1. Introduction
First-order formulas are used to express semantic and syntacticdefinable properties. They are broadly used in mathematics and other applications to study and to classify various aspects of reality. Definability is connected with the decidability, computability and complexity of mathematical objects [
1,
2,
3]. This tool has been used to solve a serious of famous mathematical problems including the quantifier elimination and the decidability of an algebraically closed field for real closed fields [
4,
5,
6], etc. Reducibilities for definable sets and their properties are studied and described in [
7].
Since in general there are more naturalthan definable properties, these formulas can express them in a partial way. In the present paper, we study the links between formulas and arbitrary properties, considering characteristics reflecting measures of their correspondence.
The paper is organized as follows. Preliminary notions, notations and results are represented in
Section 1. In
Section 2, we consider the links between formulas and properties for semantic and syntactic families. We study and characterize these links with respect to constructions of formulas, set-theoretic operations and closures. In
Section 3, we study, characterize and describe rank values and degree values for formulas with respect to the given properties. In
Section 4, we describe spectra for cardinalities of definable properties. In
Section 5, generic sentences and theories with respect to properties are introduced, and their links and ranks and described. Some illustrations of the considered links between formulas and properties are considered in
Section 6.
Throughout this work, we use the standard terminology in mathematical logic [
6], as well as the concepts, notations and results of [
8,
9,
10].
2. Preliminaries
Let be a language. If is relational, we denote by the family of all theories of the language . If contains functional symbols f, then is the family of all theories of the language , which is obtained by replacements of all n-ary symbols f with -ary predicate symbols interpreted by .
Following [
8], we define the
rank for families
, similar to the Morley rank for a fixed theory, and a hierarchy with respect to these ranks in the following way.
By , we denote the set of all formulas in the language , and by , we denote the set of all sentences in .
For a sentence , we denote by the set of all theories with .
Any set is called the φ-neighborhood, or simply a neighborhood, for , or the (-)definable subset of . The set is also called (formula or sentence-)definable (by the sentence ) with respect to , (sentence-)-definable, or simply s-definable.
Definition 1 ([
8])
. For the empty family , we put the rank ; for finite nonempty families , we put ; and for infinite families, — . For a family and an ordinal , we put if there are pairwise inconsistent -sentences , , such that , . If α is a limit ordinal, then if for any . We set if and . If for any α, we put .A family is called e-totally transcendental or totally transcendental, if is an ordinal.
If is e-totally transcendental, with , we define the degree of as the maximal number of pairwise inconsistent sentences such that .
Definition 2 ([
11])
. An infinite family is called e-minimal
if for any sentence , is finite or is finite. According to this definition, a family
is
e-minimal iff
and
[
8] and iff
has a unique accumulation point [
11].
In [
12]. the notion of
E-closure was introduced and characterized as follows:
Proposition 1. If is an infinite set and , then (i.e., T is an accumulation point for with respect to E-closure ) if and only if for any sentence the set is infinite.
The following theorem characterizes the property of e-total transcendency for countable languages.
Theorem 1. ([
8]).
For any family with , the following conditionsare equivalent:- (1)
;
- (2)
e-;
- (3)
.
Theorem 2 ([
9])
. For any language Σ, either is finite, if Σ consists of finitely many 0-ary and unary predicates, and finitely many constant symbols, or otherwise. For a language , we denote by the family of all theories in having n-element models, , as well as by the family of all theories in having infinite models.
Theorem 3 ([
9])
. For any language Σ, either , if Σ is finite, or and Σ has finitely many predicate symbols, or otherwise. Theorem 4 ([
9])
. For any language Σ, either is finite, if Σ is finite and without predicate symbols of arities as well as without functional symbols of arities , or otherwise. According to this definition, the families , , are E-closed. Thus, combining Theorem 1 with Theorems 2–4, we obtain the following possibilities of cardinalities for the families , , depending on and :
Proposition 2. For any language Σ, either is countable, if Σ consistsof finitely many 0-ary and unary predicates and finitely many constant symbols, or otherwise.
Proposition 3. For any language Σ, either is finite, if Σ is finite or and Σ has finitely many predicate symbols, or otherwise.
Proposition 4. For any language Σ, either is at most countable, if Σ is finite and without predicate symbols of arities as well as without functional symbols of arities , or otherwise.
