ) ∩ **M***<sup>ξ</sup>* is **P***<sup>ξ</sup>* -full then it remains **P**-full.

#### *4.4. Basic Generic Extension*

The proof of Theorem 1 makes use of a generic extension of the form **L**[*G z*], where *G* ⊆ **Q** is a set **<sup>Q</sup>**-generic over **<sup>L</sup>**, and *<sup>z</sup>* ⊆ I<sup>+</sup> , *<sup>z</sup>* <sup>∈</sup>/ **<sup>L</sup>**. The following two theorems will play the key role in the proof. Define formulas **Γ***<sup>ν</sup>* (*ν* ∈ I ) as follows:

$$\mathbb{T}\_{\nu}(\mathcal{S}) \ := \text{def } \mathcal{S} \subseteq \mathbf{S} \mathtt{Eq}^{\mathcal{L}} \land \forall f \in \mathbf{FUN}^{\mathcal{L}} \left( f \in \mathbb{U}(\nu) \iff \mathcal{S} \text{ does not cover } f \right).$$

**Lemma 22.** *Suppose that a set G* <sup>⊆</sup> **<sup>Q</sup>** *is* **<sup>Q</sup>***-generic over* **<sup>L</sup>***, and <sup>ν</sup>* ∈ I *, c* <sup>∈</sup> **<sup>L</sup>**[*G*]*,* <sup>∅</sup> <sup>=</sup> *<sup>c</sup>* ⊆ I<sup>+</sup> *. Then*


**Proof.** To prove (i) apply Lemma 9 (ii); (ii) is easy. Furthermore, Lemma 9 (vi) immediately implies (iii).

To prove (iv), we need more work. Let *X* = **SEQL**. Suppose towards the contrary that some *<sup>S</sup>* <sup>∈</sup> **<sup>L</sup>**[*G* =*ν*], *<sup>S</sup>* <sup>⊆</sup> *<sup>X</sup>* <sup>=</sup> **SEQ<sup>L</sup>** satisfies **<sup>Γ</sup>***ν*(*S*). It follows from Lemma <sup>10</sup> (with *<sup>U</sup>* <sup>=</sup> **<sup>U</sup>** and *<sup>c</sup>* <sup>=</sup> <sup>I</sup><sup>+</sup> {*ν*}), that there is a name *<sup>τ</sup>* <sup>∈</sup> **SN***X*(**Q**) =*<sup>ν</sup>* in **<sup>L</sup>** such that *<sup>S</sup>* <sup>=</sup> *<sup>τ</sup>*[*G* =*ν*]. There is an ordinal *<sup>ξ</sup>* <sup>&</sup>lt; *<sup>ω</sup>*<sup>1</sup> satisfying *<sup>τ</sup>* <sup>∈</sup> **<sup>M</sup>***<sup>ξ</sup>* and *<sup>τ</sup>* <sup>∈</sup> **SN***X*(**Q***<sup>ξ</sup>* =*ν*). Then *<sup>S</sup>* <sup>=</sup> *<sup>τ</sup>*[*G<sup>ξ</sup>* =*ν*], where *<sup>G</sup><sup>ξ</sup>* <sup>=</sup> *<sup>G</sup>* <sup>∩</sup> **<sup>P</sup>***<sup>ξ</sup>* is **<sup>P</sup>***<sup>ξ</sup>* generic over **<sup>M</sup>***<sup>ξ</sup>* by Corollary 4 (ii), and by the way *<sup>S</sup>* belongs to **<sup>M</sup>***<sup>ξ</sup>* [*G<sup>ξ</sup>* =*ν*] by the choice of *<sup>ξ</sup>* .

Please note that *F* = **U**(*ν*) **U***<sup>ξ</sup>* (*ν*) = ∅ by Corollary 4 (i). Let *f* ∈ *F*. Then *f* is Cohen generic over the model **<sup>M</sup>***<sup>ξ</sup>* by Corollary 4. On the other hand, *<sup>G</sup><sup>ξ</sup>* =*<sup>ν</sup>* is **<sup>P</sup>***<sup>ξ</sup>* =*ν*-generic over **<sup>M</sup>***<sup>ξ</sup>* [ *<sup>f</sup>* ] by Theorem 2 (iii). Therefore *<sup>f</sup>* is Cohen generic over **<sup>M</sup>***<sup>ξ</sup>* [*G<sup>ξ</sup>* =*ν*] as well.

Recall that *<sup>S</sup>* <sup>∈</sup> **<sup>M</sup>***<sup>ξ</sup>* [*G<sup>ξ</sup>* =*ν*] and **<sup>Γ</sup>***ν*(*S*) holds, hence *<sup>S</sup>* does not cover *<sup>f</sup>* . As *<sup>f</sup>* is Cohen generic over **<sup>M</sup>***<sup>ξ</sup>* [*G<sup>ξ</sup>* =*ν*], it follows that there is a sequence *<sup>s</sup>* <sup>∈</sup> **SEQL**, *<sup>s</sup>* <sup>⊂</sup> *<sup>f</sup>* , such that *<sup>S</sup>* contains no subsequences of *f* extending *s*. Take any *μ* ∈ I , *μ* = *ν*. By Corollary 4 (i), there exists a function *g* ∈ **U**(*μ*) **U***<sup>ξ</sup>* (*μ*), *g* ∈/ **U**(*ν*), satisfying *s* ⊂ *g*. Then, *S* covers *g* by **Γ***ν*(*S*). However, this is absurd by the choice of *s*.

The proof of the next important *elementary equivalence theorem* will be given below in Section 6.3.

**Theorem 4** (elementary equivalence theorem)**.** *Assume that in* **<sup>L</sup>***,* <sup>−</sup><sup>1</sup> <sup>∈</sup> *<sup>d</sup>* ⊆ I<sup>+</sup> *, sets <sup>Z</sup>* , *Z* ⊆ I *d satisfy* card (<sup>I</sup> *<sup>Z</sup>*) <sup>≤</sup> *<sup>ω</sup>*<sup>1</sup> *and* card (<sup>I</sup> *<sup>Z</sup>* ) ≤ *ω*1*, the symmetric difference Z* Δ *Z is at most countable, and the complementary set* I (*d* ∪ *Z* ∪ *Z* ) *is infinite.*

*Let <sup>G</sup>* <sup>⊆</sup> **<sup>Q</sup>** *be* **<sup>Q</sup>***-generic over* **<sup>L</sup>***, and <sup>x</sup>*<sup>0</sup> <sup>∈</sup> **<sup>L</sup>**[*G <sup>d</sup>*] *be any real. Then any closed <sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>** *formula ϕ, with real parameters in* **L**[*x*0]*, is simultaneously true in* **L**[*x*0, *G Z*] *and in* **L**[*x*0, *G Z* ]*.*

#### *4.5. The Main Theorem Modulo the Elementary Equivalence Theorem: The Model*

Here we begin **the proof of Theorem 1 on the base of Theorem 4 of Section 4.4**. We fix a number **n** ≥ 2 during the proof. The goal is to define a generic extension of **L** in which for any set *x* ⊆ *ω* the following is true: *<sup>x</sup>* <sup>∈</sup> **<sup>L</sup>** iff *<sup>x</sup>* <sup>∈</sup> *<sup>Δ</sup>*<sup>1</sup> **<sup>n</sup>**+1. The model is a part of the basic generic extension defined in Section 4.4.

In the notation of Definition 2 in Section 4.3, consider a set *G* ⊆ **Q**, **Q**-generic over **L**. Then *b<sup>G</sup>* = -*G*(−1) is a **C**-generic map from *ω* onto P(*ω*) ∩ **L** by Lemma 9 (i). We define

$$w[G] = \{\omega k + 2^{\circ} : k < \omega \land j \in \mathfrak{b}\_G(k)\} \cup \{\omega k + 3^{\circ} : j, k < \omega\} \subseteq \omega^2,\tag{12}$$

and *<sup>w</sup>*+[*G*] = {−1} ∪ *<sup>w</sup>*[*G*]. We also define, for any *<sup>m</sup>* <sup>&</sup>lt; *<sup>ω</sup>*,

$$w\_{\geq m}[G] = \left\{ \omega k + \ell \in w[G] : k \geq m \right\}, \quad w\_{$$

and accordingly *w*<sup>+</sup> <sup>≥</sup>*m*[*G*] = {−1} ∪ *<sup>w</sup>*≥*m*[*G*] and *<sup>w</sup>*<sup>+</sup> <sup>&</sup>lt;*m*[*G*] = {−1} ∪ *w*<*m*[*G*].

With these definitions, each *k*th slice

$$w\_k[\mathcal{G}] = \{\omega k + 2^j : j \in \mathfrak{b}\_{\mathcal{G}}(k)\} \cup \{\omega k + 3^j : j < \omega\} \tag{13}$$

of *w*[*G*] is necessarily infinite and coinfinite, and it codes the target set *bG*(*k*) since

$$\mathbf{b}\_{\mathbf{G}}(k) = \{ j < \omega : \omega k + 2^j \in w\_k[\mathbf{G}] \} = \{ j < \omega : \omega k + 2^j \in w^+[\mathbf{G}] \}. \tag{14}$$

It will be important below that definition (12) is *monotone w. r. t. b<sup>G</sup>* , i.e., if *bG*(*k*) ⊆ *b<sup>G</sup>* (*k*) for all *k*, then *w*[*G*] ⊆ *w*[*G* ] and *<sup>w</sup>*+[*G*] <sup>⊆</sup> *<sup>w</sup>*+[*<sup>G</sup>* ]. Non-monotone modifications, like e.g.,

$$w[G] = \{\omega k + 2^j : j \in \mathfrak{b}\_G(k)\} \cup \{\omega k + 3^j : j \notin \mathfrak{b}\_G(k)\}.$$

would not work. Finally, let

$$\mathcal{W} = [\omega^2, \omega\_2) = \{\check{\zeta} : \omega^2 \le \check{\zeta} < \omega\_2\} \dots$$

Anyway, *<sup>w</sup>*+[*G*] <sup>⊆</sup> *<sup>ω</sup>*<sup>2</sup> <sup>=</sup> *<sup>ω</sup>* · *<sup>ω</sup>* (the ordinal product) is a set in the model **<sup>L</sup>**[*bG*] = **<sup>L</sup>**[*w*+[*G*]] = **<sup>L</sup>**[*w*[*G*]] = **<sup>L</sup>**[*w*≥*m*[*G*]] for each *<sup>m</sup>*, containing <sup>−</sup>1, while *<sup>w</sup>*<*m*[*G*] <sup>∈</sup> **<sup>L</sup>** for all *m*. We are going to prove the following lemma:

**Lemma 23.** *The model* **<sup>L</sup>**[*G* (*w*+[*G*] <sup>∪</sup> *<sup>W</sup>*)] *witnesses Theorem 1. That is, let a set <sup>G</sup>* <sup>⊆</sup> **<sup>Q</sup>** *be* **<sup>Q</sup>***-generic over* **<sup>L</sup>***. Then it holds in* **<sup>L</sup>**[*G* (*w*+[*G*] <sup>∪</sup> *<sup>W</sup>*)] *that*


Recall that if *<sup>Z</sup>* ⊆ I<sup>+</sup> then *<sup>G</sup> <sup>Z</sup>* <sup>=</sup> {*<sup>p</sup>* <sup>∈</sup> *<sup>G</sup>* : <sup>|</sup>*p*<sup>|</sup> <sup>+</sup> <sup>⊆</sup> *<sup>Z</sup>*}.

**Proof** (Claim (i) of the lemma). Consider an arbitrary ordinal *ν* = *ωk* + ; *k*, < *ω*. We claim that

$$\nu \in \mathcal{w}[\mathcal{G}] \iff \exists \, \mathcal{S} \, \mathbb{F}\_{\nu}(\mathcal{S}) \tag{15}$$

holds in **<sup>L</sup>**[*G* (*w*+[*G*] <sup>∪</sup> *<sup>W</sup>*)]. Indeed, assume that *<sup>ν</sup>* <sup>∈</sup> *<sup>w</sup>*[*G*]. Then *<sup>S</sup>* <sup>=</sup> *SG*(*ν*) <sup>∈</sup> **<sup>L</sup>**[*G <sup>w</sup>*+[*G*]], and we have **<sup>Γ</sup>***ν*(*S*) in **<sup>L</sup>**[*G* (*w*+[*G*] <sup>∪</sup> *<sup>W</sup>*)] by Lemma 22 (ii), (iii). Conversely assume that *<sup>ν</sup>* <sup>∈</sup>/ *<sup>w</sup>*[*G*]. Then we have *<sup>w</sup>*+[*G*] <sup>∈</sup> **<sup>L</sup>**[*bG*] <sup>⊆</sup> **<sup>L</sup>**[*G <sup>w</sup>*+[*G*]] <sup>⊆</sup> **<sup>L</sup>**[*G* =*ν*], but **<sup>L</sup>**[*G* =*ν*] contains no *<sup>S</sup>* with **<sup>Γ</sup>***ν*(*S*) by Lemma 22 (iv).

However, the right-hand side of (15) defines a *<sup>Σ</sup>*H*ω*<sup>2</sup> **<sup>n</sup>** ({*ω*<sup>1</sup> **<sup>L</sup>**, **SEQL**}) relation in **<sup>L</sup>**[*G* (*w*+[*G*] <sup>∪</sup> *<sup>W</sup>*)] by Lemma 21. (Indeed, (H*ω*2)**<sup>L</sup>** <sup>=</sup> **<sup>L</sup>***ω*<sup>2</sup> **<sup>L</sup>** <sup>=</sup> **<sup>L</sup>***ω*<sup>1</sup> in **<sup>L</sup>**[*G* (*w*+[*G*] <sup>∪</sup> *<sup>W</sup>*)], therefore (H*ω*2)**<sup>L</sup>** is *<sup>Σ</sup>*H*ω*<sup>2</sup> 1 in **<sup>L</sup>**[*G* (*w*+[*G*] <sup>∪</sup> *<sup>W</sup>*)].) On the other hand, the sets {*ω*<sup>1</sup> **<sup>L</sup>**} and {**SEQL**} remain *<sup>Δ</sup>*H*ω*<sup>2</sup> <sup>2</sup> singletons in **<sup>L</sup>**[*G* (*w*+[*G*] <sup>∪</sup> *<sup>W</sup>*)], so they can be eliminated since **<sup>n</sup>** <sup>≥</sup> 2. This yields *<sup>w</sup>*[*G*] <sup>∈</sup> *<sup>Σ</sup>*HC **<sup>n</sup>** in **<sup>L</sup>**[*G* (*w*+[*G*] <sup>∪</sup> *<sup>W</sup>*)]. It follows that *<sup>w</sup>*[*G*] <sup>∈</sup> *<sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> by ([10] (Lemma 25.25)), as required.

Consider an arbitrary set *x* ∈ **L**, *x* ⊆ *ω*. By genericity there exists *k* < *ω* such that *bG*(*k*) = *x*. Then *<sup>x</sup>* <sup>=</sup> {*<sup>j</sup>* : *<sup>ω</sup><sup>k</sup>* <sup>+</sup> <sup>2</sup>*<sup>j</sup>* <sup>∈</sup> *<sup>w</sup>*[*G*]} by (12), therefore *<sup>x</sup>* is *<sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> as well. However, *<sup>ω</sup> <sup>x</sup>* <sup>∈</sup> *<sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> by the same argument. Thus, *x* is *Δ*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> in **<sup>L</sup>**[*G* (*w*+[*G*] <sup>∪</sup> *<sup>W</sup>*)], as required. (Claim (i) of Lemma 23)

#### *4.6. Proof of the Key Claim of Lemma 23*

The **proof of Lemma 23** (ii) is based on several intermediate lemmas. Recall that *<sup>W</sup>* = [*ω*2, *<sup>ω</sup>*2) = {*<sup>ξ</sup>* : *<sup>ω</sup>*<sup>2</sup> <sup>≤</sup> *<sup>ξ</sup>* <sup>&</sup>lt; *<sup>ω</sup>*2}.

**Lemma 24** (compare with Lemma 33 in [7])**.** *Suppose that G* ⊆ **Q** *is* **Q***-generic over* **L***, and m* < *ω. Let c* <sup>⊆</sup> *<sup>w</sup>*<*m*[*G*] *be any set in* **<sup>L</sup>***. Then any closed <sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>** *formula* <sup>Φ</sup>*, with reals in* **<sup>L</sup>**[*G* (*<sup>c</sup>* <sup>∪</sup> *<sup>w</sup>*<sup>+</sup> <sup>≥</sup>*m*[*G*] <sup>∪</sup> *<sup>W</sup>*)] *as parameters, is simultaneously true in* **<sup>L</sup>**[*G* (*<sup>c</sup>* <sup>∪</sup> *<sup>w</sup>*<sup>+</sup> <sup>≥</sup>*m*[*G*] <sup>∪</sup> *<sup>W</sup>*)] *and in* **<sup>L</sup>**[*G* (*w*+[*G*] <sup>∪</sup> *<sup>W</sup>*)]*.*

*It follows that if <sup>c</sup>* <sup>⊆</sup> *<sup>c</sup>* <sup>⊆</sup> *<sup>w</sup>*<*m*[*G*] *in* **<sup>L</sup>***, then any closed <sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> *formula* Ψ*, with parameters in* **<sup>L</sup>**[*G* (*<sup>c</sup>* <sup>∪</sup> *<sup>w</sup>*<sup>+</sup> <sup>≥</sup>*m*[*G*] <sup>∪</sup> *<sup>W</sup>*)]*, true in* **<sup>L</sup>**[*G* (*<sup>c</sup>* <sup>∪</sup> *<sup>w</sup>*<sup>+</sup> <sup>≥</sup>*m*[*G*] <sup>∪</sup> *<sup>W</sup>*)]*, is true in* **<sup>L</sup>**[*G* (*<sup>c</sup>* <sup>∪</sup> *<sup>w</sup>*<sup>+</sup> <sup>≥</sup>*m*[*G*] <sup>∪</sup> *<sup>W</sup>*)] *as well.*

**Proof** (Lemma 24). There is an ordinal *ξ* < *ω*<sup>2</sup> such that all parameters in *ϕ* belong to **L**[*GY*], where *<sup>Y</sup>* <sup>=</sup> *<sup>c</sup>* <sup>∪</sup> *<sup>w</sup>*<sup>+</sup> <sup>≥</sup>*m*[*G*] <sup>∪</sup> *<sup>X</sup>* and *<sup>X</sup>* = [*ω*2, *<sup>ξ</sup>*) = {*<sup>γ</sup>* : *<sup>ω</sup>*<sup>2</sup> <sup>≤</sup> *<sup>γ</sup>* <sup>&</sup>lt; *<sup>ξ</sup>*}. The set *<sup>Y</sup>* belongs to **<sup>L</sup>**[*bG*], in fact, **<sup>L</sup>**[*Y*] = **<sup>L</sup>**[*bG*]. Therefore *<sup>G</sup><sup>Y</sup>* is equi-constructible with the pair *<sup>b</sup>G*, {*SG*(*ν*)}*ν*∈*<sup>X</sup>* , where *<sup>b</sup><sup>G</sup>* is a map from *ω* onto, essentially, *ω*<sup>1</sup> **<sup>L</sup>**. It follows that there is a real *x*<sup>0</sup> with **L**[*GY*] = **L**[*x*0]. Then all parameters of *ϕ* belong to **L**[*x*0].

To prepare for Theorem 4 of Section 4.4, put *Z* = [*ξ*, *ω*2), *e* = *w*<*m*[*G*] *c*, *Z* = *e* ∪ *Z* ,

$$d = \{-1\} \cup \{\omega k + j : k \ge m \land j < \omega\} \cup X \dots$$

As *w*<sup>+</sup> <sup>≥</sup>*m*[*G*] ⊆ {−1}∪{*ω<sup>k</sup>* <sup>+</sup> *<sup>j</sup>* : *<sup>k</sup>* <sup>≥</sup> *<sup>m</sup>* <sup>∧</sup> *<sup>j</sup>* <sup>&</sup>lt; *<sup>ω</sup>*}, we have *<sup>Y</sup>* <sup>=</sup> *<sup>c</sup>* <sup>∪</sup> *<sup>w</sup>*<sup>+</sup> <sup>≥</sup>*m*[*G*] <sup>∪</sup> *<sup>X</sup>* <sup>⊆</sup> *<sup>d</sup>*, and hence *x*<sup>0</sup> ∈ **L**[*G d*]. It follows by Theorem 4 that *ϕ* is simultaneously true in **L**[*x*0, *G Z*] and in **L**[*x*0, *G Z* ]. However, **L**[*x*0, *G Z* ] = **L**[*G* (*Y* ∪ *Z* )] = **<sup>L</sup>**[*G* (*<sup>c</sup>* <sup>∪</sup> *<sup>w</sup>*<sup>+</sup> <sup>≥</sup>*m*[*G*] <sup>∪</sup> *<sup>W</sup>*)] by construction, while **<sup>L</sup>**[*x*0, *<sup>G</sup> <sup>Z</sup>*] = **<sup>L</sup>**[*G* (*w*+[*G*] <sup>∪</sup> *<sup>W</sup>*)], and we are done.

In continuation of the **proof of Lemma 23** (ii), suppose that

(†) *<sup>ϕ</sup>*(·) and *<sup>ψ</sup>*(·) are parameter-free *<sup>Σ</sup>*<sup>1</sup> *<sup>n</sup>*+<sup>1</sup> formulas that provide a *<sup>Δ</sup>*<sup>1</sup> *<sup>n</sup>*+<sup>1</sup> definition for a set *x* ⊆ *ω*, *<sup>x</sup>* <sup>∈</sup> **<sup>L</sup>**[*G* (*w*+[*G*] <sup>∪</sup> *<sup>W</sup>*)], i.e., we have

$$\mathfrak{a} = \{\ell < \omega : \mathfrak{p}(\ell) \} = \{\ell < \omega : \neg \mathfrak{p}(\ell) \}.$$

in **<sup>L</sup>**[*G* (*w*+[*G*] <sup>∪</sup> *<sup>W</sup>*)]. Thus, the equivalence <sup>∀</sup> (*ϕ*() ⇐⇒ ¬ *<sup>ψ</sup>*()) is forced to be true in **<sup>L</sup>**[*G* (*w*+[*G*] <sup>∪</sup> *<sup>W</sup>*<sup>ˇ</sup> )] by a condition *<sup>p</sup>*<sup>0</sup> <sup>∈</sup> *<sup>G</sup>*.

Here, *<sup>G</sup>* is the canonical **<sup>Q</sup>**-name for the generic set *<sup>G</sup>* <sup>⊆</sup> **<sup>Q</sup>**, as usual, while *<sup>W</sup>*<sup>ˇ</sup> is a name for *<sup>W</sup>* <sup>∈</sup> **<sup>L</sup>**.

**Lemma 25.** *Assume* (†)*. If* < *<sup>ω</sup> then the sentence* "**L**[*G* (*w*+[*G*] <sup>∪</sup> *<sup>W</sup>*<sup>ˇ</sup> )] <sup>|</sup><sup>=</sup> *<sup>ϕ</sup>*()" *is* **<sup>Q</sup>***-decided by p*<sup>0</sup> *.*

**Proof.** Suppose, for the sake of simplicity, that *p*<sup>0</sup> is the empty condition (i.e., |*p*0| <sup>+</sup> = ∅); the general case does not differ much. Then <sup>∀</sup> (*ϕ*() ⇐⇒ ¬ *<sup>ψ</sup>*()) holds in **<sup>L</sup>**[*G* (*w*+[*G*] <sup>∪</sup> *<sup>W</sup>*)] for **any** generic set *G* ⊆ **Q**.

Say that conditions *p*, *q* ∈ **Q** = **Q**[**U**] are *close neighbours* iff −1 ∈ |*p*| <sup>+</sup> ∩ |*q*<sup>|</sup> <sup>+</sup> and one of the following holds:


**Proposition 1.** *If conditions p*, *q* ∈ **Q** *are close neighbours, satisfying* (6) *in Section 3.8,* < *ω, and p* **Q***forces the sentence* "**L**[*G* (*w*+[*G*] <sup>∪</sup> *<sup>W</sup>*<sup>ˇ</sup> )] <sup>|</sup><sup>=</sup> *<sup>ϕ</sup>*()"*, then so does q.*

**Proof** (Proposition). Suppose on the contrary that *<sup>q</sup>* does not force "**L**[*G* (*w*+[*G*] <sup>∪</sup> *<sup>W</sup>*<sup>ˇ</sup> )] <sup>|</sup><sup>=</sup> *<sup>ϕ</sup>*()". As *p*, *q* satisfy (6), the associated transformation *Hpq* maps the set **Q***<sup>p</sup>* = {*p* ∈ **Q** : *p p*} onto **Q***<sup>q</sup>* = {*q* ∈ **Q** : *q q*} order-preserving by Lemma 17 (with *U* = **U**). By the choice of *q*, there is a set *Gq* <sup>⊆</sup> **<sup>Q</sup>***<sup>q</sup>* , generic over **<sup>L</sup>**, containing *<sup>q</sup>*, and such that *<sup>ϕ</sup>*() is false in **<sup>L</sup>**[*Gq* (*w*+[*Gq*] <sup>∪</sup> *<sup>W</sup>*)]. Then *<sup>ψ</sup>*() is true in **<sup>L</sup>**[*Gq* (*w*+[*Gq*] <sup>∪</sup> *<sup>W</sup>*)] by (†) (and the assumption that *<sup>p</sup>*<sup>0</sup> <sup>=</sup> ).

The set *Gp* <sup>=</sup> {(*Hpq*)−1(*<sup>q</sup>* ) : *q* ∈ *Gq* ∧ *q q*} ⊆ **Q***<sup>p</sup>* is **Q**-generic over **L** as well (as *Hpq* is an order isomorphism), and contains *<sup>p</sup>*, and hence *<sup>ϕ</sup>*() is true and *<sup>ψ</sup>*() false in **<sup>L</sup>**[*Gp* (*w*+[*Gp*] <sup>∪</sup> *<sup>W</sup>*)].

*Case 1*: (I) holds, i.e., *<sup>b</sup><sup>p</sup>* = *<sup>b</sup><sup>q</sup>* . Then by definition *<sup>b</sup>Gp* = *<sup>b</sup>Gq* , so that *<sup>w</sup>*+[*Gp*] = *<sup>w</sup>*+[*Gq*]. On the other hand, the sets *Gp* and *Gq* are equi-constructible by means of the application of *Hpq* , and hence *Gp* (*w*+[*Gp*] <sup>∪</sup> *<sup>W</sup>*) and *Gq* (*w*+[*Gq*] <sup>∪</sup> *<sup>W</sup>*) are equi-constructible, that is, the classes **<sup>L</sup>**[*Gp* (*w*+[*Gp*] <sup>∪</sup> *<sup>W</sup>*)] and **<sup>L</sup>**[*Gq* (*w*+[*Gq*] <sup>∪</sup> *<sup>W</sup>*)] coincide. However, *<sup>ϕ</sup>*() is true in one of them and false in the other one, a contradiction.

*Case 2*: (II) holds. Let *<sup>m</sup>* <sup>=</sup> lh *<sup>b</sup><sup>p</sup>* <sup>=</sup> lh *<sup>b</sup><sup>q</sup>* . Then *<sup>b</sup>Gp* (*k*) = *<sup>b</sup>Gq* (*k*) for all *<sup>k</sup>* <sup>≥</sup> *<sup>m</sup>* via *Hpq* . This implies **<sup>L</sup>**[*bGp* ] = **<sup>L</sup>**[*bGq* ], and also implies *<sup>w</sup>*<sup>+</sup> <sup>≥</sup>*m*[*Gp*] = *<sup>w</sup>*<sup>+</sup> <sup>≥</sup>*m*[*Gq*], while the difference between the sets *w*<*m*[*Gp*], *w*<*m*[*Gq*] is that for any *k* < *m* and any *j*,

$$
\omega k + 2^l \in w\_{$$

Moreover, (II) implies *Gp* =−<sup>1</sup> = *Gq* =−<sup>1</sup> , and hence *SGp* (*ν*) = *SGq* (*ν*) for all *<sup>ν</sup>* ∈ I via *Hpq* . We conclude that **<sup>L</sup>**[*Gp <sup>Z</sup>*] = **<sup>L</sup>**[*Gq <sup>Z</sup>*] for any set *<sup>Z</sup>* <sup>∈</sup> **<sup>L</sup>**[*bGp* ], *<sup>Z</sup>* ⊆ I<sup>+</sup> , in particular, **<sup>L</sup>**[*Gp* (*w*+[*Gq*] <sup>∪</sup> *<sup>W</sup>*)] = **<sup>L</sup>**[*Gq* (*w*+[*Gq*] <sup>∪</sup> *<sup>W</sup>*)].

If now (II) (a) holds, then *c* = *w*<*m*[*Gp*] ⊆ *c* = *w*<*m*[*Gq*] = *c* ∪ *z* by (16), where

*<sup>z</sup>* <sup>=</sup> {*ω<sup>k</sup>* <sup>+</sup> <sup>2</sup>*<sup>j</sup>* : *<sup>k</sup>* <sup>&</sup>lt; *<sup>m</sup>* <sup>∧</sup> *<sup>j</sup>* <sup>∈</sup> *<sup>b</sup>q*(*k*) *<sup>b</sup>p*(*k*)} ∈ **<sup>L</sup>**.

However, *<sup>ϕ</sup>*() holds in **<sup>L</sup>**[*Gp* (*w*+[*Gp*] <sup>∪</sup> *<sup>W</sup>*)], see above. It follows by Lemma <sup>24</sup> that *<sup>ϕ</sup>*() holds in **<sup>L</sup>**[*Gp* (*w*+[*Gq*] <sup>∪</sup> *<sup>W</sup>*)]. However, we know that **<sup>L</sup>**[*Gp* (*w*+[*Gq*] <sup>∪</sup> *<sup>W</sup>*)] = **<sup>L</sup>**[*Gq* (*w*+[*Gq*] <sup>∪</sup> *<sup>W</sup>*)]. Thus, *<sup>ϕ</sup>*() holds in **<sup>L</sup>**[*Gq* (*w*+[*Gq*] <sup>∪</sup> *<sup>W</sup>*)], which is a contradiction to the above. If (II) (b) holds, then argue similarly using the formula *ψ*(). (Proposition 1)

Coming back to Lemma 25, suppose towards the contrary that "**L**[*G* (*w*+[*G*] <sup>∪</sup> *<sup>W</sup>*<sup>ˇ</sup> )] <sup>|</sup><sup>=</sup> *<sup>ϕ</sup>*()" is not **Q**-decided by *p*<sup>0</sup> = . There are two conditions *p*, *q* ∈ **Q** such that *p* **Q**-forces "**L**[*G* (*w*+[*G*] <sup>∪</sup> *<sup>W</sup>*<sup>ˇ</sup> )] <sup>|</sup><sup>=</sup> *<sup>ϕ</sup>*()" while *<sup>q</sup>* **<sup>Q</sup>**-forces the negation. We may w.l. o. g. assume, by Lemma <sup>17</sup> (i), that *p*, *q* satisfy (6) of Section 3.8. We claim that *p*, *q* can be connected by a finite chain of conditions in **Q** in which each two consecutive terms are close neighbours in the sense above, satisfying (6) in Section 3.8— then Proposition 1 implies a contradiction and concludes the proof of Lemma 25.

Thus, it remains to prove the connection claim. Let *p* ∈ **Q** be defined by *b<sup>p</sup>* = *b<sup>p</sup>* and *p* =−<sup>1</sup> = *<sup>q</sup>* =−<sup>1</sup> . Then *<sup>p</sup>*, *<sup>p</sup>* are close neighbours and (6) holds for this pair as it holds for *<sup>p</sup>*, *<sup>q</sup>*. Let *<sup>r</sup>* <sup>∈</sup> **<sup>Q</sup>** be defined by *<sup>b</sup>r*(*k*) = *<sup>b</sup>p*(*k*) <sup>∪</sup> *<sup>b</sup>q*(*k*) for all *<sup>k</sup>* < <sup>=</sup> lh *<sup>b</sup><sup>p</sup>* <sup>=</sup> lh *<sup>b</sup><sup>q</sup>* and *<sup>p</sup>* =−<sup>1</sup> = *<sup>q</sup>* =−<sup>1</sup> . Still *r* is a close neighbour to both *p* and *q*, and (6) holds for *p* ,*r* and *q*,*r*. Thus, the chain *p*—*p* —*r*—*q* proves the connection claim. (Lemma 25)

Now, to accomplish the **proof of Lemma 23** (ii), apply Lemma 25.

(Lemma 23 (ii))

(Theorem 1 modulo Theorem 4 of Section 4.4)

#### **5. Forcing Approximation**

To prove Theorem 4 of Section 4.4 and thus complete the proof of Theorem 1 in the next Section 6, we define here a forcing-like relation **forc**, and exploit certain symmetries of objects related to **forc**. This similarity will allow us to only outline really analogous issues but concentrate on several things which bear some difference.

We argue under Blanket Assumption 1.

Recall that **ZFC**− is **ZFC** minus the Power Set axiom, with the schema of Collection instead of Replacement, with the axiom "*ω*<sup>1</sup> exists", and with **AC** in the form of wellorderability of every set, and **ZFC**− <sup>2</sup> is **ZFC**<sup>−</sup> plus the axioms: **<sup>V</sup>** <sup>=</sup> **<sup>L</sup>**, and "every set *<sup>x</sup>* satisfies card *<sup>x</sup>* <sup>≤</sup> *<sup>ω</sup>*1".

#### *5.1. Formulas*

Here we introduce a language that will help us to study analytic definability in **Q**[*U*]-generic extensions, for different systems *U*, and their submodels.

Let <sup>L</sup> be the 2nd order Peano language, with variables of type 1 over *<sup>ω</sup><sup>ω</sup>* . If *<sup>K</sup>* <sup>⊆</sup> **<sup>Q</sup>**<sup>∗</sup> then an <sup>L</sup>(*K*) *formula* is any formula of L, with some free variables of types 0, 1 replaced by resp. numbers in *ω* and names in **SN***<sup>ω</sup> <sup>ω</sup>*(*K*), and some type 1 quantifiers are allowed to have *bounding indices <sup>B</sup>* (i.e., <sup>∃</sup>*<sup>B</sup>* , <sup>∀</sup>*<sup>B</sup>* ) such that *<sup>B</sup>* ⊆ I<sup>+</sup> satisfies either card *<sup>B</sup>* <sup>≤</sup> *<sup>ω</sup>*<sup>1</sup> or card(<sup>I</sup> *<sup>B</sup>*) <sup>≤</sup> *<sup>ω</sup>*<sup>1</sup> (in **<sup>L</sup>**). In particular, <sup>I</sup><sup>+</sup> itself can serve as an index, and the absence

If *ϕ* is a L(**Q**∗) formula, then let

NAM *ϕ* = the set of all names *τ* that occur in *ϕ*; IND *ϕ* = the set of all quantifier indices *B* which occur in *ϕ*; |*ϕ*| <sup>+</sup> = - *<sup>τ</sup>*∈NAM *<sup>ϕ</sup>* |*τ*| <sup>+</sup> (a set of *ω*1-size); ||*ϕ*|| = |*ϕ*| <sup>+</sup> <sup>∪</sup> - IND *ϕ* <sup>−</sup> so that <sup>|</sup>*ϕ*<sup>|</sup> <sup>+</sup> ⊆ ||*ϕ*|| ⊆ I+.

If a set *G* ⊆ **Q**<sup>∗</sup> is *minimally ϕ-generic* (that is, minimally *τ*-generic w. r. t. every name *τ* ∈ NAM *ϕ*, in the sense of Section 3.5), then the *valuation ϕ*[*G*] is the result of substitution of *τ*[*G*] for any name *<sup>τ</sup>* <sup>∈</sup> NAM *<sup>ϕ</sup>*, and changing each quantifier <sup>∃</sup>*Bx*, <sup>∀</sup>*Bx* to resp. <sup>∃</sup> (<sup>∀</sup> ) *<sup>x</sup>* <sup>∈</sup> *<sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>**[*G <sup>B</sup>*], while index-free type 1 quantifiers are relativized to *<sup>ω</sup><sup>ω</sup>* ; *<sup>ϕ</sup>*[*G*] is a formula of <sup>L</sup> with real parameters, and *some* quantifiers of type 1 relativized to certain submodels of **L**[*G*].

An *arithmetic* formula in <sup>L</sup>(*K*) is a formula with no quantifiers of type 1 (names in **SN***<sup>ω</sup> <sup>ω</sup>*(*K*) are allowed). If *<sup>n</sup>* <sup>&</sup>lt; *<sup>ω</sup>* then let a <sup>L</sup>*Σ*<sup>1</sup> *<sup>n</sup>*(*K*), resp., <sup>L</sup>*Π*<sup>1</sup> *<sup>n</sup>*(*K*) formula be a formula of the form

$$
\exists^{\diamond} \mathbf{x}\_1 \forall^{\diamond} \mathbf{x}\_2 \dots \forall^{\diamond} (\exists^{\diamond}) \mathbf{x}\_{n-1} \exists (\forall) \,\mathbf{x}\_n \,\forall \,\,\,\forall^{\diamond} \mathbf{x}\_1 \exists^{\diamond} \mathbf{x}\_2 \dots \exists^{\diamond} (\forall^{\diamond}) \,\mathbf{x}\_{n-1} \,\forall \,(\exists) \,\mathbf{x}\_n \,\forall^{\diamond}
$$

respectively, where *<sup>ψ</sup>* is an arithmetic formula in <sup>L</sup>(*K*), all variables *xi* are of type 1 (over *<sup>ω</sup><sup>ω</sup>* ), the sign ◦ means that this quantifier can have a bounding index as above, and it is required that the rightmost (closest to the kernel *ψ*) quantifier does not have a bounding index.

If in addition *M* |= **ZFC**<sup>−</sup> is a transitive model and *K* ⊆ **Q**<sup>∗</sup> then define

<sup>L</sup>*Σ*<sup>1</sup> *<sup>n</sup>*(*K*, *<sup>M</sup>*) = all <sup>L</sup>*Σ*<sup>1</sup> *<sup>n</sup>*(*K*) formulas *<sup>ϕ</sup>* such that NAM *<sup>ϕ</sup>* <sup>⊆</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(*K*) ∩ *M* and each index *B* ∈ IND *ϕ* satisfies the requirement: either *B* ∈ *M* or I *B* ∈ *M*.

Define <sup>L</sup>*Π*<sup>1</sup> *<sup>n</sup>*(*K*, *M*) similarly.

#### *5.2. Forcing Approximation*

We introduce a convenient forcing-type relation *p* **forc***<sup>M</sup> <sup>U</sup> ϕ* for pairs *M*, *U* in **sJS** and formulas *ϕ* in L(*K*), associated with the truth in *K*-generic extensions of **L**, where *K* = **Q**[*U*] ⊆ **Q**<sup>∗</sup> and *U* ∈ **L** is a system.

	- (a) *M*, *U* ∈ **sJS** and *p* belongs to **Q**[*U*],
	- (b) *<sup>ϕ</sup>* is a closed formula in <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* (**Q**[*U*], *<sup>M</sup>*) ∪ L*Σ*<sup>1</sup> *<sup>k</sup>*+1(**Q**[*U*], *M*) for some *k* ≥ 1, and each name *τ* ∈ NAM *ϕ* is **Q**[*U*]-full below *p*.

Under these assumptions, the sets *U*, **Q**[*U*], *p*, NAM *ϕ* belong to *M*.

The definition of **forc** goes on by induction on the complexity of formulas.

