An Analysis of the Continuum Hypothesis

Abstract

This paper analyzes the Continuum Hypothesis, that the cardinality of a set of real numbers is either finite, countably infinite, or the same as the cardinality of the set of all real numbers. It argues (i) that the real numbers are as similar to the natural numbers as possible in the sense that the relationship between any general method of deciding membership of a set of real numbers and the cardinality of the set should be a natural generalization of the case of the same relationship in the case of a set of natural numbers; and (ii) that CH is a very strong choice principle that is maximally efficient as a principle for deciding whether a real number is in a set of real numbers in the sense that it is uniform in deciding membership for every real number in a countable number of steps. The approach taken is to formulate principles equivalent to or weaker than the Continuum Hypothesis and to use techniques from computer science (infinite binary search), information theory, and set theory to prove theorems that support theses (i) and (ii).

1. Introduction

This paper analyzes the Continuum Hypothesis (CH), which states that the cardinality of a set of real numbers is either finite, countably infinite, or the same as the cardinality of the set of all real numbers. The approach taken is to use techniques from computer science, information theory, and set theory to prove theorems that support two key theses. The first thesis (i) is that the real numbers are as similar to the natural numbers as possible in the sense that the relationship between any general method of deciding membership of a set of real numbers and the cardinality of the set should be a natural generalization of the case of the same relationship in the case of a set of natural numbers. The second thesis (ii) is that CH is a very strong choice principle that is maximally efficient as a principle for deciding whether a real number is in a set of real numbers in the sense that it is uniform in deciding membership for every real number in a countable number of steps. The novelty of the paper lies in its use of techniques from computer science and the theorems that support the two theses alluded to above, which lead to some insights about CH. It is the hope of the author that acceptance of the truth of theses (i) and (ii) will support the case for CH, despite the independence of CH from the standard first-order axiomatization of the cumulative hierarchy of sets, Zermelo Fraenkel (ZF) set theory (See [1] for an intuitive motivation of ZF, and see [2,3] for independence proofs).

Thesis (i) as it stands needs clarification. Informally, the idea is based on the view that all infinite sets are like the set of natural numbers, at least to the extent that they are definite and can be enumerated (albeit, in general, by infinitary functions, i.e., functions that cannot be represented by a finite algorithm). That is, the number of enumerations (well-orderings) of any infinite set is uncountable, and thus the notion of an arbitrary enumeration of an infinite set cannot be specified in a finite way. Any enumeration of an uncountable set cannot be performed by a finitely computable function with finite inputs, because only a countable infinity of outputs will result; see [4] for characterization of the first non-computable ordinal as a countable ordinal).

In more detail, any subset X of the set of natural numbers, $\mathbb{N}$, has a property that one can decide in finitely many steps. Whether a given $x\in \mathbb{N}$ is also a member of X can be determined at simplest by (well) ordering X in order of strictly increasing natural numbers and then checking the entries one by one until either $x\in X$ or $x\notin X$ because x is finite and the greatest value of $y\in X$ is $<x$ or x does not appear in the linear ordering of X but some $z>x$ is a member of X. The idea of checking a well-ordering of X can be generalized by choosing a member of X and then a member of $\mathbb{N}-X$ and iterating until $x\in X$ has been decided, which is called an interleaved cardinal enumeration (see Definition 9 for a definition of cardinal enumeration and Definition 13 for a definition of an interleaved enumeration) If $X\subseteq \mathbb{N}$, all interleaved cardinal enumerations must have $\le \omega$ steps. For $X\subseteq \mathbb{R}$, all interleaved cardinal enumerations must have $\le {\omega }_1$ steps. For $X\subseteq \mathbb{N}$, every interleaved cardinal enumeration decides $x\in X$ in finitely many steps because at some finite step, x will appear from either X or $\mathbb{N}-X$.