Definition 3 ([
10])
. If is a family of theories and Φ is a set of sentences, then we put and the set is called (type) or ( diagram-)definable (by the set Φ) with respect to , (diagram-) -definable or simply d-definable. Clearly, finite unions of d-definable sets are again d-definable. Considering infinite unions of d-definable sets , , one can represent them by sets of sentences with infinite disjunctions , . We call these unions -definable sets.
Definition 4 ([
10])
. Let be a family of theories, Φ be a set of sentences and α be an ordinal or . The set Φ is called an α-ranking for if . A sentence φ is called an α-ranking for if is an α-ranking for . The set (the sentence ) is called a ranking for if it is an -ranking for with some .
Proposition 5 ([
10])
. For any ordinals , if , then for some(
α-ranking)
sentence φ. Moreover, there are pairwise -inconsistent β-ranking sentences for , and if , then there are infinitely many pairwise -inconsistent α-ranking sentences for . Theorem 5 ([
10])
. Let be a family of a countable language Σ and with , and let α be a countable ordinal, . Then, there is a -definable subfamily such that and . Theorem 6 ([
13])
. For any two disjoint subfamilies and of an E-closed family , the following conditions are equivalent:- (1)
and are separated by some sentence φ: and ;
- (2)
E-closures of and are disjoint in : ;
- (3)
E-closures of and are disjoint: .
Definition 5 ([
12])
. Let be a family of theories. A subset is said to be generating if . The generating set (for ) is minimal if does not contain proper generating subsets. A minimal generating set is least if is contained in each generating set for . Theorem 7 ([
12])
. If is a generating set for a E-closed set , then the following conditions are equivalent:- (1)
is the least generating set for ;
- (2)
is a minimal generating set for ;
- (3)
Any theory in is isolated by some set ; i.e., for any there is such that ;
- (4)
Any theory in is isolated by some set ; i.e., for any there is such that .
3. Relations between Formulas and Properties
Definition 6. Let Σ be a language, be a formula in and be a subclass of the class of all structures in the language Σ. We say that partially (respectively, totally) satisfies , denoted by or ( or ), if there are and (for any there is ) such that .
If is a subclass of the class of isomorphism types for the class , then we say that partially (respectively, totally) satisfies , denoted by or ( or ) if (), where consists of allstructures whose isomorphism types belong to .
If is a subset of the set of all complete theories in the language Σ, then we say that partially (respectively, totally) satisfies , denoted by or ( or ), if there are , , and (for any there are and ) such that .
We write if a -relation does not hold.
Remark 1. According to this definition, we have the following obvious properties:
If and , then . Similarly, if for nonempty , then , and if for nonempty P, then .
For any singleton , implies . Similarly, implies for any singleton , and implies for any singleton .
If and , then . This implies that the relations and do not depend on the choice of models for .
(Reflexivity) For any sentence φ and a (nonempty) family , we have (and ).
(Monotony) If , and then . If , and , then . If , and then . If , and then . If , and , then . If , and then .
For a property , we denote by the class of isomorphism types for structures in and by the set for some .
For a property , we denote by the class of all structures whose isomorphism types are represented in and by the set .
For a property , we denote by the class of all models of theories in and by the class .
In terms of these notations, by definition, we have the following natural links between semantic properties and and syntactic properties :
Proposition 6. For any formula and properties , and , the following conditions hold:
- (1)
iff , and iff ;
- (2)
iff , and iff ;
- (3)
iff , and iff ;
- (4)
iff , and iff ;
- (5)
iff , and iff ;
- (6)
iff , and iff .
In items and , the class can be replaced by a subclass such that . Similarly, in items and , the class can be replaced by a subclass such that , and independently can be replaced by a subclass such that .
According to Proposition 6, semantic properties and can be naturally transformed into syntactic ones and vice versa. This means that natural model-theoretic properties such as -categoricity, stability, simplicity, etc. can be formulated for theories, for structures and for their isomorphism types.
The links between ⊳-relations highlighted in Proposition 6 allow us to reduce our consideration to the relations and . Besides, for simplicity, we principally consider sentences instead of formulas in general. Reductions of formulas to sentences use the operators and .
Proposition 7. For any sentences and properties , the following conditions hold:
- (1)
If , then and ; the converse implication does not hold. There are and such that , , and ;
- (2)
iff and ;
- (3)
If and , then ; the converse implication does not hold. There are and such that , whereas and .