	- (a) *p* **forc***<sup>M</sup> <sup>U</sup>* <sup>∃</sup>*Bx <sup>ϕ</sup>*(*x*) iff there is a name *<sup>τ</sup>* <sup>∈</sup> *<sup>M</sup>* <sup>∩</sup>**SN***<sup>ω</sup> <sup>ω</sup>*(**Q**[*U*]) *B*, **Q**[*U*]-full below *p* (by (F1)b) and such that *p* **forc***<sup>M</sup> <sup>U</sup> ϕ*(*τ*).
	- (b) *p* **forc***<sup>M</sup> <sup>U</sup>* <sup>∃</sup> *<sup>x</sup> <sup>ϕ</sup>*(*x*) iff there is a name *<sup>τ</sup>* <sup>∈</sup> *<sup>M</sup>* <sup>∩</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**[*U*]), **Q**[*U*]-full below *p* (by (F1)b) and such that *p* **forc***<sup>M</sup> <sup>U</sup> ϕ*(*τ*).

(F4) If *<sup>k</sup>* <sup>≥</sup> 2, *<sup>ϕ</sup>* is a closed <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* (**Q**[*U*], *<sup>M</sup>*) formula, *<sup>p</sup>* <sup>∈</sup> **<sup>Q</sup>**[*U*], and (F1) holds, then *<sup>p</sup>* **forc***<sup>M</sup> <sup>U</sup> ϕ* iff we have <sup>¬</sup> *<sup>q</sup>* **forc***<sup>M</sup> <sup>U</sup> ϕ*<sup>¬</sup> for every pair *M* , *U*  ∈ **sJS** extending *M*, *U*, and every condition *q* ∈ **Q**[*U* ], *<sup>q</sup> <sup>p</sup>*, where *<sup>ϕ</sup>*<sup>¬</sup> is the result of canonical conversion of <sup>¬</sup> *<sup>ϕ</sup>* to <sup>L</sup>*Σ*<sup>1</sup> *<sup>k</sup>* (**Q**[*U*], *M*).

The next theorem classifies the complexity of **forc** in terms of projective hierarchy. Please note that if *<sup>M</sup>*, *<sup>U</sup>*<sup>∈</sup> **sJS** and *<sup>k</sup>* <sup>≥</sup> 1 then any formula *<sup>ϕ</sup>* in <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* (**Q**[*U*], *<sup>M</sup>*) ∪ L*Σ*<sup>1</sup> *<sup>k</sup>*+1(**Q**[*U*], *M*) belongs to *<sup>M</sup>* if we somehow "label" any large index *<sup>B</sup>* <sup>∈</sup> IND *<sup>ϕ</sup>* (such that card(<sup>I</sup> *<sup>B</sup>*) <sup>≤</sup> *<sup>ω</sup>*1) by its small complement I *B* ∈ *M*. Therefore, the sets

$$\begin{aligned} \mathsf{Fore}(\varPi\_k^1) &= \{ \langle M, \mathsf{U}, p, \varphi \rangle : \langle M, \mathsf{U} \rangle \in \mathsf{s} \mathsf{JS} \land p \in \mathsf{Q}[\mathsf{U}] \land \\ &\land \ \varphi \text{ is a closed formula in } \mathcal{L}\varPi\_k^1(\mathsf{Q}[\mathsf{U}], \mathsf{M}) \land p \text{ } \mathsf{forc}\_{\mathsf{U}}^{\mathsf{M}} \neq \emptyset \}, \end{aligned}$$

and **Forc**(*Σ*<sup>1</sup> *<sup>k</sup>* ) similarly defined, are subsets of H*ω*<sup>2</sup> (in **L**).

**Lemma 26** (in **L**)**.** *The sets* **Forc**(*Π*<sup>1</sup> <sup>1</sup> ) *and* **Forc**(*Σ*<sup>1</sup> <sup>2</sup>) *belong to <sup>Δ</sup>*H*ω*<sup>2</sup> <sup>1</sup> *. If k* <sup>≥</sup> <sup>2</sup> *then the sets* **Forc**(*Π*<sup>1</sup> *<sup>k</sup>* ) *and* **Forc**(*Σ*<sup>1</sup> *<sup>k</sup>*+1) *belong to <sup>Π</sup>*H*ω*<sup>2</sup> *<sup>k</sup>*−<sup>1</sup> *.*

**Proof** (sketch). Suppose that *<sup>ϕ</sup>* is <sup>L</sup>*Π*<sup>1</sup> <sup>1</sup> . Under the assumptions of the theorem, items (F1)a, (F1)b of (F1) are *Δ*H*ω*<sup>2</sup> <sup>1</sup> relations, while (F2) is reducible to a forcing relation over *M* that we can relativize to *M*. The inductive step goes on straightforwardly using (F3), (F4). Please note that the quantifier over names in (F3) is a bounded quantifier (bounded by *M*), hence it does not add any extra complexity.

#### *5.3. Further Properties of Forcing Approximations*

The notion of names *<sup>ν</sup>*, *<sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**∗) being equivalent *below* some *p* ∈ **Q**<sup>∗</sup> , is introduced in Subsection 3.5. We continue with a couple of routine lemmas.

**Lemma 27.** *Suppose that M*, *U*, *p*, *ϕ satisfy* (F1) *of Section 5.2, and* NAM *ϕ* = {*τ*1, ... , *τ<sup>m</sup>* }*. Let μ*1, ... , *μ<sup>m</sup> be another list of names in* **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**[*U*])*,* **Q**[*U*]*-full below p, and such that τ<sup>j</sup> and μ<sup>j</sup> are equivalent below p for each j* = 1, . . . , *m. Then p* **forc***<sup>M</sup> <sup>U</sup> <sup>ϕ</sup>*(*τ*1, ..., *<sup>τ</sup>m*) *iff p* **forc***<sup>M</sup> <sup>U</sup> ϕ*(*μ*1, ..., *μm*)*.*

**Proof.** Suppose that *<sup>ϕ</sup>* is <sup>L</sup>*Π*<sup>1</sup> <sup>1</sup> . Let *G* ⊆ **Q**[*U*] be a set **Q**[*U*]-generic over *M*, and containing *p*. Then *τ*[*G*] = *μ*[*G*] for all by Lemma 12. This implies the result required, by (F2) of Section 5.2.

The induction steps <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* → L*Σ*<sup>1</sup> *<sup>k</sup>*+<sup>1</sup> and <sup>L</sup>*Σ*<sup>1</sup> *<sup>k</sup>* → L*Π*<sup>1</sup> *<sup>k</sup>* are carried out by an easy reduction to items (F3), (F4) of Section 5.2.

**Lemma 28** (in **L**)**.** *Let M*, *U*, *p*, *ϕ satisfy* (F1) *of Section 5.2. Then*:

(i) *if k* <sup>≥</sup> <sup>2</sup>*, <sup>ϕ</sup> is* <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* (**Q**[*U*], *<sup>M</sup>*)*, and p* **forc***<sup>M</sup> <sup>U</sup> <sup>ϕ</sup>, then p* **forc***<sup>M</sup> <sup>U</sup> ϕ*<sup>¬</sup> *fails*;

(ii) *if p* **forc***<sup>M</sup> <sup>U</sup> ϕ, M*, *U M* , *U*  ∈ **sJS***, and q* ∈ **Q**[*U* ]*, q p, then q* **forc***<sup>M</sup> <sup>U</sup> ϕ* .

**Proof.** Claim (i) immediately follows from (F4) of Section 5.2.

To prove (ii) let *<sup>ϕ</sup>* <sup>=</sup> *<sup>ϕ</sup>*(*τ*1, ... , *<sup>τ</sup>m*) be a closed formula in <sup>L</sup>*Π*<sup>1</sup> <sup>1</sup> (**Q**[*U*], *M*), where all **Q**[*U*] names *τ<sup>j</sup>* belong to *M* and are **Q**[*U*]-full below *p*. Then all names *τ<sup>j</sup>* remain **Q**[*U* ]-full below *p* by Corollary 3 in Section 4.2, therefore below *q* as well since *q p*. Consider a set *G* ⊆ **Q**[*U* ], **Q**[*U* ] generic over *M* and containing *q*. We have to prove that *ϕ*[*G* ] is true in *M* [*G* ]. Please note that the set *G* = *G* ∩ **Q**[*U*] is **Q**[*U*]-generic over *M* by Corollary 2 in Section 4.2, and we have *p* ∈ *G*. Moreover, the valuation *ϕ*[*G* ] coincides with *ϕ*[*G*] since all names in *ϕ* belong to **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**[*U*]). And *ϕ*[*G*] is true in *M*[*G*] as *p* **forc***<sup>M</sup> <sup>U</sup> ϕ*. It remains to apply Mostowski's absoluteness (see [10] (p. 484) or [11]) between the models *M*[*G*] ⊆ *M* [*G* ].

The induction steps related to (F3), (F4) of Section 5.2 are easy.

#### *5.4. Transformations and Invariance*

To prove Theorem 4 of Section 4.4, we make use of the transformations considered in Sections 3.6–3.8. In addition to the definitions given there, define, in **L**, the action of any transformation *π* ∈ **BIJ** (permutation), *λ* ∈ **LIP**<sup>I</sup> (multi-Lipschitz), or one of the form *Hpq* (multisubstitution), on <sup>L</sup>-formulas with quantifier indices and names in **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**∗) as parameters.


**Lemma 29** (in **L**)**.** *Suppose that M*, *U* ∈ **sJS***, p* ∈ **Q**[*U*]*, k* ≥ 1*, ϕ is a formula in* <sup>L</sup>*Σ*<sup>1</sup> *<sup>k</sup>*+1(**Q**[*U*], *<sup>M</sup>*) ∪ L*Π*<sup>1</sup> *<sup>k</sup>* (**Q**[*U*], *M*)*, and π* ∈ **BIJ** *is* **coded in** *M in the sense that* |*π*| ∈ *M and <sup>π</sup>* <sup>|</sup>*π*| ∈ *M. Then*: *<sup>p</sup>* **forc***<sup>M</sup> <sup>U</sup> <sup>ϕ</sup> iff* (*π· <sup>p</sup>*) **forc***<sup>M</sup> <sup>π</sup> ·<sup>U</sup> πϕ.*

**Proof.** Under the conditions of the lemma, *π* acts as an isomorphism on all relevant domains and preserves all relevant relations between the objects involved. Thus, *M*, *π·U*, *π· p*, *πϕ* still satisfy (F1) in Section 5.2. This allows proof of the lemma by induction on the complexity of *ϕ*.

**Base.** Suppose that *<sup>ϕ</sup>* is a closed formula in <sup>L</sup>*Π*<sup>1</sup> <sup>1</sup> (**Q**[*U*], *M*). Then *πϕ* is a closed formula in <sup>L</sup>*Π*<sup>1</sup> <sup>1</sup> (**Q**[*π·U*], *<sup>M</sup>*). Moreover, the map *<sup>p</sup>* −→ *<sup>π</sup>· <sup>p</sup>* is an order isomorphism (in *<sup>M</sup>*) **<sup>Q</sup>**[*U*] onto −→ **Q**[*π·U*] by Lemma 14. We conclude that a set *G* ⊆ *P* is **Q**[*U*]-generic over *M* iff *π·G* is, accordingly, **Q**[*π·U*] generic over *<sup>M</sup>*, and the valuated formulas *<sup>ϕ</sup>*[*G*] and (*πϕ*)[*π·G*] coincide. Now the result for *<sup>Π</sup>*<sup>1</sup> 1 formulas follows from (F2) in Section 5.2.

**Step** *Π*<sup>1</sup> *<sup>n</sup>* <sup>→</sup> *<sup>Σ</sup>*<sup>1</sup> *<sup>n</sup>*+<sup>1</sup> **,** *<sup>n</sup>* <sup>≥</sup> 1. Let *<sup>ψ</sup>*(*x*) be a <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* (**Q**[*U*], *M*) formula, and *ϕ* be ∃ *x ψ*(*x*). Assume *p* **forc***<sup>M</sup> <sup>U</sup> <sup>ϕ</sup>*. By definition there is a name *<sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**[*U*]) ∩ *M*, **Q**[*U*]-full below the given *<sup>p</sup>* <sup>∈</sup> **<sup>Q</sup>**[*U*], such that *<sup>p</sup>* **forc***<sup>M</sup> <sup>U</sup> ψ*(*τ*). Then, by the inductive hypothesis, we have *<sup>π</sup>· <sup>p</sup>* **forc***<sup>M</sup> <sup>π</sup> ·<sup>U</sup>* (*πψ*)(*π·τ*), and hence by definition *<sup>π</sup>· <sup>p</sup>* **forc***<sup>M</sup> <sup>π</sup> ·<sup>U</sup> πϕ*.

The case of *<sup>ϕ</sup>* being <sup>∃</sup>*Bx <sup>ψ</sup>*(*x*) is similar.

**Step** *Σ*<sup>1</sup> *<sup>n</sup>* <sup>→</sup> *<sup>Π</sup>*<sup>1</sup> *<sup>n</sup>* **,** *<sup>n</sup>* <sup>≥</sup> 2. This is somewhat less trivial. Assume that *<sup>ϕ</sup>* is a closed <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* (**Q**[*U*], *M*) formula; all names in *ϕ* belong to **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**[*U*]) ∩ *M* and are **Q**[*U*]-full below *p*. Then *πϕ* is a closed <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* (**Q**[*π·U*], *<sup>M</sup>*) formula, whose all names belong to **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**[*π·U*]) ∩ *M* and are **Q**[*π·U*]-full below *<sup>π</sup>· <sup>p</sup>*. Suppose that *<sup>p</sup>* **forc***<sup>M</sup> <sup>U</sup> ϕ* **fails**.

By definition there exist a pair *M*1, *U*1 ∈ **sJS** with *M*, *U M*1, *U*<sup>1</sup>, and a condition *<sup>q</sup>* <sup>∈</sup> **<sup>Q</sup>**[*U*1], *<sup>q</sup> <sup>p</sup>*, such that *<sup>q</sup>* **forc***M*<sup>1</sup> *<sup>U</sup>*<sup>1</sup> *<sup>ϕ</sup>*<sup>¬</sup> . Then (*π· <sup>q</sup>*) **forc***M*<sup>1</sup> *<sup>π</sup> ·U*<sup>1</sup> *πϕ*<sup>¬</sup> by the inductive hypothesis. Yet the pair *M*1, *π·U*<sup>1</sup> belongs to **sJS** and extends *M*, *π·U*. (Recall that *U* ∈ *M* and *π* is coded in *<sup>M</sup>*.) In addition, *<sup>π</sup>· <sup>q</sup>* <sup>∈</sup> **<sup>Q</sup>**[*π·U*1], and *<sup>π</sup>· <sup>q</sup> <sup>π</sup>· <sup>p</sup>*. Therefore, the statement (*π· <sup>p</sup>*) **forc***<sup>M</sup> <sup>π</sup> ·<sup>U</sup> πϕ* fails, as required.

**Lemma 30** (in **L**)**.** *Suppose that M*, *U* ∈ **sJS***, p* ∈ **Q**[*U*]*, k* ≥ 1*, ϕ is a formula in* <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* (**Q**[*U*], *<sup>M</sup>*) ∪ L*Σ*<sup>1</sup> *<sup>k</sup>*+1(**Q**[*U*], *<sup>M</sup>*)*, and <sup>α</sup>* <sup>∈</sup> **LIP**<sup>I</sup> <sup>∩</sup> *M. Then*: *<sup>p</sup>* **forc***<sup>M</sup> <sup>U</sup> <sup>ϕ</sup> iff* (*α· <sup>p</sup>*) **forc***<sup>M</sup> <sup>α</sup> ·<sup>U</sup> <sup>α</sup>ϕ.*

**Proof.** Similar to the previous one, but with a reference to Lemma 15 rather than Lemma 14.

**Lemma 31** (in **L**)**.** *Assume that M*, *U* ∈ **sJS***, conditions p*, *q* ∈ **Q**[*U*] *satisfy* (6) *of Section 3.8, k* ≥ 1*, ϕ is a closed formula in* <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* (**Q**[*U*], *<sup>M</sup>*) ∪ L*Σ*<sup>1</sup> *<sup>k</sup>*+1(**Q**[*U*], *<sup>M</sup>*) *with all names in* **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**∗)*pq* (see Section 3.8)*, and r* <sup>∈</sup> **<sup>Q</sup>**[*U*]*, r p. Then*: *<sup>r</sup>* **forc***<sup>M</sup> <sup>U</sup> <sup>ϕ</sup> iff Hpq ·<sup>r</sup>* **forc***<sup>M</sup> <sup>U</sup> Hpq ϕ.*

**Proof.** Similar to the proof of Lemma 29, except for the step *Π*<sup>1</sup> *<sup>k</sup>* <sup>→</sup> *<sup>Σ</sup>*<sup>1</sup> *<sup>k</sup>*+<sup>1</sup> , *k* ≥ 1, where we need to take additional care to keep the names involved in **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**[*U*])*pq* . Thus, let *<sup>ψ</sup>*(*x*) be a <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* (**Q**[*U*], *M*) formula, with names in **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**[*U*])*pq* , and let *<sup>ϕ</sup>* be <sup>∃</sup> *<sup>x</sup> <sup>ψ</sup>*(*x*). Assume that *<sup>r</sup>* **forc***<sup>M</sup> <sup>U</sup> ϕ*.

By definition there is a name *<sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**[*U*]) <sup>∩</sup> *<sup>M</sup>*, **<sup>Q</sup>**[*U*]-full below *<sup>r</sup>*, such that *<sup>r</sup>* **forc***<sup>M</sup> <sup>U</sup> ψ*(*τ*). Please note that *τ* does not necessarily belong to **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**[*U*])*pq* . However, the restricted name *τ* = *τ<sup>r</sup>* (see Lemma 13 in Section 3.8) is still a name in **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**[*U*]) because *r* ∈ **Q**[*U*], and we have *<sup>r</sup>* <sup>∈</sup> dom *<sup>τ</sup>* <sup>=</sup><sup>⇒</sup> *<sup>r</sup> <sup>r</sup> <sup>p</sup>*, so that *<sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**[*U*])*pq* . Moreover, *τ* is equivalent to *τ* below *r* by Lemma 13. We conclude that *r* **forc***<sup>M</sup> <sup>U</sup> ψ*(*τ* ), by Lemma 27.

Then, by the inductive hypothesis, we have *Hpq ·<sup>r</sup>* **forc***<sup>M</sup> <sup>U</sup>* (*Hpq ψ*)(*Hpq ·τ* ), and hence by definition *Hpq ·<sup>r</sup>* **forc***<sup>M</sup> <sup>U</sup> Hpq ϕ* via *Hpq ·τ* .

#### **6. Elementary Equivalence Theorem**

The goal of this section is to prove Theorem 4 of Section 4.4, and accomplish the proof of Theorem 1. We make use of the relation **forc** defined above, and exploit certain symmetries in **forc** studied in Section 5.4.

#### *6.1. Hidden Invariance*

To explain the idea, one may note first that elementary equivalence of subextensions of a given generic extension is usually a corollary of the fact that the forcing notion considered is enough homogeneous, or in different words, invariant w. r. t. a sufficiently large system of order-preserving transformations. The forcing notion **Q** = **Q**[**U**] we consider, as well as basically any **Q**[*U*], is invariant w. r. t. multi-substitutions by Lemma 17. However, for a straightaway proof of Theorem 4 we would naturally need the invariance under permutations of Section 3.6—to interchange the domains *Z* and *Z* , whereas **Q** is definitely not invariant w. r. t. permutations.

On the other hand, the relation **forc** is invariant w. r. t. both permutations (Lemma 29) and multi-Lipschitz (Lemma 30), as well as still w. r. t. multi-substitutions by Lemma 31. To bridge the gap between **forc** (not explicitly connected with **Q** in any way) and **Q**-generic extensions, we prove Lemma 33, which ensures that **forc** admits a forcing-style association with the truth in **Q**-generic extensions, bounded to formulas of type *Σ*<sup>1</sup> **<sup>n</sup>** and below. This key result will be based on the **n**-completeness property (Definition 2 in Section 4.3). Speaking loosely, one may say that some transformations, i.e., permutations and multi-Lipschitz, are *hidden* in construction of **Q**, so that they do not act per se, but their influence up to **n**th level, is preserved.

This method of *hidden invariance*, i.e., invariance properties (of an auxiliary forcing-type relationship like **forc**) hidden in **Q** by a suitable generic-style construction of **Q**, was introduced in Harrington's notes [3] in a somewhat different terminology. We may note that the hidden invariance technique is well known in some other fields of mathematics, including more applied fields, see e.g., [12,13].

#### *6.2. Approximations of the* **n***-Complete Forcing Notion*

We return to the forcing notion **Q** = **Q**[**U**] defined in **L** as in Definition 2 in Section 4.3 for a given number **n** ≥ 2 of Theorem 1. **Arguing in L**, we let the pairs **M***<sup>ξ</sup>* , **U***<sup>ξ</sup>* , *ξ* < *ω*2, also be as in Definition 2. Let **forc***<sup>ξ</sup>* denote the relation **forcM***<sup>ξ</sup>* **<sup>U</sup>***<sup>ξ</sup>* , and let *<sup>p</sup>* **forc**<sup>∞</sup> *<sup>ϕ</sup>* mean: ∃ *<sup>ξ</sup>* < *<sup>ω</sup>*<sup>2</sup> (*<sup>p</sup>* **forc***<sup>ξ</sup> <sup>ϕ</sup>*).

Claims (i), (ii) of Lemma 28 take the form:


The next lemma shows that **forc**∞ satisfies a key property of forcing relations up to the level of *Π*<sup>1</sup> **<sup>n</sup>**−<sup>1</sup> formulas.

**Lemma 32.** *If <sup>ϕ</sup> is a closed formula in* <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* (**Q**)*,* 2 ≤ *k* < **n***, p* ∈ **Q***, and all names in ϕ are* **Q***-full below p, then there is a condition q* ∈ **Q***, q p, such that either q* **forc**<sup>∞</sup> *ϕ, or q* **forc**<sup>∞</sup> *ϕ*<sup>¬</sup> *.*

**Proof.** As the names considered are *ω*1-sizeobjects, there is an ordinal *η* < *ω*<sup>2</sup> such that *p* ∈ **Q***<sup>η</sup>* , and all names in *<sup>ϕ</sup>* belong to **<sup>M</sup>***<sup>η</sup>* <sup>∩</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q***η*); then all names in *ϕ* are **Q***η*-full below *p*, of course. As *k* < **n**, the set *D* of all pairs *M*, *U* ∈ **sJS** that extend **M***η*, **U***<sup>η</sup>* and there is a condition *q* ∈ **Q**[*U*], *q p*, satisfying *q* **forc***<sup>M</sup> <sup>U</sup> ϕ*<sup>¬</sup> , belongs to **Σ**HC **<sup>n</sup>**−<sup>2</sup> by Lemma 26. Therefore, by the **<sup>n</sup>**-completeness of the sequence {**M***<sup>ξ</sup>* , **<sup>U</sup>***<sup>ξ</sup>* }*ξ*<*ω*<sup>1</sup> , there is an ordinal *<sup>ζ</sup>* , *<sup>η</sup>* <sup>≤</sup> *<sup>ζ</sup>* <sup>&</sup>lt; *<sup>ω</sup>*1, such that **<sup>M</sup>***<sup>ζ</sup>* , **<sup>U</sup>***<sup>ζ</sup>* <sup>∈</sup> *<sup>D</sup>*solv .

We have two cases.

*Case 1*: **<sup>M</sup>***<sup>ζ</sup>* , **<sup>U</sup>***<sup>ζ</sup>* <sup>∈</sup> *<sup>D</sup>*. Then there is a condition *<sup>q</sup>* <sup>∈</sup> *<sup>K</sup>*[**U***<sup>ζ</sup>* ], *<sup>q</sup> <sup>p</sup>*, satisfying *<sup>q</sup>* **forcM***<sup>ζ</sup>* **<sup>U</sup>***<sup>ζ</sup> <sup>ϕ</sup>*<sup>¬</sup> , that is, *q* **forc**<sup>∞</sup> *ϕ*<sup>¬</sup> . However, obviously *q* ∈ **Q**.

*Case 2*: there is no pair *M*, *U* ∈ *D* extending **M***<sup>ζ</sup>* , **U***<sup>ζ</sup>* . Prove *p* **forc***<sup>ζ</sup> ϕ*. Suppose otherwise. Then by the choice of *η* and (F4) in Section 5.2, there exist: a pair *M*, *U* ∈ **sJS** extending **M***<sup>ζ</sup>* , **U***<sup>ζ</sup>* , and a condition *<sup>q</sup>* <sup>∈</sup> **<sup>Q</sup>**[*U*], *<sup>q</sup> <sup>p</sup>*, such that *<sup>q</sup>* **forc***<sup>M</sup> <sup>U</sup> ϕ*<sup>¬</sup> . Then *M*, *U* ∈ *D*, a contradiction.

Now we prove another key lemma which connects, in a forcing-style way, the relation **forc**∞ and the truth in **Q**-generic extensions of **L**, up to the level of *Σ*<sup>1</sup> **<sup>n</sup>** formulas.

**Lemma 33.** *Suppose that <sup>ϕ</sup> is a formula in* <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* (**Q**) ∪ L*Σ*<sup>1</sup> *<sup>k</sup>*+1(**Q**)*,* 1 ≤ *k* < **n***, and all names in ϕ are* **Q***-full. Let G* ⊆ **Q** *be* **Q***-generic over* **L***. Then ϕ*[*G*] *is true in* **L**[*G*] *iff there is a condition p* ∈ *G such that p* **forc**∞ *ϕ.*

**Proof.** We proceed by induction and begin with **the case of** <sup>L</sup>*Π*<sup>1</sup> <sup>1</sup> **formulas**. Consider a closed formula *<sup>ϕ</sup>* in <sup>L</sup>*Π*<sup>1</sup> <sup>1</sup> (**Q**). As names in the formulas considered are *<sup>ω</sup>*1-sizenames in **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**), there is an ordinal *<sup>ξ</sup>* <sup>&</sup>lt; *<sup>ω</sup>*<sup>2</sup> such that *<sup>ϕ</sup>* is a <sup>L</sup>*Π*<sup>1</sup> <sup>1</sup> (**Q***<sup>ξ</sup>* ) formula. Please note that since *G* ⊆ **P** is **Q**-generic over **L**, the smaller set *G<sup>ξ</sup>* = *G* ∩ **Q***<sup>ξ</sup>* is **Q***<sup>ξ</sup>* -generic over **M***<sup>ξ</sup>* by Corollary 2 in Section 4.2, and the formulas *ϕ*[*G*], *ϕ*[*G<sup>ξ</sup>* ] coincide by the choice of *ξ* . Therefore

*ϕ*[*G*] holds in **L**[*G*]:

iff *ϕ*[*G<sup>ξ</sup>* ] holds in **M***<sup>ξ</sup>* [*G<sup>ξ</sup>* ] by the Mostowski absoluteness [10] (p. 484),

iff there is *p* ∈ *G<sup>ξ</sup>* which **Q***<sup>ξ</sup>* -forces *ϕ* over **M***<sup>ξ</sup>* ,

iff ∃ *p* ∈ *G<sup>ξ</sup>* (*p* **forc***<sup>ξ</sup> ϕ*) by (F2) in Section 5.2,

easily getting the result required since *ξ* is arbitrary.

**The step from** <sup>L</sup>*Σ*<sup>1</sup> *<sup>k</sup>* **to** <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* **,** *<sup>k</sup>* <sup>≥</sup> <sup>2</sup>**.** Prove the theorem for a <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* (**Q**) formula *ϕ*, assuming that the result holds for *ϕ*¬ . Suppose that *ϕ*[*G*] is false in **L**[*G*]. Then *ϕ*¬[*G*] is true, and hence by the inductive hypothesis, there is a condition *p* ∈ *G c* such that *p* **forc**<sup>∞</sup> *ϕ*<sup>¬</sup> . Then it follows from (I) and (II) above that *q* **forc**<sup>∞</sup> *ϕ* fails for all *q* ∈ *G*.

Conversely let *p* **forc**<sup>∞</sup> *ϕ* fail for all *p* ∈ *G*. Then by Lemma 32 there exists *q* ∈ *G* satisfying *q* **forc**<sup>∞</sup> *ϕ*<sup>¬</sup> . It follows that *ϕ*¬[*G*] is true by the inductive hypothesis, therefore *ϕ*[*G*] is false.

**The step from** <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* **to** <sup>L</sup>*Σ*<sup>1</sup> *<sup>k</sup>*+<sup>1</sup> **.** Let *<sup>ϕ</sup>*(*x*) be a <sup>L</sup>*Π*<sup>1</sup> *<sup>k</sup>* (**Q**) formula; prove the result for a formula <sup>∃</sup>*Bx <sup>ϕ</sup>*(*x*). If *<sup>p</sup>* <sup>∈</sup> *<sup>G</sup>* and *<sup>p</sup>* **forc***<sup>ξ</sup>* <sup>∃</sup>*Bx <sup>ϕ</sup>*(*x*) then by definition there is a name *<sup>τ</sup>* <sup>∈</sup> **<sup>M</sup>***<sup>ξ</sup>* <sup>∩</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q***<sup>ξ</sup>* ) *B*, **Q***<sup>ξ</sup>* -full below *p*, and such that *p* **forc***<sup>ξ</sup> ϕ*(*τ*). Then *ϕ*(*τ*)[*G*] holds by the inductive hypothesis, and this implies (∃*Bx <sup>ϕ</sup>*(*x*))[*G*] since obviously *<sup>τ</sup>*[*G*] <sup>∈</sup> **<sup>L</sup>**[*G <sup>B</sup>*].

If conversely (∃*Bx <sup>ϕ</sup>*(*x*))[*G*] is true, then by Lemma <sup>11</sup> there is a **<sup>Q</sup>**-full name *<sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**) *B* such that *ϕ*(*τ*)[*G*] is true. Then, by the inductive hypothesis, there is a condition *p* ∈ *G* such that *<sup>p</sup>* **forc**<sup>∞</sup> *<sup>ϕ</sup>*(*τ*). Therefore *<sup>p</sup>* **forc**<sup>∞</sup> <sup>∃</sup>*Bx <sup>ϕ</sup>*(*x*) by the choice of *<sup>τ</sup>*.

The case of ∃ *x ϕ*(*x*) is treated similarly.

#### *6.3. The Elementary Equivalence Theorem*

We begin **the proof of Theorem 4 of Section 4.4**, so let *d*, *Z*, *Z* , *x*<sup>0</sup> be as in the theorem.

**Step 1.** We assume w.l. o. g. that *x*<sup>0</sup> itself is the only parameter in the *Σ*<sup>1</sup> **<sup>n</sup>** formula Φ of Theorem 4. By Lemma 11, there exists, in **<sup>L</sup>**, a **<sup>Q</sup>**-full name *<sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**) such that *x*<sup>0</sup> = *τ*[*G*] and |*τ*| <sup>+</sup> <sup>⊆</sup> *<sup>d</sup>*. Thus, <sup>Φ</sup> is *<sup>ϕ</sup>*(*τ*[*G*]), where *<sup>ϕ</sup>*(·) is a parameter-free *<sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>** formula with a single free variable. Then |*ϕ*(*τ*)| <sup>+</sup> <sup>=</sup> <sup>|</sup>*τ*<sup>|</sup> <sup>+</sup> <sup>⊆</sup> *<sup>d</sup>*.

We also assume w.l. o. g. that the sets *Z*, *Z* satisfy the requirement that *Z Z and Z Z are infinite (countable) sets*. Indeed, otherwise, under the assumptions of Theorem 4, one easily defines a third set *Z* such that each of the pairs *Z*, *Z* and *Z* , *Z* still satisfies the assumptions of the theorem, and in addition, all four sets *Z Z* , *Z Z*, *Z Z* and *Z Z* are infinite. Please note that this argument necessarily requires that the complementary set I (*d* ∪ *Z* ∪ *Z* ) is infinite.

**Step 2.** We are going to reorganize the quantifier prefix of *ϕ*, in particular, by assigning the indices *Z* and *Z* to certain quantifiers, to reflect the relativization to classes **L**[*x*0, *G Z*] and **L**[*x*0, *G Z* ]. This is not an easy task because generally speaking there is no set *Z*<sup>0</sup> ⊆ I in **L** satisfying **L**[*x*0] = **L**[*G Z*0]. However, nevertheless we will define an <sup>L</sup>*Σ*<sup>1</sup> **<sup>n</sup>** formula, say *ψZ*(*v*), and then *ψ<sup>Z</sup>* (*v*) by the substitution of *Z* for *Z*, such that the following will hold:

(A) For any set *G* ⊆ **Q**, **Q**-generic over **L** :

$$\begin{aligned} &\varphi(\pi[G]) \text{ is true in } \mathbf{L}[\pi[G], G \upharpoonright Z] \quad \text{iff} \quad \psi^Z(\pi)[G] \text{ is true in } \mathbf{L}[G], \quad \text{and} \\ &\varphi(\pi[G]) \text{ is true in } \mathbf{L}[\pi[G], G \upharpoonright Z'] \quad \text{iff} \quad \psi^{Z'}(\pi)[G] \text{ is true in } \mathbf{L}[G]. \end{aligned}$$

(See Section 5.1 on the interpretation *ψ*[*G*] for any L-formula *ψ*.)

To explain this transformation, assume that **n** = 4 for the sake of brevity, and hence *ϕ*(*v*) has the form <sup>∃</sup> *<sup>x</sup>* <sup>∀</sup> *<sup>y</sup> <sup>ϑ</sup>*(*v*, *<sup>x</sup>*, *<sup>y</sup>*), where *<sup>ϑ</sup>* is a *<sup>Σ</sup>*<sup>1</sup> <sup>2</sup> formula. To begin with, we define

$$\psi\_1^Z(v) := \exists^Z \mathbf{x}' \exists \mathbf{x} \in \mathbf{L}[\mathbf{x}', v] \; \forall^Z y' \forall \, y \in \mathbf{L}[v, y'] \; \theta(v, \mathbf{x}, y) \; \tag{17}$$

and define *ψ<sup>Z</sup>* <sup>1</sup> (*v*) accordingly.

**Lemma 34.** *The formulas ψ<sup>Z</sup>* <sup>1</sup> *, <sup>ψ</sup><sup>Z</sup>* <sup>1</sup> *satisfy* (A)*.*

**Proof.** To prove the implication =⇒ , suppose that *ϕ*(*τ*[*G*]) holds in **L**[*τ*[*G*], *G Z*], so that there is a real *<sup>x</sup>*<sup>1</sup> <sup>∈</sup> *<sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>**[*τ*[*G*], *<sup>G</sup> <sup>Z</sup>*] satisfying <sup>∀</sup> *<sup>y</sup> <sup>ϑ</sup>*(*τ*[*G*], *<sup>x</sup>*1, *<sup>y</sup>*) in **<sup>L</sup>**[*τ*[*G*], *<sup>G</sup> <sup>Z</sup>*]. By a standard argument there is a real *<sup>x</sup>* <sup>∈</sup> *<sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>**[*G <sup>Z</sup>*] with *<sup>x</sup>*<sup>1</sup> <sup>∈</sup> *<sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>**[*τ*[*G*], *<sup>x</sup>* ]. We claim that these reals *x* and *x*<sup>1</sup> witness that *ψ<sup>Z</sup>* <sup>1</sup> (*τ*)[*G*] holds in **<sup>L</sup>**[*G*], that is, we have <sup>∀</sup>*Zy* <sup>∀</sup> *<sup>y</sup>* <sup>∈</sup> **<sup>L</sup>**[*τ*[*G*], *<sup>y</sup>* ] *ϑ*(*τ*[*G*], *x*1, *y*) in **L**[*G*].

Indeed, suppose that *<sup>y</sup>* <sup>∈</sup> *<sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>**[*G <sup>Z</sup>*] and *<sup>y</sup>* <sup>∈</sup> *<sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>**[*τ*[*G*], *<sup>y</sup>* ]. Then *y* ∈ **L**[*τ*[*G*], *G Z*], of course. Therefore *ϑ*(*τ*[*G*], *x*1, *y*) is true in **L**[*τ*[*G*], *G Z*] by the choice of *x*<sup>1</sup> . We conclude that *ϑ*(*τ*[*G*], *x*1, *y*) is true in **L**[*G*] as well by the Shoenfield absoluteness theorem, as *ϑ* is a *Σ*<sup>1</sup> <sup>2</sup> formula. The inverse implication is proved similarly. (Lemma)

Thus, the formulas *ψ<sup>Z</sup>* <sup>1</sup> , *<sup>ψ</sup><sup>Z</sup>* <sup>1</sup> do satisfy (A), but they are not <sup>L</sup>*Σ*<sup>1</sup>

**<sup>n</sup>** formulas as defined in Section 5.1, of course. It will take some effort to convert them to a <sup>L</sup>*Σ*<sup>1</sup> **<sup>n</sup>** form. We must recall some instrumentarium known in Gödel's theory of constructability of reals.


As a one more pre-requisite, we make use of a system of maps *<sup>f</sup> <sup>ξ</sup>* : *<sup>ω</sup><sup>ω</sup>* <sup>→</sup> *<sup>ω</sup><sup>ω</sup>* , *<sup>ξ</sup>* <sup>&</sup>lt; *<sup>ω</sup>*1, such that:

(a) if *<sup>x</sup>* <sup>∈</sup> *<sup>ω</sup><sup>ω</sup>* then **<sup>L</sup>**[*x*] <sup>∩</sup> *<sup>ω</sup><sup>ω</sup>* <sup>=</sup> { *<sup>f</sup> <sup>ξ</sup>* (*x*) : *<sup>ξ</sup>* <sup>&</sup>lt; *<sup>ω</sup>*1}, and

(b) there exist a *Σ*<sup>1</sup> <sup>1</sup> formula *<sup>S</sup>*(*x*, *<sup>y</sup>*, *<sup>w</sup>*) and a *<sup>Π</sup>*<sup>1</sup> <sup>1</sup> formula *P*(*x*, *y*, *w*) such that if *w* ∈ **WO** then *<sup>f</sup>*|*w*<sup>|</sup> (*x*) = *<sup>y</sup>* ⇐⇒ *<sup>S</sup>*(*x*, *<sup>y</sup>*, *<sup>w</sup>*) ⇐⇒ *<sup>P</sup>*(*x*, *<sup>y</sup>*, *<sup>w</sup>*) for all *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> *<sup>ω</sup><sup>ω</sup>* ,

see e.g., ([14] (Theorem 2.6)). Recall that *ω*<sup>1</sup> **<sup>L</sup>**[*G*] = *ω*<sup>1</sup> **<sup>L</sup>**[*G <sup>Z</sup>*] = *ω*<sup>2</sup> **<sup>L</sup>** by Lemma 22.

Now consider the formula

$$\begin{split} \psi\_2^Z(v) := \exists^Z \mathbf{x} \Big( \mathbf{w} \mathbf{o}((\mathbf{x})\_{\text{ev}}) \land \forall^Z y \, [\mathbf{w} \mathbf{o}((y)\_{\text{ev}}) \implies \\ \implies \theta(v, f\_{|(\mathbf{x})\_{\text{ev}}|}(v \ast (\mathbf{x})\_{\text{odd}}), f\_{|(y)\_{\text{ev}}|}(v \ast (y)\_{\text{odd}}))] \Big), \end{split} \tag{18}$$

and define *ψ<sup>Z</sup>* <sup>2</sup> (*v*) similarly.

We keep the global understanding that the quantifiers <sup>∃</sup>*<sup>Z</sup>* , <sup>∀</sup>*<sup>Z</sup>* are relativized to **<sup>L</sup>**[*G <sup>Z</sup>*] <sup>∩</sup> *<sup>ω</sup><sup>ω</sup>* .