Theorem 5 shows that CH is equivalent to the statement that for $X\subseteq \mathbb{R}$, any interleaved cardinal enumeration of X and $\mathbb{R}-X$ decides $x\in X$ in countably many steps. Thus, CH implies that the countability of any interleaved cardinal enumeration process for a membership of a subset of real numbers generalizes the finiteness of any (interleaved) cardinal enumeration process for a membership of a subset of the natural numbers. The same relationship between finiteness and countability comes out of the use of binary search to find a $x\in X$ depending on whether $X\subseteq \mathbb{N}$ or $X\subseteq \mathbb{R}$ in Corollary 1. The sense of similarity between $\mathbb{N}$ and $\mathbb{R}$ refers to generalization from finiteness to countability. Since a countable set is one that can be mapped one-to-one and onto $\mathbb{N}$ and finiteness is a bound $\le n$ for some $n\in \mathbb{N}$, we can also say that CH says that: as finiteness of membership decisions of subsets of $\mathbb{N}$ is to countability of membership decisions of subsets of $\mathbb{R}$, so countability of $\mathbb{N}$ is to the least strict cardinal bound on countability as the cardinality of $\mathbb{R}$. This is summarized in

Table 1. Shows the minimum number of steps to decide membership by enumeration.

Set	Steps	Cardinality	Default Cardinality
$\mathbb{N}$	<$\omega$	${\aleph }_0$	${\aleph }_0$
$\mathbb{R}$	<${\omega }_1$ (CH)	${\aleph }_1$ (CH)	$2^{{\aleph }_0}$

.

Thesis (ii) is supported by a number of principles that are either equivalent to CH or weaker (Properties 1 to 5). Remark 8 draws out the role of CH as a choice principle. Theorem 5 draws out the countability of the decision process of membership of a subset of $\mathbb{R}$ by interleaved enumeration, and Theorem 4 draws out uniformity of a countable enumeration that is particular to CH and does not apply to binary search in general.

The results in this paper have been kept simple (limited and “bottom up”) because it is easy to lose a sense of what claims CH is making by following the debate on whether new axioms of set theory decide the truth of CH (see [5,6,7,8] for accessible discussions) or whether or not CH is compatible with the truth of a combinatorial problem (see [9,10,11,12] for example). There is much more to say, such as the relationship between size complexity and topology. Historically, the earliest approach to CH from Cantor’s time has been to classify the topological complexity of sets, a subject known as descriptive set theory (see [13,14,15] for example), which remains a powerful stimulus to the foundations of real analysis and set theory to this day. However, the key message of this paper is that if you accept choice principles (including the Axiom of Choice), theses (i) and (ii) are very reasonable assertions to make.

2. Materials and Methods

The technical background assumed by this paper is a basic familiarity with Zermelo Fraenkel set theory, including the Axiom of Choice (AC), and algorithmic complexity and binary search from computer science. Definitions of all terms used in the paper are provided in the Results Section 3, mainly in the Preliminaries Section 3.1. This paper does assume the truth of AC, since in its standard formulation CH is a stronger choice principle than AC when applied to sets of real numbers. (It is possible to formulate CH without using AC; see [16]. In this paper, AC and well-ordering of all sets is assumed). The only original definition is the definition of the number of bits of information in a real number and in a set of real numbers in Definition 10, which is used in the formulation of Property 3, in the proof of Theorem 3, and in the discussion of the results.

This paper provides a mathematical analysis of the Continuum Hypothesis (Property 1) by formulating two principles that are proved equivalent to the Continuum Hypothesis (Properties 2 (CH=), 3 (CH*)) and two principles that are weaker than the Continuum Hypothesis (Properties 4 (Consequence of the Axiom of Choice) and 5 (CH*-)). Theorem 1 establishes the equivalence of the Continuum Hypothesis with Property 2 (CH=), and Theorem 3 establishes the equivalence of the Continuum Hypothesis with Property 3 (CH*). Property 4 and Property 5 (CH*-) are provable using the Axiom of Choice.