Proof. - (1)
If , then there is with . Since , and , we obtain and . Therefore, it is sufficient to note for that sentences asserting distinct finite cardinalities m and n for universes partially satisfy a property containing a theory with an m-element model and a theory with an n-element model. At the same time, since is inconsistent.
- (2)
If , then , and so and belong to all theories in implying and . Conversely, if and , then , , and so belongs to all theories in , implying .
- (3)
If and , then and , implying ; i.e., . Finally, if and are nonempty with and and are inconsistent sentences, then and and , implying and .
□
Proposition 8. For any sentences and properties the following conditions hold:
- (1)
If or , then ; the converse implication does not hold. There are and such that , and or ;
- (2)
iff or ;
- (3)
If and , then ; the converse implication does not hold. There are and such that whereas and .
Proof. - (1)
If or , then for some or for some . Therefore T or witness that . If is a tautology and is an inconsistent sentence, then for any nonempty , we have , , and .
- (2)
It holds that a sentence belongs to a complete theory T if and only if or .
- (3)
If and , then and . Therefore, and , implying ; thus, .
Now, let , be a sentence belonging to some but not all theories in . For the sentence , we have since is a tautology, and by the choice of . □
Proposition 9. For any sentence and a property , the following conditions hold:
- (1)
iff ;
- (2)
iff .
Proof. - (1)
If , then there is such that . Since T is complete, then , implying . Conversely, if , then does not belong to some theory . Since T is complete, then , implying .
- (2)
The second proposition immediately follows from .
□
Proposition 10. For any formula and a property , the following conditions hold:
- (1)
If and , then ;
- (2)
If and , then ;
- (3)
If and , then ;
- (4)
If and , then .
Proof. - (1)
Let and . Then, there are , , such that . This implies ; therefore, there is with —i.e., , and thus .
- (2)
We repeat the arguments presented in , replacing with an arbitrary value.
- (3)
Let and . Then, there are , , such that . This implies —i.e., , and thus .
- (4)
As for , we repeat the arguments presented in , replacing with an arbitrary value.
□
The following two theorems assert that the relations and are preserved under E-closures.
Theorem 8. For any sentence and a property , the following conditions are equivalent:
- (1)
;
- (2)
;
- (3)
for any/some with .
Proof. holds in view of and the monotony of the relation .
. It suffices to show that for any with . Since , there is a theory with . If , we have . Otherwise, T is an accumulation point of , implying, in view of Proposition 1.1, that belongs to infinitely many theories in . Therefore, .
is obvious.
□
Theorem 9. For any sentence and a property , the following conditions are equivalent:
- (1)
;
- (2)
;
- (3)
for any/some with .
Proof. . Let . If is finite, then , and we have . If is finite, then by Proposition 1.1, consists of theories in and of theories such that for any sentence , the set is infinite. Since , belongs to each such theory T. Thus, ;
and are obvious;
follows assuming for any/some with repeating the arguments presented in .
□
For a property , we denote by the set of all sentences with and by the set of all sentences with .
According to this definition, , , consists of all consistent sentences and consists of all tautologies .
Proposition 11. For any property the following conditions hold:
- (1)
;
- (2)
is consistent iff , and is a complete theory iff is a singleton;
- (3)
;
- (4)
is a consistent theory iff , and is complete iff is a singleton;
- (5)
For any , , and iff is a singleton.
Proof. - (1)
If , then and for some , implying . Conversely, if , then for some , implying and therefore ;
- (2)
Since , it is consistent. If , then ; i.e., is consistent and complete. If contains two distinct theories and , then , implying that is inconsistent as there are sentences such that and ;
- (3)
If , then and for any , implying . Conversely, if , then for any , implying and therefore ;
- (4)
Since , it is inconsistent. If , then by , implying that is a consistent theory as an intersection of complete theories. If is complete, then is both nonempty and does not contain two distinct theories; i.e., is a singleton. Conversely, if then which is a complete theory;
- (5)
If , then by and , we have . If , then . If for some , then .
□
Theorems 6, 8 and 9 and Proposition 11 immediately imply the following corollary on the separability of properties with respect to the relations and .
Corollary 1. For any properties , the following conditions hold:
- (1)
There exists such that and iff and are nonempty and ; in particular, there exists such that and iff ;
- (2)
There exists such that and iff .
Corollary 2. For any nonempty property the following conditions hold:
- (1)
The set forms a filter on with respect to ⊢;
- (2)
The filter is principal iff is forced by some sentence; i.e., is a finitely axiomatizable theory, which is incomplete for ;
- (3)
The filter is an ultrafilter iff is a singleton.