**Lemma 35.** *The formulas ψ<sup>Z</sup>* <sup>1</sup> (*τ*[*G*]) *and <sup>ψ</sup><sup>Z</sup>* <sup>2</sup> (*τ*[*G*]) *are equivalent in* **<sup>L</sup>**[*G*]*, and the same for <sup>ψ</sup><sup>Z</sup>* <sup>1</sup> *and <sup>ψ</sup><sup>Z</sup>* 2 .

**Proof** (Lemma). To prove the implication <sup>=</sup><sup>⇒</sup> , assume that *<sup>ψ</sup><sup>Z</sup>* <sup>1</sup> (*τ*[*G*]) holds in **L**[*G*], and this is witnessed by reals *<sup>x</sup>* <sup>∈</sup> *<sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>**[*G <sup>Z</sup>*] and *<sup>x</sup>*<sup>1</sup> <sup>∈</sup> *<sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>**[*τ*[*G*], *<sup>x</sup>* ] = *<sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>**[*τ*[*G*]∗*<sup>x</sup>* ] satisfying <sup>∀</sup>*Zy* <sup>∀</sup> *<sup>y</sup>* <sup>∈</sup> **<sup>L</sup>**[*τ*[*G*], *<sup>y</sup>* ] *ϑ*(*τ*[*G*], *x*1, *y*) in **L**[*G*]. Please note that *ω*<sup>1</sup> **<sup>L</sup>**[*G <sup>Z</sup>*] = *ω*<sup>1</sup> **<sup>L</sup>**[*G*] = *ω***<sup>L</sup>** <sup>2</sup> by Lemma 9 (ii). It follows by (a) that there is an ordinal *ξ* < *ω*<sup>1</sup> **<sup>L</sup>**[*G <sup>Z</sup>*] with *<sup>x</sup>*<sup>1</sup> <sup>=</sup> *<sup>f</sup> <sup>ξ</sup>* (*τ*[*G*]∗*<sup>x</sup>* ), and then there is a real *w* ∈ **WO** ∩ **L**[*G Z*] with *ξ* = |*w*|.

Now let *<sup>x</sup>*˜ <sup>=</sup> *<sup>w</sup>*∗*<sup>x</sup>* , so that *<sup>w</sup>* = (*x*˜)ev , *<sup>x</sup>* = (*x*˜)odd , and *<sup>x</sup>*<sup>1</sup> <sup>=</sup> *<sup>f</sup>*|(*x*˜)ev<sup>|</sup> (*τ*[*G*]∗(*x*˜)odd). We claim that *x*˜ witnesses *ψ<sup>Z</sup>* <sup>2</sup> (*τ*[*G*]) in **<sup>L</sup>**[*G*]. Indeed, assume that *<sup>y</sup>*˜ <sup>∈</sup> *<sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>**[*G <sup>Z</sup>*] and *<sup>w</sup>* = (*y*˜)ev <sup>∈</sup> **WO**, *η* = |(*y*˜)ev|, and *y*<sup>1</sup> = *f <sup>η</sup>*(*τ*[*G*]∗(*y*˜)odd); we must prove that *ϑ*(*τ*[*G*], *x*1, *y*1) is true in **L**[*G*].

However, we have *y*<sup>1</sup> ∈ **L**[*τ*[*G*], *y* ] by construction, where *y* = (*y*˜)odd ∈ **L**[*G Z*] by the choice of *y*˜. Now it follows by the choice of *x*<sup>1</sup> that *ϑ*(*τ*[*G*], *x*1, *y*1) indeed holds, as required.

The proof of the inverse implication is similar. (Lemma)

Please note that the formula *ψ<sup>Z</sup>* <sup>2</sup> (*v*) can be converted to the following logically equivalent form:

$$\begin{split} \psi\_3^Z(v) := \exists^Z \mathbf{x} \forall^Z y \left[ \mathbf{w} \mathbf{o}((\mathbf{x})\_{\text{ev}}) \land \left( \mathbf{w} \mathbf{o}((y)\_{\text{ev}}) \implies \\ \implies \vartheta(v, f\_{|(\mathbf{x})\_{\text{ev}}|}(v\*(\mathbf{x})\_{\text{odd}}), f\_{|(y)\_{\text{ev}}|}(v\*(y)\_{\text{odd}})) \right) \right] . \end{split} \tag{19}$$

And here the kernel ... can be converted to a true *Σ*<sup>1</sup> <sup>2</sup> form, say *χ*(*v*, *x*, *y*), with the help of the formulas *<sup>S</sup>* and *<sup>P</sup>* of (b), and because **wo**(·) is *<sup>Π</sup>*<sup>1</sup> <sup>1</sup> and *<sup>ϑ</sup>* is *<sup>Σ</sup>*<sup>1</sup> <sup>2</sup> . This yields a <sup>L</sup>*Σ*<sup>1</sup> <sup>4</sup> formula *<sup>ψ</sup>Z*(*v*) :<sup>=</sup> <sup>∃</sup>*Zx* <sup>∀</sup>*Zy <sup>χ</sup>*(*v*, *<sup>x</sup>*, *<sup>y</sup>*), equivalent to *<sup>ψ</sup><sup>Z</sup>* <sup>1</sup> , and hence satisfying (A) by Lemmas 34 and 35, as required.

**Step 3.** Assuming that the formula Φ := *ϕ*(*τ*[*G*]) is true in **L**[*x*0, *G Z*], the transformed formula *<sup>ψ</sup>Z*(*τ*)[*G*] holds in **<sup>L</sup>**[*G*] by (A). By Lemma <sup>33</sup> there is a condition *<sup>p</sup>* <sup>∈</sup> *<sup>G</sup>* such that *<sup>p</sup>* **forc**<sup>∞</sup> *<sup>ψ</sup>Z*(*τ*) that is, there is an ordinal *<sup>ξ</sup>* <sup>&</sup>lt; *<sup>ω</sup>*<sup>2</sup> such that *<sup>p</sup>* **forc***<sup>ξ</sup> <sup>ψ</sup>Z*(*τ*)—then by definition *<sup>p</sup>* <sup>∈</sup> **<sup>Q</sup>**[**U***<sup>ξ</sup>* ]. We w.l. o. g. assume that *p* and *ζ* satisfy the following two requirements:


Please note that if *ξ* < *η* < *ω*<sup>2</sup> then still *p* **forc***<sup>η</sup> ψZ*(*τ*) by Lemma 28. Therefore, we can increase *ξ* below *ω*<sup>2</sup> so that the following holds:

(D) the sets *d*, *Z Z* , *Z Z* belong to **M***<sup>ξ</sup>* and are subsets of |**U***<sup>ξ</sup>* |.

**Step 4.** Now, to finalize the proof of Theorem 4, it suffices (by Lemma 33) to prove:

**Lemma 36.** *We have p* **forc**<sup>∞</sup> *ψ<sup>Z</sup>* (*τ*) *as well.* **Proof** (Lemma). Let *δ* = *d* ∪ (*Z* Δ *Z* ); then *δ* ∈ **M***<sup>ξ</sup>* by (D), and *δ* ⊆ |**U***<sup>ξ</sup>* |. There is a bijection *f* ∈ *M*, *f* : *δ* onto −→ *δ*, such that

(E) *f d* is the identity, *f* maps *Z Z* onto *Z Z* and vice versa.

Then, by (B), *f* maps |*p*| onto |*p*|. Let *π* be the trivial extension of *f* onto I : *π*(*ν*) = *ν* for *ν* ∈/ *δ*. Thus, *<sup>π</sup>* is coded in **<sup>M</sup>***<sup>ξ</sup>* in the sense of Lemma 29, and <sup>|</sup>*π*| ⊆ *<sup>δ</sup>* ⊆ |**U***<sup>ξ</sup>* <sup>|</sup>. We have *<sup>p</sup>* **forcM***<sup>ξ</sup>* **<sup>U</sup>***<sup>ξ</sup> <sup>ψ</sup>Z*(*τ*) by the choice of *ξ* , hence **U***<sup>ξ</sup>* ∈ **M***<sup>ξ</sup>* and *p* ∈ **P***<sup>ξ</sup>* = **Q**[**U***<sup>ξ</sup>* ] ∈ **M***<sup>ξ</sup>* . Moreover, *π·τ* = *τ* because |*τ*| <sup>+</sup> <sup>⊆</sup> *<sup>d</sup>* and *<sup>π</sup> <sup>d</sup>* is the identity by (E). It follows that *<sup>p</sup>* **forcM***<sup>ξ</sup> <sup>U</sup> <sup>ϕ</sup><sup>Z</sup>* (*τ*) by Lemma 29, where *U* = *π·***U***<sup>ξ</sup>* , *p* = *π· p*. Please note that *p* ∈ **Q**[*U* ], |*p* | <sup>+</sup> <sup>=</sup> <sup>|</sup>*p*<sup>|</sup> <sup>+</sup> , <sup>|</sup>*<sup>U</sup>* | = |**U***<sup>ξ</sup>* |, *U d* = **U***<sup>ξ</sup> d*, *p d* = *p d*. Also note that

(F) if *ν* ∈ |*p* | = |*p*| then the sets *F*<sup>∨</sup> *<sup>p</sup>* (*ν*), *F*<sup>∨</sup> *<sup>p</sup>* (*ν*) are i-similar by (C), (E).

We conclude, by Lemma 16, that there is a transformation *<sup>λ</sup>* = {*λν* }*ν*∈|**U***<sup>ξ</sup>* <sup>|</sup> ∈ **LIP**<sup>I</sup> ∩ **<sup>M</sup>***<sup>ξ</sup>* , such that *λ·U* = **U***<sup>ξ</sup>* , *λν* = the identity for all *ν* ∈ *d*, and *F*<sup>∨</sup> *<sup>p</sup>* (*ν*) = *F*<sup>∨</sup> *<sup>q</sup>* (*ν*) for all *ν* ∈ |*p*| = |*p* | = |*q*|, where *<sup>q</sup>* <sup>=</sup> *<sup>λ</sup>· <sup>p</sup>* <sup>∈</sup> **<sup>Q</sup>**[**U***<sup>ξ</sup>* ]. Then we have *<sup>q</sup>* **forcM***<sup>ξ</sup>* **<sup>U</sup>***<sup>ξ</sup> <sup>ψ</sup><sup>Z</sup>* (*τ*) by Lemma 30. Here *<sup>λ</sup>·ψ<sup>Z</sup>* (*τ*) = *ψ<sup>Z</sup>* (*τ*) by the choice of *λ*, because |*τ*| <sup>+</sup> <sup>⊆</sup> *<sup>d</sup>*. And *<sup>q</sup> <sup>d</sup>* <sup>=</sup> *<sup>p</sup> <sup>d</sup>* holds by the same reason.

It remains to derive *<sup>p</sup>* **forcM***<sup>ξ</sup>* **<sup>U</sup>***<sup>ξ</sup> <sup>ψ</sup><sup>Z</sup>* (*τ*) from *<sup>q</sup>* **forcM***<sup>ξ</sup>* **<sup>U</sup>***<sup>ξ</sup> <sup>ψ</sup><sup>Z</sup>* (*τ*). Please note that *p*, *q* satisfy (6) of Section 3.8 by construction, hence the transformation *Hqp* is defined. Moreover, the only name *τ* occurring in *ψ<sup>Z</sup>* (*τ*) satisfies |*τ*| <sup>+</sup> <sup>⊆</sup> *<sup>d</sup>*, and *<sup>π</sup> <sup>d</sup>* is the identity by (E). It follows that *<sup>τ</sup>* <sup>∈</sup> **SN***<sup>ω</sup> <sup>ω</sup>*(**Q**∗)*qp* , and *<sup>π</sup>·<sup>τ</sup>* <sup>=</sup> *<sup>τ</sup>*. We conclude that Lemma <sup>31</sup> is applicable. This yields *<sup>p</sup>* **forcM***<sup>ξ</sup>* **<sup>U</sup>***<sup>ξ</sup> <sup>ψ</sup><sup>Z</sup>* (*τ*), as required. (Lemma 36)

(Theorem 4 of Section 4.4)

(Theorem 1, see Section 4.5)

#### **7. Conclusions and Discussion**

In this study, the method of almost-disjoint forcing was employed to the problem of getting a model of **ZFC** in which the constructible reals are precisely the *Δ*<sup>1</sup> *<sup>n</sup>* reals, for different values *n* > 2. The problem appeared under no 87 in Harvey Friedman's treatise *One hundred and two problems in mathematical logic* [1], and was generally known in the early years of forcing, see, e.g., problems 3110, 3111, 3112 in an early survey [2] by A. Mathias. The problem was solved by Leo Harrington, as mentioned in [1,2] and a sketch of the proof mainly related to the case *n* = 3 in Harrington's own handwritten notes [3].

From this study, it is concluded that the hidden invariance technique (as outlined in Section 6.1) allows the solution of the general case of the problem (an arbitrary *n* ≥ 3 ), by providing a generic extension of **L** in which the constructible reals are precisely the *Δ*<sup>1</sup> *<sup>n</sup>* reals, for a chosen value *n* ≥ 3, as sketched by Harrington. The hidden invariance technique has been applied in recent papers [7,15–17] for the problem of getting a set theoretic structure of this or another kind at a pre-selected projective level. We may note here that the hidden invariance technique, as a true mathematical technique, also has multiple applications both in the physical and engineering fields. In this regard, we cite works [18,19] that have exploited this technique (albeit simplified) for engineering applications.

We continue with a brief discussion with a few possible future research lines.

1. Harvey Friedman completes [1] with a modified version of the above problem, defined as Problem 87 : find a model of

$$\textbf{ZFC} + \text{'' for any reals } \textbf{x}, \textbf{y}, \text{we have: } \textbf{x} \in \textbf{L}[\underline{y}] \implies \textbf{x} \text{ is } \Lambda\_3^1 \text{ in } \textbf{y}''. \tag{20}$$

This problem was also known in the early years of forcing, see, e.g., problem 3111 in [2]. Problem (20) was solved in the positive by René David [20], where the question is attributed to Harrington. So far it

is unknown whether this result generalizes to higher classes *Δ*<sup>1</sup> *<sup>n</sup>*, *<sup>n</sup>* <sup>≥</sup> 4, or *<sup>Δ</sup>*<sup>1</sup> <sup>∞</sup>, and whether it can be strengthened towards ⇐⇒ instead of =⇒ . This is a very interesting and perhaps difficult question.

2. Another question to be mentioned here is the following. Please note that in any extension of **L** satisfying Theorem 1, it is true that every universal *Σ*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> set *<sup>u</sup>* <sup>⊆</sup> *<sup>ω</sup>* <sup>×</sup> *<sup>ω</sup>* is by necessity *<sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> but non-*Δ*<sup>1</sup> **<sup>n</sup>**+1, and hence nonconstructible. This gives another proof of Theorem 3 in [7]. (It claims, for any **<sup>n</sup>** <sup>≥</sup> 2, the existence of a generic extension of **<sup>L</sup>** in which there is a nonconstructible *<sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> set *a* ⊆ *ω* whereas all *Δ*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> sets are constructible.) And the problem is, given **n** ≥ 2, to find a model in which

all *Δ*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> reals are constructible, but there exists a *<sup>Σ</sup>*<sup>1</sup> **<sup>n</sup>**+<sup>1</sup> nonconstructible real *u* ⊆ *ω*, *which satisfies* **V** = **L**[*u*].

Neither the model considered in Section 4.5 above, nor the model for ([7] (Theorem 3)), suffice to solve the problem, because these models in principle are incompatible with **V** = **L**[*u*] for a real *u*.

3. For any *n* < *ω*, let *D*1*<sup>n</sup>* be the set of all reals (here subsets of *ω* = {0, 1, 2, ...}), definable by a type-theoretic parameter-free formula whose quantifiers have types bounded by *n* from above. In particular, *D*10= arithmetically definable reals and *D*11= analytically definable reals. Alfred Tarski asked in [6] whether it is true that for a given *n* ≥ 1, the set *D*1*<sup>n</sup>* belongs to *D*2*<sup>n</sup>* , that is, is itself definable by a type-theoretic parameter-free formula whose quantifiers have types bounded by *n*. The axiom of constructibility **V** = **L** implies that *D*1*<sup>n</sup>* ∈/ *D*2*<sup>n</sup>* , so the problem is to find a generic model in which *D*1*<sup>n</sup>* ∈ *D*2*<sup>n</sup>* holds, and moreso the equality *D*1*<sup>n</sup>* = **L**∩ P(*ω*) holds. We believe that such a model can be constructed by an appropriate modification of the methods developed in this paper.

4. It will be interesting to apply the hidden invariance technique to some other forcing notions and coding systems (those not of the almost-disjoint type), such as in [21,22].

5. This is a rather technical question. One may want to consider a smaller extension **L**[*w*+[*G*]] instead of **L**[*w*+[*G*], *GW*] in Lemma 23. Claim (i) of Lemma 23 then holds for such a smaller model in virtue of the same argument as above. However, the proof of Claim (ii) of Lemma 23, as given above for **L**[*w*+[*G*], *GW*], does not go through for **L**[*w*+[*G*]]. The obstacle is that if we try to carry out the proof of Lemma 24 for **L**[*w*+[*G*]], then it may well happen that say *Z* = ∅, and then Theorem 4 is not applicable. It is an interesting problem to figure out whether in fact Claim (ii) of Lemma 23 holds in **L**[*w*+[*G*]].

**Supplementary Materials:** Table of contents and Index are available online at http://www.mdpi.com/2227- 7390/8/9/1477/s1.

**Author Contributions:** Conceptualization, V.K. and V.L.; methodology, V.K. and V.L.; validation, V.K. and V.L.; formal analysis, V.K. and V.L.; investigation, V.K. and V.L.; writing original draft preparation, V.K.; writing review and editing, V.K.; project administration, V.L.; funding acquisition, V.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Russian Foundation for Basic Research RFBR grant number 18-29-13037.

**Acknowledgments:** We thank the anonymous reviewers for their thorough review and highly appreciate the comments and suggestions, which significantly contributed to improving the quality of the publication.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **On the 'Definability of Definable' Problem of Alfred Tarski †**

## **Vladimir Kanovei \*,‡ and Vassily Lyubetsky \*,‡**

Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, Russia

**\*** Correspondence: kanovei@iitp.ru (V.K.); lyubetsk@iitp.ru (V.L.)

† The supplementary materials to this paper, published separately, include the Index.

‡ These authors contributed equally to this work.

Received: 14 November 2020; Accepted: 7 December 2020; Published: 14 December 2020 -

**Abstract:** In this paper we prove that for any *m* ≥ 1 there exists a generic extension of **L**, the constructible universe, in which it is true that the set of all constructible reals (here subsets of *ω*) is equal to the set **D**1*<sup>m</sup>* of all reals definable by a parameter free type-theoretic formula with types bounded by *m*, and hence the Tarski 'definability of definable' sentence **D**1*<sup>m</sup>* ∈ **D**2*<sup>m</sup>* (even in the form **D**1*<sup>m</sup>* ∈ **D**<sup>21</sup> ) holds for this particular *m*. This solves an old problem of Alfred Tarski (1948). Our methods, based on the almost-disjoint forcing of Jensen and Solovay, are significant modifications and further development of the methods presented in our two previous papers in this Journal.

**Keywords:** definability of definable; tarski problem; type theoretic hierarchy; generic models; almost disjoint forcing

## **MSC:** 03E15; 03E35

Dedicated to the 70-th Anniversary of A. L. Semenov.

#### **Contents**



#### **References 110**

#### **1. Introduction**

This paper continues our research project on the issues of definability in models of set theory, that was started in [1–3] among other papers, and most recently in [4,5] in this Journal. Questions of definability of mathematical objects were raised in the course of discussions on the foundations of mathematics, set theory, and the axiom of choice in the early twentieth century, such as, for instance, the famous discussion between Baire, Borel, Hadamard, and Lebesgue published in *Sinq lettres* [6]. Various aspects of definability in models of set theory have since remained the focus of work on the foundations of mathematics, see, for example, [7–13] among many important recent studies.

The topic of this paper goes back to the profound research by Alfred Tarski, who demonstrated in [14] that 'being definable' (in most general, unrestricted sense) is not a mathematically well-defined notion (see Murawski [15] on the history of this discovery and the role of Gödel, and Addison [16] on the modern perspective of the Tarski definability theory). More specificly, restricted notions of definability, in particular, type-theoretic definability, were considered by Tarski in [17] and later work in [18].

**Definition 1** (Tarski)**.** *If m*, *k* < *ω then* **D***km is the set of all elements of order k , definable by a parameter free type-theoretic formula of order m.*

Here elements of order 0 are just natural numbers (members of the set *ω* = {0, 1, 2, ...}), elements of order 1 are sets of natural numbers (commonly called *reals* in modern set theory), and generally, elements of order *k* + 1 ( *k* < *ω*) are arbitrary sets of elements of order *k* (see details in

Section 2.1 below). The order of a type-theoretic formula is the largest order of all its quantified and free variables. The notion of definability is taken in the form:

$$\mathbf{x}^{k} = \{ y^{k-1} \text{ of order } k-1 : \boldsymbol{\varrho}(y^{k-1}) \},\tag{1}$$

where the upper index routinely denotes the order of a variable or element.

#### *1.1. The Problem*

Investigating the definability properties of sets **D***km* , Tarski notes in [18] that **D***km* ∈ **D***k*+1,*m*+<sup>1</sup> . To prove this result, one can exploit the fact that the truth of all formulas of order *m* can be suitably expressed by a single formula of order *m* + 1. Using such a formula, one easily gets **D***km* ∈ **D***k*+1,*m*+<sup>1</sup> . Then Tarski turns to the question whether a stronger sentence **D***km* ∈ **D***k*+1,*<sup>m</sup>* holds. Tarski comes to the following conclusion (verbatim):

*the solution of the problem is (trivially) positive if k* = 0*; the solution is negative if k* ≥ 2*; in the (perhaps most interesting) case k* = 1 *the problem remains open.*

The negative result for *k* ≥ 2 (and *m* ≥ *k* − 1, to avoid trivialities) is obtained in [18] (page 110) essentially by virtue of the fact that countable ordinals admit a definable embedding into the set of all elements of order 2. This leaves:

$$\mathbf{D}\_{1m} \in \mathbf{D}\_{2m} \quad (m \ge 1) \tag{2}$$

as a major open problem in [18].

Tarski notes in [18], with a reference to Gödel's work on constructibility [19], that it seems:

#### *very unlikely that an affirmative solution of the problem is possible.*

Tarski does not elaborate on this point, but it is quite clear that the axiom of constructibility **V** = **L** (and even a weaker hypothesis, see Lemma 2 below) implies **D**1*<sup>m</sup>* ∈/ **D**2*<sup>m</sup>* for all *m* ≥ 1, and hence no proof of **D**1*<sup>m</sup>* ∈ **D**2*<sup>m</sup>* for even one single *m* ≥ 1 (the "affirmative solution" in Tarski's words), can be maintained in **ZFC**. In other words, the hypothesis:

$$\mathbf{D}\_{1m} \notin \mathbf{D}\_{2m} \text{ holds for all } m \ge 1$$

(the negative solution of (2) for all *m* ≥ 1 simultaneously) does not contradict the **ZFC** axioms. The problem of consistency of the affirmative sentences **D**1*<sup>m</sup>* ∈ **D**2*<sup>m</sup>* was left open in [18].

This paper is devoted to this problem of Alfred Tarski.

#### *1.2. Further Reformulations and Harrington's Statement*

The problem emerged once again in the early years of forcing, especially in the case *m* = 1 corresponding to analytic definability in second-order arithmetic. The early survey [20] by A. R. D. Mathias (the original typescript has been known to set theorists since 1968) contains Problem 3112, that requires finding a model of **ZFC** in which it is true that:

#### *the set of analytically definable reals is analytically definable*

that is, **D**<sup>11</sup> ∈ **D**<sup>21</sup> . Recall that *reals* in this context mean subsets of *ω*. Another problem there, P 3110, suggests a sharper form of this statement, namely; find a model in which it is true that

#### *analytically definable reals are precisely the constructible reals*

that is, **<sup>D</sup>**<sup>11</sup> <sup>=</sup> <sup>P</sup>(*ω*) <sup>∩</sup> **<sup>L</sup>**. The set <sup>P</sup>(*ω*) <sup>∩</sup> **<sup>L</sup>** of all constructible reals is (lightface) *<sup>Σ</sup>*<sup>1</sup> <sup>2</sup> , and hence **D**<sup>21</sup> , so that the equality **D**<sup>11</sup> = P(*ω*) ∩ **L** implies **D**<sup>11</sup> ∈ **D**<sup>21</sup> , that is the case *m* = 1 of the sentence (2).

Somewhat later, Problem 87 in Harvey Friedman's survey *One hundred and two problems in mathematical logic* [21] requires to prove that for each *n* in the domain 2 < *n* ≤ *ω* there is a model of:

$$\textbf{ZFC} + \text{\textquotedblleft the constructible reals are precisely the } \boldsymbol{\Delta}\_n^1 \text{ reals\textquotedblright.} \tag{3}$$

For *<sup>n</sup>* <sup>≤</sup> 2 this is definitely impossible by the Shoenfield absoluteness theorem. As *<sup>Δ</sup>*<sup>1</sup> *<sup>ω</sup>* is the same as **D**<sup>11</sup> = all analytically definable reals, the case *n* = *ω* in (3) is just a reformulation of **D**<sup>11</sup> = P(*ω*) ∩ **L**.

At the very end of [21], it is noted that Leo Harrington had solved problem (3) affirmatively. A similar remark, see in [20] (p. 166), a comment to P 3110. And indeed, Harrington's handwritten notes [22] present the following major result quoted here verbatim:

**Theorem 1** (Harrington [22] (p. 1))**.** *There are models of* **ZFC** *in which the set of constructible reals is, respectively, exactly the following set of reals*:

$$
\begin{array}{cccc}
\Delta^1\_{3\prime} & \Delta^1\_{4\prime} & \dots & \Delta^1\_{\omega} \\
\end{array} = \text{projective}, \ \Delta^m\_{n\prime} \ 1 \le n \le \omega, \ 2 \le m \le \omega \\
\dots
$$

We may note that *Δ*<sup>1</sup> *<sup>ω</sup>* = **D**<sup>11</sup> and generally *Δ<sup>m</sup> <sup>ω</sup>* = **D**1*<sup>m</sup>* for any *m* ≥ 2 in the context of Theorem 1. On the other hand the set <sup>P</sup>(*ω*) <sup>∩</sup> **<sup>L</sup>** of constructible reals is *<sup>Σ</sup>*<sup>1</sup> <sup>2</sup> , and hence **D**<sup>21</sup> . Therefore Theorem 1 implies the consistency of the affirmative sentences **D**<sup>1</sup> ∈ **D**<sup>2</sup> and **D**1*<sup>m</sup>* ∈ **D**2*<sup>m</sup>* for any particular value *m* ≥ 1, and hence shows that the Tarski problems considered are independent of **ZFC**.

Based on the almost-disjoint forcing tool of Jensen and Solovay [23], a sketch of a generic extension of **<sup>L</sup>**, in which it is true that *<sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>** <sup>=</sup> *<sup>Δ</sup>*<sup>1</sup> <sup>3</sup> , follows in [22] (pp. 2–4). Then a few sentences are added on page 5 of [22], which explain, without much going into details, as how Harrington planned to get some other models claimed by the theorem, in particular, a model in which *<sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>** <sup>=</sup> *<sup>Δ</sup>*<sup>1</sup> *<sup>n</sup>* holds for a given (arbitrary) natural index *<sup>n</sup>* <sup>&</sup>gt; 3, and a model in which *<sup>ω</sup><sup>ω</sup>* <sup>∩</sup> **<sup>L</sup>** <sup>=</sup> *<sup>Δ</sup>*<sup>1</sup> *<sup>ω</sup>* , where *Δ*<sup>1</sup> *<sup>ω</sup>* = - *<sup>n</sup> Δ*<sup>1</sup> *<sup>n</sup>* = **D**<sup>11</sup> (all analytically definable reals). This positively solves Problem 87 of [21], including the case *n* = *ω*, of course. Different cases of higher order definability are briefly observed in [22] (p. 5) as well.

Yet, for all we know, no detailed proofs have ever emerged in Harrington's published works. An article by Harrington, entitled "Consistency and independence results in descriptive set theory", which apparently might have contained these results among others, was announced in the References list in Peter Hinman's book [24] (p. 462) to appear in *Ann. of Math.*, 1978, but in fact this or a similar article has never been published in *Annals of Mathematics* or any other journal. Some methods sketched in [22] were later used in [25], but with respect to different questions and only in relation to the definability classes of the 2nd and 3rd projective level.

#### *1.3. The Main Theorem*

The goal of this paper is to present a complete proof of the following part of Harrington's statement in Theorem 1, related to the consistency of the Tarski sentence **D**1*<sup>m</sup>* ∈ **D**2*<sup>m</sup>* and the equality **D**1*<sup>m</sup>* = P(*ω*) ∩ **L**, strengthened by extra claims (ii) and (iii). This is **the main result** of this paper.

**Theorem 2.** *Let* **M** ≥ 1. *There is a generic extension of* **L** *in which it is true that*


Thus, for every particular **M** ≥ 1, there exists a generic extension of **L** in which the Tarski sentence **D**1**<sup>M</sup>** ∈ **D**2**<sup>M</sup>** *holds* whereas **D**1*<sup>n</sup>* ∈/ **D**2*<sup>n</sup>* for all other values *n* = **M**. We recall that **D**1**<sup>M</sup>** ∈ **D**2**<sup>M</sup>** *fails* in **L** itself for all **M**, see above.

**Corollary 1.** *If* **M** ≥ 1 *then the sentence* **D**1**<sup>M</sup>** ∈ **D**2**<sup>M</sup>** *is undecidable in* **ZFC***, even in the presence of* ∀ *n* = **M** (**D**1*<sup>n</sup>* ∈/ **D**2*n*)*.*

This paper is dedicated to the proof of Theorem 2. This will be another application of the methods sketched by Harrington and developed in detail in our previous papers [4,5] in this Journal, but here modified and further developed for the purpose of a solution to the Tarski problem.

We may note that problems of construction of models of set theory in which this or another effect is obtained at a certain prescribed definability level (not necessarily the least possible one) are considered in modern set theory, see e.g., Problem 9 in [26] (Section 9) or Problem 11 in [27] (page 209). Some results of this type have recently been obtained in set theory, namely:


Theorem 2 of this paper naturally extends this research line.

#### *1.4. Structure of the Proof*

To define a model for Theorem 2, we employ the product of two forcing notions. The first forcing **<sup>C</sup>** is a Cohen-style collapse forcing that adjoins a generic collapse map *<sup>ζ</sup>* : *<sup>ω</sup>* onto −→ **Ξ** = P(*ω*) ∩ **L**, Section 2.7. The collapse is necessary since any model for Theorem 2 has to satisfy the inequality *ω***<sup>L</sup>** <sup>1</sup> < *ω*<sup>1</sup> .

The second forcing notion has the form of the product **<sup>P</sup><sup>Ω</sup>** <sup>=</sup> <sup>∏</sup>*n*,*i*<*<sup>ω</sup>* **<sup>P</sup>Ω**(*n*, *<sup>i</sup>*) <sup>∈</sup> **<sup>L</sup>**, where each factor **PΩ**(*n*, *i*) is an almost-disjoint type forcing determined by a set:

$$\mathbb{U}^{\square}(n,i) \in \mathbf{L}\_{\prime} \quad \mathbb{U}^{\square}(n,i) \subseteq \mathbf{Fun}\_{\square} = (\mathbb{U}^{\square}) \cap \mathbf{L}\_{\prime}$$

dense in **Fun<sup>Ω</sup>** , where **Ω** = *ω***<sup>L</sup> <sup>M</sup>** and **M** ≥ 1 is the number we are dealing with in Theorem 2. This forcing **<sup>P</sup><sup>Ω</sup>** adjoins an according system of generic sets *<sup>S</sup>*(*n*, *<sup>i</sup>*) <sup>⊆</sup> **Seq<sup>Ω</sup>** = (**Ω**<**Ω**) <sup>∩</sup> **<sup>L</sup>**, such that:

(∗) if *f* ∈ **Fun<sup>Ω</sup>** in **L** then *S*(*n*, *i*) covers *f* (that is, *f ξ* ∈ *S*(*n*, *i*) for unbounded-many *ξ* < **Ω**) iff *<sup>f</sup>* <sup>∈</sup>/ **<sup>U</sup>Ω**(*n*, *<sup>i</sup>*) (Lemma 15).

Basically any *system U* ∈ **L** of dense sets *U*(*n*, *i*) ⊆ **Fun<sup>Ω</sup>** defines a similar product forcing **P**[*U*] = ∏*n*,*i*<*<sup>ω</sup> P*[*U*(*n*, *i*)] ∈ **L** (see Section 3.2). Forcing notions of the form **P**[*U*] satisfy certain chain and distributivity conditions in **L** (Lemma 14), that imply some general properties of related generic extensions (Lemmas 15 and 16).

The key system **U<sup>Ω</sup>** is defined in Section 4.4 (Definition 6, on the base of Theorem 6 in Section 4.2), in the form of componentwise union **U<sup>Ω</sup>** = *<sup>α</sup>*<**Ω**<sup>⊕</sup> **U<sup>Ω</sup>** *<sup>α</sup>* , i.e., **UΩ**(*n*, *i*) = - *<sup>α</sup>*<**Ω**<sup>⊕</sup> **U<sup>Ω</sup>** *<sup>α</sup>* (*n*, *i*) for all *n*, *i* < *ω*, where **Ω**<sup>⊕</sup> = *ω***<sup>L</sup> <sup>M</sup>**+<sup>1</sup> is the **<sup>L</sup>**-cardinal next to **<sup>Ω</sup>**, and the systems **<sup>U</sup><sup>Ω</sup>** *<sup>α</sup>* ∈ **L** are:


We apply a diamond-based argument in Section <sup>4</sup> to ensure that the resulting system **<sup>U</sup><sup>Ω</sup>** <sup>∈</sup> **<sup>L</sup>** has its different *slices* {**UΩ**(*n*, *<sup>i</sup>*)}*i*<*<sup>ω</sup>* (*<sup>n</sup>* <sup>&</sup>lt; *<sup>ω</sup>*) satisfying different definability and inner genericity requirements (Theorem 6 in Section 4.2), so that the descriptive complexity and the level of inner genericity (or completeness) of *n*th 'slice' tends to infinity with *n* → ∞. This is a major novelty of the construction.

Then we consider the key product forcing notion **P<sup>Ω</sup>** = **P**[**UΩ**] = ∏*n*,*i*<*<sup>ω</sup>* **PΩ**(*n*, *i*). We extend **L** by a collapse-generic map *ζ* : *ω* onto −→ P(*ω*) ∩ **L** to **L**, as above, and define the partial product **<sup>P</sup><sup>Ω</sup>** *<sup>w</sup>* <sup>=</sup> <sup>∏</sup>*<sup>n</sup>*,*i*∈*<sup>w</sup>* **<sup>P</sup>Ω**(*n*, *<sup>i</sup>*) <sup>∈</sup> **<sup>L</sup>**[*ζ*] as a forcing notion in **<sup>L</sup>**[*ζ*], where:

$$\mathfrak{w} = \mathfrak{w}[\mathfrak{f}] = \{ \langle n, i \rangle : n \in \omega \land i \in \mathfrak{f}(n) \}.$$

Adjoining a (**P<sup>Ω</sup>** *w*)-generic set *G* to **L**[*ζ*], we get a model **L**[*ζ*, *G*] for Theorem 2. In particular, if *x* = *ζ*(*n*) ∈ P(*ω*) ∩ **L**, then *x* is definable in **L**[*ζ*, *G*] by means of the equivalence:

$$\exists i \in \mathbf{x} \iff \exists S \subseteq \mathbf{Seq}\_{\Pi} \forall f \in \mathbf{Fun}\_{\Pi} \left( S \text{ covers } f \text{ iff } f \notin \mathbb{U}^{\Box}(n, i) \right), \tag{4}$$

in which the implication =⇒ follows from (∗) via *S* = *S*(*n*, *i*) (note that *S*(*n*, *i*) ∈ **L**[*ζ*, *G*] since *n*, *i* ∈ *w* in case *i* ∈ *x* = *ζ*(*n*)), whereas the inverse implication ⇐= is based on the completeness properties of the system **U<sup>Ω</sup>** . It also takes some effort to check that the right-hand side of (4) really defines a **D**1**<sup>M</sup>** relation in **L**[*ζ*, *G*]; for that purpose Theorem 3 is proved beforehand in Section 2.3.

To prove that, conversely, every *x* ∈ **D**1**<sup>M</sup>** in **L**[*ζ*, *G*] belongs to **L**, we introduce *forcing approximations* in Section 5, a forcing-like relation used to prove the elementary equivalence theorem. Its key advantage is the invariance under some transformations, including the permutations of the index set <sup>I</sup> , see Section 6.5. The actual forcing notion **<sup>P</sup><sup>Ω</sup>** <sup>=</sup> **<sup>P</sup>**[**UΩ**] is absolutely not invariant under permutations of I , but the **M**-completeness property, maintained through the inductive construction of **U<sup>Ω</sup>** in **L**, allows us to prove that the auxiliary forcing is in the same relation to the truth in **P<sup>Ω</sup>** -generic extensions, as the true **P<sup>Ω</sup>** -forcing relation (Theorem 10). We call this construction *hidden invariance* (see Section 6.1), and this is the other major novelty of this paper.

Finally, Section 6 presents the proof of the invariance theorem (Theorem 11), with the help of forcing approximations, and thereby completes the proof of Theorem 2.

The flowchart of the proof can be seen in Figure 1 on page 6.

**Figure 1.** Flowchart of the proof of Theorem 2.

#### **2. Preliminaries**

This Section contains several definitions and results that will be very instrumental in the proof of Theorem 2.

#### *2.1. Definability Issues*

Beginning with the type-theoretic definability, we recall some details of Tarski's constructions from [18]. The type-theoretic language deals with variables *xk*, *yk*, ... of orders *k* < *ω*, and includes the Peano arithmetic language for order 0 and the atomic predicate <sup>∈</sup> of membership used as *<sup>x</sup><sup>k</sup>* <sup>∈</sup> *<sup>y</sup>k*+<sup>1</sup> . The *order* of a formula *ϕ* is equal to the highest order of all variables in *ϕ*. Variables of each order *k* can be substituted with elements of the corresponding iteration:

> P*k*(*ω*) = P(P(... P(*ω*)...)) *k* times the powerset operation P(·) , the set of all elements of order *k*

of the powerset operation. In particular, P0(*ω*) = *ω* (natural numbers), P1(*ω*) = P(*ω*) (the reals), <sup>P</sup>2(*ω*) = <sup>P</sup>(P(*ω*)) (sets of reals), and so on. Accordingly each quantifier <sup>∃</sup> *<sup>x</sup><sup>k</sup>* , <sup>∀</sup> *<sup>x</sup><sup>k</sup>* in a type-theoretic formula is naturally relativized to P*k*(*ω*), and the truth of a closed type-theoretic formula (with or without parameters) is understood in the sense of such a relativization.