There are two theorems (Theorem 4 Corollary 1 and Theorem 5) that have force because they explicitly bring out an analogy between the finiteness of the search for a member of the search for a natural number (in a linearly ordered set) and the countability of a search for a real number (in a linearly ordered set assuming the Continuum Hypothesis). The discussion that follows the Results section notes the implausibility of a real number (that contains countably many bits of information) requiring uncountably many bits of information to find via a search (binary search or cardinal enumeration). The difference in the logical form of binary search with the Continuum Hypothesis is noted in Theorem 4, the conclusion being that the Continuum Hypothesis is a uniform choice principle that is stronger than the Axiom of Choice for sets of real numbers (or sets of mathematical objects that contain countably many bits of information).

The proof techniques used in the theorems are either first-order logic, transfinite induction, or construction of functions by transfinite recursion.

There are no data associated with this paper, as the results are theoretical, incorporating techniques from computer science and basic set theory.

3. Results

3.1. Preliminaries

In the following definitions, standard logic symbols are used as a means of notational compactness: ∨ for “or”, ∧ for “and”, ⇒ for “implies”, ∃ for “there exists” and ∀ for “for all”. Definitions are taken where possible from a standard text on set theory [17].

Definition 1 (Definition of linear ordering).

A set X is linearly ordered by linear ordering < if $(\forall x\in X)(\forall y\in X)(x<y\vee x=y\vee y>x)$, where $x\nless x$, $x<y\Rightarrow y\nless x$, $(x<y\wedge y<z)\Rightarrow x<z$ for all $x,:y,:z$ (see [17] Definition 2.1).

Definition 2 (Definition of lexicographical linear ordering).

There is a natural lexicographical linear ordering on the real numbers given by ${\langle x_i\rangle }_{i<\omega }<{\langle y_i\rangle }_{i<\omega }$ if $(\exists k<\omega )[(\forall l<k)(x_l=y_l)\wedge (x_k<y_k)]$,${\langle x_i\rangle }_{i<\omega }={\langle y_i\rangle }_{i<\omega }$if $(\forall l<\omega )(x_l=y_l)$ and ${\langle y_i\rangle }_{i<\omega }<{\langle x_i\rangle }_{i<\omega }$ otherwise (see [18] Section 14).

Definition 3 (Definition of well-ordering).

A set X is well-ordered by well-ordering < if it x is linearly ordered by < and every subset of X has a least element, that is, for all $Y\subseteq X$$(\exists z\in Y)(\forall y\in Y)(z<y\vee z=y)$ (see [17] Definition 2.3).

Remark 1.

While a linearly ordered set is one that can be viewed as a line with a notion of <, = and >, a well-ordered set is a set that can be enumerated in such a way that no member of the set is omitted. The enumeration in general can be very long and can be at least as long as (the ordinal corresponding to) the cardinal of the set.

Definition 4 (Definition of enumeration).

An enumeration of a set X is a well-ordering of X.

Definition 5 (Definition of ordinal).

A well-ordered set $\langle X,<\rangle$ has an order type or ordinal that is a measure of order complexity that applies to any well-ordered set $\langle Y,\prec \rangle$ for which there is a one-to-one function of such that $f\left(X\right)=Y$ and $x<y$ if and only if $f\left(x\right)\prec f\left(y\right)$ (compare [17] Definition 2.2). For definiteness, we identify an ordinal $\alpha \ge 0$ inductively with the set of all ordinals less than α, that is $\left\{\beta :\beta <\alpha \right\},$ where the ordinal 0 is identified with the empty set.

Remark 2.

The inductive definition of ordinal in Definition 5 provides a standard label for measuring the complexity of the well-ordering of a set, but any other well-ordered set could be chosen as the standard. There is another definition of an ordinal as the proper class of all well-ordered sets of that order type, but that definition will not be used here because a proper class should not be needed when dealing with the order type of well-ordered sets.

Definition 6 (Definition of a real number).