Proof. - (1)
The first proposition holds by monotony and Proposition 7, (2);
- (2)
The second immediately follows from Proposition 11, (3);
- (3)
The third is satisfied in view of Proposition 11, (4).
□
4. Ranks of Sentences and Spectra with Respect to Properties
In this section, we introduce a measure of complexity for sentences satisfying a property using the
-rank for families of theories [
8,
9,
10].
Definition 7. For a sentence and a property , we put , and if is defined.
If , then we omit P and write , instead of and , respectively.
Clearly, if and , then , and if , then .
Proposition 12. - (1)
iff .
- (2)
iff .
Definition 8. For a sentence and a property , we say that φ is P-totally transcendental if is an ordinal.
A sentence φ is co-(P)-totally transcendental if is P-totally transcendental.
We omit P and consider totally transcendental and co-totally transcendental sentences if .
According to this definition, each sentence obtains the characteristics and , considering that is (co)-rich enough with respect to the property P. The characteristics and , if they are defined, give additional information regarding the “P-richness” of .
For instance, if and , then is P-finite, exactly satisfying n theories in P. Respectively, if and , then is P-cofinite; i.e., it does not satisfy exactly n theories in P.
Clearly, is both P-finite and P-cofinite iff P is nonempty and is finite.
Theorem 10. For a language Σ, there is a totally transcendental sentence iff Σ has finitely many predicate symbols.
Proof. If has finitely many predicate symbols, we choose a sentence , assuming that the universe is a singleton. Since functional symbols have unique interpretations and there are finitely many possibilities for (non)empty language predicates, we obtain ; that is, is totally transcendental.
Conversely, if has infinitely many predicate symbols, then each consistent sentence obtains a 2-tree with respect to (non)empty predicates . This 2-tree evinces that ; i.e., is not totally transcendental. □
Remark 2. If Σ is finite, then for the proof of Theorem 10, it suffices to choose a sentence φ assuming that a universe is finite, since there are finitely manypossibilities, up to isomorphism, for interpretations of language symbols implying .
The following definition introduces values for the richness of a sentence with respect to a property.
Definition 9. For a language Σ, a property , an ordinal α and a natural number , a sentence is called-(co-)rich if and (respectively, and ).
A sentence is called-(co-)rich if (respectively, ).
If , we write that φ is -(co-)rich instead of -(co-)rich and ∞-(co-)rich instead of -(co-)rich.
If for a property P there is a -(co-)rich sentence φ, we say that P has a-(co-)rich sentence, where or .
According to this definition, if a sentence is -rich, then , .
Theorem 11. - (1)
If a property has a -rich sentence φ which is -co-rich, then , for , for , and for .
- (2)
If for a property , and , then for each sentence , the following assertions hold:
- (i)
;
- (ii)
If , then φ is -rich for some , and for , either or φ is -co-rich for some and , and if , then φ is -co-rich.
Proof. - (1)
For the sentence , we have , , , . This means that P is divided into two disjointed parts and with given characteristics , , , ;
If , then , , ;
If , then a tree evincing the value can be transformed step-by-step using theories either in or in : in each step evincing , there are infinitely many branches of previous values related to or to .
In the first case, related to , we have , and in the second case, related to , . If , a tree for shows that . If , a tree for shows that . If , then both trees for and for show that and , since there are exactly s-definable subsets of P having the rank and the degree 1.
- (2)
For (i), we can see that by the monotony of rank (if then ) and inclusion, . For (ii), we can see that it holds by the monotony of degree for a fixed rank (if and then ) and the inclusion that . Besides, if , then P can not have a tree in , showing that , since otherwise should be more than n. Therefore, either is P-inconsistent—i.e., —or is -co-rich for some and . If , then is -co-rich in view of (1).
□
By Theorem 11 for any e-totally transcendental property P and any , there are s-definable subfamilies with . Similarly, all values are also realized by appropriate s-definable subfamilies.
Thus, the
spectrum for the pairs
with nonempty
forms the set
which is an initial segment
consisting of all pairs
with
and
for
,
,
.
Remark 3. If , then s-definable subfamilies can have only values or both the value and pairs, forming some initial segment .