If *<sup>k</sup>*, *<sup>m</sup>* <sup>&</sup>lt; *<sup>ω</sup>*, *<sup>k</sup>* <sup>≥</sup> 1, then, by Definition 1, **<sup>D</sup>***km* is the set of all *<sup>x</sup><sup>k</sup>* <sup>∈</sup> <sup>P</sup>*k*(*ω*), definable in the form:

$$\mathfrak{a}^k = \{ y^{k-1} \in \mathcal{P}^{k-1}(\omega) : \mathfrak{q}(y^{k-1}) \},$$

by a parameter free formula *<sup>ϕ</sup>* of order <sup>≤</sup> *<sup>m</sup>*; thus **<sup>D</sup>***km* <sup>⊆</sup> <sup>P</sup>*k*(*ω*).

**Remark 1.** *We will occasionally extend the definition of* **D***km to binary relations, especially in the case k* = 1*. Namely a set <sup>X</sup>* <sup>⊆</sup> <sup>P</sup>*k*−1(*ω*) <sup>×</sup> <sup>P</sup>*k*−1(*ω*) *belongs to* **<sup>D</sup>***km if it is definable by a parameter free formula of order* ≤ *m with two free variables.*

In matters of ∈**-definability**, we refer to e.g., [31] (Part B, 5.4), or [32] (Chapter 13) on the Lévy hierarchy of <sup>∈</sup>-formulas and definability classes *<sup>Σ</sup><sup>H</sup> <sup>n</sup>* , *Π<sup>H</sup> <sup>n</sup>* , *Δ<sup>H</sup> <sup>n</sup>* for any transitive set *H*. In particular,

*Σ<sup>H</sup> <sup>n</sup>* = all sets *X* ⊆ *H*, definable in *H* by a parameter-free *Σ<sup>n</sup>* formula; *Σn*(*H*) = all sets *X* ⊆ *H* definable in *H* by a *Σ<sup>n</sup>* formula with any sets in *H* as parameters.

Something like *Σ<sup>H</sup> <sup>n</sup>* (*x*), *<sup>x</sup>* <sup>∈</sup> *<sup>H</sup>*, means that only *<sup>x</sup>* is admitted as a parameter, while *<sup>Σ</sup><sup>H</sup> <sup>n</sup>* (*P*), where *<sup>P</sup>* <sup>⊆</sup> *<sup>H</sup>*, means that all *<sup>x</sup>* <sup>∈</sup> *<sup>P</sup>* can be parameters. Collections like *<sup>Π</sup><sup>H</sup> <sup>n</sup>* , *Π<sup>H</sup> <sup>n</sup>* (*x*), *Π<sup>H</sup> <sup>n</sup>* (*P*) are defined similarly, and *Δ<sup>H</sup> <sup>n</sup>* = *Σ<sup>H</sup> <sup>n</sup>* <sup>∩</sup> *<sup>Π</sup><sup>H</sup> <sup>n</sup>* , etc. These definitions usually work with transitive sets of the form:

*<sup>H</sup>* <sup>=</sup> **<sup>H</sup>**<sup>κ</sup> <sup>=</sup> {*<sup>x</sup>* : card (TC (*x*)) <sup>&</sup>lt; <sup>κ</sup>}, where <sup>κ</sup> is an infinite cardinal,

and TC is *the transitive closure*. In particular, **HC** = **H***ω*<sup>1</sup> , all heredidarily-countable sets.

#### *2.2. Constructibility Issues*

As usual, **L** is the constructible universe, and <**<sup>L</sup>** will denote the Gödel wellordering of **L**. Let κ be an infinite regular cardinal. The following are well-known facts in the theory of constructibility, see e.g., [33] and Lemma 6.3 ff in [31] (Section B.5):

<sup>1</sup>◦. The set **<sup>H</sup>**<sup>κ</sup> <sup>∩</sup> **<sup>L</sup>** belongs to *<sup>Σ</sup>***H**<sup>κ</sup> <sup>1</sup> and is equal to (**H**κ)**<sup>L</sup>** = **<sup>L</sup>**<sup>κ</sup> .


<sup>4</sup>◦. The map *<sup>x</sup>* −→ pr *<sup>x</sup>* <sup>=</sup> {*<sup>y</sup>* : *<sup>y</sup>* <sup>&</sup>lt;**<sup>L</sup>** *<sup>x</sup>*} : (**H**κ)**<sup>L</sup>** <sup>→</sup> (**H**κ)**<sup>L</sup>** is *<sup>Δ</sup>*(**H**κ)**<sup>L</sup>** <sup>1</sup> as well.

The last statement implies the following useful definability estimation.

<sup>5</sup>◦. Assume that *<sup>m</sup>* <sup>≥</sup> 1 and *<sup>P</sup>* <sup>⊆</sup> (**H**κ)**<sup>L</sup>** <sup>×</sup> (**H**κ)**<sup>L</sup>** is *<sup>Δ</sup>*(**H**κ)**<sup>L</sup>** *<sup>m</sup>* . If *<sup>x</sup>* <sup>∈</sup> *<sup>D</sup>* <sup>=</sup> {*<sup>x</sup>* <sup>∈</sup> (**H**κ)**<sup>L</sup>** : <sup>∃</sup> *y P*(*x*, *<sup>y</sup>*)}, then let *yx* <sup>∈</sup> (**H**κ)**<sup>L</sup>** be the <sup>&</sup>lt;**L**-least witness. Then *<sup>P</sup>* <sup>=</sup> {*x*, *yx* : *<sup>x</sup>* <sup>∈</sup> *<sup>D</sup>*} ⊆ *<sup>P</sup>* is *<sup>Δ</sup>*(**H**κ)**<sup>L</sup>** *<sup>m</sup>* as well.

Indeed *y* = *yx* is equivalent to *P*(*x*, *y*) ∧ ∀ *z* ∈ pr *y* ¬ *P*(*x*, *z*), where:

$$\begin{aligned} \forall z \in \operatorname{pr} y \rightharpoonup P(x, z) &\quad \Longleftrightarrow \quad \exists Z \left( Z = \operatorname{pr} y \land \forall z \in Z \neg P(x, z) \right) \\ &\quad \Longleftrightarrow \quad \forall Z \left( Z = \operatorname{pr} y \implies \forall z \in Z \neg P(x, z) \right) \end{aligned}$$

and the bounded quantifiers ∀ *z* ∈ *Z* do not influence the definability class.

We proceed with several easy and rather known lemmas.

**Lemma 1.** *Assume that <sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> <sup>P</sup>(*ω*) <sup>∩</sup> **<sup>L</sup>** *and <sup>y</sup>* <sup>&</sup>lt;**<sup>L</sup>** *<sup>x</sup> . Then <sup>y</sup>* <sup>∈</sup> *<sup>Δ</sup>*<sup>1</sup> <sup>2</sup>(*x*)*, and hence if y* ∈ **D**1*<sup>n</sup> , n* ≥ 1*, or y* ∈ **D**<sup>1</sup> *, then x* ∈ **D**1*<sup>n</sup> , resp., x* ∈ **D**<sup>1</sup> *as well.*

**Proof.** By the Shoenfield absoluteness, it suffices to prove that *<sup>y</sup>* <sup>∈</sup> *<sup>Δ</sup>*<sup>1</sup> <sup>2</sup>(*x*) is true in **L**.

**We argue in L.** Let κ = *ω*<sup>1</sup> , so that **H**κ = (**H**κ)**<sup>L</sup>** = **HC** (hereditarily countable). The set:

$$P = \left\{ \langle z, f \rangle : z \subseteq \omega \land f : \omega \to \mathcal{P}(\omega) \land \mathtt{ran} \, f = \mathtt{pr} \, z \right\}.$$

belongs to *Δ***HC** <sup>1</sup> by 4◦ since:

$$\begin{aligned} \mathsf{ran}\,f = \mathsf{pr}\,z \quad \Longleftrightarrow \quad \exists\,u\Big(u = \mathsf{pr}\,z \land \forall\,n\,(f(n)\in u) \land \forall\,z' \in u\,\exists\,n\,(f(n)=z')\Big) \\ \Longleftrightarrow \quad \forall\,u\Big(u = \mathsf{pr}\,z \implies \forall\,n\,(f(n)\in u) \land \forall\,z' \in u\,\exists\,n\,(f(n)=z')\Big). \end{aligned}$$

Let *fz* be the <sup>&</sup>lt;**L**-least *<sup>f</sup>* such that *<sup>z</sup>*, *<sup>f</sup>*<sup>∈</sup> *<sup>P</sup>*; then *<sup>P</sup>* <sup>=</sup> {*z*, *fz* : *<sup>z</sup>* <sup>⊆</sup> *<sup>ω</sup>*} is *<sup>Δ</sup>***HC** <sup>1</sup> by 5◦. It follows that *fx* is *Δ***HC** <sup>1</sup> (*x*) (with *<sup>x</sup>* as the only parameter). Therefore, as *<sup>y</sup>* <sup>&</sup>lt;**<sup>L</sup>** *<sup>x</sup>* , we have *<sup>y</sup>* <sup>∈</sup> *<sup>Δ</sup>***HC** <sup>1</sup> (*x*) because *<sup>y</sup>* <sup>=</sup> *fx*(*n*) for some *<sup>n</sup>*. It follows that *<sup>y</sup>* <sup>∈</sup> *<sup>Δ</sup>*<sup>1</sup> <sup>2</sup>(*x*). (See e.g., [34] (p. 281) on this translation result.)

**Remark 2** (Essentially Tarski [18])**.** *If n* <sup>≥</sup> <sup>1</sup> *and <sup>ω</sup>***<sup>L</sup>** <sup>1</sup> = *ω*<sup>1</sup> *then* **D**1*<sup>n</sup>* ∈/ **D**2*<sup>n</sup> .*

**Proof.** If *ω***<sup>L</sup>** <sup>1</sup> = *ω*<sup>1</sup> then the set *Y* = P(*ω*) ∩ **L** is uncountable. On the other hand *X* = **D**1*<sup>n</sup>* is countable, hence *Z* = *Y X* = ∅. Note that *Y* ∈ **D**<sup>21</sup> by 3◦ above. It follows that if *X* ∈ **D**2*<sup>n</sup>* then *Z* belongs to **D**2*<sup>n</sup>* , too, and then the <**L**-least element *z*<sup>0</sup> of the set *Z* belongs to **D**1*<sup>n</sup>* because <**<sup>L</sup>** is **D**<sup>21</sup> on *Y* still by 3◦. However *z*<sup>0</sup> ∈/ *X* = **D**1*<sup>n</sup>* by construction. This is a contradiction.

**Lemma 2.** *If* 1 ≤ *n* < *m* < *ω and* **D**1*<sup>m</sup>* ⊆ **L***, then* **D**1*<sup>n</sup>* ∈/ **D**2*<sup>n</sup> .*

**Proof.** We have **D**1*<sup>n</sup>* - **D**1*<sup>m</sup>* since *n* < *m*. Therefore **D**1*<sup>n</sup>* - **D**1*<sup>m</sup>* ⊆ *Y* = P(*ω*) ∩ **L**. If, to the contrary, **D**1*<sup>n</sup>* ∈ **D**2*<sup>n</sup>* , then the set *Y* **D**1*<sup>n</sup>* belongs to **D**2*<sup>n</sup>* as well since *Y* ∈ **D**<sup>21</sup> by 3◦ above. We conclude that the <**L**-least element *y*<sup>0</sup> ∈ *Y* **D**1*<sup>n</sup>* belongs to **D**1*<sup>n</sup>* , because <**<sup>L</sup>** is **D**<sup>21</sup> on *Y* by 3◦. This is a contradiction since *z*<sup>0</sup> ∈/ **D**1*<sup>n</sup>* by construction.

## *2.3. Type-Theoretic Definability vs.* ∈*-Definability*

It occurs that the definability classes in sets of the form **H**κ correspond to the Tarski definability classes, in the sense of the following theorem:

**Theorem 3.** *Assume that the generalized continuum hypothesis* 2*<sup>ϑ</sup>* = *ϑ*<sup>+</sup> *holds for all infinite cardinals ϑ* < *ωm*−<sup>1</sup> *. If m* ≥ 1 *and x* ⊆ *ω, then x is* **D**1*<sup>m</sup> if x is* ∈*-definable in* **H***ω<sup>m</sup> .*

In case *m* = 1 (then **H***ω<sup>m</sup>* = **H***ω*<sup>1</sup> = **HC** and the GCH premice is vacuous), this result was explicitly mentioned, in [34] (p. 281), a detailed proof see e.g., [32] (Lemma 25.25).

**Proof.** The GCH premice of the theorem is equivalent to <sup>P</sup>*m*(*ω*) <sup>⊆</sup> **<sup>H</sup>***ω<sup>m</sup>* . This implies <sup>=</sup><sup>⇒</sup> : if *x* ∈ **D**1*<sup>m</sup>* then *x* is surely ∈-definable in **H***ω<sup>m</sup>* .

The inverse implication takes more effort. We have to somehow *model* the ∈-structure of **H***ω<sup>m</sup>* in **<sup>D</sup>**1*<sup>m</sup>* . For this purpose, if *<sup>k</sup>* <sup>&</sup>lt; *<sup>ω</sup>* and *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> <sup>P</sup>*k*(*ω*) then define a quasi-pair *<sup>x</sup>*, *<sup>y</sup> <sup>k</sup>* <sup>∈</sup> <sup>P</sup>*k*(*ω*) by induction as follows. If *<sup>k</sup>* <sup>=</sup> 0, so that *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> *<sup>ω</sup>*, then put *<sup>x</sup>*, *<sup>y</sup>* <sup>0</sup> <sup>=</sup> <sup>2</sup>*<sup>x</sup>* · <sup>3</sup>*<sup>y</sup>* <sup>∈</sup> *<sup>ω</sup>*. If *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> <sup>P</sup>*k*+1(*ω*) then put *<sup>x</sup>*, *<sup>y</sup> <sup>k</sup>*+<sup>1</sup> <sup>=</sup> {0, *<sup>x</sup> <sup>k</sup>* : *<sup>x</sup>* <sup>∈</sup> *<sup>x</sup>*} ∪ {1, *<sup>y</sup> <sup>k</sup>* : *<sup>y</sup>* <sup>∈</sup> *<sup>y</sup>*} ∈ <sup>P</sup>*k*+1(*ω*). Note that elements 0 <sup>=</sup> <sup>∅</sup> and 1 <sup>=</sup> {∅} belong to every type-theoretic level <sup>P</sup>*k*(*ω*). It can be easily established by induction that if *<sup>x</sup>*, *<sup>y</sup>*, *<sup>a</sup>*, *<sup>b</sup>* <sup>∈</sup> <sup>P</sup>*k*(*ω*) and *<sup>x</sup>*, *<sup>y</sup> <sup>k</sup>* <sup>=</sup> *<sup>a</sup>*, *<sup>b</sup> <sup>k</sup>* then *<sup>x</sup>* <sup>=</sup> *<sup>a</sup>* and *<sup>y</sup>* <sup>=</sup> *<sup>b</sup>* .

Following [32] (25.13), we associate, with each *<sup>r</sup>* <sup>∈</sup> <sup>P</sup>*m*(*ω*), a binary relation <sup>E</sup>*<sup>r</sup>* defined so that:

$$\propto \mathbb{E}\_r \, y \quad \text{iff} \quad \mathbf{x}, y \in M = \mathcal{P}^{m-1}(\omega) \text{ and } \langle \mathbf{x}, y \rangle^{m-1} \in r. \text{ } $$

on the set *<sup>M</sup>* <sup>=</sup> <sup>P</sup>*m*−1(*ω*). Let WFE0 contain all sets *<sup>r</sup>* <sup>∈</sup> <sup>P</sup>*m*(*ω*) such that <sup>E</sup>*<sup>r</sup>* is an extensional well-founded relation on |*r*| = {0}∪{*x* ∈ *M* : ∃ *y* ∈ *M* (*x* E*<sup>r</sup> y* ∨ *y* E*<sup>r</sup> x*)}, with the additional property that 0 is the only *top element* of |*r*|, that is, 0 E*<sup>r</sup> x* holds for no *x* ∈ |*r*|. If *r* ∈ WFE0 then let *π<sup>r</sup>* be the unique 1-1 map defined on |*r*| and satisfying *πr*(*x*) = {*πr*(*y*) : *y* E*<sup>r</sup> x*} for all *x* ∈ |*r*| — the *transitive collapse*. We put *F*(*r*) = *πr*(0).

Under our assumptions, *F* is a map from WFE0 onto **H***ω<sup>m</sup>* , ∈-definable in **H***ω<sup>m</sup>* .

One easily proves that WFE0 belongs to **D***mm* , that is, it is type-theoretically definable with quantifiers only over order levels ≤ *m*. Moreover the binary relations EQ, IN defined on WFE0 by:

$$r \uplus \mathbb{Q} \neq q \text{ iff } \ F(r) = F(q) \, , \quad \text{and} \quad r \uplus \mathbb{N} \neq q \text{ iff } \ F(r) \in F(q) \, ,$$

belong to **D***mm* as well. Namely, let a *bisimulation for r*, *q* ∈ WFE0 be any binary relation B ⊆ |*r*|×|*q*| satisfying 0 B 0 and, for all *x* ∈ |*r*| and *y* ∈ |*q*|,

$$\text{x } \mathsf{B} \text{ y } \quad \text{iff} \quad \forall \text{x'} \exists \text{y'} \left(\text{x'} \to \text{x} \implies \text{y'} \to\_{\mathsf{F}} \mathsf{y} \land \text{x'} \to \text{y'}\right) \\ \quad \land \forall \text{y'} \exists \text{x'} \left(\text{y'} \to\_{\mathsf{F}} \mathsf{y} \implies \text{x'} \to\_{\mathsf{F}} \mathsf{x} \land \text{x'} \to \text{y'}\right) \\ \quad \dots$$

Then, on the one hand, *<sup>F</sup>*(*r*) = *<sup>F</sup>*(*q*) iff there exists a bisimulation for *<sup>r</sup>*, *<sup>q</sup>* iff there exists *<sup>b</sup>* <sup>∈</sup> <sup>P</sup>*m*(*ω*) such that E*<sup>b</sup>* is a bisimulation for *r*, *q* . On the other hand, we can express the property "E*<sup>b</sup>* is a bisimulation for *r*, *q*" by a type-theoretic formula with quantifiers only over orders ≤ *m*, by suitably replacing pairs ·, · with quasipairs ·, ·*m*−<sup>1</sup> .

To treat IN, we have to only change 0 B 0 above to ∃ *y*<sup>0</sup> ∈ |*q*|(0 B *y*<sup>0</sup> ∧ *y*<sup>0</sup> E*<sup>q</sup>* 0).

Finally if *<sup>n</sup>* <sup>&</sup>lt; *<sup>ω</sup>* then let *rn* <sup>=</sup> {*i*, *<sup>j</sup> <sup>m</sup>*−<sup>1</sup> : <sup>1</sup> <sup>≤</sup> *<sup>i</sup>* <sup>&</sup>lt; *<sup>j</sup>* <sup>≤</sup> *<sup>n</sup>*} ∪ {*i*, 0*<sup>m</sup>*−<sup>1</sup> : <sup>1</sup> <sup>≤</sup> *<sup>i</sup>* <sup>≤</sup> *<sup>n</sup>*}, so that *rn* ∈ WFE0 and *F*(*rn*) = *n*.

And now let *x* = {*n* < *ω* : **H***ω<sup>m</sup>* |= *ϕ*(*n*)} ⊆ *ω* be ∈-definable in **H***ω<sup>m</sup>* by a parameter free formula *ϕ*(·). Then we have *x* = {*n* < *ω* : Φ(*rn*)}, where Φ is obtained from *ϕ* by substitution of EQ for = and IN for ∈ and relativization of all quantifiers to WFE0 . This proves *x* ∈ **D**1*<sup>m</sup>* .

#### *2.4. Reduction to the Powerset Definability*

Let be the wellordering of **Ord** × **Ord** defined so that *ξ*, *η ξ* , *η*  iff:

$$\langle \max \{ \zeta^{\iota}, \eta \} , \zeta^{\iota}, \eta \rangle \lessapprox\_{\mathsf{T} \mathsf{a} \mathsf{x}} \langle \max \{ \zeta^{\iota}, \eta^{\iota} \} , \zeta^{\iota}, \eta^{\iota} \rangle$$

lexicographically. Let **<sup>p</sup>** : **Ord** <sup>×</sup> **Ord** onto −→ **Ord** be the order preserving map: *ξ*, *η ξ* , *η*  iff **p**(*ξ*, *η*) ≤ **p**(*ξ* , *η* )—the canonical pairing function. Let **p**<sup>1</sup> and **p**<sup>2</sup> be the inverse functions, so that *α* = **p**(**p**1(*α*), **p**2(*α*)) for all *α*.

**Lemma 3** (routine)**.** *If* <sup>Ω</sup> *is an infinite cardinal and <sup>κ</sup>* <sup>=</sup> <sup>Ω</sup><sup>+</sup> *, then* **<sup>p</sup>** *maps* <sup>Ω</sup> <sup>×</sup> <sup>Ω</sup> *onto* <sup>Ω</sup> *bijectively, and the restriction* **<sup>p</sup>** (<sup>Ω</sup> <sup>×</sup> <sup>Ω</sup>) *is constructible and <sup>Δ</sup>***H***<sup>κ</sup>* <sup>1</sup> *.*

Now we prove another reduction-type definability theorem.

**Theorem 4.** *If* <sup>Ω</sup> *is a regular cardinal,* <sup>κ</sup> <sup>=</sup> <sup>Ω</sup><sup>+</sup> *, <sup>X</sup>*,*<sup>Y</sup>* <sup>⊆</sup> *<sup>ω</sup>, and <sup>X</sup> is* <sup>∈</sup>*-definable in* **<sup>H</sup>**<sup>κ</sup> *with <sup>Y</sup> as the only parameter, then X is* ∈*-definable in the structure* P(Ω); ∈, **p** *with Y as the only parameter.*

**Proof (sketch).** If *x* ⊆ Ω then let E *<sup>x</sup>* = {*ξ*, *η* : *ξ*, *η* < Ω ∧ **p**(*ξ*, *η*) ∈ *x*} be a binary relation on its domain <sup>|</sup>*x*<sup>|</sup> <sup>=</sup> dom <sup>E</sup> *<sup>x</sup>* <sup>∪</sup> ran <sup>E</sup> *<sup>x</sup>* . Following the proof of Theorem 3, let WFE <sup>0</sup> contain all sets *x* ⊆ Ω such that E *<sup>x</sup>* is an extensional well-founded relation on |*x*|, with the additional property that 0 ∈ |*x*| and 0 is the only *top element* of |*x*|, that is, 0 E *<sup>x</sup> ξ* holds for no *ξ* ∈ |*x*|. If *x* ∈ WFE <sup>0</sup> then let *ϕ<sup>x</sup>* be the unique 1-1 map defined on |*x*| and satisfying *ϕx*(*ξ*) = {*ϕx*(*η*) : *η* E *<sup>x</sup> ξ*} for all *ξ* ∈ |*x*|—the *transitive collapse*. We put *F* (*x*) = *ϕx*(0); *F* is a map from WFE <sup>0</sup> onto **H**κ, ∈-definable in **H**κ.

Both WFE <sup>0</sup> and the binary relations EQ , IN defined on WFE <sup>0</sup> by:

$$\text{ax } \mathbb{E} \mathbb{Q}' y \text{ iff } F'(\mathbf{x}) = F'(y), \quad \text{and} \quad \mathbf{x} \, \mathbb{M}' y \text{ iff } F'(\mathbf{x}) \in F'(y) \,\,\,\,\,\,$$

are ∈-definable in P(Ω); ∈, **p** by the same bisimulation argument as in the proof of Theorem 3. Finally if *n* < *ω* then let *xn* = {**p**(*i*, *j*) : 1 ≤ *i* < *j* ≤ *n*}∪{**p**(*i*, 0) : 1 ≤ *i* ≤ *n*}, so that *xn* ∈ WFE <sup>0</sup> and *F* (*xn*) = *n*.

Now let *X* = {*n* < *ω* : **H**κ |= Φ(*n*,*Y*)} ⊆ *ω* be ∈-definable in **H**κ by a formula *ϕ*(·,*Y*). Then we have *X* = {*n* < *ω* : Φ (*xn*)}, where Φ is obtained from Φ by the substitution of EQ for = and IN for ∈ and relativization of all quantifiers to WFE <sup>0</sup> . This proves the theorem.

#### *2.5. A Useful Result in Forcing Theory*

We remind that, by [32] (Chapter 15), if κ is an infinite ordinal, then a forcing notion *P* = *P* ; ≤:


We will make use of the following general result in forcing theory.

**Lemma 4.** *Assume that, in* **<sup>L</sup>***, <sup>ϑ</sup>* <sup>&</sup>lt; **<sup>Ω</sup>** <sup>=</sup> *<sup>ϑ</sup>*<sup>+</sup> *are regular infinite cardinals, and <sup>Q</sup>*, *<sup>P</sup>* <sup>∈</sup> **<sup>L</sup>** *are forcing notions, Q satisfies* **Ω***-CC in* **L***, and P is ϑ-closed in* **L***. Assume that F*, *G is a pair* (*Q* × *P*)*-generic over* **L***. Then,*


**Proof.** (i) Consider any sequence {*D<sup>α</sup>* }*α*<*<sup>ϑ</sup>* in **L**[*F*] of open dense sets *D<sup>α</sup>* ⊆ *P*. Prove that their intersection is dense. Let *<sup>p</sup>* <sup>∈</sup> *<sup>P</sup>*. Then *<sup>D</sup>* <sup>=</sup> {*α*, *<sup>p</sup>* : *<sup>α</sup>* <sup>&</sup>lt; *<sup>ϑ</sup>* <sup>∧</sup> *<sup>p</sup>* <sup>∈</sup> *<sup>D</sup><sup>α</sup>* } belongs to **<sup>L</sup>**[*F*]. Therefore there is a name *t* ∈ **L**, *t* ⊆ *Q* × (*ϑ* × *P*), satisfying *D* = *t*[*F*]. Then *D<sup>α</sup>* = *tα*[*F*] for all *α*, where *t<sup>α</sup>* = {*q*, *p* : *q*,*α*, *p* ∈ *t*}. There exists a condition *q*<sup>0</sup> ∈ *F* which *Q*-forces

(A) "*tα*[*F*] is open dense in *P*"

over **L** for every *α* < *ϑ*. We can w.l. o. g. assume that 1*<sup>Q</sup>* forces (A), otherwise replace *Q* by *Q* = {*q* ∈ *Q* : *q q*<sup>0</sup> }. Under this assumption, we have the following:

(B) If *α* < *ϑ*, *p* ∈ *P*, and *q* ∈ *Q* then there exist *q* ∈ *Q* and *p* ∈ *P* such that *q q* , *p p*, and *q Q*-forces *p* ∈ *tα*[*F*] over **L**.

Now we prove a stronger fact:

(C) If *γ* < *ϑ* and *p* ∈ *P*, then there is *p* ∈ *P*, *p p*, such that 1*<sup>Q</sup>* forces *p* ∈ *tγ*[*F*] over **L**.

Indeed, arguing in **L**, and using (B) and the assumption that *P* is *ϑ*-closed, we can define a decreasing sequence {*p<sup>α</sup>* }*α*<*<sup>η</sup>* of conditions in *P*, where *η* < **Ω**, and a sequence {*q<sup>α</sup>* }*α*<*<sup>η</sup>* of conditions in *Q*, such that *q*<sup>0</sup> = *q* , *q<sup>α</sup>* is incompatible with *q<sup>β</sup>* whenever *α* = *β*, and each *q<sup>α</sup> Q*-forces *pα*+<sup>1</sup> ∈ *tγ*[*F*]. Note that the construction really has to stop at some *η* < **Ω** otherwise we have an antichain in *Q* of cardinality **Ω**. Thus *A* = {*q<sup>α</sup>* : *α* < *η*} is a maximal antichain, and on the other hand, as *P* is *ϑ*-closed and *<sup>η</sup>* <sup>&</sup>lt; **<sup>Ω</sup>** <sup>=</sup> *<sup>ϑ</sup>*<sup>+</sup> , there is a condition *<sup>p</sup>* <sup>∈</sup> *<sup>P</sup>* satisfying *<sup>p</sup> <sup>p</sup><sup>α</sup>* for all *<sup>α</sup>* <sup>&</sup>lt; *<sup>η</sup>* . Then every *<sup>q</sup>* <sup>∈</sup> *A Q*forces *p* ∈ *tγ*[*F*] by construction, therefore, as *A* is a maximal antichain, *q* witnesses (C).

To accomplish the proof of (i), we define, using (C), a decreasing sequence {*p<sup>γ</sup>* }*γ*<*<sup>ϑ</sup>* ∈ **L** of conditions in *<sup>P</sup>*, such that *<sup>p</sup>*<sup>0</sup> *<sup>p</sup>* and, for any *<sup>γ</sup>* <sup>&</sup>lt; *<sup>ϑ</sup>*, 1*<sup>Q</sup>* forces *<sup>p</sup>γ*+<sup>1</sup> <sup>∈</sup> *<sup>t</sup>γ*[*F*] over **<sup>L</sup>**. Once again, there is a condition *<sup>p</sup>* ∈ *<sup>P</sup>*, *<sup>p</sup> <sup>p</sup><sup>γ</sup>* for all *<sup>γ</sup>*. Then 1*<sup>Q</sup>* forces *<sup>p</sup>* ∈ *<sup>t</sup>γ*[*F*] for all *<sup>γ</sup>*, hence *<sup>p</sup>* ∈ *<sup>γ</sup> D<sup>γ</sup>* , as required.

Finally, as *Q* is **Ω**-CC in **L**, **Ω** remains a cardinal in **L**[*F*]. Then, as *P* is *ϑ*-distributive in **L**[*F*], we obtain (ii) and (iii) by standard arguments.

#### *2.6. Definable Names*

Let *Q* ∈ **L** be any forcing notion. It is well known (see, e.g., Lemma 2.5 in Chapter B.4 of [31]) that if *F* ⊆ *Q* is a *Q*-generic filter over **L**, *X* ∈ **L**, and *Y* ∈ **L**[*F*], *Y* ⊆ *X*, then there is a set *t* ∈ **L**, *t* ⊆ *Q* × *X*, such that:

$$\mathcal{Y} = \mathfrak{t}[\mathcal{Q}] := \{ \mathfrak{x} \in X : \exists \, q \in F \, (\langle q, \mathfrak{x} \rangle \in t) \} \; ; \; \mathcal{Y}$$

such a *t* is called a *Q*-*name* (for *Y*), whereas *t*[*G*] is the *G*-*valuation*, or *G*-*interpretation* of *t*. There is a more comprehensive system of names and valuations, which involves all sets *Y* in generic extensions, not only those included in the groung model, see e.g., Chapter IV in [35], but it will not be used in this paper. The next theorem claims that in certain cases such a name *t* as above can be chosen of nearly the same definability level as the set *Y* itself.

**Theorem 5.** *Assume that Q* ∈ **L** *is any forcing, F* ⊆ *Q is Q-generic over* **L***,* κ > *ω is a cardinal in* **L**[*F*] (hence, in **<sup>L</sup>**, too)*, n* <sup>≥</sup> <sup>1</sup>*, H* = (**H**κ)**<sup>L</sup>** *, H*[*F*]=(**H**κ)**L**[*F*] *, and Y* <sup>∈</sup> **<sup>L</sup>**[*F*]*, Y* <sup>⊆</sup> *H . Then,*


**Proof.** To prove (i) note that *H* = *H*[*F*] ∩ **L**. But the formula " *x* is contructible" is *Σ*<sup>1</sup> [31] (Part B, 5.4). It follows that *H* is *ΣH*[*F*] <sup>1</sup> . Now the result is clear: We formally relativize, to the *<sup>Σ</sup>H*[*F*] <sup>1</sup> set *H*, all quantifiers in the *Σ<sup>n</sup>* definition of *Y* in *H*, getting a *Σ<sup>n</sup>* definition of *Y* in *H*[*F*].

To prove (ii), assume that *Q* ∈ *H*. We utilize a more complex system of representation of sets in **L**[*F*], affecting all these sets, not just subsets of sets in **L**. We take it from [36]. Inductively on the <sup>∈</sup>-rank rk (*a*), each set *<sup>a</sup>* is mapped to the set *<sup>K</sup>*(*a*) = {*K*(*b*) : <sup>∃</sup> *<sup>q</sup>* <sup>∈</sup> *<sup>F</sup>* (*<sup>q</sup>*, *<sup>b</sup>*<sup>∈</sup> *<sup>a</sup>*)} (depends on *<sup>F</sup>*!). The next lemma continues the proof of Theorem 5.

**Lemma 5.** *H*[*F*] = {*K*(*a*) : *a* ∈ *H*}*.*

**Proof.** From right to left, an elementary induction argument works. Prove it from left to right. Induction by the <sup>∈</sup>-rank rk (*x*), for each *<sup>x</sup>* <sup>∈</sup> *<sup>H</sup>*[*F*] we define a set *ax* <sup>∈</sup> *<sup>H</sup>* such that *<sup>x</sup>* <sup>=</sup> *<sup>K</sup>*(*ax*). If *<sup>x</sup>* <sup>=</sup> <sup>∅</sup>, then *ax* <sup>=</sup> <sup>∅</sup> will do. Assume that rk (*x*) <sup>&</sup>gt; 0 and *ay* is already defined for each *<sup>y</sup>* <sup>∈</sup> *<sup>x</sup>* . The set *A* = {*ay* : *y* ∈ *x*} ∈ *L*[*F*], *A* ⊆ *H* has cardinality < κ in *L*[*F*]. Moreover, there is a set *B* ∈ **L**, *B* ⊆ *H*, of cardinality ≤ κ in **L**, such that *A* ⊆ *B*. (Indeed, *H* ∈ **L** has cardinality κ in **L**. Let {*t<sup>α</sup>* }*α*<<sup>κ</sup> be a constructible enumeration of elements of *<sup>H</sup>*. As card *<sup>A</sup>* <sup>&</sup>lt; <sup>κ</sup> strictly, there is *<sup>γ</sup>* <sup>&</sup>lt; <sup>κ</sup> such that *A* ⊆ *B* = {*t<sup>α</sup>* : *α* < *γ*}. The set *B* is as required.)

According to the above, we have *A* = *τ*[*F*] for some *τ* ∈ **L**, *τ* ⊆ *Q* × *B*. Then *τ* ∈ *H*. On the other hand, it is easy to check that *x* = {*K*(*b*) : *b* ∈ *A*} = *K*(*τ*), that is, you can take *ax* = *τ* . This ends the proof of the lemma.

In continuation of the proof of Theorem 5(ii), we introduce, following [36], the forcing relation *q ϕ* (where *q* ∈ *Q*) by induction on the logical complexity of the formula *ϕ* (a closed formula with parameters in *H*); it corresponds to *H*[*F*] as a *Q*-generic extension of *H*. Below is the partial order on *Q*, and *q q* means that *q* is a stronger condition.

(I) *q a* ∈ *b* iff ∃ *q* , *c* ∈ *b* (*q q* ∧ *q a* = *c*); (II) *q a* = *b* iff ∃ *q* , *c* ∈ *b* (*q q* ∧ *q c* ∈/ *a*) or ∃ *q* , *c* ∈ *a* (*q q* ∧ *q c* ∈/ *b*); (III) *q* ¬ *ϕ* iff ¬ ∃ *q* (*q q* ∧ *q ϕ*); (IV) *q ϕ* ∨ *ψ* iff *q ϕ* or *q ψ*; (V) *q* ∃ *x* ∈ *b ϕ*(*x*) iff ∃ *q* , *c* ∈ *b* (*q q* ∧ *q ϕ*(*c*)); (VI) *q* ∃ *x ϕ*(*x*) iff ∃ *c* ∈ *H* (*q ϕ*(*c*)).

This definition assumes that some logical connectives are expressed in a certain way via other connectives. For each parameter free formula *ϕ*(*x*1,..., *xk*), define a set:

$$F\_{\emptyset} = \left\{ (\emptyset, a\_1, \dots, a\_k) : a\_1, \dots, a\_k \in H \land q \in Q \land q \Vdash \emptyset \,\,\middle|\, q(a\_1, \dots, a\_k) \right\}.$$

**Lemma 6.** *If k* > <sup>1</sup> *and <sup>ϕ</sup> is a <sup>Σ</sup><sup>k</sup> formula, then F<sup>ϕ</sup> is <sup>Σ</sup><sup>H</sup> <sup>k</sup>* ({*Q*}) (*Q* is allowed as a sole parameter)*.*

**Proof.** All quantifiers of definitions (I)–(V) are bounded either by the set *Q* ∈ *H*, or by a set of the form *<sup>Q</sup>* <sup>×</sup> *<sup>a</sup>* , where still *<sup>a</sup>* <sup>∈</sup> *<sup>H</sup>*. Therefore it is not difficult to show that *<sup>F</sup><sup>ϕ</sup>* <sup>∈</sup> *<sup>Σ</sup><sup>H</sup>* <sup>1</sup> for any bounded formula *ϕ*. (The sole unbounded quantifier will express the existence of a full description of all subformulas of the form *a* ∈ *b* , *a* = *b* , that appear in accordance with (I)–(III).) Induction on *k* proves the result.

The next lemma is similar to the Truth Lemma as in [36], so the proof is omitted.

**Lemma 7.** *Let* Φ *be a closed formula with parameters in H, and* Φ *obtained from* Φ *so that each a* ∈ *H is replaced by K*(*a*)*. Then* Φ *is true in H*[*F*] *iff there exists q* ∈ *F such that q* Φ*.*

Let us finish the proof of Theorem 5(ii). Let *Y* ∈ *Σn*(*H*[*F*]), *Y* ⊆ *H*. There is a parameter free *Σ<sup>n</sup>* formula *ϕ*(·, ·), and a parameter *y* ∈ *H*[*F*], such that *X* = {*x* ∈ *H* : *ϕ*(*x*, *y*) holds in *H*[*F*]}. For each *<sup>x</sup>* ∈ *<sup>H</sup>*, we define the set *<sup>x</sup>*˘ ∈ *<sup>H</sup>* by induction, so that ∅˘ = ∅, and if *<sup>x</sup>* = ∅ then *<sup>x</sup>*˘ = {*q*, *<sup>z</sup>*˘ : *q* ∈ *Q* ∧ *z* ∈ *x*}. Then *K*(*x*˘) = *x* for all *x* . It follows by Lemma 7 that:

$$X = \{ \mathbf{x} \in H : \exists \, q \in F \, (q \Vdash \!q (\check{\mathbf{x}}, b)) \} = t[F] \,,$$

where *b* ∈ *H* is such that *y* = *K*(*b*) (exists by Lemma 5), whereas:

$$\mathcal{I} = \left\{ \langle q, \mathbf{x} \rangle : q \in Q \land \mathbf{x} \in H \land q \Vdash \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/\!\/\!\/\!\/\!\/\!\/\,\,\big|$$

Finally, note that the function *<sup>x</sup>* −→ *<sup>x</sup>*˘ belongs to *<sup>Δ</sup><sup>H</sup>* <sup>1</sup> ({*Q*}). We conclude that *t* ∈ *Σn*(*H*) by Lemma 6, as required. This completes the proof of Theorem 5.