We can identify a real number as a binary ${\langle f_i\rangle }_{0<i<\omega }$-sequence, written as ${\langle f_i\rangle }_{0<i<\omega }$; that is to say a function $f:\mathbb{N}\to \{0,1\}$, where $\mathbb{N}$ is the set of all natural numbers and ω is the order type of the natural numbers in their standard strictly increasing ordering 0,1,2, … (see [19]).

Remark 3.

There are many equivalent formulations of real numbers, but the set of real numbers refers to any instantiation of a linear ordered field that is Dedekind complete, that is, every < bounded set of real numbers has a real number that is a < least upper bound (see, for example, [20]). It is easy to see that ${\langle f_i\rangle }_{0<i<\omega }$ can be mapped onto the binary sum ${\sum }_{0<i<\omega }{f}_i\left(x\right)\times 2^{-i}$ (This mapping is not one-to-one as all the eventually constant ${\langle f_i\rangle }_{0<i<\omega }$ sequences have two representation as an eventually constant 0 sequence and an eventually constant 1 sequence), so that the set of all binary ω-sequences maps onto the closed interval of real numbers from 0 to 1, $[0,1]$, but has a one-to-one refinement by choosing the lesser of the at most two representations. Larger real numbers (1.0101, for example) can be coded as real numbers in a binary $omega$-sequence by using even bits (index > 0) to code whether the previous odd bit is to the left or right of the binary point, and then coding the bits to the right of the binary point as a fine initial sub-sequence of the binary ω-sequence. Hence, given that there is a one-to-one function from $\mathbb{R}$ to the set of binary sequences, B, a one-to-one and onto function from B to $[0,1]$, and there is a one-to-one mapping from [0,1] to $\mathbb{R}$, it follows that there is a one-to-one and onto map from B to$\mathbb{R}$.

Definition 7 (Definition of a binary tree).

A set of real numbers is identified with a binary tree, where a binary tree T comprises a set of branches, that is to say, a set of binary ω-sequences (see [19]). The set of all real numbers is denoted by $\mathbb{R}$.

Definition 8 (Definition of a cardinal).

A set X has a cardinal that is a measure of size that applies to any set Y such that there is a one-to-one function f such that $f\left(X\right)=Y$ (see [17] 3.1). We denote the cardinal of a countably infinite set, i.e., the cardinal of the set of natural numbers, as ${\aleph }_0$, which in fact is the smallest infinite cardinal. ${\aleph }_1$ is the least cardinal strictly greater than ${\aleph }_0$, which is equivalent to saying that ${\aleph }_1$ is the least uncountable cardinal. The cardinal of a set X, often called its cardinality, is written $\left|X\right|$. For definiteness, we identify $\left|X\right|$ with the least ordinal that corresponds to, i.e., maps one-to-one onto, X.

Remark 4.

For infinite cardinals, there is a many-to-one relation between ordinals and cardinals, and any ordinal can be mapped to a cardinal by ignoring the well-ordering of the ordinal. There is a least ordinal that corresponds to any given cardinal. For example, the least ordinal that corresponds to ${\aleph }_0$ is called ω, and the least ordinal that corresponds to ${\aleph }_1$ is called ${\omega }_1$. The only theorem we assume is $2^{{\aleph }_0}\ge {\aleph }_1$ due to Cantor (see ([21] for example), where $\mathbb{R}$ has cardinality $2^{{\aleph }_0}$.

Definition 9 (Definition of a cardinal enumeration).

A cardinal enumeration of a set X is an enumeration of X in a well-ordering of the least ordinal that corresponds to the cardinal of X.

Remark 5.

A cardinal enumeration is used in Theorem 5 as Theorem 5 is not true for longer enumerations than a cardinal enumeration (which is the shortest of all enumerations of X).

Definition 10 (Definition of the number of bits of information).