Indeed, let P be a family of theories in a language of independent 0-ary predicates , , , such that each sentence , , , , is P-consistent. Each P-consistent sentence is divided into 2-trees, showing that . In such a case, we say that the spectrum equals .
The family P above can be extended by a family with dependent predicates producing a given -rank and -degree for a subfamily with, e.g., . Therefore, the arguments for Theorem 11 produce an initial segment for the spectrum of the s-definable family . Thus, .
Since each nonempty s-definable subfamily has a spectrum of the form , or , or , initial segments are well-ordered with respect to the relation ⊆, and the ordinal -ranks are bounded by , all values , for nonempty properties , are exhausted by these three possibilities, and we obtain the following:
Theorem 12. For any nonempty property , one of the following possibilities holds for some and :
- (1)
;
- (2)
;
- (3)
.
All possibilities above are realized by appropriate languages Σ and properties .
Theorem 13. Any value can be naturally extended until , corresponding to the value of the empty subfamily of . It is also natural to put for an empty . In view of Theorem 3.6, we obtain the following description of extended spectra :
- (i)
;
- (ii)
;
- (iii)
;
- (iv)
.
Theorem 14 ([
10])
. Let be a family of a countable language Σ, and with , let α be a countable ordinal, . Then, there is a -definable subfamily such that and . Theorems 12 and 14 immediately imply the following:
Corollary 3. Let be a family of a countable language Σ, and with , let α be a countable ordinal, . Then, there is a -definable property such that .
5. Spectra for Cardinalities of Definable Subproperties
In this section, we study some refinements of the relation .
For a cardinality , a sentence and a property , we write if satisfies exactly theories in P; i.e., .
According to this definition, if and , then , and conversely implies for finite P. For an infinite P, the converse implication may fail. Moreover, since infinite sets can be divided into two parts of same cardinality, one can easily introduce an expansion of P by a 0-ary predicate Q such that and , implying that .
For a property P, we denote by the set for some sentence . This set is called the -spectrum of P.
According to this definition, for any nonempty P and for any ; i.e., .
A natural question arises regarding the description of -spectra.
This question is easily answered for finite P, since in such a case, all subsets of theories are separated as s-definable singletons from their complements, and we obtain the following:
Proposition 13. For any finite property , .
The following assertion generalizes Proposition in terms of isolated points due to Theorem 7:
Proposition 14. If P has exactly isolated points, then .
Proof. Let be isolated points in P. If —i.e., there is a sentence with —then consists of isolated points since elements of the finite set are separated as s-definable singletons from their complements in . Then, . Conversely, if , then , since each equals for some sentence , implying . □
Proposition 15. For any nonempty property , either equals an initial segment for some or .
Proof. If there are sentences with a finite , then either there is a with the greatest finite cardinality, implying by the arguments of Proposition 14, or the finite cardinalities are unbounded, which means . □
Proposition 16. For any infinite property , there is a nonempty set of infinite cardinalities such that either there is with , or . All values , and Y, for a nonempty set Y of infinite cardinalities and , are realized as for appropriate properties P.
Proof. Since and Proposition 15 describes all possibilities for , it suffices, for a nonempty set Y of infinite cardinalities and , to find a property with , a property with and a property with . For the property , one can take a finite n-element family , expanded by a 0-ary predicate marking all theories in , and extend by families , for each , of theories of independent 0-ary predicates expanded by a 0-ary predicate marking all theories in . Any -consistent sentence satisfies either theories in or many theories in and possibly many theories in for finitely many . This means that the cardinalities evince the equality .
For the property , we repeat the process for , replacing the part with an e-minimal family consisting of some theories, all of which are marked by the new 0-ary predicate . Realizing this process, we find that s-definable sets are finite, cofinite or consist of theories for . Thus, .
For the property , we repeat the process for without the part , obtaining . □
Summarizing Propositions 13–16, we obtain the following theorem describing tje -spectra of properties.
Theorem 15. For any nonempty property , one of the following conditions holds:
- (1)
for some ; it is satisfied iff P is finite with ;
- (2)
for some nonempty set of infinite cardinalities and ;
- (3)
for some nonempty set of infinite cardinalities;
- (4)
for some nonempty set of infinite cardinalities.
All values , , and Y, for a nonempty set Y of infinite cardinalities and , are realized as for an appropriate property P.
The following assertion shows that Y in Theorem 15 is finite for a property P with .
Proposition 17. If then for some finite nonempty set of infinite cardinalities with .