#### *2.7. Collapse Forcing*

We conclude from Lemma 2 that the construction of any generic extension of **L**, in which **<sup>D</sup>**1*<sup>n</sup>* <sup>∈</sup> **<sup>D</sup>**2*<sup>n</sup>* holds for some *<sup>n</sup>* <sup>≥</sup> 1, has to involve a collapse of *<sup>ω</sup>***<sup>L</sup>** <sup>1</sup> down to *ω*, explicitly or implicitly. To set up such a collapse in a technically convenient form, we let **Ξ** = P(*ω*) ∩ **L** be the set of all constructible sets *<sup>x</sup>* <sup>⊆</sup> *<sup>ω</sup>*, and let **<sup>C</sup>** <sup>=</sup> **<sup>Ξ</sup>**<*<sup>ω</sup>* . Thus **<sup>C</sup>** <sup>∈</sup> **<sup>L</sup>** is the ordinary Cohen-style collapse forcing that makes **Ξ** (and *ω***<sup>L</sup>** <sup>1</sup> as well) countable in **C**-generic extensions. The choice of **Ξ** as the collapse domain, instead of *ω***<sup>L</sup>** <sup>1</sup> , is made by technical reasons that will be clear below. Note that **C** adjoins generic maps *ζ* : *ω* onto −→ **<sup>Ξ</sup>** to **<sup>L</sup>**. A map *<sup>ζ</sup>* <sup>∈</sup> **<sup>Ξ</sup>***<sup>ω</sup>* is **<sup>C</sup>**-generic over **<sup>L</sup>** iff the set *<sup>G</sup><sup>ζ</sup>* <sup>=</sup> {*<sup>e</sup>* <sup>∈</sup> **<sup>C</sup>** : *<sup>e</sup>* <sup>⊂</sup> *<sup>ζ</sup>*} is **C**-generic in the usual sense.

**Lemma 8** (Routine)**.** *If <sup>ζ</sup>* <sup>∈</sup> **<sup>Ξ</sup>***<sup>ω</sup> is* **<sup>C</sup>***-generic over* **<sup>L</sup>** *then <sup>ω</sup>***L**[*ζ*] *<sup>ξ</sup>* = *<sup>ω</sup>***<sup>L</sup>** *<sup>ξ</sup>*+<sup>1</sup> *for all ξ* ∈ **Ord***.*

The representation result, as in the beginning of Section 2.6, takes the following form: If *<sup>ζ</sup>* <sup>∈</sup> **<sup>Ξ</sup>***<sup>ω</sup>* is **C**-generic over **L**, *X* ∈ **L**, and *Y* ∈ **L**[*ζ*], *Y* ⊆ *X*, then there is a set *t* ∈ **L**, *t* ⊆ **C** × *X*, such that:

$$\mathcal{Y} = \mathfrak{t}[\mathfrak{f}] := \{ \mathfrak{x} \in X : \exists \, \mathfrak{e} \in G\_{\mathfrak{f}} \, (\langle \mathfrak{e}, \mathfrak{x} \rangle \in t) \} \,\, \forall \, \mathfrak{f}$$

such a *t* is called a **C**-*name* (for *Y*).

Theorem <sup>5</sup> is applicable for *<sup>Q</sup>* <sup>=</sup> **<sup>C</sup>** and any **<sup>L</sup>**-cardinal <sup>κ</sup> <sup>≥</sup> *<sup>ω</sup>***<sup>L</sup>** <sup>2</sup> , whereas if *ξ* ∈ **Ord**, *ξ* ≥ 1, then Lemma 4 is applicable for *Q* = **C**, *ϑ* = *ω***<sup>L</sup>** *<sup>ξ</sup>* , **<sup>Ω</sup>** = *<sup>ω</sup>***<sup>L</sup>** *<sup>ξ</sup>*+<sup>1</sup> , and any forcing *P* ∈ **L**, *ϑ*-complete in **L**.

#### **3. Almost Disjoint Forcing, Uncountable Version**

Here we introduce the main coding tool used in the proof of Theorem 2, an uncountable version of almost disjoint forcing of Jensen–Solovay [23].

#### *3.1. Introduction to almost Disjoint Forcing*

**Definition 2.** *Fix an uncountable successor* **L***-cardinal* **Ω** = *ω***<sup>L</sup>** *<sup>μ</sup>*+<sup>1</sup> *. The value of* **Ω** *will be specified in Section 4.5 with respect to the integer* **M** *of Theorem 2, namely,* **Ω** = *ω***<sup>L</sup> <sup>M</sup>** *, but until then we will view* **Ω** *as an arbitrary successor* **L***-cardinal.*

*We put* **Ω** = *ω***<sup>L</sup>** *<sup>μ</sup> and* **Ω**<sup>⊕</sup> = *ω***<sup>L</sup>** *<sup>μ</sup>*+<sup>2</sup> *. Here* <sup>⊕</sup> *, resp., mean the next, resp., previous* **L***-cardinal, which may not be true cardinals in generic extensions of* **L***.*

*We finally put:*

$$\mathbb{H} = (\mathbf{H} \mathbb{I} \mathbb{I}^{\oplus})^{\mathbf{L}} = \{ \mathbf{x} \in \mathbf{L} \colon \mathbf{card} \left( \mathbf{T} \mathbb{C} \left( \mathbf{x} \right) \right) < \mathbb{I}^{\oplus} \text{ in } \mathbf{L} \}. \tag{5}$$

*Moreover if* **L**[*G*] *is a generic extension of* **L** *then we define:*

$$\mathbb{H}[\mathbf{G}] = (\mathbf{H} \mathbb{I}^{\oplus})^{\mathbf{L}[\mathbf{G}]} = \{ \mathbf{x} \in \mathbf{L}[\mathbf{G}] : \mathbf{card}\left(\mathbf{TC}\left(\mathbf{x}\right)\right) < \mathbb{I}^{\oplus} \text{ in } \mathbf{L}[\mathbf{G}] \}. \tag{6}$$

*provided* **Ω**⊕ *remains a cardinal in* **L**[*G*]*.*


**Definition 3** (in **L**)**.** ∗ *P***<sup>Ω</sup>** *is the set of all pairs p* = *Sp* ; *Fp* ∈ **L** *of sets Fp* ⊆ **Fun<sup>Ω</sup>** *, Sp* ⊆ **Seq<sup>Ω</sup>** *of cardinality strictly less than* **Ω** *in* **L***. Elements of* ∗ *P***<sup>Ω</sup>** *will be called (forcing) conditions.*

*If p*, *q* ∈ <sup>∗</sup> *P***<sup>Ω</sup>** *then p* ∧ *q* = *Sp* ∪ *Sq* ; *Fp* ∪ *Fq ; a condition in* <sup>∗</sup> *P***<sup>Ω</sup>** *.*

*Let p*, *q* ∈ <sup>∗</sup> *P***<sup>Ω</sup>** *. Define q p ( q is stronger as a forcing condition) iff Sp* ⊆ *Sq* , *Fp* ⊆ *Fq , and the difference Sq Sp does not intersect F*∨ *<sup>p</sup> , that is, Sq* ∩ *F*<sup>∨</sup> *<sup>p</sup>* = *Sp* ∩ *F*<sup>∨</sup> *<sup>p</sup> . Here F*<sup>∨</sup> *<sup>p</sup>* = (*Fp*)<sup>∨</sup> *.*

**Lemma 9** (in **L**)**.** *The sets* **Seq<sup>Ω</sup>** *,* **Fun<sup>Ω</sup>** *,* <sup>∗</sup> *<sup>P</sup>***<sup>Ω</sup>** *belong to* **<sup>L</sup>** *and* card (**SeqΩ**) = **<sup>Ω</sup>** *while* card (**FunΩ**) = card ∗ *P***<sup>Ω</sup>** = **Ω**<sup>⊕</sup> *in* **L***.*

Clearly *q p* iff *Sp* ⊆ *Sq* , *Fp* ⊆ *Fq* , and *Sq* ∩ *F*<sup>∨</sup> *<sup>p</sup>* = *Sp* ∩ *F*<sup>∨</sup> *p* .

**Lemma 10** (in **L**)**.** *Conditions p*, *q* ∈ <sup>∗</sup> *P***<sup>Ω</sup>** *are compatible in* <sup>∗</sup> *P***<sup>Ω</sup>** *iff* 1) *Sq Sp does not intersect F*<sup>∨</sup> *<sup>p</sup> , and* 2) *Sp Sq does not intersect F*∨ *<sup>q</sup> . Therefore any p*, *q* ∈ *P*<sup>∗</sup> *are compatible in P*<sup>∗</sup> *iff p* ∧ *q p and p* ∧ *q q.*

**Proof.** If (1), (2) hold then *p* ∧ *q p* and *p* ∧ *q q* , thus *p*, *q* are compatible.

If *u* ⊆ **Fun<sup>Ω</sup>** then put *P*[*u*] = {*p* ∈ <sup>∗</sup> *P***<sup>Ω</sup>** : *Fp* ⊆ *u*}. Thus if *u* ∈ **L** then *P*[*u*] ∈ **L**.

Any conditions *p*, *q* ∈ *P*[*u*] are compatible in *P*[*u*] iff they are compatible in <sup>∗</sup> *P***<sup>Ω</sup>** iff *p* ∧ *q* = *Sp* ∪ *Sq* ; *Fp* ∪ *Fq* ∈ *P*[*u*] satisfies both (*p* ∧ *q*) *p* and (*p* ∧ *q*) *q* . Thus we say that conditions *p*, *q* ∈ <sup>∗</sup> *P***<sup>Ω</sup>** are compatible (or incompatible) without an indication which set *P*[*u*] containing *p*, *q* is considered.

**Lemma 11** (in **<sup>L</sup>**)**.** *Let* <sup>∅</sup> <sup>=</sup> *<sup>u</sup>* <sup>⊆</sup> **Fun<sup>Ω</sup>** *. Then it is true in* **<sup>L</sup>** *that* card *<sup>P</sup>*[*u*] <sup>≤</sup> **<sup>Ω</sup>**<sup>⊕</sup> *, and the forcing notion P*[*u*] *satisfies* **Ω**⊕*-CC, and is* **Ω***-closed, hence* **Ω***-distributive. Moreover P*[*u*] *satisfies* **Ω**⊕*-CC in any generic extension* **L**[*H*] *of* **L***, in which* **Ω**⊕ *remains a cardinal.*

**Proof.** The closed/distributive claim is obvious on the base of the cardinality restrictions in Definition 3. To prove the **Ω**⊕-CC claim, argue in **L**[*H*]. If *p* = *q* belong to an antichain *A* ⊆ *P*[*u*] then *Sp* = *Sq* by Lemma 10. Let *M* = {*Sp* : *p* ∈ <sup>∗</sup> *<sup>P</sup>***Ω**} <sup>=</sup> all subsets *<sup>S</sup>* <sup>⊆</sup> **Seq<sup>Ω</sup>** , *<sup>S</sup>* <sup>∈</sup> **<sup>L</sup>**, with card *<sup>S</sup>* <sup>&</sup>lt; **<sup>Ω</sup>** in **<sup>L</sup>**. Then *<sup>M</sup>* is a set of cardinality **Ω** in **L**, hence in **L**[*H*] as well.

If *<sup>u</sup>* ⊆ **Fun<sup>Ω</sup>** in **<sup>L</sup>**, and *<sup>G</sup>* ⊆ *<sup>P</sup>*[*u*] is a *<sup>P</sup>*[*u*]-generic set, then put *SG* = - *<sup>p</sup>*∈*<sup>G</sup> Sp* ; thus *SG* ⊆ **Seq<sup>Ω</sup>** . The next lemma witnesses that forcing notions of the form *P*[*u*] belong to the type of *almost disjoint* (AD, for brevity) forcing, invented in [23] (§ 5).

**Lemma 12.** *Suppose that, in* **L***, u* ⊆ **Fun<sup>Ω</sup>** *is dense. Let G* ⊆ *P*[*u*] *be a set P*[*u*]*-generic over* **L***. Then:*


**Proof.** (i) Let *f* ∈ *u*. The set *Df* = {*p* ∈ *P*[*u*] : *f* ∈ *Fp* } is dense in *P*[*u*]. (Let *q* ∈ *P*[*u*]. Define *p* ∈ *P*[*u*] so that *Sp* = *Sq* and *Fp* = *Fq* ∪ { *f* }. Then *p* ∈ *Df* and *p q* .) Therefore *Df* ∩ *G* = ∅. Pick any *p* ∈ *Df* ∩ *G*. Then *f* ∈ *Fp* . Now every *r* ∈ *G* is compatible with *p*, and hence *Sr*/ *f* ⊆ *Sp*/ *f* by Lemma 10. Thus *SG*/ *f* = *Sp*/ *f* is bounded in **Ω**. Let *f* ∈/ *u*. If *ξ* < **Ω** then the set *Df <sup>ξ</sup>* = {*p* ∈ *P*[*u*] : sup(*Sp*/ *<sup>f</sup>*) <sup>&</sup>gt; *<sup>ξ</sup>*} is dense in *<sup>P</sup>*[*u*]. (If *<sup>q</sup>* <sup>∈</sup> *<sup>P</sup>*[*u*] then card (*F*<sup>∨</sup> *<sup>q</sup>* ) < **Ω**. As *f* ∈/ *u*, there is *η* > *ξ* , *η* < **Ω**, with *f η* ∈/ *F*<sup>∨</sup> *<sup>q</sup>* . Define *p* so that *Fp* = *Fq* and *Sp* = *Sq* ∪ { *f η*}. Then *p* ∈ *Df <sup>ξ</sup>* and *p q* .) Let *<sup>p</sup>* <sup>∈</sup> *Df <sup>ξ</sup>* <sup>∩</sup> *<sup>G</sup>*. Then sup(*SG*/ *<sup>f</sup>*) <sup>&</sup>gt; *<sup>ξ</sup>* . As *<sup>ξ</sup>* <sup>&</sup>lt; **<sup>Ω</sup>** is arbitrary, *SG*/ *<sup>f</sup>* is unbounded.

(ii) Consider any *p* ∈ *P*[*u*]. Suppose *p* ∈ *G*. Then *Sp* ⊆ *SG* . If there exists *s* ∈ (*SG Sp*) ∩ *F*<sup>∨</sup> *p* then by definition we have *s* ∈ *Sq* for some *q* ∈ *G*. However, then *p*, *q* are incompatible by Lemma 10, a contradiction. Now suppose *p* ∈/ *G*. Then there exists *q* ∈ *G* incompatible with *p*. By Lemma 10, there are two cases. First, there exists *s* ∈ (*Sq Sp*) ∩ *F*<sup>∨</sup> *<sup>p</sup>* . Then *s* ∈ *SG Sp* , so *p* is not compatible with *SG* . Second, there exists *s* ∈ (*Sp Sq*) ∩ *F*<sup>∨</sup> *<sup>q</sup>* . Then any condition *r q* satisfies *s* ∈/ *Sr* . Therefore *s* ∈/ *SG* , so *Sp* ⊆ *SG* , and *p* is not compatible with *SG* .

#### *3.2. Product Almost Disjoint Forcing*

Arguing under the assumptions and notation of Definition 2, we consider I = *ω* × *ω*, the cartesian product, as the index set for a product forcing.

**Definition 4** (in **L**)**.** <sup>∗</sup>**P<sup>Ω</sup>** *(note the boldface upright form) is the* **L***-product of* I *copies of* <sup>∗</sup> *P***<sup>Ω</sup>** *(Definition 3 in Section 3.1), ordered componentwise: p q ( p is stronger) iff p*(*n*, *i*) *q*(*n*, *i*) *in* ∗ *P***<sup>Ω</sup>** *for all n*, *i* < *ω.*

*That is,* <sup>∗</sup>**P<sup>Ω</sup>** ∈ **L** *and* <sup>∗</sup>**P<sup>Ω</sup>** *consists of all maps p* ∈ **L***, p* : I → <sup>∗</sup> *P***<sup>Ω</sup>** *. If p* ∈ <sup>∗</sup>**P<sup>Ω</sup>** *then put* **F***p*(*n*, *i*) = *Fp*(*n*,*i*) *and* **S***p*(*n*, *i*) = *Sp*(*n*,*i*) *for all n*, *i* < *ω, so that p*(*n*, *i*) = **S***p*(*n*, *i*); **F***p*(*n*, *i*)*, where* **S***<sup>p</sup>* : I → P<**Ω**(**SeqΩ**) *and* **F***<sup>p</sup>* : I → P<**Ω**(**FunΩ**) *are arbitrary, and* P<**<sup>Ω</sup>** *means all subsets of cardinality* < **Ω** *strictly.*


$$\mathbb{F}\_p^\vee(n, i) = F\_{p(n, i)}^\vee = \{ f \upharpoonright \xi : f \in \mathbb{F}\_p(n, i) \land 1 \le \xi < \mathbb{1} \};$$

• If *p*, *q* ∈ <sup>∗</sup>**P<sup>Ω</sup>** then define *p* ∧ *q* ∈ <sup>∗</sup>**P<sup>Ω</sup>** by (*p* ∧ *q*)(*n*, *i*) = *p*(*n*, *i*) ∧ *q*(*n*, *i*), in the sense of Definition 3 in Section 3.1, for all *n*, *i* < *ω*.

**Lemma 13.** *Conditions p*, *q* ∈ <sup>∗</sup>**P<sup>Ω</sup>** *are compatible in* <sup>∗</sup>**P<sup>Ω</sup>** *iff* (*p* ∧ *q*) *p and* (*p* ∧ *q*) *q.*

Let an **Ω-system** be any map *U* ∈ **L**, *U* : I → P(**FunΩ**) such that each set *U*(*n*, *i*) is empty or dense in **Fun<sup>Ω</sup>** . In this case, let |*U*| = {*n*, *i* : *U*(*n*, *i*) = ∅}.

• If *U* is an **Ω**-system then **P**[*U*] = {*p* ∈ <sup>∗</sup>**P<sup>Ω</sup>** : ∀ *n*, *i*∈|*p*| (**F***p*(*n*, *i*) ⊆ *U*(*n*, *i*))} is the **L**-product of the sets *P*[*U*(*n*, *i*)], *n*, *i* < *ω*.

**Lemma 14** (in **L**)**.** *Let U be an* **Ω***-system. Then it is true in* **L** *that* card **P**[*U*] = **Ω**<sup>⊕</sup> *, and the forcing notion* **P**[*U*] *is* **Ω***-closed, hence* **Ω***-distributive, and satisfies* **Ω**⊕*-CC, and the product* **C** × **P**[*U*] *satisfies* **Ω**⊕*-CC as well. Moreover* **P**[*U*] *satisfies* **Ω**⊕*-CC in any generic extension of* **L** *in which* **Ω**⊕ *remains a cardinal.*

**Proof.** The closed/distributive claims follow from Lemma 11. To prove the antichain claim we observe that if *p*, *q* ∈ <sup>∗</sup>**P<sup>Ω</sup>** satisfy **S***<sup>p</sup>* = **S***<sup>q</sup>* then *p*, *q* are compatible. However the set Δ**<sup>S</sup>** = {**S***<sup>p</sup>* : *p* ∈ <sup>∗</sup>**PΩ**} has cardinality ≤ **Ω** < **Ω**<sup>⊕</sup> in **L** as it consists of all functions **S***<sup>p</sup>* : I → P<**Ω**(**SeqΩ**). To extend the result to the product **<sup>C</sup>** <sup>×</sup> **<sup>P</sup>**[*U*], note that card**<sup>C</sup>** <sup>=</sup> *<sup>ω</sup>***<sup>L</sup>** <sup>1</sup> ≤ **Ω**.

**Definition 5.** *Suppose that z* ⊆ I *. If p* ∈ <sup>∗</sup>**P<sup>Ω</sup>** *then define p* = *p z to be the usual restriction, so that* dom (*p z*) = *z and p* (*n*, *<sup>i</sup>*) = *<sup>p</sup>*(*n*, *<sup>i</sup>*) *for all <sup>n</sup>*, *<sup>i</sup>* ∈ *<sup>z</sup> . A special case: If <sup>n</sup>*, *<sup>i</sup>* < *<sup>ω</sup> then let <sup>p</sup>* <sup>=</sup>*<sup>n</sup>*,*<sup>i</sup>* = *<sup>p</sup> <sup>z</sup> , where z* = (I {*n*, *i*})*. If U is an* **Ω***-system then define U z to be the ordinary restriction as well. Furthermore, if m* < *ω then define:*

> *<sup>p</sup>* <sup>&</sup>lt;*<sup>m</sup>* <sup>=</sup> *<sup>p</sup> z and U* <sup>&</sup>lt;*<sup>m</sup>* <sup>=</sup> *<sup>U</sup> <sup>z</sup>*, *where z* <sup>=</sup> {*<sup>k</sup>* : *<sup>k</sup>* <sup>&</sup>lt; *<sup>m</sup>*} × *<sup>ω</sup>*, *<sup>p</sup>* <sup>≥</sup>*<sup>m</sup>* <sup>=</sup> *<sup>p</sup> z and U* <sup>≥</sup>*<sup>m</sup>* <sup>=</sup> *<sup>U</sup> <sup>z</sup>*, *where z* <sup>=</sup> {*<sup>k</sup>* : *<sup>k</sup>* <sup>≥</sup> *<sup>m</sup>*} × *<sup>ω</sup>*, *<sup>p</sup> <sup>m</sup>* <sup>=</sup> *<sup>p</sup> z and U <sup>m</sup>* <sup>=</sup> *<sup>U</sup> <sup>z</sup>*, *where z* <sup>=</sup> {*m*} × *<sup>ω</sup>*.

*Finally, if Q* ⊆ <sup>∗</sup>**P<sup>Ω</sup>** *then let Q z* = {*p z* : *p* ∈ *Q*}*; Q z* ⊆ <sup>∗</sup>**P<sup>Ω</sup>** *z . This will be applied, e.g., in case <sup>Q</sup>* = **<sup>P</sup>**[*U*]*, where <sup>U</sup>* ∈ **<sup>L</sup>** *is a* **<sup>Ω</sup>***-system, and then we get* **<sup>P</sup>**[*U*] *<sup>z</sup>* = {*p <sup>z</sup>* : *<sup>p</sup>* ∈ **<sup>P</sup>**[*U*]}*,* **<sup>P</sup>**[*U*] <sup>=</sup>*<sup>n</sup>*,*<sup>i</sup> ,* **P**[*U*] <sup>≥</sup>*<sup>m</sup> , etc.*

**Remark 3.** *Suppose that z* ∈ **L** *in Definition 5. If p* ∈ <sup>∗</sup>**P<sup>Ω</sup>** *, then p z can be identified with a condition q* ∈ <sup>∗</sup>**P<sup>Ω</sup>** *such that q z* = *p z and q*(*n*, *i*) = ∅; ∅ *for all n*, *i*∈I *z . For instance, this applies w. r. t. <sup>p</sup>* <sup>=</sup>*<sup>n</sup>*,*<sup>i</sup> , p* <sup>≥</sup>*<sup>m</sup> , p* <sup>&</sup>lt;*<sup>m</sup> , p <sup>m</sup> .*

*With such an identification, we have* <sup>∗</sup>**P<sup>Ω</sup>** *z* ⊆ <sup>∗</sup>**P<sup>Ω</sup>** *, and Q z* ⊆ <sup>∗</sup>**P<sup>Ω</sup>** *for Q* ⊆ <sup>∗</sup>**P<sup>Ω</sup>** *(in case z* ∈ **L***). However, if z* ∈/ **L** *then such an identification fails. This is a consequence of our deviation from the*

*finite-support product approach taken in [4,5], which would not work in the setting of this paper.*

*The same applies for the restrictions U z of* **Ω***-systems U .*

#### *3.3. Structure of Product almost Disjoint Generic Extensions*

Arguing under the assumptions and notation of Definition 2, we let *U* be an **Ω**-system in **L**. Consider **P**[*U*] as a forcing notion. We will study **P**[*U*]-generic extensions **L**[*G*] of the ground universe **L**. Define some elements of these extensions. Suppose that *G* ⊆ **P**[*U*] is a generic set. Let,

$$\mathbf{S}\_G(n, i) = \mathbf{S}\_{G(n, i)} = \bigcup\_{p \in G} \mathbf{S}\_p(n, i) \text{ for any } n, i < \omega, \epsilon$$

where *G*(*n*, *i*) = {*p*(*n*, *i*) : *p* ∈ *G*} ⊆ *P*[*U*(*n*, *i*)]; thus **S***G*(*n*, *i*) ⊆ **Seq<sup>Ω</sup>** and *G* ⊆ **P**[*U*] splits into the family of sets *G*(*n*, *i*), *n*, *i* < *ω*. This defines a sequence *SG* = {**S***G*(*n*, *i*)}*n*,*i*<*<sup>ω</sup>* of subsets of **Seq<sup>Ω</sup>** .

If *z* ⊆ I then let *G z* = {*p z* : *p* ∈ *G*}. If *z* ∈ **L** then *G z* can be identified with {*p* ∈ *G* : |*p*| ⊆ *z*}.

$$\text{Put } G \upharpoonright\_{\neq} \!/ \_{\neq} \!/ \_{\langle n, i \rangle} = \{ p \in G : \langle n, i \rangle \notin |p| \} = G \upharpoonright (\mathcal{Z} \sim \{ \langle n, i \rangle \}) \dots$$

**Lemma 15.** *Let U be an* **Ω***-system in* **L***, and G* ⊆ **P**[*U*] *be a set* **P**[*U*]*-generic over* **L***. Then*:


**Proof.** To prove (i) apply Lemma 12(ii).

The genericity in (ii) holds by the product forcing theorem, then use Lemma 12(i).

Claim (iii) follows from the **Ω**-closure claim of Lemma 14.

(iv) We conclude from (iii) that all **L**-cardinals ≤ **Ω** remain cardinals in **L**[*G*], and GCH holds for all **L**-cardinals < **Ω** strictly. It follows from the **Ω**⊕-CC claim of Lemma 14 that all **L**-cardinals <sup>≥</sup> **<sup>Ω</sup>**<sup>⊕</sup> remain cardinals in **<sup>L</sup>**[*G*], and since card **<sup>P</sup>**[*U*] <sup>≤</sup> **<sup>Ω</sup>**<sup>⊕</sup> in **<sup>L</sup>**, GCH holds for all of them in **<sup>L</sup>**[*G*]. And finally we still have exp(**Ω**) = **Ω**⊕ in **L**[*G*] since by (i) the model **L**[*G*] is an extension of **L** by adjoining a subset of **Ω** obtained by a suitable wrapping of *SG* .

The next lemma is useful in dealing with combined (**C** × **P**[*U*])-generic extensions **L**[*ζ*, *G*] of **L**, where, by the product forcing theorem, *<sup>ζ</sup>* <sup>∈</sup> **<sup>Ξ</sup>***<sup>ω</sup>* is **<sup>C</sup>**-generic over **<sup>L</sup>** and *<sup>G</sup>* is **<sup>P</sup>**[*U*]-generic over **<sup>L</sup>**[*ζ*], or equivalently, *G* is **P**[*U*]-generic over **L** and *ζ* is **C**-generic over **L**[*G*].

**Lemma 16.** *Let U be an* **Ω***-system in* **L***, and a pair ζ*, *G is* (**C** × **P**[*U*])*-generic over* **L***. Then*:


Note that Claims (iii), (iv) are not applicable in case **Ω** = *ω***<sup>L</sup>** 1 . **Proof.** To prove (i), (ii) recall that all **L**-cardinals remain cardinals in **L**[*G*], and GCH holds in **L**[*G*], by Lemma 15(iv). It remains to note that *ζ* is **C**-generic over **L**[*G*] and make use of Lemma 8. To prove (iii) apply Lemma 4 with *ϑ* = **Ω** , *P* = **P**[*U*], *Q* = **C**. Note that card**C** = *ω***<sup>L</sup>** <sup>1</sup> <sup>≤</sup> **<sup>Ω</sup>** in case **<sup>Ω</sup>** <sup>≥</sup> *<sup>ω</sup>***<sup>L</sup>** 2 .

Finally Claim (iv) is a routine corollary of (i)–(iii).

#### **4. The Forcing Notion and the Model**

In this Section, we prove Theorem 2 on the base of another result, Theorem 8, see Remark 4 on page 23. The proof of Theorem 8 will follow in the remainder of the paper. The structure of the extension will be presented in Section 4.6, after the definition of the forcing notion involved in Section 4.5. Recall that the **L**-cardinals:

$$
\Omega^{\ominus} = \omega\_{\mu}^{\text{L}} < \upharpoonright = \omega\_{\mu+1}^{\text{L}} < \upharpoonright^{\ominus} = \omega\_{\mu+2}^{\text{L}}
$$

were introduced by Definition 2 on page 13. They remain to be fixed until Section 4.5, where their value will be specified in terms of the number **M** ≥ 1 we are dealing with in Theorem 2.

#### *4.1. Systems, Definability Aspects*

We argue in **L** under the assumptions and notation of Definition 2 on page 13. In continuation of our notation related to **Ω**-systems in Section 3.2, define the following.


Let **DS<sup>Ω</sup>** (disjoint systems) be the set of all disjoint **Ω**-systems, and let **sDS<sup>Ω</sup>** (*small* disjoint systems) be the set of all small disjoint **Ω**-systems *U* ∈ **DS<sup>Ω</sup>** .

Define **sDS<sup>Ω</sup>** <sup>≥</sup>*<sup>m</sup>* <sup>=</sup> {*U* <sup>≥</sup>*<sup>m</sup>* : *<sup>U</sup>* <sup>∈</sup> **sDSΩ**}, and similarly **sDS<sup>Ω</sup>** <sup>&</sup>lt;*<sup>m</sup>* etc. by Definition 5.

The sets **DS<sup>Ω</sup>** , **sDS<sup>Ω</sup>** , **sDS<sup>Ω</sup>** <sup>≥</sup>*<sup>m</sup>* , **DS<sup>Ω</sup>** <sup>&</sup>lt;*<sup>m</sup>* etc., and the order relation , **belong to L**, of course. Recall that, by (5),

$$\mathbb{H} = (\mathbf{H} \mathbb{I}^{\oplus})^{\mathbf{L}} = \{ \mathbf{x} \in \mathbf{L} \colon \mathbf{card}\left(\mathbf{TC}\left(\mathbf{x}\right)\right) \le \mathbb{I} \text{ in } \mathbf{L}\}\dots$$

**Lemma 17** (in **L**)**.** *The following sets belong to Δ***<sup>H</sup>** <sup>1</sup> ({**Ω**}) *and to <sup>Δ</sup>***<sup>H</sup>** <sup>3</sup> : {**Ω**}*,* {**SeqΩ**}*,* **Fun<sup>Ω</sup>** *,* <sup>∗</sup>**P<sup>Ω</sup>** *,* **sDS<sup>Ω</sup>** *,* **sDS<sup>Ω</sup>** <sup>≥</sup>*<sup>m</sup> ,* **sDS<sup>Ω</sup>** <sup>&</sup>lt;*<sup>m</sup> , the set* {*U*, *<sup>p</sup>* : *<sup>U</sup>* <sup>∈</sup> **sDS<sup>Ω</sup>** <sup>∧</sup> *<sup>p</sup>* <sup>∈</sup> **<sup>P</sup>**[*U*]}*, the relation .*

**Proof.** All these sets have rather straightforward *Δ***<sup>H</sup>** <sup>1</sup> ({**Ω**}) definitions, with **Ω** ∈ **H** as the only parameter. To eliminate **<sup>Ω</sup>**, it suffices to prove that {**Ω**} ∈ *<sup>Δ</sup>***<sup>H</sup>** <sup>3</sup> . Note first of all that "*ϑ* is a cardinal (initial ordinal)" is a *Π*<sup>1</sup> formula:

$$
\forall \theta \in \mathbf{Ord} \land \forall \mathfrak{a} < \theta \,\forall f \,(f: \mathfrak{a} \to \theta \implies \mathbf{ran} \, f \neq \theta) \,\,\mathsf{a}
$$

On the other hand, **Ω** is the largest cardinal in **H**, hence it holds in **H** that:

$$
\vartheta = \mathbb{O} \iff \forall \varkappa \left( \varkappa \text{ is a cardinal } \implies \varkappa \le \mathbb{O} \right) .
$$

We conclude that {**Ω**} ∈ *<sup>Π</sup>***<sup>H</sup>** <sup>2</sup> <sup>⊆</sup> *<sup>Δ</sup>***<sup>H</sup>** <sup>3</sup> . Finally, the conversion *<sup>Δ</sup>***<sup>H</sup>** <sup>1</sup> ({**Ω**}) <sup>→</sup> *<sup>Δ</sup>***<sup>H</sup>** <sup>3</sup> is routine.

#### *4.2. Complete Sequences*

We prove a major theorem (Theorem 6) in this Subsection. It deals with -increasing transfinite sequences in **sDS<sup>Ω</sup>** , satisfying some genericity/definability requirements. This is similar to some constructions in [4] and especially in [5] (Theorem 3). Yet there is a principal difference. Here the notion of extension is just the componentwise set theoretic extension, unlike [4,5], and originally [23], where the extension method was designed so that increments had to be finitewise Cohen-style generic over associated transitive models of a certain fragment of **ZFC**. Here the only restriction is that extensions have to obey the disjointness condition as defined in Section 4.1. In other words, if *U V* are **Ω**-systems in **sDS<sup>Ω</sup>** , then, beside *U*(*n*, *i*) ⊆ *V*(*n*, *i*), the increments Δ(*n*, *i*) = *V*(*n*, *i*) *U*(*n*, *i*) have to be pairwise disjoint and each Δ(*n*, *i*) to be disjoint with the union - *<sup>k</sup>*,*j* <sup>=</sup>*<sup>n</sup>*,*<sup>i</sup> <sup>U</sup>*(*k*, *<sup>j</sup>*).

Such a simplification is made possible here largely because the definability classes of the form **D**1*<sup>m</sup>* depend only on the highest quantifier order and do not depend on the number and type of the quantifiers involved in the definition of the set considered—unlike e.g., [5], where we dealt with the definability classes *Δ*<sup>1</sup> *<sup>n</sup>* , which obviously depend on the number of the quantifiers involved.

We begin with an auxiliary lemma.

Recall that, by (5), **<sup>H</sup>** = (**HΩ**⊕)**<sup>L</sup>** <sup>=</sup> {*<sup>x</sup>* <sup>∈</sup> **<sup>L</sup>** : card (TC (*x*)) <sup>≤</sup> **<sup>Ω</sup>** in **<sup>L</sup>**} <sup>=</sup> **<sup>L</sup>Ω**<sup>⊕</sup> .

**Lemma 18** (in **L**)**.** *Under the assumptions and notation of Definition 2, for any α* < **Ω**<sup>⊕</sup> *there exist m<sup>α</sup>* < *ω, <sup>t</sup><sup>α</sup>* <sup>∈</sup> **<sup>H</sup>***, and <sup>U</sup><sup>α</sup>* <sup>∈</sup> **sDS<sup>Ω</sup>** *such that the sequences* {*m<sup>α</sup>* }*α*<**Ω**<sup>⊕</sup> *,* {*t<sup>α</sup>* }*α*<**Ω**<sup>⊕</sup> *,* {*U<sup>α</sup>* }*α*<**Ω**<sup>⊕</sup> *belong to <sup>Δ</sup>***<sup>H</sup>** <sup>3</sup> *and, if m* < *ω, t* ∈ **H***, and* {*U<sup>α</sup>* }*α*<**Ω**<sup>⊕</sup> *is a -increasing continuous sequence of* **Ω***-systems in* **sDS<sup>Ω</sup>** *, then any closed unbounded set C* <sup>⊆</sup> **<sup>Ω</sup>**<sup>⊕</sup> *contains an ordinal <sup>α</sup>* <sup>∈</sup> *C such that m* <sup>=</sup> *<sup>m</sup><sup>α</sup> , t* <sup>=</sup> *<sup>t</sup><sup>α</sup> , U<sup>α</sup>* <sup>=</sup> *<sup>U</sup><sup>α</sup> .*

**Proof.** We argue in **L**, that is, under the assumption of **V** = **L**, the axiom of constructibility. It is known that the diamond principle ♦<sup>κ</sup> holds in **L** for any regular cardinal κ, in particular, for κ = **Ω**<sup>⊕</sup> , see, e.g., Theorem 13.21 and page 442 in [32]. The principle ♦<sup>κ</sup> asserts that there is a sequence {*S<sup>α</sup>* }*α*<**Ω**<sup>⊕</sup> <sup>∈</sup> **<sup>L</sup>** of sets *<sup>S</sup><sup>α</sup>* <sup>⊆</sup> *<sup>α</sup>*, of definability class *<sup>Δ</sup>***<sup>H</sup>** <sup>1</sup> , and such that:

(∗) If *X* ⊆ **Ω**<sup>⊕</sup> and *C* ⊆ **Ω**<sup>⊕</sup> is a closed unbounded set then there is *α* ∈ *C* such that *X* ∩ *α* = *S<sup>α</sup>* .

Let *<sup>h</sup>* : **<sup>Ω</sup>**<sup>⊕</sup> onto −→ **<sup>H</sup>** be any *<sup>Δ</sup>***<sup>H</sup>** <sup>1</sup> bijection. Put *<sup>Y</sup><sup>α</sup>* <sup>=</sup> {*h*(*ξ*) : *<sup>ξ</sup>* <sup>∈</sup> *<sup>S</sup><sup>α</sup>* }. Clearly {*Y<sup>α</sup>* }*α*<**Ω**<sup>⊕</sup> is still a *<sup>Δ</sup>***<sup>H</sup>** 1 sequence. Moreover the following is true:

(†) If {*B<sup>α</sup>* }*α*<**Ω**<sup>⊕</sup> is a sequence of sets in **H** and *C* ⊆ **Ω**<sup>⊕</sup> is a closed unbounded set then there is *α* ∈ *C* with {*B<sup>ξ</sup>* }*ξ*<*<sup>α</sup>* = *Y<sup>α</sup>* .

Using the sets *Y<sup>α</sup>* , we accomplish the proof of the lemma as follows. Assume that *α* < **Ω**<sup>⊕</sup> . If *Y<sup>α</sup>* is a sequence of the form {*y<sup>ξ</sup>* }*ξ*<*<sup>α</sup>* , such that each *<sup>y</sup><sup>ξ</sup>* is a triple *<sup>m</sup>*, *<sup>t</sup>*, *<sup>U</sup><sup>α</sup> <sup>ξ</sup>* , where both *m* ∈ *ω* and *t* ∈ **H** do not depend on *ξ* whereas *U<sup>α</sup> <sup>ξ</sup>* <sup>∈</sup> **sDS<sup>Ω</sup>** for each *<sup>ξ</sup>* and {*U<sup>α</sup> <sup>ξ</sup>* }*ξ*<*<sup>α</sup>* is a -increasing and continuous sequence, then put *m<sup>α</sup>* = *m*, *t<sup>α</sup>* = *t*, and *U<sup>α</sup>* = *<sup>ξ</sup>*<*<sup>α</sup> U<sup>α</sup> <sup>ξ</sup>* . Otherwise put *<sup>m</sup><sup>α</sup>* = *<sup>t</sup><sup>α</sup>* = 0 and let *<sup>U</sup><sup>α</sup>* be the null **Ω**-system, that is, *Uα*(*n*, *i*) = ∅ for all *n*, *i*. It follows from (†) (plus a routine analysis of definability based on Lemma 17) that this construction leads to the result required.

**Theorem 6** (in **L**)**.** *Under the assumptions and notation of Definition 2, there is a -increasing sequence* {**U***<sup>α</sup>* }*α*<**Ω**<sup>⊕</sup> *of* **Ω***-systems in* **sDS<sup>Ω</sup>** *, such that:*

	- <sup>−</sup> *either* **<sup>U</sup>***<sup>ξ</sup>* <sup>≥</sup>*<sup>m</sup>* <sup>∈</sup> *<sup>D</sup>* ;
	- <sup>−</sup> *or there is no* **<sup>Ω</sup>***-system U* <sup>∈</sup> *D with* **<sup>U</sup>***<sup>ξ</sup>* <sup>≥</sup>*<sup>m</sup> <sup>U</sup>* ;

Here the "slice" *<sup>U</sup> <sup>n</sup>* of a system *<sup>U</sup>* is essentially equal to the "column" {*U*(*n*, *<sup>i</sup>*)}*i*<*<sup>ω</sup>* of the whole "matrix" *<sup>U</sup>* <sup>=</sup> {*U*(*n*, *<sup>i</sup>*)}*n*,*i*<*<sup>ω</sup>* , while the "tail" *<sup>U</sup>* <sup>≥</sup>*<sup>m</sup>* can be viewed in the union of all columns to the right of *m* inclusively, see Definition 5.

**Proof. We argue in L.** One of the difficulties here is that we have to account for different levels of genericity and completeness for different slices of the construction. To cope with this issue, we make use of Lemma 18. Let us fix the sequences of terms *m<sup>α</sup>* , *t<sup>α</sup>* , *U<sup>α</sup>* such as in the lemma.

Let <**<sup>L</sup>** be Gödel's wellordering of **L**, as in Section 2.2.

For any *<sup>m</sup>* <sup>&</sup>lt; *<sup>ω</sup>*, let <sup>Θ</sup>*<sup>m</sup>* <sup>⊆</sup> **<sup>H</sup>** <sup>×</sup> **<sup>H</sup>** be a fixed universal *<sup>Σ</sup>***<sup>H</sup>** *<sup>m</sup>*+<sup>3</sup> set, that is, <sup>Θ</sup>*<sup>m</sup>* itself is *<sup>Σ</sup>***<sup>H</sup>** *<sup>m</sup>*+<sup>3</sup> , and if *X* ⊆ **H** is *Σm*+3(**H**) (parameters in **H** allowed), then there is *t* ∈ **H** such that *X* = {*x* : *t*, *x* ∈ Θ*<sup>m</sup>* }. If *m* < *ω* and *α* < **Ω**<sup>⊕</sup> , then let *Um<sup>α</sup>* be the <**L**-least **Ω**-system in **sDS<sup>Ω</sup>** satisfying *U<sup>m</sup> Um<sup>α</sup>* and:

(a) *Um<sup>α</sup>* <sup>&</sup>lt;*<sup>m</sup>* = *U<sup>α</sup>* <sup>&</sup>lt;*<sup>m</sup>* , and

(b) The **<sup>Ω</sup>**-system *<sup>U</sup>m<sup>α</sup>* <sup>≥</sup>*<sup>m</sup> <sup>m</sup>*-solves the set *<sup>D</sup><sup>α</sup>* <sup>=</sup> {*<sup>V</sup>* <sup>∈</sup> **sDS<sup>Ω</sup>** : *<sup>t</sup>α*, *<sup>V</sup>*<sup>∈</sup> <sup>Θ</sup>*<sup>m</sup>* }.

Making use of <sup>5</sup>◦ of Section 2.2, we conclude that the sequence {*Um<sup>α</sup>* }*α*<**Ω**<sup>⊕</sup> is *<sup>Δ</sup>***<sup>H</sup>** *<sup>m</sup>*+<sup>4</sup> .

Now we define a sequence of **Ω**-systems **U***<sup>ξ</sup>* , as required by Theorem 6, by induction.

Put **U**0(*n*, *i*) = ∅ for all *n*, *i*.

If *λ* < **Ω**<sup>⊕</sup> is the limit then by (i) define **U***<sup>λ</sup>* = *<sup>α</sup>*<*<sup>λ</sup>* **U***<sup>α</sup>* .

Suppose that a **Ω**-system **U***<sup>α</sup>* is defined, and the goal is to define the next one **U***α*+<sup>1</sup> . Fix *n*, *i* and define the components **U***α*+1(*n*, *i*). Note that this definition will depend on the components **U***α*(*n*, *i*) (with the same *n*, *i*) only, but not on the **Ω**-system **U***α* as a whole.

If it is true that:

$$m\_{\mathfrak{a}} \le n \quad \text{and} \quad \mathbb{U}\_{\mathfrak{a}}(n, i) = \mathcal{U}^{\mathfrak{a}}(n, i) \tag{7}$$

(where *U<sup>α</sup>* is the **Ω**-system given by Lemma 18), then put *m* = *m<sup>α</sup>* and **U***α*+1(*n*, *i*) = *Umα*(*n*, *i*). Otherwise, i.e., if (7) fails, just keep it with **U***α*+1(*n*, *i*) = **U***α*(*n*, *i*).

We assert that this inductive construction of **Ω**-systems **U***α* leads to Theorem 6.

Requirement (i) of the theorem is satisfied by construction.

The definability requirement (ii) of the theorem is subject to routine verification on the base of Lemma 17, which we leave to the reader.

To prove (iii), fix a number *<sup>m</sup>* and a *<sup>Σ</sup>m*+3(**H**) set *<sup>D</sup>* <sup>⊆</sup> **sDS<sup>Ω</sup>** <sup>≥</sup>*<sup>m</sup>* . We have to find an index *<sup>ξ</sup>* <sup>&</sup>lt; **<sup>Ω</sup>**<sup>⊕</sup> such that the **<sup>Ω</sup>**-system **<sup>U</sup>***<sup>ξ</sup>* <sup>≥</sup>*<sup>m</sup> <sup>m</sup>*-solves *<sup>D</sup>*. There is an element *<sup>t</sup>* <sup>∈</sup> **<sup>H</sup>** satisfying:

$$D = \{ V \in \mathbf{sDS}\_{\mathsf{D}} \restriction{\cong}^{m} : \langle t, V \rangle \in \Theta\_{m} \},$$

where Θ*<sup>m</sup>* is the universal set as above. Pick, by Lemma 18, an ordinal *α* < **Ω**<sup>⊕</sup> satisfying *m* = *m<sup>α</sup>* , *<sup>t</sup>* <sup>=</sup> *<sup>t</sup><sup>α</sup>* , **<sup>U</sup>***<sup>α</sup>* <sup>=</sup> *<sup>U</sup><sup>α</sup>* . Then (7) holds for all *<sup>n</sup>* <sup>≥</sup> *<sup>m</sup>*, and hence by definition we have **<sup>U</sup>***α*+<sup>1</sup> <sup>≥</sup>*<sup>m</sup>* <sup>=</sup> *<sup>U</sup>m<sup>α</sup>* <sup>≥</sup>*<sup>m</sup>* . Therefore the **Ω**-system **U***α*+<sup>1</sup> <sup>≥</sup>*<sup>m</sup> m*-solves the set *D* by (b), as required.

(iv) Coming back to the choice of universal sets Θ*<sup>m</sup>* in (b), it can be w.l. o. g. assumed that there is a recursive sequence of parameter free ∈-formulas *ϑn*(*t*, *x*) such that each *ϑ<sup>n</sup>* is a *Σn*+<sup>3</sup> formula and Θ*<sup>m</sup>* = {*t*, *x* ∈ **H** : **H** |= *ϑn*(*t*, *x*)}. This routinely leads to ∈-formulas *χn*(*α*, *x*) required. It can be observed that in fact each *χ<sup>n</sup>* is a *Σn*+<sup>4</sup> formula (not important and will not be used).

This completes the proof of Theorem 6.

#### *4.3. Preservation of the Completeness*

The next lemma says that the completeness property (iii) of Theorem 6, of the sequence {**U***<sup>ξ</sup>* }*ξ*<**Ω**<sup>⊕</sup> , still holds, to some extent, in rather mild generic extensions of **L**.

**Lemma 19.** *Under the assumptions and notation of Definition 2, suppose that* {**U***<sup>α</sup>* }*α*<**Ω**<sup>⊕</sup> ∈ **L** *is a increasing sequence of* **Ω***-systems in* **sDS<sup>Ω</sup>** *satisfying* (i)*–*(iv) *of Theorem 6.*

*Let <sup>Q</sup>* <sup>∈</sup> **<sup>L</sup>** *be a forcing notion with* card *<sup>Q</sup>* <sup>≤</sup> **<sup>Ω</sup>** *in* **<sup>L</sup>***, e.g., <sup>Q</sup>* <sup>=</sup> **<sup>C</sup>***. Let <sup>F</sup>* <sup>⊆</sup> *<sup>Q</sup> be a set Q-generic over* **L***.*

*Assume that <sup>m</sup>* <sup>&</sup>lt; *<sup>ω</sup>, <sup>δ</sup>* <sup>&</sup>lt; **<sup>Ω</sup>**<sup>⊕</sup> *, and a set <sup>D</sup>* <sup>∈</sup> **<sup>L</sup>**[*F*]*, <sup>D</sup>* <sup>⊆</sup> **sDS<sup>Ω</sup>** <sup>≥</sup>*<sup>m</sup> , belongs to <sup>Σ</sup>m*+3(**H**[*F*])*, and is open in* **sDS<sup>Ω</sup>** <sup>≥</sup>*<sup>m</sup> so that any extension of a* **<sup>Ω</sup>***-system U* <sup>∈</sup> *D in* **sDS<sup>Ω</sup>** <sup>≥</sup>*<sup>m</sup> belongs to D itself.*

*Then there is an ordinal <sup>α</sup>, <sup>δ</sup>* <sup>≤</sup> *<sup>α</sup>* <sup>&</sup>lt; **<sup>Ω</sup>**<sup>⊕</sup> *, such that* **<sup>U</sup>***<sup>α</sup>* <sup>≥</sup>*<sup>m</sup> <sup>m</sup>***-solves** *D , as in Theorem <sup>6</sup>*(iii)*.*

We recall that **H** = (**HΩ**⊕)**<sup>L</sup>** and **H**[*F*]=(**HΩ**⊕)**L**[*F*] by (5), (6).

**Proof.** As obviously **sDS<sup>Ω</sup>** <sup>≥</sup>*<sup>m</sup>* <sup>⊆</sup> **<sup>H</sup>**, we conclude by Theorem 5(ii) that there is a *<sup>Σ</sup>m*+3(**H**) name *t* ∈ **L**, *t* ⊆ *Q* × **H**, such that *D* = *t*[*F*].

We argue in **<sup>L</sup>**. If *<sup>q</sup>* <sup>∈</sup> *<sup>Q</sup>*, *<sup>U</sup>* <sup>∈</sup> **sDS<sup>Ω</sup>** <sup>≥</sup>*<sup>m</sup>* , and there is such a condition *<sup>h</sup>* <sup>∈</sup> *<sup>Q</sup>* that *<sup>h</sup> <sup>q</sup>* (meaning *h* is stronger) and *h*, *U* ∈ *t*, then write *A*(*q*, *U*). If *b* ∈ *Q* then we define:

$$D(b) = \left\{ \mathcal{U} \in \mathbf{sDS}\_{\Omega} \restriction^{\geq m} : \exists \, q \in \mathcal{Q} (q \lessdot A(q, \mathcal{U})) \right\}.$$

Each of the sets *D*(*b*) ⊆ **H** belongs to *Σm*+3(**H**) by virtue of Lemma 17 and the choice of *t*. Therefore, by the choice of the sequence of **Ω**-systems, for every *b* ∈ *Q* there is an ordinal *α*(*b*), *δ* < *α*(*b*) < **Ω**<sup>⊕</sup> , such that the **Ω**-system **U***α*(*b*) <sup>≥</sup>*<sup>m</sup> m*-solves the set *D*(*b*).

Note that *<sup>δ</sup>* <sup>=</sup> sup*b*∈*<sup>Q</sup> <sup>α</sup>*(*b*) <sup>&</sup>lt; **<sup>Ω</sup>**<sup>⊕</sup> by the cardinality argument.

We claim that the **<sup>Ω</sup>**-system **<sup>U</sup>***<sup>δ</sup>* <sup>≥</sup>*<sup>m</sup> <sup>m</sup>*-solves *<sup>D</sup>*. It suffices to prove that if a **<sup>Ω</sup>**-system *<sup>U</sup>* <sup>∈</sup> *<sup>D</sup>* extends **U***<sup>δ</sup>* <sup>≥</sup>*<sup>m</sup>* , then the **Ω**-system **U***<sup>δ</sup>* <sup>≥</sup>*<sup>m</sup>* itself belongs to *D*. Moreover, as *D* is open, it suffices to find *<sup>b</sup>* <sup>∈</sup> *<sup>Q</sup>*, satisfying **<sup>U</sup>***α*(*b*) <sup>≥</sup>*<sup>m</sup>* <sup>∈</sup> *<sup>D</sup>*.

We argue in **<sup>L</sup>**. Consider the set *<sup>B</sup>* <sup>=</sup> {*<sup>b</sup>* <sup>∈</sup> *<sup>Q</sup>* : **<sup>U</sup>***α*(*b*) <sup>≥</sup>*<sup>m</sup>* <sup>∈</sup> *<sup>D</sup>*(*b*)}. If *<sup>b</sup>* <sup>∈</sup> *<sup>B</sup>* then pick a particular *<sup>q</sup>* <sup>=</sup> *<sup>q</sup>*(*b*) <sup>∈</sup> *<sup>Q</sup>* such that *<sup>q</sup> <sup>b</sup>* and *<sup>A</sup>*(*q*, **<sup>U</sup>***α*(*b*) <sup>≥</sup>*m*) holds. If *<sup>b</sup>* <sup>∈</sup> *<sup>Q</sup> <sup>B</sup>* then put *<sup>q</sup>*(*b*) = *<sup>b</sup>* . The set *Q* = {*q*(*b*) : *b* ∈ *Q*} is dense in *Q*. It follows that there is *b* ∈ *Q* ∩ *F*. On the other hand, as *U* ∈ *D*, there is a condition *h* ∈ *Q* with *h*, *U* ∈ *t*.

Then there exists some *q* ∈ *F* satisfying *q h* and *q h*(*b*) *b* . This implies *U* ∈ *D*(*b*). It follows, by the choice of *<sup>α</sup>*(*b*), that **<sup>U</sup>***α*(*b*) <sup>≥</sup>*<sup>m</sup>* <sup>∈</sup> *<sup>D</sup>*(*b*), too. However then *<sup>b</sup>* <sup>∈</sup> *<sup>B</sup>*, and hence we have *<sup>A</sup>*(*q*(*b*), **<sup>U</sup>***α*(*b*) <sup>≥</sup>*m*). By definition there is a condition *<sup>h</sup>* <sup>∈</sup> *<sup>Q</sup>* with *<sup>q</sup>*(*b*) *<sup>h</sup>* , such that *h* , **<sup>U</sup>***α*(*b*) <sup>≥</sup>*m*<sup>∈</sup> *<sup>t</sup>* . However *<sup>h</sup>* <sup>∈</sup> *<sup>F</sup>* (since *<sup>f</sup>*(*b*) <sup>∈</sup> *<sup>F</sup>*). We conclude that **<sup>U</sup>***<sup>δ</sup>* <sup>≥</sup>*<sup>m</sup>* <sup>∈</sup> *<sup>D</sup>*, as required.

#### *4.4. Key Definability Engine*

We argue under the assumptions and notation of Definition 2 on page 13. In particular, a successor **L**-cardinal **Ω** > *ω* is fixed. We make the following arrangements.

**Definition 6** (in **<sup>L</sup>**)**.** *We fix a -increasing sequence of* **<sup>Ω</sup>***-systems* {**U<sup>Ω</sup>** *<sup>ξ</sup>* }*ξ*<**Ω**<sup>⊕</sup> *satisfying conditions* (i)*–*(iv) *of Theorem 6 for the particular* **L***-cardinal* **Ω** *introduced by Definition 2.*

*We define the limit* **Ω***-system* **U<sup>Ω</sup>** = *<sup>ξ</sup>*<**Ω**<sup>⊕</sup> **U<sup>Ω</sup>** *<sup>ξ</sup> , the basic forcing notion* **<sup>P</sup><sup>Ω</sup>** = **<sup>P</sup>**[**UΩ**]*, and the subforcings* **PΩ** *<sup>γ</sup>* = **P**[**U<sup>Ω</sup>** *<sup>γ</sup>* ]*, γ* < **Ω**<sup>⊕</sup> *.*

*Define restrictions* **<sup>P</sup><sup>Ω</sup>** *z, G z (z* ⊆ I *, G* <sup>⊆</sup> **<sup>P</sup><sup>Ω</sup>** *),* **<sup>P</sup><sup>Ω</sup>** <sup>=</sup>*<sup>n</sup>*,*<sup>i</sup> etc. as in Section 3.2.*

Thus by construction **<sup>P</sup><sup>Ω</sup>** <sup>∈</sup> **<sup>L</sup>** is the **<sup>L</sup>**-product of sets **<sup>P</sup>Ω**(*n*, *<sup>i</sup>*) = *<sup>P</sup>*[**UΩ**(*n*, *<sup>i</sup>*)], *<sup>n</sup>*, *<sup>i</sup>* <sup>∈</sup> *<sup>ω</sup>*. Lemma <sup>14</sup> implies some cardinal characterictics of **P<sup>Ω</sup>** , namely:


**Corollary 2. P<sup>Ω</sup>** *does not adjoin new reals to* **L***.*

**Proof.** The result follows from (III) because **Ω** ≥ *ω* by Definition 2.

As for definability, the set **U<sup>Ω</sup>** is not parameter free definable in **H** = (**HΩ**⊕)**<sup>L</sup>** , yet its slices are:

**Lemma 20** (in **<sup>L</sup>**)**.** *Let <sup>n</sup>* <sup>&</sup>lt; *<sup>ω</sup>. Then the set* **<sup>U</sup><sup>Ω</sup>** *<sup>n</sup>* <sup>=</sup> {*i*, *<sup>f</sup>* : *<sup>f</sup>* <sup>∈</sup> **<sup>U</sup>Ω**(*n*, *<sup>i</sup>*)} *belongs to <sup>Σ</sup>***<sup>H</sup>** *<sup>n</sup>*+<sup>4</sup> *. In addition there is a recursive sequence of parameter free* ∈*-formulas un*(*i*, *f*) *such that, for any n* < *ω, if i* < *ω and <sup>f</sup>* <sup>∈</sup> **Fun<sup>Ω</sup>** *then f* <sup>∈</sup> **<sup>U</sup>Ω**(*n*, *<sup>i</sup>*) *iff* **<sup>H</sup>** <sup>|</sup><sup>=</sup> *<sup>u</sup>n*(*i*, *<sup>f</sup>*)*.*

**Proof.** To prove the first claim, apply (ii) of Theorem 6. To prove the additional claim define:

$$\mathfrak{u}\_n(i, f) := \exists \mathfrak{a} \exists \mathfrak{x} \left(\chi\_n(\mathfrak{a}, \mathfrak{x}) \land f \in \mathfrak{x}(n, i)\right),$$

where *χ<sup>n</sup>* are formulas given by (iv) of Theorem 6.

We further let formulas **Γ<sup>Ω</sup>** *ni* (*n*, *i* ∈ *ω*) be defined as follows:

$$\mathbb{T}^{\square}\_{\text{nil}}(\mathbb{S}) \ := \mathsf{Left} \ \mathbb{S} \subseteq \mathbf{Seq}\_{\square} \land \forall f \in \mathbf{Fun}\_{\square} \left( f \in \cup^{\square}(\mathbb{n}, i) \iff \mathbb{S} \text{ does not cover } f \right) .$$

The next theorem shows that any real in **L** and even in some generic extensions of **L** can be made parameter free definable in appropriate subextensions of **PΩ**-generic extensions, basically by means of the formulas **Γ<sup>Ω</sup>** *ni*(*S*). We prove this result in a rather general form, which includes the case of a forcing notion *Q* = **C**, actually used in this paper, as just a particular case. The proof of the particular case *Q* = **C** would not be any simpler though.

**Theorem 7.** *Assume that <sup>Q</sup>* <sup>∈</sup> **<sup>L</sup>** *is a forcing notion,* card *<sup>Q</sup>* <sup>≤</sup> **<sup>Ω</sup>** *in* **<sup>L</sup>***, a pair <sup>W</sup>*, *<sup>G</sup> is* (*<sup>Q</sup>* <sup>×</sup> **<sup>P</sup>Ω**)*-generic over* **L***, Y* ∈ **L**[*W*]*, and z* ∈ **L**[*Y*]*, z* ⊆ I = *ω* × *ω. Then,*


**Proof.** (i) **Ω**<sup>⊕</sup> remains a cardinal in **L**[*G*] by Lemma 15(iv), hence *Q* still satisfies card *Q* < **Ω**<sup>⊕</sup> in **L**[*G*]. As *W* is *Q*-generic over **L**[*G*], **Ω**<sup>⊕</sup> remains a cardinal in **L**[*W*, *G*] and in **L**[*Y*, *G*] ⊆ **L**[*W*, *G*].

(ii) If *n*, *i* ∈ *z* then by construction:

$$\mathcal{G}(n,i) := \{ p(n,i) : p \in \mathcal{G} \} = \{ p'(n,i) : p' \in \mathcal{G} \mid \mathbf{z} \} \in \mathbf{L}[\mathcal{G} \mid \mathbf{z}],$$

and hence *SG*(*n*, *<sup>i</sup>*) <sup>∈</sup> **<sup>L</sup>**[*G <sup>z</sup>*] as well. Now **<sup>Γ</sup><sup>Ω</sup>** *ni*(*SG*(*n*, *i*)) follows from Lemma 15(ii).

(iii) We w.l. o. g. assume that *<sup>z</sup>* <sup>=</sup> <sup>I</sup> {*n*, *<sup>i</sup>*} and *<sup>Y</sup>* <sup>=</sup> *<sup>W</sup>* . Then **<sup>P</sup><sup>Ω</sup>** *<sup>z</sup>* <sup>=</sup> **<sup>P</sup><sup>Ω</sup>** <sup>=</sup>*<sup>n</sup>*,*<sup>i</sup>* can be identified with {*<sup>p</sup>* <sup>∈</sup> **<sup>P</sup><sup>Ω</sup>** : *<sup>p</sup>*(*n*, *<sup>i</sup>*) = <sup>∅</sup>, <sup>∅</sup>}, see Remark 3. Suppose towards the contrary that *<sup>S</sup>* <sup>∈</sup> **<sup>L</sup>**[*W*, *<sup>G</sup>* <sup>=</sup>*<sup>n</sup>*,*<sup>i</sup>*] = **<sup>L</sup>**[*W*][*G* <sup>=</sup>*<sup>n</sup>*,*<sup>i</sup>*] satisfies **<sup>Γ</sup><sup>Ω</sup>** *ni*(*S*). There is a name *<sup>τ</sup>* <sup>∈</sup> **<sup>L</sup>**[*W*], *<sup>τ</sup>* <sup>⊆</sup> **<sup>P</sup><sup>Ω</sup>** <sup>=</sup>*<sup>n</sup>*,*<sup>i</sup>* <sup>×</sup> **Seq<sup>Ω</sup>** , such that:

$$S = \pi[G\upharpoonright\_{\neq} \upharpoonright\_{\neq} \upharpoonright\_{\neq} := \{ s \in \mathbf{Seq}\_{\Omega} : \exists \ p \in G \upharpoonright\_{\neq} \upharpoonright\_{\neq} (\nw{}, \text{s}) \in \tau \} \text{ .} $$

The forcing **P<sup>Ω</sup>** remains **Ω**⊕-CC in **L**[*W*] by Lemma 14. This allows us to w.l. o. g. assume that card *<sup>τ</sup>* <sup>&</sup>lt; **<sup>Ω</sup>**<sup>⊕</sup> in **<sup>L</sup>**[*W*], and then *<sup>τ</sup>* <sup>∈</sup> **<sup>H</sup>**[*W*]=(**HΩ**⊕)**L**[*W*] .

There is a condition *<sup>p</sup>*<sup>0</sup> <sup>∈</sup> *<sup>G</sup>* which (**P<sup>Ω</sup>** <sup>=</sup>*<sup>n</sup>*,*<sup>i</sup>*)-forces **<sup>Γ</sup>***ni*(*τ*[*G* <sup>=</sup>*<sup>n</sup>*,*<sup>i</sup>*]) over **<sup>L</sup>**[*W*]. If *<sup>s</sup>* <sup>∈</sup> **Seq<sup>Ω</sup>** then put *As* <sup>=</sup> {*<sup>p</sup>* : *<sup>p</sup>*,*s*<sup>∈</sup> *<sup>τ</sup>*}; *As* <sup>⊆</sup> **<sup>P</sup><sup>Ω</sup>** <sup>=</sup>*<sup>n</sup>*,*<sup>i</sup>* .

We argue in **<sup>L</sup>**. As card *<sup>τ</sup>* <sup>&</sup>lt; **<sup>Ω</sup>**<sup>⊕</sup> , there is an ordinal *<sup>γ</sup>* <sup>&</sup>lt; **<sup>Ω</sup>**<sup>⊕</sup> such that *<sup>τ</sup>* <sup>⊆</sup> (**P<sup>Ω</sup>** *<sup>γ</sup>* <sup>=</sup>*<sup>n</sup>*,*<sup>i</sup>*) × **Seq<sup>Ω</sup>** and *<sup>p</sup>*<sup>0</sup> <sup>∈</sup> **<sup>P</sup><sup>Ω</sup>** *<sup>γ</sup>* <sup>=</sup>*<sup>n</sup>*,*<sup>i</sup>* . Consider the set *<sup>D</sup>* of all **<sup>Ω</sup>**-systems *<sup>U</sup>* <sup>∈</sup> **sDS<sup>Ω</sup>** extending **<sup>U</sup><sup>Ω</sup>** *<sup>γ</sup>* and such that there exists a condition *<sup>p</sup>* <sup>∈</sup> **<sup>P</sup>**[*U*] <sup>=</sup>*<sup>n</sup>*,*<sup>i</sup>* , *<sup>p</sup> <sup>p</sup>*<sup>0</sup> , an element *<sup>f</sup>* <sup>∈</sup> *<sup>U</sup>*(*n*, *<sup>i</sup>*) = - *<sup>k</sup>*,*j* <sup>=</sup>*<sup>n</sup>*,*<sup>i</sup> <sup>U</sup>*(*k*, *<sup>j</sup>*), and an ordinal *<sup>μ</sup>* < **<sup>Ω</sup>**, such that *<sup>p</sup>* contradicts to every *<sup>p</sup>* ∈ - *<sup>μ</sup>*≤*α*<**<sup>Ω</sup>** *Af <sup>α</sup>* . Then *<sup>D</sup>* is *<sup>Δ</sup>***<sup>H</sup>** <sup>3</sup> by Lemma 17 (and Theorem 5(i), to transfer the definability properties from **H** to **H**[*W*]), with *τ* ∈ **H**[*W*] as a parameter. Therefore, by Lemma 19, there is an ordinal *η* < **Ω**<sup>⊕</sup> such that the pair **U<sup>Ω</sup>** *<sup>η</sup>* 0-solves *D* as in Theorem 6(iii). We have two cases.

*Case 1*: **U<sup>Ω</sup>** *<sup>η</sup>* <sup>∈</sup> *<sup>D</sup>*. Let this be witnessed by *<sup>p</sup>* , *<sup>f</sup>* , *<sup>μ</sup>* as indicated. Then *<sup>f</sup>* <sup>∈</sup> (**U<sup>Ω</sup>** *<sup>η</sup>* )(*n*, *i*), therefore *<sup>f</sup>* <sup>∈</sup>/ **<sup>U</sup>Ω**(*n*, *<sup>i</sup>*). By definition **<sup>U</sup><sup>Ω</sup>** *<sup>γ</sup>* **U<sup>Ω</sup>** *<sup>η</sup>* , hence *γ* ≤ *η* . Furthermore, if *s* = *f ξ* , *μ* ≤ *ξ* < *ω*<sup>1</sup> , then the condition *<sup>p</sup>* (**P<sup>Ω</sup>** <sup>=</sup>*<sup>n</sup>*,*<sup>i</sup>*)-forces *<sup>s</sup>* <sup>∈</sup>/ *<sup>τ</sup>*[*G* <sup>=</sup>*<sup>n</sup>*,*<sup>i</sup>*] over **<sup>L</sup>**[*W*]. We conclude that *<sup>p</sup>* forces *<sup>τ</sup>*[*G*]/ *<sup>f</sup>* < *<sup>μ</sup>* < **<sup>Ω</sup>** over **<sup>L</sup>**[*W*]. Note that *<sup>p</sup>*<sup>0</sup> forces *<sup>τ</sup>*[*G* <sup>=</sup>*<sup>n</sup>*,*<sup>i</sup>*]/ *<sup>f</sup>* = **<sup>Ω</sup>** because *<sup>f</sup>* ∈/ **<sup>U</sup>**(*n*, *<sup>i</sup>*). However *p p*<sup>0</sup> . This is a contradiction.

*Case 2*: There is no **<sup>Ω</sup>**-system *<sup>U</sup>* <sup>∈</sup> *<sup>D</sup>* extending **<sup>U</sup><sup>Ω</sup>** *<sup>η</sup>* . We can assume that *γ* ≤ *η* , since if *η* < *γ* then the **Ω**-system **U<sup>Ω</sup>** *<sup>γ</sup>* has the same property. Easily there exists *δ* , *η* < *δ* < *ω*<sup>1</sup> , such that **U<sup>Ω</sup>** *<sup>δ</sup>* (*n*, *<sup>i</sup>*) **U<sup>Ω</sup>** *<sup>η</sup>* (*n*, *i*) = ∅. (To prove this claim note that the set *D* of all **Ω**-systems *U* ∈ **sDS<sup>Ω</sup>** satisfying *U*(*n*, *i*) **U<sup>Ω</sup>** *<sup>η</sup>* (*n*, *i*) = ∅ is dense in **sDS<sup>Ω</sup>** therefore, any *U* that 0-solves *D* belongs to *D* .)

Take any *<sup>f</sup>* <sup>∈</sup> **<sup>U</sup><sup>Ω</sup>** *<sup>δ</sup>* (*n*, *<sup>i</sup>*) **<sup>U</sup><sup>Ω</sup>** *<sup>η</sup>* (*n*, *<sup>i</sup>*). Then *<sup>f</sup>* <sup>∈</sup> **<sup>U</sup>Ω**(*n*, *<sup>i</sup>*), and hence *<sup>p</sup>*<sup>0</sup> forces *<sup>τ</sup>*[*G*]/ *<sup>f</sup>* <sup>&</sup>lt; **<sup>Ω</sup>** over **<sup>L</sup>**[*W*] by the choice of *<sup>p</sup>*<sup>0</sup> . It follows that there exists a condition *<sup>p</sup>* <sup>∈</sup> **<sup>P</sup><sup>Ω</sup>** <sup>=</sup>*<sup>n</sup>*,*<sup>i</sup>* , *<sup>p</sup> <sup>p</sup>*<sup>0</sup> , and an ordinal *<sup>μ</sup>* < *<sup>ω</sup>*<sup>1</sup> , such that for any *<sup>α</sup>* ≥ *<sup>μ</sup>*, *<sup>p</sup>* forces *<sup>s</sup>* ∈/ *<sup>τ</sup>*[*G* <sup>=</sup>*<sup>n</sup>*,*<sup>i</sup>*] over **<sup>L</sup>**[*W*], where *<sup>s</sup>* = *<sup>f</sup> <sup>α</sup>*. Thus *<sup>p</sup>* contradicts to each condition *<sup>p</sup>* ∈ - *<sup>μ</sup>*≤*α*<**<sup>Ω</sup>** *Af <sup>α</sup>* . We may w.l. o. g. assume that *<sup>p</sup>* <sup>∈</sup> **<sup>P</sup><sup>Ω</sup>** *<sup>δ</sup>* <sup>=</sup>*<sup>n</sup>*,*<sup>i</sup>* (otherwise increase *δ* appropriately). Under these assumptions, define a **Ω**-system *U* so that:

$$\mathcal{U}(n,i) = \mathbb{U}\_\delta^\Omega(n,i) \smile \{f\}, \quad \mathcal{U}(n,i+1) = \mathbb{U}\_\delta^\Omega(n,i+1) \cup \{f\} \lrcorner$$

and *U*(*k*, *j*) = **U<sup>Ω</sup>** *<sup>δ</sup>* (*k*, *j*) for all pairs of indices *k*, *j* other than *n*, *i* and *n*, *i* + 1. Obviously *U* extends **U<sup>Ω</sup>** *<sup>η</sup>* , and *p* ∈ **P**[*U*]. Therefore *U* ∈ *D*. But this contradicts the Case 2 hypothesis.

Claim (iv) is an immediate corollary of (ii) and (iii).

To prove (v), note that (\*) (*z*)*<sup>n</sup>* <sup>=</sup> {*<sup>i</sup>* : <sup>∃</sup> *<sup>S</sup>* <sup>⊆</sup> **Seq<sup>Ω</sup> <sup>Γ</sup><sup>Ω</sup>** *ni*(*S*)} by (iv). However with *n* fixed the relation *<sup>f</sup>* <sup>∈</sup> **<sup>U</sup>Ω**(*n*, *<sup>i</sup>*) with *<sup>i</sup>*, *<sup>f</sup>* as arguments is *<sup>Σ</sup>***<sup>H</sup>** *<sup>n</sup>*+<sup>4</sup> by Lemma 20, hence *<sup>Σ</sup>***<sup>T</sup>** *<sup>n</sup>*+<sup>4</sup> by Theorem 5(i). Now (*z*)*<sup>n</sup>* <sup>∈</sup> *<sup>Σ</sup>***<sup>T</sup>** *<sup>n</sup>*+<sup>6</sup> follows by (\*).

To prove (vi), make use of (v) and Theorem 3.

Let us finally prove (vii). Detalizing the proof of (v) and (vi) on the base of formulas *un*(*f* , *i*) of Lemma 20, we obtain a recursive sequence of parameter free ∈-formulas *ϕn*(*i*) such that if *n*, *i* < *ω* then *i* ∈ (*z*)*<sup>n</sup>* iff **T** |= *ϕn*(*i*). The proof of Theorem 3 is obviously effective enough to obtain another recursive sequence of parameter free type-theoretic formulas *ψn*(*i*) of order ≤ such that it holds in **L**[*Y*, *G z*] that: *i* ∈ (*z*)*<sup>n</sup>* iff *ψn*(*i*), that is, *z* = {*n*, *i* : *ψn*(*i*)}.

However it is known that the truth of formulas of order ≤ can be uniformly expressed by a suitable formula of order + 1, see e.g., [18]. In other words, there is a parameter free type theoretic formula Ψ(*n*, *i*) of order ≤ + 1 such that it holds in **L**[*Y*, *G z*] that: *i* ∈ (*z*)*<sup>n</sup>* iff Ψ(*n*, *i*), that is, *z* = {*n*, *i* : Ψ(*n*, *i*)}. We conclude that *z* is definable in **L**[*Y*, *G z*] by a type-theoretic formula of order ≤ + 1. In other words, *z* ∈ **D**1,+<sup>1</sup> in **L**[*Y*, *G z*], as required.

#### *4.5. We Specify* **Ω**

We come back to Theorem 2. Now it is time to specify the value of the **L**-cardinal **Ω**, so far left rather arbitrary by Definition 2 on page 13.

**Definition 7** (in **L**)**.** *Recall that* 1 ≤ **M** < *ω is a number considered in Theorem 2.*

*We let* **Ω** = *ω***<sup>L</sup> <sup>M</sup>** *, and accordingly define* **<sup>Ω</sup>** = *<sup>ω</sup>***<sup>L</sup> <sup>M</sup>**−<sup>1</sup> *,* **<sup>Ω</sup>**<sup>⊕</sup> <sup>=</sup> *<sup>ω</sup>***<sup>L</sup> <sup>M</sup>**+<sup>1</sup> *,*

$$\mathbb{H} = (\mathbf{H} \mathbb{U}^{\oplus})^{\mathbf{L}} = (\mathbf{H} \omega^{\mathbf{L}}\_{\mathbb{H}+1})^{\mathbf{L}} = \{ \mathbf{x} \in \mathbf{L} : \text{card}\left(\mathbf{T} \mathbb{C}\left(\mathbf{x}\right)\right) < \omega^{\mathbf{L}}\_{\mathbb{H}+1} \text{ in } \mathbf{L} \}$$

*by Definition 2. Applying Definition 6 with* **Ω** = *ω***<sup>L</sup> <sup>M</sup>** *, we accordingly fix:*


*and define restrictions* **<sup>P</sup><sup>Ω</sup>** *z (z* ⊆ I *),* **<sup>P</sup><sup>Ω</sup>** <sup>≥</sup>*<sup>n</sup> ,* **<sup>P</sup><sup>Ω</sup>** <sup>&</sup>lt;*<sup>n</sup> ,* **<sup>P</sup><sup>Ω</sup>** <sup>=</sup>*<sup>n</sup>*,*<sup>i</sup> etc. as in Section 3.2.*

#### *4.6. The Model*

To prove Theorem <sup>2</sup> we make use of a certain submodel of a (**<sup>C</sup>** <sup>×</sup> **<sup>P</sup>Ω**)-generic extension of **<sup>L</sup>**. First of all, if *g* : *ω* → P(*ω*) is any function then we put:

$$\mathfrak{w}[\mathcal{g}] = \{ \langle k, j \rangle : k < \omega \land j \in \mathcal{g}(k) \}. \tag{8}$$

Now consider a pair *<sup>ζ</sup>*, *<sup>G</sup>*, (**<sup>C</sup>** <sup>×</sup> **<sup>P</sup>Ω**)-generic over **<sup>L</sup>**. Thus *<sup>ζ</sup>* : *<sup>ω</sup>* onto −→ **Ξ** is a generic collapse function, while the set *<sup>G</sup>* <sup>⊆</sup> **<sup>P</sup><sup>Ω</sup>** is **<sup>P</sup>Ω**-generic over **<sup>L</sup>**[*ζ*]. The set:

$$\text{sw}[\mathbb{J}] = \{ \langle k, j \rangle : k < \omega \land j \in \mathbb{J}(k) \} \subseteq \mathbb{Z} = \omega \times \omega \tag{9}$$

obviously belongs to the model **L**[*ζ*] = **L**[*w*[*ζ*]], but not to **L**. Therefore the restrictions **P<sup>Ω</sup>** *w*[*ζ*], *G w*[*ζ*] in the next theorem have to be understood in the sense of Definition 5 on page 15, ignoring Remark <sup>3</sup> since, definitely *<sup>w</sup>*[*ζ*] <sup>∈</sup>/ **<sup>L</sup>**. Thus **<sup>P</sup><sup>Ω</sup>** *<sup>w</sup>*[*ζ*] is a forcing notion in **<sup>L</sup>**[*ζ*], not in **<sup>L</sup>**.

The following theorem describes the structure of such generic models.

**Theorem 8.** *Under the assumptions of Definition 7, let a pair <sup>ζ</sup>*, *<sup>G</sup> be* (**<sup>C</sup>** <sup>×</sup> **<sup>P</sup>Ω**)*-generic over* **<sup>L</sup>***. Then:*


*and it is true in the model* **L**[*ζ*, *G w*[*ζ*]] *that*


**Remark 4.** *Theorem 8 implies Theorem 2 via the model* **L**[*ζ*, *G w*[*ζ*]]*, of course. As for Theorem 8 itself, its proof follows below in this paper. Claims* (i)*–*(vi) *will be established right now, and Claim* (vii) *is accomplished in Section 6.6, based on the substantial work in Sections 5 and 6.*

**Proof (**Claims (i)–(vi) of Theorem 8). To prove that *G w*[*ζ*] is (**P<sup>Ω</sup>** *w*[*ζ*])-generic over **L**[*ζ*], note that *<sup>G</sup>* <sup>⊆</sup> **<sup>P</sup><sup>Ω</sup>** is **<sup>P</sup>Ω**-generic over **<sup>L</sup>**[*ζ*] by the product forcing theorem w. r. t. the product **<sup>C</sup>** <sup>×</sup> **<sup>P</sup><sup>Ω</sup>** . However **<sup>P</sup><sup>Ω</sup>** can be naturally identified with the product (**P<sup>Ω</sup>** *<sup>w</sup>*[*ζ*]) <sup>×</sup> (**P<sup>Ω</sup>** *<sup>z</sup>*) in **<sup>L</sup>**[*ζ*], where *<sup>z</sup>* <sup>=</sup> <sup>I</sup> *<sup>w</sup>*[*ζ*]. This implies the result by another application of the product forcing theorem.