Proof. Since , P is divided into infinite s-definable e-minimal parts , each part has only finite and cofinite s-definable subsets producing . Since each s-definable subset of P is a Boolean combination of s-definable subsets of and for infinite s-definable , , , we obtain . □
Remark 4. Describing possibilities for -spectra , we admit that properties P may not be E-closed. If we assume that P is infinite and E-closed, then we have two cases: either P is e-totally transcendental with the least generating set of a cardinality and with accumulation points, or with by Theorem 1.2. In the first case, values for are exhausted by all cardinalities in and by some infinite cardinalities . In particular, for a countable Σ, since is a countable ordinal, we have . In the second case, values for are exhausted by cardinalities in or by cardinalities of part of its initial segment, depending on tje existence of the least generating set for P, and by some infinite cardinalities and cardinalities . In particular, for a countable Σ, includes , and possibly ω, depending on the existence of an infinite, totally transcendental, definable part. Thus, in Theorem 4.5, for an E-closed P, some cases are not realized: , , , , etc.
From Remark 4 and using Theorem 1, we have the following:
Theorem 16. For any nonempty E-closed property with at most the countable language Σ, one of the following possibilities holds:
- (1)
for some , if P is finite with ;
- (2)
for some , if P is infinite and has n isolated points;
- (3)
, if P is infinite and totally transcendental;
- (4)
, if P has an infinite, totally transcendental, definable subfamily but P itself is not totally transcendental;
- (5)
, if P has infinitely many isolated points but does not have infinite, totally transcendental, definable subfamilies.
Remark 5. Possibilities in 16 give low bounds forthe correspondening cases in uncountable languages.
6. P-Generic Sentences and P-Generic Theories
Definition 10 ([
14,
15,
16])
. For a property , a sentence is calledP-generic
if , and if is defined.If , then we omit P, and a P-generic sentence is called generic.
According to this definition, we have the following:
Proposition 18. Any P-generic sentence φ is -rich if is an ordinal and -rich if . In contrast, each -rich sentence, for an ordinal , is P-generic, and each -rich sentence, for , is P-generic.
Proposition 19. If , then φ is P-generic.
In view of Proposition 19, any property has P-generic sentences.
Corollary 4. If a property is finite and , then iff φ is P-generic.
Proof. For Proposition 19, it suffices to show that if is P-generic, then . If , then both and . If P consists of theories, then , . Assuming that is P-generic, we have , , implying that belongs to all n theories in P and . □
Remark 6. In view of Corollary 4, the converse implication for Proposition 19 holds iff P is finite. Indeed, if for a countable language Σ with , which is characterized in Theorem 1, then we can construct a 2-tree of sentences , , showing the value of . This means that P is divided into two disjointed definable parts and with . Thus, and are generic, whereas and . Moreover, this effect works both for an arbitrary property P with and for an arbitrary property P with and . In the latter case, we can remove a branch in the tree, resulting in the values and when only considering a sentence , where is a tautology and is nonempty with . In such a case, is P-generic and .
In view of Remark 6, we have the following:
Proposition 20. For a property , there is a P-generic sentence with minimal/least iff P is finite. If that φ exists, then .
By Proposition 20 for a property with , P-generic sentences produce infinite descending chains of s-definable subfamilies .
Proposition 21. If for a property , , , then for any sentence , either and , or and .
Proof. According to the conjecture for P and the monotony for pairs of values of and , we have and for any , and if the -rank equals , then the -degree equals 1. We cannot obtain and for by Theorem 11. Thus, and , or and . The latter conditions cannot be satisfied simultaneously, as otherwise . □
Since any property with is represented as a disjointed union of s-definable subfamilies with and , Proposition 21 immediately implies the following:
Corollary 5. For any property with and any sentence , either φ is P-generic or is P-generic, or, for with non-P-generic φ and , φ is represented as a disjunction of k -rich sentences and is represented as a disjunction of m -rich sentences such that , , .
Remark 7. By Proposition 21 for any property with and , there is a unique ultrafilter consisting of P-generic sentences. By Proposition 20, this ultrafilter is principal if and only if P is finite; i.e., in such a case, it is a singleton. In any case, produces a theory consisting of P-generic sentences only. This theory T is calledP-generic.
If P is infinite, then T belongs to the E-closure [8,12] of P as a unique element of the α-th Cantor–Bendixson derivative of ; i.e., an element of having the Cantor–Bendixson rank [8]. If , we can divide P into s-definable parts with and , each of which has a unique -generic theory . The set is called the set of P-generic theories.