To establish (ii), (iii), and (iv), it suffices to apply Lemma 16, as **L**[*ζ*] ⊆ **L**[*ζ*, *G w*[*ζ*]] ⊆ **L**[*ζ*, *G*].

To prove Claim (v), let *x* ∈ **L**, *x* ⊆ *ω*. By the genericity of *ζ* , there is a number *n*<sup>0</sup> < *ω* such that *x* = *ζ*(*n*0). Then, for any *i*, we have *n*0, *i* ∈ *w*[*ζ*] iff *i* ∈ *x* . By Theorem 7(vi) (with *Q* = **C**, *z* = *w*[*ζ*], *Y* = *ζ* , = **M**), it is true in **L**[*ζ*, *G w*[*ζ*]] that *x* belongs to **D**1**<sup>M</sup>** , as required.

To prove Claim (vi), assume that 1 ≤ *m* < *ω* and *m* = **M**; we have to show that **D**1*<sup>m</sup>* ∈/ **D**2*<sup>m</sup>* in **L**[*ζ*, *G w*[*ζ*]]. We have two cases.

*Case 1*: *m* > **M**. Consider the set *z* = *w*[*ζ*] defined by (9) in Section 4.6. By definition *z* ⊆ *ω* × *ω*, *z* ∈ **L**[*ζ*]. It follows from Theorem 7(vii) (with *Q* = **C**, *z* = *w*[*ζ*], *Y* = *ζ* , = **M**), that *z* ∈ **D**1,**M**+<sup>1</sup> , hence *z* ∈ **D**1,**M**+<sup>1</sup> as **M** + 1 ≤ *m*. Now suppose to the contrary that **D**1*<sup>m</sup>* ∈ **D**2*<sup>m</sup>* in **L**[*ζ*, *G z*]. As *ω***L**[*z*] <sup>1</sup> <sup>=</sup> *<sup>ω</sup>***L**[*ζ*,*<sup>G</sup> <sup>z</sup>*] <sup>1</sup> = *<sup>ω</sup>***<sup>L</sup>** <sup>2</sup> , there exist real *x* ∈ **L**[*z*], *x* ⊆ *ω*, which do not belong to **D**1*<sup>m</sup>* ; let *x*<sup>0</sup> be the least of them in the sense of the Gödel well ordering of **L**[*z*]. Then *x*<sup>0</sup> itself belongs to **D**1*<sup>m</sup>* by 5◦ of Section 2.2, since so does *z* by the above, which is a contradiction.

*Case 2*: 1 ≤ *m* < **M**. It suffices to apply Lemma 2 on page 8 because *m* < **M** and **D**1**<sup>M</sup>** = P(*ω*) ∩ **L** holds in **L**[*ζ*, *G w*[*ζ*]] by Claims (v) and (vii). We may note that this short argument refers to Claim (vii) that will be conclusively established only in Section 6.6.

An independent proof is as follows. If 1 ≤ *m* < **M**, then **M** ≥ 2, and hence Theorem 8(iii) implies:

$$\mathcal{O}^{\otimes m}(\omega) \cap \mathbf{L}[\mathfrak{J}] = \mathcal{O}^{\otimes m}(\omega) \cap \mathbf{L}[\mathfrak{J}, G \upharpoonright w] = \mathcal{O}^{\otimes m}(\omega) \cap \mathbf{L}[\mathfrak{J}, G].$$

We conclude that the sets **D**1*<sup>m</sup>* and **D**2*<sup>m</sup>* are the same in these models, and hence it suffices to prove that **D**1*<sup>m</sup>* ∈/ **D**2*<sup>m</sup>* in the **C**-generic extension **L**[*ζ*]. Now we apply the fact that collapse forcing notions similar to **C** are homogeneous enough for any parameter free formula either be forced by every condition, or be negated by every condition. In our case, it follows that (**D**1*m*)**L**[*ζ*] <sup>∈</sup> **<sup>L</sup>** and (**D**1*m*)**L**[*ζ*] is countable in **L**. Therefore if, to the contrary, **D**1*<sup>m</sup>* ∈ **D**2*<sup>m</sup>* in **L**[*ζ*], then taking the Gödel-least *x* ∈ (P(*ω*) ∩ **L**) **D**1*<sup>m</sup>* in **L**[*ζ*], we routinely get *x* ∈ **D**1*<sup>m</sup>* in **L**[*ζ*] via 5◦ of Section 2.2, with a contradiction.

This completes the proof of Claims (i)–(vi) of Theorem 8.

#### **5. Forcing Approximation**

We argue under the assumptions and notation of Definition 7 on page 22.

Beginning here a lengthy proof of Claim (vii) of Theorem 8, our plan will be to establish the following, somewhat unexpected result. Recall that, by Theorem 8(ii), it is true in **L**[*ζ*, *G w*[*ζ*]] that **<sup>Ω</sup>** <sup>=</sup> *<sup>ω</sup>***M**−<sup>1</sup> and **<sup>Ω</sup>**<sup>⊕</sup> <sup>=</sup> **<sup>Ω</sup>**<sup>+</sup> <sup>=</sup> *<sup>ω</sup>***<sup>M</sup>** in case **<sup>M</sup>** <sup>≥</sup> 2, whereas *<sup>ω</sup>* <sup>&</sup>lt; **<sup>Ω</sup>** <sup>=</sup> **<sup>Ω</sup>**<sup>⊕</sup> <sup>=</sup> *<sup>ω</sup>*<sup>1</sup> in case **<sup>M</sup>** <sup>=</sup> 1.

**Theorem 9.** *Assume that a pair <sup>ζ</sup>*, *<sup>G</sup> is* (**<sup>C</sup>** <sup>×</sup> **<sup>P</sup>Ω**)*-generic over* **<sup>L</sup>***, and <sup>a</sup>* <sup>∈</sup> **<sup>L</sup>**[*ζ*, *<sup>G</sup> <sup>w</sup>*[*ζ*]]*, <sup>a</sup>* <sup>⊆</sup> *<sup>ω</sup>, and it is true in* **L**[*ζ*, *G w*[*ζ*]] *that:*

**either M** ≥ 2 *and a is* ∈*-definable in* P(**Ω**); ∈, **p** (see Section 2.4)*;* **or M** = 1 *and a is* ∈*-definable in* P(*ω*); ∈*.*

*Then a* ∈ **L**[*G*]*.*

**Remark 5.** *Theorem 9 implies Claim* (vii) *of Theorem 8.*

*Indeed,* **arguing in L**[*ζ*, *G w*[*ζ*]]*, suppose that a* ⊆ *ω, a* ∈ **D**1**<sup>M</sup>** *. If* **M** = 1 *then we immediately have the "or" case of Theorem 9. Thus suppose that* **M** ≥ 2*. Theorem 3 is applicable by Theorem 8*(iv)*, therefore x is* ∈*-definable in* **H***ω***<sup>M</sup>** *, that is, in* **HΩ**<sup>⊕</sup> *by Theorem 8*(iii)*. Then Theorem 4 is applicable as well, and hence we have the "either" case of Theorem 9. We conclude that a* ∈ **L**[*G*] *by Theorem 9. However, by Lemma 14, the forcing notion* **P** *is* **Ω***-closed in* **L***, and this property is sufficient for* **P***-generic sets not to add new subsets of ω, so a* ∈ **L***, as required by* (vii) *of Theorem 8.*

*Thus Theorem 9 completes the proof of Theorem 8 as a whole because other claims of Theorem 8 have been already established, see Section 4.6.*

To prove Theorem 9, we are going to define a forcing-like relation **forc** similar to approximate forcing relations considered in [4,5], and earlier in [3] and some other papers on the base of forcing notions not of an almost-disjoint type. Then we exploit certain symmetries of objects related to **forc** .

**Definition 8.** *Extending Definition <sup>7</sup> on page 22, let us fix a pair <sup>ζ</sup>*, **<sup>G</sup>** *,* (**<sup>C</sup>** <sup>×</sup> **<sup>P</sup>Ω**)*-generic over* **<sup>L</sup>** *for the remainder of the text. We consider generic extensions:*

$$\mathbf{L}[\boldsymbol{\zeta}] \subseteq \mathbf{L}[\boldsymbol{\zeta}, \mathbf{G} \mid \boldsymbol{w}[\boldsymbol{\zeta}]] \subseteq \mathbf{L}[\boldsymbol{\zeta}, \mathbf{G}] \dots$$

*We shall assume that* **M** ≥ 2 *(the "either" case of Theorem 9). The "or" case* **M** = 1 *is pretty similar:* **Ω** *is changed to ω during the course of the proof.*

*5.1. Language*

We argue under the assumptions and notation of Definitions 7 and 8.

• Assume that *<sup>z</sup>* <sup>∈</sup> **<sup>L</sup>**[*ζ*], *<sup>z</sup>* ⊆ I <sup>=</sup> *<sup>ω</sup>* <sup>×</sup> *<sup>ω</sup>*. Then let **Nam***<sup>z</sup> <sup>ζ</sup>* ∈ **L**[*ζ*] be the set of all sets *τ* ∈ **L**[*ζ*], *<sup>τ</sup>* <sup>⊆</sup> (∗**P<sup>Ω</sup>** *<sup>z</sup>*) <sup>×</sup> **<sup>Ω</sup>**, with card *<sup>τ</sup>* <sup>&</sup>lt; **<sup>Ω</sup>**<sup>⊕</sup> in **<sup>L</sup>**[*ζ*].

Note that <sup>∗</sup>**P<sup>Ω</sup>** , a bigger forcing notion, is used instead of **P<sup>Ω</sup>** in this definition. One of the advantages is that <sup>∗</sup>**P<sup>Ω</sup>** is ∈-definable in **H** by Lemma 17.

If *<sup>τ</sup>* <sup>∈</sup> **Nam***<sup>z</sup> <sup>ζ</sup>* and *G* ⊆ <sup>∗</sup>**P<sup>Ω</sup>** *z* then put *τ*[*G*] = {*α* < **Ω** : ∃ *p* ∈ *G* (*p*, *α* ∈ *τ*)}.

$$\mathbf{\color{red}{L}}\mathbf{\color{red}{L}}\mathbf{\texttt{Lemma 21.}}\quad \mathcal{F}^{p}(\mathbb{O})\cap\mathbf{L}[\mathbf{\color{red}{L}}\mathbf{\color{red}{G}\cap w[\mathbf{\color{red}{\mathcal{G}}]}]] = \{\texttt{\color{red}{\tau}[\mathbf{G}\cap w[\mathbf{\color{red}{\mathcal{G}}]}]:\tau\in\mathbf{Nam}\_{\mathsf{\zeta}}^{w[\mathbf{\color{red}{\mathcal{G}}]}}\}.\text{ }\mathbf{\color{red}{\mathcal{G}}\mathbf{\color{red}{\mathcal{G}}\mathbf{\color{red}{\mathcal{G}}}\mathbf{\color{red}{\mathcal{G}}}\mathbf{\color{red}{\mathcal{G}}\mathbf{\color{red}{\mathcal{G}}}\mathbf{\color{red}{\mathcal{G}}}\mathbf{\color{red}{\mathcal{G}}}\mathbf{\color{red}{\mathcal{G}}}\mathbf{\color{red}{\mathcal{G}}}\mathbf{\color{red}{\mathcal{G}}}\}.$$

**Proof.** Let *<sup>X</sup>* <sup>∈</sup> **<sup>L</sup>**[*ζ*, *<sup>G</sup> <sup>w</sup>*[*ζ*]], *<sup>X</sup>* <sup>⊆</sup> **<sup>Ω</sup>**. The set **<sup>G</sup>** *<sup>w</sup>*[*ζ*] is (**P<sup>Ω</sup>** *<sup>w</sup>*[*ζ*])-generic over **<sup>L</sup>**[*ζ*] by the product forcing theory. Therefore, by a well-known property of generic extensions (see, e.g., [32]), there is a name *<sup>t</sup>* <sup>∈</sup> **<sup>L</sup>**[*ζ*], *<sup>t</sup>* <sup>⊆</sup> (**P<sup>Ω</sup>** *<sup>w</sup>*[*ζ*]) <sup>×</sup> **<sup>Ω</sup>**, such that *<sup>X</sup>* <sup>=</sup> *<sup>t</sup>*[**G** *<sup>w</sup>*[*ζ*]]. To reduce *<sup>t</sup>* to a name *<sup>τ</sup>* with the same property, satisfying card *τ* < **Ω**<sup>⊕</sup> , apply Lemma 14.

Now, arguing in **L**[*ζ*], we introduce a language that will help us to study analytic definability in the generic extensions considered. We argue under the assumptions and notation of Definition 8.

Let L be the 2nd order language, with variables *α*, *β*, ... , assumed to vary over ordinals < **Ω**, and *X*,*Y*, . . . , varying over the subsets of **Ω**. Atomic formulas of the following types are allowed:

$$
\alpha < \beta \,, \quad \alpha = \beta \, , \quad \alpha \in X \, \quad \mathbb{P}(\alpha, \beta) = \gamma \, .
$$

(See Section 2.4 on **p**.) Only the connectives ∧ and ¬ and quantifiers ∃ *α* and ∃ *X* are allowed, the other connectives and ∀ are treated as shortcuts, and, to reduce the number of cases, the equality *X* = *Y* will be treated as a shortcut for ∀ *α*(*α* ∈ *X* ⇐⇒ *α* ∈ *Y*).

The *complexity* #(*ϕ*) of an L-formula *ϕ* is defined by induction so that:


Note that the complexity of quantifier-free formulas can be as high as one wants. If *z* ∈ **L**[*ζ*], *z* ⊆ *ω* × *ω*, then let L(*z*) be the extension of L by:


If *G* ⊆ <sup>∗</sup>**P***<sup>M</sup> z* , then the *valuation ϕ*[*G*] of such a formula *ϕ* is defined by substitution of *τ*[*G*] for any name *<sup>τ</sup>* <sup>∈</sup> **Nam***<sup>z</sup> <sup>ζ</sup>* that occurs in *ϕ*, and relativizing each quantifier ∃ *α* or ∃ *X* to resp. **Ω**, P(**Ω**). Thus *ϕ*[*G*] is a formula of L with parameters in P(**Ω**) ∩ **L**[*ζ*, *G*] and quantifiers relativized as above, that is, to **Ω** and to P(**Ω**), and *ϕ*[*G*] can contain **p** interpreted as **p** (**Ω** × **Ω**). (See Section 2.4 on **p**.)

#### *5.2. Forcing Approximation*

We still argue under the assumptions and notation of Definitions 7 and 8.

Our next goal is to define, in **L**[*ζ*], a forcing-style relation *p* **forc***<sup>z</sup> <sup>U</sup> ϕ*. In case *z* = *w*[*ζ*] and *U* = **U<sup>Ω</sup>** , the relation **forc***<sup>z</sup> <sup>U</sup>* will be compatible with the truth in the model **L**[*ζ*, **G** *w*[*ζ*]] = **L**[*ζ*][**G** *w*[*ζ*]], viewed as a (**P<sup>Ω</sup>** *w*[*ζ*])-generic extension of **L**[*ζ*]. But, perhaps unlike the true forcing relation associated with **P<sup>Ω</sup>** *w*[*ζ*], the relation **forc***<sup>z</sup> <sup>U</sup>* will be invariant under certain transformations.

The definition goes on in **L**[*ζ*] by induction on the complexity of *ϕ*.


We precede the last item with another definition. If *n* < *ω* then let **sDS**[*n*] be the set of all **Ω**-systems *<sup>U</sup>* <sup>∈</sup> **sDS<sup>Ω</sup>** such that *<sup>U</sup>* <sup>&</sup>lt;*<sup>n</sup>* <sup>=</sup> **<sup>U</sup><sup>Ω</sup>** *<sup>ξ</sup>* <sup>&</sup>lt;*<sup>n</sup>* for some *<sup>ξ</sup>* <sup>&</sup>lt; **<sup>Ω</sup>**<sup>⊕</sup> . Thus **sDS**[0] = **sDS<sup>Ω</sup>** .

(F8) If *<sup>U</sup>*, *<sup>p</sup>*, *<sup>z</sup>* are as in (F1), *<sup>ϕ</sup>* is a closed <sup>L</sup>*<sup>z</sup>* formula, *<sup>n</sup>* <sup>=</sup> #(*ϕ*), then *<sup>p</sup>* **forc***<sup>z</sup> <sup>U</sup>* ¬ *ϕ* iff there is no **Ω**system *U* ∈ **sDS**[*n*] extending *U*, and no *q* ∈ **P**[*U* ] *z* , *q p*, such that *q* **forc***<sup>z</sup> <sup>U</sup> ϕ*.

**Lemma 22** (in **L**[*ζ*])**.** *Let U*, *p*, *z* , *ϕ satisfy* (F1) *above. Then*:


**Proof.** The proof of (i) by straightforward induction is elementary. As for (ii), make use of (F8).

Now let us evaluate the complexity of the relation **forc** . Given a parameter free L-formula *ϕ*(*α*, *β*,..., *X*,*Y*,...) with any set of free variables allowed in L, we define, in **L**[*ζ*], the set:

$$\begin{array}{rcl} \mathsf{Fcor}(\mathfrak{q}) & = & \left\{ \langle z, \mathsf{U}, p, a, \beta, \ldots, \mathsf{r} \mathsf{x}, \mathsf{r}\_{Y}, \ldots \rangle : \!\!\mathcal{U} \in \mathsf{s} \mathsf{DS}\_{\mathsf{D}} \wedge z \subseteq \omega \times \omega \right\} \\ & \wedge \ p \in \mathsf{P}[\mathsf{U}] \, | \, z \wedge a, \beta, \choose \cdot \mathrel{\scalebox{1.0pt}{ $\psi$ }} \,\, \wedge \, \tau\_{\mathsf{X}}, \tau\_{\mathsf{Y}}, \ldots \in \mathsf{Nam}^{z}\_{\mathsf{U}} \\ & \wedge \, p \, \mathsf{for} \mathsf{c}^{z}\_{\mathsf{U}} \,\, \!\!\!\!\!\!\!/ \left( a, \beta, \ldots, \tau\_{\mathsf{X}}, \tau\_{\mathsf{Y}}, \ldots \right) \,\, \right\}. \end{array}$$

**Lemma 23** (in **<sup>L</sup>**[*ζ*])**.** *If <sup>ϕ</sup> is a parameter free* <sup>L</sup>*-formula and n* <sup>=</sup> #(*ϕ*)*, then* **Forc**(*ϕ*) *is <sup>Σ</sup>***H**[*ζ*] *<sup>n</sup>*+<sup>3</sup> *.*

**Proof.** The set **sDS<sup>Ω</sup>** is *Δ***<sup>H</sup>** <sup>3</sup> by Lemma 17, and hence *<sup>Δ</sup>***H**[*ζ*] <sup>3</sup> as well by Theorem 5(i) in Section 2.6. The relations *<sup>p</sup>* <sup>∈</sup> **<sup>P</sup>**[*U*] *<sup>z</sup>* , *<sup>α</sup>* <sup>&</sup>lt; **<sup>Ω</sup>** , *<sup>τ</sup>* <sup>∈</sup> **Nam***<sup>z</sup> <sup>ζ</sup>* , with arguments resp. *p*, *U*, *z*; *α*; *τ*, *z* , are routinely checked to be *Δ***H**[*ζ*] <sup>3</sup> , too. (Note that bounded quantifiers preserve *<sup>Δ</sup>***H**[*ζ*] <sup>3</sup> .) After this remark, prove the lemma by induction on the structure of *ϕ*.

The case of atomic formulas of type (F2) is immediately clear. (The pairing function **p** (**Ω** × **Ω**) in (F2) is *Δ***<sup>H</sup>** <sup>1</sup> by Lemma 3.) The result for atomic formulas of type (F3) amounts to the formula <sup>∃</sup> *<sup>q</sup>* <sup>∈</sup> <sup>∗</sup>**P<sup>Ω</sup>** *<sup>z</sup>* (*<sup>q</sup>*, *<sup>α</sup>*<sup>∈</sup> *<sup>Y</sup>* <sup>∧</sup> *<sup>p</sup> <sup>q</sup>*), which is *<sup>Σ</sup>***H**[*ζ*] <sup>3</sup> by the above. The step (F4) amounts to the intersection of two sets is quite obvious. And so are steps (F5) and (F6) (a ∃-quantification on the top of a given *Σ***H**[*ζ*] #(*ϕ*)+<sup>3</sup> ).

To carry out the step (F8), note that **sDS**[*n*] is *Σ***<sup>H</sup>** *<sup>n</sup>*+<sup>3</sup> by Lemma 20, therefore *<sup>Σ</sup>***H**[*ζ*] *<sup>n</sup>*+<sup>3</sup> by Theorem 5(i) in Section 2.6. This if **Forc**(*ϕ*) is *<sup>Σ</sup>***H**[*ζ*] *<sup>n</sup>*+<sup>3</sup> then **Forc**(<sup>¬</sup> *<sup>ϕ</sup>*) is *<sup>Π</sup>***H**[*ζ*] *<sup>n</sup>*+<sup>3</sup> , hence *<sup>Σ</sup>***H**[*ζ*] *<sup>n</sup>*+<sup>4</sup> , as required.

#### *5.3. Consequences for the Complete Forcing Notions*

We continue to argue under the assumptions and notation of Definitions 7 on page 22 and 8 on page 25. Coming back to the sequence of **Ω**-systems **U<sup>Ω</sup>** *<sup>ξ</sup>* ∈ **sDS<sup>Ω</sup>** given by Definition 7, we note that every **Ω**-system **U<sup>Ω</sup>** *<sup>ξ</sup>* belongs to *<sup>m</sup>* **sDS**[*m*].

Let **forc***<sup>z</sup> <sup>ξ</sup>* be **forc***<sup>z</sup>* **U<sup>Ω</sup>** *ξ* , and let *p* **forc***<sup>z</sup>* <sup>∞</sup> *<sup>ϕ</sup>* mean: <sup>∃</sup> *<sup>ξ</sup>* <sup>&</sup>lt; **<sup>Ω</sup>**<sup>⊕</sup> (*<sup>p</sup>* **forc***<sup>z</sup> <sup>ξ</sup> <sup>ϕ</sup>*). Note that *<sup>p</sup>* **forc***<sup>z</sup> <sup>ξ</sup> ϕ* implies *<sup>p</sup>* <sup>∈</sup> **<sup>P</sup><sup>Ω</sup>** *<sup>ξ</sup> <sup>z</sup>* , whereas *<sup>p</sup>* **forc***<sup>z</sup>* <sup>∞</sup> *<sup>ϕ</sup>* implies *<sup>p</sup>* <sup>∈</sup> **<sup>P</sup><sup>Ω</sup>** *<sup>z</sup>* . Lemma <sup>22</sup> takes the following form:

**Lemma 24** (in **<sup>L</sup>**[*ζ*])**.** *Assume that z* <sup>⊆</sup> *<sup>ω</sup>* <sup>×</sup> *<sup>ω</sup>, <sup>ϕ</sup> is a closed* <sup>L</sup>*<sup>z</sup> formula, p* <sup>∈</sup> **<sup>P</sup><sup>Ω</sup>** *z. Then*:


The following result will be very important.

**Lemma 25** (in **<sup>L</sup>**[*ζ*])**.** *If <sup>z</sup>* <sup>⊆</sup> *<sup>ω</sup>* <sup>×</sup> *<sup>ω</sup>, <sup>ϕ</sup> is a closed* <sup>L</sup>*<sup>z</sup> formula, <sup>p</sup>* <sup>∈</sup> **<sup>P</sup><sup>Ω</sup>** *<sup>z</sup> , then there is a condition <sup>q</sup>* <sup>∈</sup> **<sup>P</sup><sup>Ω</sup>** *z, q p, such that either q* **forc***<sup>z</sup>* <sup>∞</sup> *ϕ, or q* **forc***<sup>z</sup>* <sup>∞</sup> ¬ *ϕ.*

**Proof.** Let *<sup>n</sup>* <sup>=</sup> #(*ϕ*). There is an ordinal *<sup>η</sup>* <sup>&</sup>lt; **<sup>Ω</sup>**<sup>⊕</sup> such that *<sup>p</sup>* <sup>∈</sup> **<sup>P</sup><sup>Ω</sup>** *<sup>η</sup> z* . Consider the set *D* of all **<sup>Ω</sup>**-systems *<sup>U</sup>* <sup>∈</sup> **sDS<sup>Ω</sup>** <sup>≥</sup>*<sup>n</sup>* such that there is a **<sup>Ω</sup>**-system *<sup>U</sup>* <sup>∈</sup> **sDS**[*n*] that extends **<sup>U</sup><sup>Ω</sup>** *<sup>η</sup>* and satisfies *<sup>U</sup>* <sup>≥</sup>*<sup>n</sup>* <sup>=</sup> *<sup>U</sup>* , and there is also a condition *<sup>q</sup>* <sup>∈</sup> **<sup>P</sup>**[*U*] *<sup>z</sup>* , *<sup>q</sup> <sup>p</sup>*, satisfying *<sup>q</sup>* **forc***<sup>z</sup> <sup>U</sup> ϕ*. The set *D* belongs to *<sup>Σ</sup>n*+3(**H**[*ζ*]) (with **<sup>U</sup><sup>Ω</sup>** *<sup>η</sup>* , **V<sup>Ω</sup>** *<sup>η</sup>* , *p* as definability parameters) by Lemma 23. Therefore by Lemma 19 there is an ordinal *<sup>ξ</sup>* , *<sup>η</sup>* <sup>≤</sup> *<sup>ξ</sup>* <sup>&</sup>lt; **<sup>Ω</sup>**<sup>⊕</sup> , such that the **<sup>Ω</sup>**-system **<sup>U</sup><sup>Ω</sup>** *<sup>ξ</sup>* <sup>≥</sup>*<sup>n</sup> <sup>n</sup>*-solves *<sup>D</sup>*. We have two cases.

*Case 1*: **U<sup>Ω</sup>** *<sup>ζ</sup>* <sup>≥</sup>*<sup>m</sup>* <sup>∈</sup> *<sup>D</sup>*. Then there exist: a **<sup>Ω</sup>**-system *<sup>U</sup>* <sup>∈</sup> **sDS**[*n*] extending **<sup>U</sup><sup>Ω</sup>** *<sup>η</sup>* and satisfying *U* <sup>≥</sup>*<sup>n</sup>* = **U<sup>Ω</sup>** *<sup>ζ</sup>* <sup>≥</sup>*<sup>n</sup>* , and a condition *<sup>q</sup>* <sup>∈</sup> **<sup>P</sup>**[*U*] *<sup>z</sup>* , *<sup>q</sup> <sup>p</sup>*, with *<sup>q</sup>* **forc***<sup>z</sup> <sup>U</sup> ϕ*. By definition there is an ordinal *ϑ* < **Ω**<sup>⊕</sup> such that *U* <sup>&</sup>lt;*<sup>n</sup>* = **U<sup>Ω</sup>** *<sup>ϑ</sup>* <sup>&</sup>lt;*<sup>n</sup>* . Now let *<sup>μ</sup>* <sup>=</sup> max{*ξ*, *<sup>ϑ</sup>*}. Then *<sup>U</sup>* **<sup>U</sup><sup>Ω</sup>** *<sup>μ</sup>* , hence *q* **forc***<sup>z</sup> <sup>μ</sup> ϕ* and *q* **forc***<sup>z</sup>* <sup>∞</sup> *ϕ*.

*Case 2*: There is no **<sup>Ω</sup>**-system *<sup>U</sup>* <sup>∈</sup> *<sup>D</sup>* that extends **<sup>U</sup><sup>Ω</sup>** *<sup>ξ</sup>* <sup>≥</sup>*<sup>n</sup>* . Prove that *<sup>p</sup>* **forc***<sup>z</sup> <sup>ξ</sup>* ¬ *ϕ*. Suppose towards the contrary that this fails. Then, by (F8) in Section 5.2, there exists a **Ω**-system *U* ∈ **sDS**[*n*] extending **U<sup>Ω</sup>** *<sup>ξ</sup>* , and a condition *<sup>q</sup>* <sup>∈</sup> **<sup>P</sup>**[*U*], *<sup>q</sup> <sup>p</sup>*, such that *<sup>q</sup>* **forc***<sup>z</sup> <sup>U</sup> ϕ*. Define *U* = *U* <sup>≥</sup>*<sup>n</sup>* . Then by definition the **Ω**-system *U* belongs to **sDS<sup>Ω</sup>** <sup>≥</sup>*<sup>n</sup>* , and moreover the **Ω**-system *U* witnesses that *U* ∈ *D*. But this contradicts the Case 2 assumption.

#### *5.4. Truth Lemma*

According to the next theorem ("the truth lemma"), the truth in the generic extensions considered is connected in the usual way with the relation **forc**<sup>∞</sup> . We continue to argue under the assumptions and notation of Definitions 7 on page 22 and 8 on page 25.

**Theorem 10.** *Assume that z* = *w*[*ζ*] *and ϕ is a* L*<sup>z</sup> -formula. Then ϕ*[**G** *z*] *is true in* **L**[*ζ*, **G** *z*] *iff there is a condition p* <sup>∈</sup> **<sup>G</sup>** *<sup>z</sup> such that p* **forc***<sup>z</sup>* <sup>∞</sup> *ϕ.*

**Proof.** We proceed by induction. Suppose that *ϕ* is an atomic formula of type (F3) of Section 5.2. (The case of formulas as in (F2) is pretty elementary.) To prove the implication ⇐= , assume that *<sup>p</sup>* <sup>∈</sup> **<sup>G</sup>** *<sup>z</sup>* and *<sup>p</sup>* **forc***<sup>z</sup>* <sup>∞</sup> *<sup>α</sup>* <sup>∈</sup> *<sup>τ</sup>* , where *<sup>α</sup>* <sup>&</sup>lt; **<sup>Ω</sup>** and *<sup>τ</sup>* <sup>∈</sup> **Nam***<sup>z</sup> <sup>ζ</sup>* . Then by definition ((F3) in Section 5.2) there exists a condition *<sup>q</sup>* <sup>∈</sup> **<sup>P</sup><sup>Ω</sup>** *<sup>z</sup>* satisfying *<sup>p</sup> <sup>q</sup>* and *<sup>q</sup>*, *<sup>α</sup>*<sup>∈</sup> *<sup>τ</sup>* . There are conditions *<sup>p</sup>* , *<sup>q</sup>* <sup>∈</sup> **<sup>P</sup><sup>Ω</sup>** such that *p* = *p z* and *q* = *q z*, but not necessarily *p q* . We only know that *p* (*n*, *i*) *q* (*n*, *i*) for all *n*, *i* ∈ *z*. Therefore *z* ⊆ *Z* = {*n*, *i* : *p* (*n*, *i*) *q* (*n*, *i*)}. The set *Z* belongs to **L** since so do *p* , *<sup>q</sup>* as elements of **<sup>P</sup><sup>Ω</sup>** <sup>∈</sup> **<sup>L</sup>** (whereas about *<sup>z</sup>* we only assert that *<sup>z</sup>* <sup>∈</sup> **<sup>L</sup>**[*ζ*]). Therefore a condition *<sup>q</sup>* <sup>∈</sup> **<sup>P</sup><sup>Ω</sup>** can be defined by:

$$q'''(n,i) = \begin{cases} \begin{array}{c} q(n,i) \\\\ p(n,i) \end{array} & \text{in case} \quad \langle n,i \rangle \in Z\_{\text{-}},\\ \begin{array}{c} p(n,i) \\\\ \end{array} & \text{in case} \quad \langle n,i \rangle \notin Z\_{\text{-}}. \end{array}$$

and we still have *q z* = *q z* and *p q* . It follows that *q* ∈ **G** by genericity, hence *q z* = *q z* ∈ **G** *z*. But then *α* ∈ *τ*[**G** *z*], as required.

To prove the converse, assume that *α* ∈ *τ*[**G** *z*]. There exists a condition *p* ∈ **G** *z* such that *<sup>q</sup>*, *<sup>α</sup>*<sup>∈</sup> *<sup>τ</sup>* , and we have *<sup>p</sup>* **forc***<sup>z</sup>* <sup>∞</sup> *α* ∈ *τ* , as required.

Rather simple inductive steps (F4), (F5) of Section 5.2 are left for the reader.

Let us carry out step (F6). Let *<sup>ϕ</sup>* :<sup>=</sup> <sup>∃</sup> *<sup>X</sup> <sup>ψ</sup>*(*X*). Suppose that *<sup>p</sup>* <sup>∈</sup> **<sup>G</sup>** *<sup>z</sup>* and *<sup>p</sup>* **forc***<sup>z</sup>* <sup>∞</sup> *ϕ*. By definition there exists a name *<sup>τ</sup>* <sup>∈</sup> **Nam***<sup>z</sup> <sup>ζ</sup>* such that *<sup>p</sup>* **forc***<sup>z</sup>* <sup>∞</sup> *ψ*(*τ*). The formula *ψ*(*τ*)[**G** *z*] is then true in **L**[*ζ*, **G** *z*] by the inductive hypothesis. But *ψ*(*τ*)[**G** *z*] coincides with *ψ*[**G** *z*](*Y*), where *Y* = *τ*[**G** *z*] ∈ **L**[*ζ*, **G** *z*], *Y* ⊆ **Ω**. We conclude that ∃ *X ψ*(*X*)[**G** *z*] is true in **L**[*ζ*, **G** *z*], as required.

To prove the converse, let *ϕ*[**G** *z*], that is, ∃ *X ψ*(*X*)[**G** *z*], be true in **L**[*ζ*, **G** *z*]. As *X* is relativized to P(**Ω**), there is a set *X* ∈ P(**Ω**) in **L**[*ζ*, **G** *z*] satisfying *ϕ*(*X*)[**G** *z*] in **L**[*ζ*, **G** *z*]. By Lemma 21, there is a name *<sup>τ</sup>* <sup>∈</sup> **Nam***<sup>z</sup> <sup>ζ</sup>* with *X* = *τ*[**G** *z*], so *ψ*(*τ*)[**G** *z*] holds in **L**[*ζ*, **G** *z*]. The inductive hypothesis implies that some *<sup>p</sup>* <sup>∈</sup> **<sup>G</sup>** *<sup>z</sup>* satisfies *<sup>p</sup>* **forc***<sup>z</sup>* <sup>∞</sup> *ψ*(*τ*), hence *p* **forc***<sup>z</sup>* <sup>∞</sup> *ϕ*, as required.

Finally, let us carry out step (F8), which is somewhat less trivial. Prove the lemma for a L*<sup>z</sup>* formula ¬ *ϕ*, assuming that the result holds for *ϕ*. If ¬ *ϕ*[**G** *z*] is false in **L**[*ζ*, **G**] then *ϕ*[**G** *z*] is true. Thus by the inductive hypothesis, there is a condition *<sup>p</sup>* <sup>∈</sup> **<sup>G</sup>** *<sup>z</sup>* such that *<sup>p</sup>* **forc***<sup>z</sup>* <sup>∞</sup> *ϕ*. Then *q* **forc***<sup>z</sup>* <sup>∞</sup> ¬ *ϕ* for any *q* ∈ **G** *z* is impossible by Lemma 24 above.

Conversely suppose that *p* **forc***<sup>z</sup>* <sup>∞</sup> ¬ *ϕ* holds for no *p* ∈ **G** *z*. Then by Lemma 25 there exists *q* ∈ **G** *z* such that *q* **forc***<sup>z</sup>* <sup>∞</sup> *ϕ*. It follows that *ϕ*[**G** *z*] is true by the inductive hypothesis, therefore *ϕ*[**G** *z*] is false.

#### **6. Invariance**

The goal of this section is to prove Theorem 9 on page 24, and thereby accomplish the proof of Theorem 8, and the proof of Theorem 2 (the main theorem) itself. The proof makes use of the relation **forc** introduced in Section 5, and exploits certain symmetries in **forc** , investigated in Section 6.5.

#### *6.1. Hidden Invariance*

Theorem 9 belongs to a wide group of results on the structure of generic models which assert that such-and-such elements of a given generic extension belong to a smaller and/or better shaped extension. One of possible methods to prove such results is to exploit the homogeneity of the forcing notion considered, or in different words, its invariance w. r. t. a sufficiently large system of order-preserving transformations. In particular, for a straightforward proof of Theorem 11 below, which is our key technical step in the proof of Theorem 9, the invariance of the forcing notion **P<sup>Ω</sup>** under permutations of indices in <sup>I</sup> <sup>=</sup> *<sup>ω</sup>* <sup>×</sup> *<sup>ω</sup>* (to permute areas *<sup>z</sup>* and *z*) would be naturally required, whereas **P<sup>Ω</sup>** is definitely not invariant w. r. t. permutations.

On the other hand, the auxiliary forcing relation **forc** is invariant w. r. t. permutations. Theorem 10 in Section 5.4 conveniently binds the relation **forc** with the truth in **P<sup>Ω</sup>** -generic extensions by means of a forcing-style association. This principal association was based on the **M**-completeness property (Definition 7 on page 22 and Theorem 6). Basically it occurs that some transformations, that is, permutations, are *hidden* in construction of **P<sup>Ω</sup>** , so that they do not act explicitly, but their influence is preserved and can be recovered via the relation **forc**.

This method of *hidden invariance*, that is, invariance properties (of an auxiliary forcing-type relation like **forc**) hidden in **P<sup>Ω</sup>** by a suitable generic-style construction of **P<sup>Ω</sup>** , was introduced in Harrington's notes [22] in in the context of the almost disjoint forcing (in a somewhat different terminology from what is used here). It was introduced independently by one of the authors in [37] in the context of the Sacks forcing and its Jensen's modification in [38]; see e.g., [3,28,39] for further research in this direction based on product and iterated versions of the Sacks and Jensen forcing earlier studied in detail in [40–47].

#### *6.2. The Invariance Theorem*

We still argue under the assumptions and notation of Definitions 7 on page 22 and 8 on page 25. Let Π be the group of all *finite permutations* of *ω*, that is, all bijections *π* : *ω* onto −→ *ω* since the set |*π*| = {*k* : *π*(*k*) = *k*} is finite. If *m* < *ω* then the subgroup Π*<sup>m</sup>* consists of all *π* ∈ Π satisfying *π*(*k*) = *k* for all *k* < *m*. If *π* ∈ Π, and *z* ⊆ *ω* × *ω* then put *πz* = {*π*(*n*), *i* : *n*, *i* ∈ *z*}.

If in addition *g* : *ω* → **Ξ** then define *πg* : *ω* → **Ξ** by *πg*(*π*(*n*)) = *g*(*n*), all *n*.

Similarly if *<sup>e</sup>* <sup>∈</sup> **<sup>C</sup>** and <sup>|</sup>*π*| ⊆ lh *<sup>e</sup>*, then define *<sup>e</sup>* <sup>=</sup> *<sup>π</sup><sup>e</sup>* <sup>∈</sup> **<sup>C</sup>** such that lh *<sup>e</sup>* <sup>=</sup> lh *<sup>e</sup>* and *<sup>e</sup>* (*π*(*n*)) = *e*(*n*) for all *n* < lh *e*. The following is the invariance theorem.