Thus, we have the following:
Proposition 22. Each e-totally transcendental property P has finitely many—exactly —P-generic theories. These theories have the Cantor–Bendixson rank .
Now, we extend the results above to generic sentences and theories for properties P with .
Proposition 23. If for a property , , then for any sentence , either or .
Proof. Assume that for a sentence we have and . We can suppose that and are both P-consistent. Then, is -rich and -co-rich for some , . Applying Theorem 3.4, we obtain , contradicting . □
Proposition 23 immediately implies the following:
Corollary 6. For any property with and any sentence either φ is P-generic or is P-generic.
Remark 8. By Proposition 23, for any property with , there is an ultrafilter consisting of P-generic sentences. Since the condition implies the existence of a 2-tree of Σ-sentences, there is at least a continuum of these ultrafilters, and by Theorem 1, there are exactly a continuum of ultrafilters for the language Σ with . Each produces a theory consisting of P-generic sentences only. This theory T is calledP-generic.
The P-generic theories form the perfect kernel with respect to Cantor–Bendixson derivatives of ; i.e., the set of elements of having the Cantor–Bendixson rank .
Applying Theorem 12 and summarizing Remarks 7 and 8, we obtain the following:
Theorem 17. - (1)
For any nonempty property , there are P-generic theories if P is totally transcendental, and at least a continuum if P is not totally transcendental. In the latter case, either all theories in P are P-generic if , or P has at least non-P-generic theories if .
- (2)
The -rank of each P-generic theory equals .
Definition 11 ([
11])
. For a property , a sentence is calledP-complete
if φ isolates a unique theory T in P; i.e., is a singleton. In such a case, the theory is calledP-finitely axiomatizable
(by the sentence φ). Since P-finitely axiomatizable theories are isolated points, we obtain the following:
Proposition 24. For any nonempty property , a P-finitely axiomatizable theory T is P-generic iff P is finite.
7. Illustrations
The proposed approach connecting formulas, properties and their characteristics can be applied by classifying various classes of structures and their theories. For instance, with a natural quantifier elimination and a description by invariants, we can reduce the characteristics of families of structures and their theories to possibilities of invariant values by evaluating the complexity of given class and its generating set. This approach produces a natural description for families of theories of abelian groups via Szmielew invariants [
17,
18,
19] and for families of cubic theories via numbers and dimensions of connected components [
20], etc.
Example 1. Let be a property, where Σ consists of finitely many 0-ary and unary predicates and finitely many constant symbols. Using Theorem 2 for sentences , we have finitely many possibilities for neighborhoods . Thus, or , producing finite initial segments for the spectra.
Similarly, using Theorem 3, we have the aforementioned possibilities for , where Σ is finite, or and Σ has finitely many predicate symbols.
Example 2. Let P be a property for the family of theories of abelian groups. Following [10,19], the -rank for P can be an arbitrary countable ordinal or infinity. Taking a sentence φ for abelian groups, one can obtain, following the results above, an arbitrary at most countable value or depending on the information written by φ. This means that all possibilities for spectra described in Theorem 16 can be realized. Example 3. Let P be a property for the family of theories of graphs. Following [10,20], the -rank for P can again be an arbitrary countable ordinal or infinity. Taking a sentence φ for graphs, one can obtain, in a similar manner to Example 2, an arbitrary—at most countable—value or depending on the information written by φ. Thus, all possibilities for spectra described in Theorem 16 can be realized. 8. Conclusions
In this work, we study the links between formulas and properties and considered and described the characteristics of properties with respect to satisfying formulas, their ranks and degrees. Possibilities of spectra for ranks, degrees and cardinalities of definable properties are shown. Generic formulas and theories are introduced and characterized.
Possibilities for ranks and degrees for formulas and theories with respect to the given properties are described. Their highest values form generic sentences and theories, which are also described and characterized.
There are many natural model-theoretic and other properties that can be studied and described in this context. In this case, the relations and are preserved under natural closures. At the same time, the operator of the E-closure does not preserve a series of natural model-theoretic properties. For instance, there are families that are strongly minimal whose accumulation points have strictly ordered properties. This implies that families of -stable, superstable and stable theories cannot be E-closed in these classes. Natural questions arise regarding the characteristics and characterizations of families in classes whose E-closures are contained in .