**Theorem 11** (in **L**[*ζ*])**.** *Assume that z* = *w*[*ζ*]*, π* ∈ Π*<sup>m</sup> , z* = *πz, ϕ is a closed parameter free formula of* <sup>L</sup>*<sup>z</sup> ,* #(*ϕ*) <sup>≤</sup> *m, and p*<sup>0</sup> <sup>∈</sup> **<sup>P</sup><sup>Ω</sup>** *. Then p*<sup>0</sup> *<sup>z</sup>* **forc***<sup>z</sup>* <sup>∞</sup> <sup>¬</sup> *<sup>ϕ</sup> iff p*<sup>0</sup> *<sup>z</sup>* **forc***<sup>z</sup>* <sup>∞</sup> ¬ *ϕ.*

A lengthy proof of Theorem 11 follows below in this Section.

#### *6.3. Proof of Theorem 9 from the Invariance Theorem*

Under the assumptions of Theorem 9, consider an arbitrary set *a* ∈ **L**[*ζ*, **G** *w*[*ζ*]], *a* ⊆ *ω*, and assume that **M** ≥ 2 (see Definition 8) and it is true in **L**[*ζ*, **G** *w*[*ζ*]] that *a* is parameter free definable in P(**Ω**); ∈, **p**, i.e., *a* = {*j* < *ω* : ¬ *ϕ*(*j*)}, where *ϕ*(·) is a parameter free L*<sup>z</sup>* -formula. Let *m* = #(*ϕ*) and *w* = *w*[*ζ*]. The goal is to prove that *a* ∈ **L**[**G**]. This is based on the next lemma.

**Lemma 26.** *The set T* <sup>=</sup> {*p*, *<sup>j</sup>* : *<sup>p</sup>* <sup>∈</sup> **<sup>P</sup><sup>Ω</sup>** <sup>∧</sup> *<sup>p</sup> <sup>w</sup>* **forc***<sup>w</sup>* <sup>∞</sup> ¬ *ϕ*(*j*)} *belongs to* **L***.*

**Proof.** Note that, by Lemma 23, the set:

$$\mathcal{K} = \left\{ \langle z, p, j \rangle : p \in \mathbb{P}^{\mathbb{D}} \land z \in \mathbf{L}[\mathfrak{f}] \land z \subseteq \omega \times \omega \land j < \omega \land p \restriction z \text{ forc}^z\_{\infty} \neg \varphi(j) \right\}$$

is definable in **L**[*ζ*] by a formula with sets in **L** as parameters, say *K* = {*z*, *p*, *j* : *ϑ*(*z*, *p*, *j*, *S*)} in **L**[*ζ*], where *<sup>S</sup>* <sup>∈</sup> **<sup>L</sup>** is a sole parameter. Recall that *<sup>ζ</sup>* <sup>∈</sup> **<sup>Ξ</sup>***<sup>ω</sup>* is **<sup>C</sup>**-generic over **<sup>L</sup>**, and *<sup>w</sup>* <sup>=</sup> *<sup>w</sup>*[*ζ*] = {*n*, *<sup>j</sup>* : *<sup>j</sup>* <sup>∈</sup> *<sup>ζ</sup>*(*n*)}. Let ˘ *ζ* be a canonical **C**-name for *ζ* , and be the **C**-forcing relation over **L**. We claim that:

$$\theta(\mathfrak{w}, p, j, \mathcal{S}) \text{ holds in } \mathbf{L}[\mathfrak{J}] \quad \text{iff} \quad \mathfrak{J} \backslash \mathfrak{m} \Vdash \theta(\mathfrak{w}[\mathfrak{J}], p, j, \mathcal{S}) \; ; \tag{10}$$

*ζ m* belongs to **C**, of course. The direction ⇐= is obvious.

To establish =⇒ , assume that the right-hand side fails. Then there is a condition *e*<sup>0</sup> ∈ **C** such that *<sup>ζ</sup> <sup>m</sup>* <sup>⊆</sup> *<sup>e</sup>*<sup>0</sup> and *<sup>e</sup>*<sup>0</sup> <sup>¬</sup> *<sup>ϑ</sup>*(*w*[˘ *ζ*], *p*, *j*, *S*). We note that the set:

$$D = \{ \mathfrak{e} \in \mathbb{C} : \mathfrak{J} \upharpoonright \mathfrak{m} \subseteq \mathfrak{e} \land \exists \, \pi \in \Pi\_{\mathfrak{m}} \left( |\pi| \subseteq \mathfrak{dom} \, \mathfrak{e} \land \mathfrak{e}\_0 \subseteq \pi \mathfrak{e} \right) \}$$

is dense in **C** over *ζ m*. Therefore, by the genericity of *ζ* , there exists a number *k* > *m* such that *e* = *ζ k* ∈ *D*. Accordingly, there is a permutation *π* ∈ Π*<sup>m</sup>* satisfying |*π*| ⊆ *k* and *e*<sup>0</sup> ⊆ *πe*.

We put *ζ* = *πζ* ; this is still a **C**-generic element of **Ξ***<sup>ω</sup>* , with **L**[*ζ* ] = **L**[*ζ*] since *π* ∈ **L**, and we have *e*<sup>0</sup> ⊆ *πe* ⊂ *ζ* . It follows, by the choice of *e*<sup>0</sup> , that *ϑ*(*w*[*ζ* ], *p*, *j*, *S*) *fails* in **L**[*ζ* ] = **L**[*ζ*], and hence *w*[*ζ* ], *p*, *j* ∈/ *K* by the choice of *ϑ*. However *w*[*ζ* ] = *π* · *w*[*ζ*] = *πw*, thus we have *πw*, *p*, *j* ∈/ *K*.

We conclude that *pπw* **forc***π<sup>w</sup>* <sup>∞</sup> <sup>¬</sup> *<sup>ϕ</sup>*(*j*) *fails* by the definition of *<sup>K</sup>*. Therefore *<sup>p</sup> <sup>w</sup>* **forc***<sup>w</sup>* <sup>∞</sup> ¬ *ϕ*(*j*) *fails* as well by Theorem 11, so we have *w*, *p*, *j* ∈/ *K*, and hence *ϑ*(*w*, *p*, *j*, *S*) fails in **L**[*ζ* ] = **L**[*ζ*], as required. This completes the proof of (10). Now, coming back to the lemma, we deduce the equality *<sup>T</sup>* <sup>=</sup> {*p*, *<sup>j</sup>*<sup>∈</sup> **<sup>L</sup>** : *<sup>ζ</sup> <sup>m</sup> <sup>ϑ</sup>*(*w*[˘ *ζ*], *p*, *j*, *S*)} from (10). This implies *T* ∈ **L**.

It remains to notice that, by Theorem 10,

$$\begin{split} \exists j \in \mathcal{a} \quad \Longleftrightarrow \quad \mathbf{L}[\mathsf{J}, \mathsf{G} \mid \mathsf{w}[\mathsf{J}]] \vdash \neg \varphi(j) \quad \Longleftrightarrow \quad \exists \, p \in \mathbf{G} \mid \mathsf{w}(p \; \mathsf{forc}^{\mathsf{w}}\_{\infty} \neg \varphi(j)) \\ \Longleftrightarrow \quad \exists \, p \in \mathbf{G} \, (p \; \mathsf{if} \, \mathsf{w} \; \mathsf{forc}^{\mathsf{w}}\_{\infty} \neg \varphi(j)) \, . \end{split}$$

Therefore *j* ∈ *a* ⇐⇒ ∃ *p* ∈ **G** (*p*, *j* ∈ *T*). But *T* ∈ **L** by Lemma 26. We conclude that *a* ∈ **L**[**G**], as required.

This completes the proof of Theorem 9 from Theorem 11.

#### *6.4. The Invariance Theorem: Setup*

We still argue under the assumptions and notation of Definitions 7 on page 22 and 8 on page 25. Here we begin **the proof of Theorem 11**. It will be completed in Section 6.6.

We fix *<sup>m</sup>*, *<sup>π</sup>* <sup>∈</sup> <sup>Π</sup>*<sup>m</sup>* , *<sup>p</sup>*<sup>0</sup> , *<sup>z</sup>* <sup>=</sup> *<sup>w</sup>*[*ζ*], *<sup>z</sup>* <sup>=</sup> *<sup>π</sup>z*, and *<sup>ϕ</sup>* with #(*ϕ*) <sup>≤</sup> *<sup>m</sup>*, as in Theorem 11. Suppose towards the contrary that *<sup>p</sup>*<sup>0</sup> *<sup>z</sup>* **forc***<sup>z</sup>* <sup>∞</sup> <sup>¬</sup> *<sup>ϕ</sup>*, but *<sup>p</sup>*<sup>0</sup> *<sup>z</sup>* **forc***<sup>z</sup>* <sup>∞</sup> ¬ *ϕ fails*. By definition there is an ordinal *<sup>μ</sup>* <sup>&</sup>lt; **<sup>Ω</sup>**<sup>⊕</sup> such that *<sup>p</sup>*<sup>0</sup> *<sup>z</sup>* **forc***<sup>z</sup> <sup>μ</sup>* <sup>¬</sup> *<sup>ϕ</sup>*, but *<sup>p</sup>*<sup>0</sup> *<sup>z</sup>* **forc***<sup>z</sup> <sup>μ</sup>* ¬ *ϕ fails*. Then we have:

(A) <sup>a</sup> **<sup>Ω</sup>**-system *<sup>U</sup>*<sup>1</sup> <sup>∈</sup> **sDS**[*m*] with **<sup>U</sup><sup>Ω</sup>** *<sup>μ</sup> <sup>U</sup>*<sup>1</sup> , and a condition *<sup>p</sup>*<sup>1</sup> <sup>∈</sup> **<sup>P</sup>**[*U*1], *<sup>p</sup>*<sup>1</sup> *<sup>p</sup>*<sup>0</sup> , such that *p*<sup>1</sup> *z* **forc***<sup>z</sup> <sup>U</sup>*<sup>1</sup> *<sup>ϕ</sup>*, but *<sup>p</sup>*<sup>1</sup> *<sup>z</sup>* **forc***<sup>z</sup> <sup>U</sup>*<sup>1</sup> ¬ *ϕ* still holds by Lemma 22.

We now recall that any condition *p* ∈ <sup>∗</sup>**P<sup>Ω</sup>** is a map *p* ∈ **L**, defined on I = *ω* × *ω*, and each value *p*(*n*, *i*) = **S***p*(*n*, *i*); **F***p*(*n*, *i*) is a pair of a set **S***p*(*n*, *i*) ⊆ **Seq<sup>Ω</sup>** and **F***p*(*n*, *i*) ⊆ **Fun<sup>Ω</sup>** , with card (**S***p*(*n*, *<sup>i</sup>*) <sup>∪</sup> **<sup>F</sup>***p*(*n*, *<sup>i</sup>*)) <sup>&</sup>lt; **<sup>Ω</sup>** strictly, in **<sup>L</sup>**. We define the *support*:

$$||p|| = \bigcup\_{n,i < \omega} ||p||\_{n i'} \text{ where } ||p||\_{ni} = \{s(0) : s \in \mathbf{S}\_p(n, i)\} \cup \{f(0) : f \in \mathbf{F}\_p(n, i)\} \text{ } \neq$$

then ||*p*|| ∈ **<sup>L</sup>**, ||*p*|| ⊆ **<sup>Ω</sup>**, and card ||*p*|| <sup>&</sup>lt; **<sup>Ω</sup>** strictly, so that ||*p*|| is a bounded subset of **<sup>Ω</sup>**. In particular, ||*p*1|| is a bounded subset of **Ω** in **L**. Therefore there is:

(B) A bijection *<sup>b</sup>* <sup>∈</sup> **<sup>L</sup>**, *<sup>b</sup>* : **<sup>Ω</sup>** onto −→ **<sup>Ω</sup>**, such that ||*p*1|| ∩ (*<sup>b</sup>* "||*p*1||) = <sup>∅</sup> and *<sup>b</sup>* <sup>=</sup> *<sup>b</sup>*−<sup>1</sup> .

Furthermore, as *<sup>U</sup>*<sup>1</sup> <sup>∈</sup> **sDS**[*m*], the **<sup>Ω</sup>**-system *<sup>U</sup>*<sup>1</sup> is **<sup>Ω</sup>**-size, and hence the set *<sup>J</sup>* <sup>=</sup> - *<sup>n</sup>*,*i*<*<sup>ω</sup> <sup>U</sup>*1(*n*, *<sup>i</sup>*) <sup>∈</sup> **<sup>L</sup>** satisfies card *<sup>J</sup>* <sup>≤</sup> **<sup>Ω</sup>** in **<sup>L</sup>**. It follows that there exists:

(C) A sequence {*F<sup>α</sup>* }*α*<**<sup>Ω</sup>** <sup>∈</sup> **<sup>L</sup>** of bijections *<sup>F</sup><sup>α</sup>* : **<sup>Ω</sup>** onto −→ **Ω**, such that *F*<sup>0</sup> = *b* (see above), *F<sup>α</sup>* = *F<sup>α</sup>* <sup>−</sup><sup>1</sup> , and if *f* , *g* ∈ *J* then there is an ordinal *α* < **Ω** such that *f*(*α*) = *Fα*(*g*(*α*)).

#### *6.5. Transformation*

In continuation of **the proof of Theorem 11**, we now define an automorphism acting on several different domains in **<sup>L</sup>**. It will be based on *<sup>π</sup>* and *<sup>F</sup><sup>α</sup>* of Section 6.4 and its action will be denoted by . Along the way we will formulate properties (D)–(H) of the automorphism, a routine check of which is left to the reader.

We argue under the assumptions and notation of Definitions 7 on page 22 and 8 on page 25.

If *α* ≤ **Ω** and *f* : *α* → **Ω** then *f* : *α* → **Ω** is defined by *f* (*γ*) = *Fγ*(*f*(*γ*)) for all *γ* < *α*. In particular, *f* (0) = *<sup>F</sup>*0(*f*(0)) = *<sup>b</sup>*(*f*(0)). This defines *<sup>s</sup>* <sup>∈</sup> **Seq<sup>Ω</sup>** and *<sup>f</sup>* ∈ **Fun<sup>Ω</sup>** for all *s* ∈ **Seq<sup>Ω</sup>** and *f* ∈ **Fun<sup>Ω</sup>** .

(D) *f* −→ *f* is a bijection **Seq<sup>Ω</sup>** onto −→ **Seq<sup>Ω</sup>** and **Fun<sup>Ω</sup>** onto −→ **Fun<sup>Ω</sup>** , and if *<sup>f</sup>* , *<sup>g</sup>* ∈ *<sup>J</sup>* = - *<sup>n</sup>*,*i*<*<sup>ω</sup> U*1(*n*, *i*) then *f* = *g* by (C).

If *<sup>u</sup>* <sup>⊆</sup> **Fun<sup>Ω</sup>** then let *<sup>u</sup>* <sup>=</sup> { *<sup>f</sup>* : *<sup>f</sup>* <sup>∈</sup> *<sup>u</sup>*}. If *<sup>S</sup>* <sup>⊆</sup> **Seq<sup>Ω</sup>** then let *<sup>S</sup>* <sup>=</sup> {*<sup>s</sup>* : *<sup>s</sup>* <sup>∈</sup> *<sup>S</sup>*}. If *U* is a **Ω**-system then define a **Ω**-system *U* , such that:

$$\begin{aligned} \bar{\mathcal{U}}(n,i)&=\mathcal{U}(n,i), & \text{in case} \quad n$$

If *<sup>p</sup>* <sup>∈</sup> <sup>∗</sup>**P<sup>Ω</sup>** then let *<sup>p</sup>* <sup>∈</sup> <sup>∗</sup>**P<sup>Ω</sup>** be defined so that:

$$\begin{array}{rcl} \widehat{p}(n,i) &=& p(n,i) \, , \quad \text{in case} \quad n < m \; ;\\ \widehat{p}(\pi(n),i) &=& \langle \widehat{\mathbf{S}\_{p}(n,i)} \rangle ; \widehat{\mathbf{F}\_{p}(n,i)} \rangle \, , \quad \text{in case} \quad n \ge m \; ;\end{array}$$

where **<sup>S</sup>***p*(*n*, *<sup>i</sup>*) = {*<sup>s</sup>* : *<sup>s</sup>* <sup>∈</sup> **<sup>S</sup>***p*(*n*, *<sup>i</sup>*)} and **<sup>F</sup>***p*(*n*, *<sup>i</sup>*) = {*<sup>s</sup>* : *<sup>s</sup>* <sup>∈</sup> **<sup>F</sup>***p*(*n*, *<sup>i</sup>*)} by the above. These are consistent definitions because *π* ∈ Π*<sup>m</sup>* .


If in addition *z* ⊆ *ω* × *ω* (not necessarily *z* ∈ **L**), then if conditions *p*, *q* ∈ <sup>∗</sup>**P<sup>Ω</sup>** satisfy *p z* = *q z* , then easily *<sup>p</sup> <sup>z</sup>* <sup>=</sup> *<sup>q</sup> <sup>z</sup>* , where *<sup>z</sup>* <sup>=</sup> *<sup>π</sup>· <sup>z</sup>* <sup>=</sup> {*π*(*n*), *<sup>i</sup>* : *<sup>n</sup>*, *<sup>i</sup>*<sup>∈</sup> *<sup>z</sup>*}. This allows us to define *<sup>r</sup>* <sup>=</sup> *<sup>p</sup> <sup>z</sup>* for every *r* ∈ <sup>∗</sup>**P<sup>Ω</sup>** *z* , where *p* ∈ <sup>∗</sup>**P<sup>Ω</sup>** is any condition satisfying *r* = *p z* .

(G) If *<sup>z</sup>* <sup>⊆</sup> *<sup>ω</sup>* <sup>×</sup> *<sup>ω</sup>* then *<sup>p</sup>* −→ *<sup>p</sup>* is a -preserving bijection of **<sup>P</sup>**[*U*] *<sup>z</sup>* onto **<sup>P</sup>**[*U*] *<sup>z</sup>* .

If *<sup>z</sup>* <sup>⊆</sup> *<sup>ω</sup>* <sup>×</sup> *<sup>ω</sup>* and *<sup>τ</sup>* <sup>∈</sup> **Nam***<sup>z</sup> <sup>ζ</sup>* (see Section 5.1) then we define *<sup>τ</sup>* <sup>=</sup> {*p*, *<sup>α</sup>* : *<sup>p</sup>*, *<sup>α</sup>*<sup>∈</sup> *<sup>τ</sup>*}, and accordingly if *<sup>ϕ</sup>* is a <sup>L</sup>*<sup>z</sup>* -formula then *<sup>ϕ</sup>* is obtained by substituting *<sup>τ</sup>* for each name *<sup>τ</sup>* in *<sup>ϕ</sup>*.

(H) If *<sup>z</sup>* <sup>⊆</sup> *<sup>ω</sup>* <sup>×</sup> *<sup>ω</sup>*, *<sup>z</sup>* <sup>∈</sup> **<sup>L</sup>**[*ζ*], then the mapping *<sup>τ</sup>* −→ *<sup>τ</sup>* is a bijection of **Nam***<sup>z</sup> <sup>ζ</sup>* onto **Nam***<sup>z</sup> <sup>ζ</sup>* and a bijection of <sup>L</sup>*<sup>z</sup>* -formulas onto <sup>L</sup>*<sup>z</sup>* -formulas.

**Remark 6.** *The action of is idempotent, so that e.g., f* = *f for any f* ∈ **Fun<sup>Ω</sup>** *etc. This is because we require that b*−<sup>1</sup> = *b and F*−<sup>1</sup> *<sup>α</sup>* = *F<sup>α</sup> for all α* < **Ω***.*

*The action of is constructible on* **Seq<sup>Ω</sup>** *,* **Fun<sup>Ω</sup>** *,* **<sup>Ω</sup>***-systems,* <sup>∗</sup>**P<sup>Ω</sup>** *, since both <sup>π</sup> and the sequence of maps Fα belong to* **L** *by (B), (C).*

*If <sup>z</sup>* <sup>∈</sup> **<sup>L</sup>**[*ζ*] *then the action of on* <sup>∗</sup>**P<sup>Ω</sup>** *<sup>z</sup> and names in* **Nam***<sup>z</sup> <sup>ζ</sup> belongs to* **L**[*ζ*]*, since the extra parameter z* ∈ **L**[*ζ*] *does not necessarily belong to* **L***.*

It is not unusual that transformations of a forcing notion considered lead to this or another invariance. The next lemma is exactly of this type.

**Lemma 27** (in **<sup>L</sup>**[*ζ*])**.** *Assume that <sup>U</sup>* <sup>∈</sup> **sDS<sup>Ω</sup>** *, <sup>z</sup>* <sup>=</sup> *<sup>w</sup>*[*ζ*]*, <sup>π</sup>* <sup>∈</sup> <sup>Π</sup>*<sup>m</sup> , <sup>z</sup>* <sup>=</sup> *<sup>π</sup>z, <sup>p</sup>* <sup>∈</sup> **<sup>P</sup>**[*U*] *<sup>z</sup>, and <sup>ϕ</sup> is a closed formula of* <sup>L</sup>*<sup>z</sup> ,* #(*ϕ*) <sup>≤</sup> *<sup>m</sup>* <sup>+</sup> <sup>1</sup>*. Then p* **forc***<sup>z</sup> <sup>U</sup>* <sup>Φ</sup> *iff <sup>p</sup>* **forc***<sup>z</sup> <sup>U</sup>* <sup>Φ</sup> *.*

**Proof.** We argue by induction on the structure of Φ. Routine cases of atomic formulas (F2) and steps (F4) and (F5) of Section 5.2 by means of (D)–(H) are left to the reader. Thus we concentrate on atomic formulas of type (F3) and steps (F6) and (F8) in Section 5.2. In all cases we take care of only one direction of the equivalence of the lemma, as the other direction is entirely similar via Remark 6 just above.

**Formulas of type** (F3)**.** Let <sup>Φ</sup> be *<sup>α</sup>* <sup>∈</sup> *<sup>τ</sup>* , where *<sup>α</sup>* <sup>&</sup>lt; **<sup>Ω</sup>** and *<sup>τ</sup>* <sup>∈</sup> **Nam***<sup>z</sup> <sup>ζ</sup>* . Assume that *<sup>p</sup>* **forc***<sup>z</sup> <sup>U</sup> α* ∈ *τ* . Then by definition there is a condition *<sup>q</sup>* <sup>∈</sup> **<sup>P</sup>**[*U*] *<sup>z</sup>* such that *<sup>p</sup> <sup>q</sup>* and *<sup>q</sup>*, *<sup>α</sup>*<sup>∈</sup> *<sup>τ</sup>* . Then *<sup>q</sup>* and *<sup>p</sup>* belong to **<sup>P</sup>**[*U*]*z*, *<sup>p</sup> <sup>q</sup>*, and *<sup>q</sup>*, *<sup>α</sup>*<sup>∈</sup> *<sup>τ</sup>*, so we have *<sup>p</sup>* **forc***<sup>z</sup> <sup>U</sup> <sup>α</sup>* <sup>∈</sup> *<sup>τ</sup>*, as required.

**Step** (F6)**.** Let <sup>Φ</sup> :<sup>=</sup> <sup>∃</sup> *<sup>X</sup>* <sup>Ψ</sup>(*X*). Suppose that *<sup>p</sup>* **forc***<sup>z</sup> <sup>U</sup>* Φ. By definition there exists a name *<sup>τ</sup>* <sup>∈</sup> **Nam***<sup>z</sup> <sup>ζ</sup>* such that *<sup>p</sup>* **forc***<sup>z</sup>* <sup>∞</sup> <sup>Ψ</sup>(*τ*), Then we have *<sup>p</sup>* **forc***<sup>z</sup> <sup>U</sup>* <sup>Ψ</sup> (*τ*) by the inductive hypothesis. But Ψ (*τ*) coincides with <sup>Ψ</sup>(*τ*), where *<sup>τ</sup>* <sup>∈</sup> **Nam***<sup>z</sup> <sup>ζ</sup>* by (H) above. We conclude that *<sup>p</sup>* **forc***<sup>z</sup> <sup>U</sup>* <sup>∃</sup> *<sup>X</sup>* <sup>Ψ</sup> (*X*), that is, *<sup>p</sup>* **forc***<sup>z</sup> <sup>U</sup>* <sup>Φ</sup> , as required.

**Step** (F8)**.** Prove the lemma for a L*<sup>z</sup>* formula Φ := ¬ Ψ, assuming that the result holds for Ψ. Note that #(Φ) *m* + 1, hence #(Ψ) *m*. Suppose that *p* **forc***<sup>z</sup> <sup>U</sup>* ¬ Ψ *fails*. By definition there is a **Ω**-system *U* ∈ **sDS**[*m*] extending *U* , and a condition *q* ∈ **P**[*U* ] *z*, *q p*, such that *q* **forc***<sup>z</sup> <sup>U</sup>* Ψ. Then *<sup>q</sup>* **forc***<sup>z</sup> U* <sup>Ψ</sup> by the inductive hypothesis. Yet *<sup>U</sup>* belongs to **sDS<sup>Ω</sup>** , extends *U* , and satisfies *U* <sup>&</sup>lt;*<sup>m</sup>* <sup>=</sup> *<sup>U</sup>* <sup>&</sup>lt;*<sup>m</sup>* by (E), hence belonging even to **sDS**[*m*] by the choice of *<sup>U</sup>* , and in addition *<sup>q</sup>* <sup>∈</sup> **<sup>P</sup>**[*<sup>U</sup>* ]*<sup>z</sup>* and *<sup>q</sup> <sup>p</sup>* by (F). We conclude, by definition, that *<sup>p</sup>* **forc***<sup>z</sup> <sup>U</sup>* <sup>¬</sup> <sup>Ψ</sup> fails too, as required.

#### *6.6. Finalization*

We continue to argue under the assumptions and notation of Definitions 6 on page 20 and 8 on page 25. The goal of this Section is to accomplish the proof of Theorem 11 in Section 6.2 that was started in Section 6.4. We return to objects introduced in (A), (B), (C) of Section 6.2.

Let *<sup>q</sup>*<sup>1</sup> <sup>=</sup> *<sup>p</sup>*<sup>1</sup> *<sup>z</sup>*, so that *<sup>q</sup>*<sup>1</sup> <sup>∈</sup> **<sup>P</sup>**[*U*1] *<sup>z</sup>* and *<sup>q</sup>*<sup>1</sup> **forc***<sup>z</sup> <sup>U</sup>*<sup>1</sup> *ϕ* by (A). We have:

$$\begin{aligned} \widehat{\mathsf{U}^{\mathsf{I}}} \in \mathsf{sDS}[m] \wedge \widehat{p\_{1}} \in \mathsf{P}[\widehat{\mathsf{U}^{\mathsf{I}}}] \wedge \widehat{q\_{1}} = \widehat{p\_{1}} \wedge \widehat{\mathsf{z}} \in \mathsf{P}[\widehat{\mathsf{U}^{\mathsf{I}}}] \wedge \widehat{\mathsf{z}} \wedge \widehat{q\_{1}} \text{ for} \widehat{\mathsf{z}}\_{\widehat{\mathsf{U}^{\mathsf{I}}}} \neq \emptyset \end{aligned} \tag{11}$$

by Lemma 27. (Here *<sup>ϕ</sup>*, as a parameter free formula, coincides with *<sup>ϕ</sup>*.) Let a **<sup>Ω</sup>**-system *<sup>U</sup>* be defined by *<sup>U</sup>*(*n*, *<sup>i</sup>*) = *<sup>U</sup>*1(*n*, *<sup>i</sup>*) <sup>∪</sup> *<sup>U</sup>* 1(*n*, *i*) .

**Lemma 28.** *The* **Ω***-system U belongs to* **sDS**[*m*] *and extends both U*<sup>1</sup> *and U* <sup>1</sup> *. Conditions p*<sup>1</sup> *and <sup>p</sup>*<sup>1</sup> *belong to* **<sup>P</sup>**[*U*] *and are compatible in* **<sup>P</sup>**[*U*]*.*

**Proof (Lemma).** It follows by (D) (last claim) that *U* is a disjoint **Ω**-system. It follows by (E) that *U* <sup>&</sup>lt;*<sup>m</sup>* = *U*<sup>1</sup> <sup>&</sup>lt;*<sup>m</sup>* = *U* <sup>1</sup> <sup>&</sup>lt;*<sup>m</sup>* . Therefore *U* belongs to **sDS**[*m*] because so does *U*<sup>1</sup> .

To prove compatibility, it suffices to check that if *<sup>n</sup>*, *<sup>i</sup>* <sup>&</sup>lt; *<sup>ω</sup>* then either *<sup>p</sup>*1(*n*, *<sup>i</sup>*) = *<sup>p</sup>*1(*n*, *<sup>i</sup>*) or ||*p*1||*ni* ∩ ||*p*1||*ni* <sup>=</sup> <sup>∅</sup>. If *<sup>n</sup>* <sup>&</sup>lt; *<sup>m</sup>* then we have the 'either' case because by definition *<sup>p</sup>*<sup>1</sup> <sup>&</sup>lt;*<sup>m</sup>* <sup>=</sup> *<sup>p</sup>*<sup>1</sup> <sup>&</sup>lt;*<sup>m</sup>* . Suppose that *<sup>n</sup>* <sup>≥</sup> *<sup>m</sup>*. Let *<sup>k</sup>* <sup>=</sup> *<sup>π</sup>*−1(*n*); thus still *<sup>k</sup>* <sup>≥</sup> *<sup>m</sup>* (as *<sup>π</sup>* <sup>∈</sup> <sup>Π</sup>*<sup>m</sup>* ), *<sup>n</sup>* <sup>=</sup> *<sup>π</sup>*(*k*), and *<sup>p</sup>*1(*n*, *<sup>i</sup>*) = **<sup>S</sup>***p*(*k*, *<sup>i</sup>*); **<sup>F</sup>***p*(*k*, *<sup>i</sup>*). It follows that ||*p*1||*ni* is the *<sup>F</sup>*<sup>0</sup> -image, hence the *<sup>b</sup>* -image of the set ||*p*1||*ki* . However ||*p*1||*ki* ∪ ||*p*1||*ni* ⊆ ||*p*1||. We conclude that ||*p*1||*ni* ∩ ||*p*1||*ni* <sup>=</sup> <sup>∅</sup> by Claim (B) of Section 6.4, as required.

To finalize **the proof of Theorem <sup>11</sup>**, let, by Lemma 28, *<sup>r</sup>* <sup>∈</sup> **<sup>P</sup>**[*U*]*<sup>z</sup>* satisfy both *<sup>r</sup> <sup>p</sup>*<sup>1</sup> *<sup>z</sup>* and *<sup>r</sup> <sup>p</sup>*<sup>1</sup> *<sup>z</sup>* <sup>=</sup> *<sup>q</sup>*<sup>1</sup> . However *<sup>q</sup>*<sup>1</sup> **forc***<sup>z</sup> U* <sup>1</sup> *<sup>ϕ</sup>* by (11). We conclude that *<sup>r</sup>* **forc***<sup>z</sup> <sup>U</sup> ϕ* by Lemma 28 and Lemma 22. On the other hand, *<sup>p</sup>*<sup>1</sup> *<sup>z</sup>* **forc***<sup>z</sup> <sup>U</sup>*<sup>1</sup> <sup>¬</sup> *<sup>ϕ</sup>* by (A) of Section 6.4, therefore we have *<sup>r</sup>* **forc***<sup>z</sup> <sup>U</sup>* ¬ *ϕ*. It remains to remind that #(*ϕ*) ≤ *m* and *U* ∈ **sDS**[*m*] by Lemma 28—and we still get a contradiction by Lemma 22(ii). The contradiction completes the proof of Theorem 11.

*Finalization.*

Theorem 11 just proved implies Theorem 9, see Section 6.3.

Theorem 9 ends the proof of Theorem 8 of Section 4.6, see Remark 5 on page 24.

This completes the proof of Theorem 2, the main result of this paper, see Remark 4 on page 23.

#### **7. Conclusions and Discussion**

In this study, the method of almost-disjoint forcing was employed to the problem of getting a model of **ZFC** in which the set **D**1*<sup>m</sup>* of all reals, definable by a parameter free type-theoretic formula with the highest quantifier order not exceeding a given natural number **M** ≥ 1, belongs to **D**2**<sup>M</sup>** , that is, it is itself definable by a formula of the same quantifier order. Moreover, we have **D**1**<sup>M</sup>** = **L** ∩ **R** in the model, that is, the set **D**1**<sup>M</sup>** is equal to the set of all Gödel-constructible reals.

The problem of getting a model for **D**1**<sup>M</sup>** ∈ **D**2**<sup>M</sup>** was posed in Alfred Tarski's article [18]. Its particular case **M** = 1 (analytical definability), that is, the problem of getting models for **D**<sup>11</sup> ∈ **D**<sup>21</sup> , or stronger, **D**<sup>11</sup> = **L** ∩ **R**, has been known since the early years of forcing, see e.g., problem 87 in Harvey Friedman's survey [21], and problems 3110, 3111, and 3112 in another early survey [20] by A. Mathias. As mentioned in [20,21], the particular case **M** = 1 was solved by Leo Harrington, and a sketch of the proof, related to a model for *Δ*<sup>1</sup> <sup>3</sup> = **L** ∩ **R**, can be found in Harrington's handwritten notes [22]. Our paper presents a full proof of the comprehensive result (Theorem 2) that finally solves the Tarski problem.

From this study, it is concluded that the hidden invariance technique (as outlined in Section 6.1) allows one to solve the problem by providing a generic extension of **L** in which the constructible reals are precisely the **D**1**<sup>M</sup>** reals, for a chosen value **M** ≥ 1. The hidden invariance technique has also been applied in recent papers [3–5,28] for the problem of getting a set theoretic structure of this or another kind at a preselected projective level. We finish with a short list of related problems.

1. If *x* ⊆ *ω* then let **D***pm*(*x*) be the set of all objects of order *p*, definable by a formula with *x* as the only parameter, whose all quantified variables are over orders ≤ *m*. (Compare to Definition 1 on page 2.) One may be interested in getting a model for:

$$\forall \mathbf{x} \subseteq \omega \; (\mathbf{D}\_{1m}(\mathbf{x}) \in \mathbf{D}\_{2m}(\mathbf{x}), \text{ or stronger}, \; \mathbf{D}\_{1m}(\mathbf{x}) = \beta^p(\omega) \cap \mathbf{L}). \tag{12}$$

This is somewhat similar to Problem 87 in [21]: Find a model of:

$$\mathbf{ZFC + "\text{for any reals } x, y, \text{we have: } \ge \mathbf{c} \to \mathbf{L}[y] \implies \mathbf{x} \text{ is } \Delta\_3^1 \text{ in } y''. \tag{13}$$

Problem (13) was known in the early years of forcing, see, e.g., problem 3111 in [20] or (3) in [23] (Section 6.1). Problem (13) was positively solved by René David [48,49], where the question is attributed to Harrington. The proof makes use of a tool known as David's trick, see S. D. Friedman [27] (Chapters 6, 8).

So far it is unknown whether the result of David [48] generalizes to higher projective classes *Δ*<sup>1</sup> *n* , *<sup>n</sup>* <sup>≥</sup> 4, or *<sup>Δ</sup>*<sup>1</sup> *<sup>ω</sup>* , whether it can be strengthened towards ⇐⇒ instead of =⇒ , and whether it can lead to an even partial solution of (12). This is a very interesting and perhaps difficult question.

2. Coming back to Harvey Friedman's *Δ*<sup>1</sup> *<sup>n</sup>* problem of getting a model for the sentence:

$$\text{the set } \, d\_n = \mathcal{P}(\omega) \cap \Lambda\_n^1 \text{ is equal to } \, \mathcal{P}(\omega) \cap \mathbf{L}, \tag{14}$$

(Section 1.2), it is clear that, unlike **D**1*<sup>m</sup>* ∈ **D**2*<sup>m</sup>* , if (14) holds for some *n* ≥ 3 then it definitely fails for any *n* = *n*. But we can try to weaken (14) to just:

$$d\_{\mathcal{U}} \in \Pi\_{\mathcal{U}}^1. \tag{15}$$

and then ask whether there is a generic extension of **<sup>L</sup>** satisfying <sup>∀</sup> *<sup>n</sup>* (*dn* <sup>∈</sup> *<sup>Π</sup>*<sup>1</sup> *<sup>n</sup>*). It holds by rather routine estimations that *<sup>d</sup>*<sup>1</sup> <sup>∈</sup> *<sup>Π</sup>*<sup>1</sup> <sup>1</sup> *<sup>Σ</sup>*<sup>1</sup> <sup>1</sup> , *<sup>d</sup>*<sup>2</sup> <sup>∈</sup> *<sup>Σ</sup>*<sup>1</sup> <sup>2</sup> *<sup>Π</sup>*<sup>1</sup> <sup>2</sup> , and if all reals are constructible then *dn* ∈ *Σ*1 *<sup>n</sup> Π*<sup>1</sup> *<sup>n</sup>* for all *<sup>n</sup>* <sup>≥</sup> 3 as well, so *<sup>Π</sup>*<sup>1</sup> *<sup>n</sup>* looks rather suitable in (15).

3. Recall that Theorem 2 implies the consistency of **D**1*<sup>m</sup>* ∈ **D**2*<sup>m</sup>* for each particular *m* ≥ 1.

But what about the consistency of the sentence "**D**1*<sup>m</sup>* ∈ **D**2*<sup>m</sup>* holds for all *m* ≥ 1 "? Perhaps a method developed in [50] can be useful to solve this problem.

4. It would be interesting to define a generic extension of **L** in which, for instance, **D**1*<sup>m</sup>* ∈ **D**2*<sup>m</sup>* holds for all even *m* ≥ 1 but fails for all odd *m* ≥ 1, or vice versa.

Lemma 2 on page 8 presents a possible difficulty: If we have **D**1*<sup>n</sup>* ∈ **D**2*<sup>n</sup>* for some *n* ≥ 1 by means of the equality **D**1*<sup>n</sup>* = P(*ω*) ∩ **L**, then **D**1*<sup>m</sup>* ∈ **D**2*<sup>m</sup>* definitely *fails* for all *m* < *n*.

5. Another question considered by Tarski in [18] is related to the sets **D***<sup>k</sup>* = - *<sup>m</sup>* **D***km* (all elements of order *k* , definable by a formula of any order). Tarski proves that **D***<sup>k</sup>* ∈/ **D***k*+<sup>1</sup> for all *k* ≥ 2, and leaves open the question whether **D**<sup>1</sup> ∈ **D**<sup>2</sup> . Similarly to the problem **D**1*<sup>m</sup>* ∈ **D**2*<sup>m</sup>* in Section 1.1, the *negative* answer **D**<sup>1</sup> ∈/ **D**<sup>2</sup> follows from the axiom of constructibility **V** = **L**, and hence is consistent with **ZFC**.

Prove the consistency of the sentences **D**<sup>1</sup> ∈ **D**<sup>2</sup> and **D**<sup>1</sup> = P(*ω*) ∩ **L**.

#### **Supplementary Materials:** The following are available online at http://www.mdpi.com/2227-7390/8/12/2214/ s1 .

**Author Contributions:** Conceptualization, V.K. and V.L.; methodology, V.K. and V.L.; validation, V.K. and V.L.; formal analysis, V.K. and V.L.; investigation, V.K. and V.L.; writing original draft preparation, V.K.; writing review and editing, V.K.; project administration, V.L.; funding acquisition, V.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Russian Foundation for Basic Research RFBR grant number 18-29-13037.

**Acknowledgments:** We thank the anonymous reviewers for their thorough review and highly appreciate the comments and suggestions, which significantly contributed to improving the quality of the publication.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **References**


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