The number of bits off information in a real number is the length in bits of a binary sequence that cannot be compressed any further losslessly (compare [22] for an example of lossless compression in terms of minimum algorithm length) (Ref. [22] is a text on algorithmic complexity, while the definition given here addresses the complexity of sequences and sets of sequences. The treatment of information here is also distinct from Shannon information (see [23,24]), which concerns information used in a communication system). Here lossless compression (of a sequence of bits q to a sequence of bits p) means that p can be encoded from q where the length of p $\left|p\right|\le \left|q\right|$ and q can be decoded from p, where the length of a binary sequence $\left|p\right|$ is the cardinal of the domain of p as a function. That is, $\left|p\right|=\left|S\right|$ for $p:S\to \{0,1\}$ where $S\subseteq N$. This definition can be extended to larger well-ordered sets than N. Lossless compression is strict if $\left|p\right|<\left|q\right|$. The cardinal of the lossless compression of a sequence p is denoted by $\rangle \left|p\right|\langle$, which is the greatest cardinal $c\le \left|q\right|$ for all lossless compressions q of p. A set of binary sequences S can be encoded as a set of binary sequences T if every binary sequence in S can be mapped one-to-one to a binary sequence in T by a computable function e that is onto T, i.e., $e\left(S\right)=T$, and a binary sequence $s\in S$ is encoded as $e\left(s\right)$. S can be then decoded from T as $e^{-1}\left(T\right)$ and a binary sequence $t\in T$ can be decoded as $e^{-1}\left(t\right)$ if $e^{-1}$ is a computable function. The number of bits of information in a set S of binary sequences can be defined as $\left|S\right|\times su{p}_{p\in S}(\rangle \left|p\right|\langle )$, where × is cardinal multiplication and $su{p}_{p\in S}(\rangle \left|p\right|\langle )$ is the least cardinal c such that $c\ge \rangle \left|p\right|\langle$ for all $p\in S$. There is a better definition of the number of bits of information in a set S of binary sequences: the least number of bits in a binary sequence comprising a concatenation of all binary sequences $p\in S$ using the same compression algorithm for any set S of binary sequences. But it is harder to use than the definition in the main text because there is a length minimization operation over all possible concatenations of sequences of members of S, even if it can result in shorter lossless compressions.

Remark 6.

The definition of the number of bits in a set in Definition 10 is a weak notion in the sense that the only limiting factor for lossless compression for infinite sets is that $\left|S\right|=\left|T\right|$ of infinite sets of binary sequences S and T provided that the members of S and T have length $<\left|S\right|(=|T\left|\right)$ because $c\times d=max(c,d)$ for cardinals c, d and at least one of which is infinite, for $max(c,d)$ the maximum of c and d. However, since the argument in Theorem 2 only needs the cardinality restriction, stronger and more adequate notions of the number of bits will not be developed here. The suggested route for such a stronger notion is to code patterns and their number of repetitions, as in the example below, to encode any set as a binary sequence, and to characterize the lossless compression of such a sequence while recognizing that the encoding of the length of a sequence does not shorten the length of the sequence to the length of the code.

Example 1.

Strict lossless compression on sets of sequences of bits exists, for example, the set of all rational numbers expressed as infinite binary sequences. Any rational number corresponds to an initial finite sequence of bits. $init$ (The initial sequence can contain repeated sub-sequences of bits, but has a finite length) and a repetition of a finite binary sequence, $seq$, ω times; so we can losslessly compress a binary sequence representing a rational number by encoding $init$, $seq$, and ω as a finite sequence of bits. We can represent the binary sequence $\langle 1,0,1\rangle$ followed by $\langle 0,1\rangle$ repeated À$\breve{9}69$ times as $\langle 1,0,0,0,1,1,0,0,1,1,1\rangle$, where 0 is used as an inter-bit marker for a sequence and an inter-bit 1 indicates a boundary to the next element of the code, whether to a finite sequence or to the code for the number of repetitions. For the number of repetitions, 1 indicates an infinite number of repetitions and 1 is the code for the ordinal exponential ${\omega }^1$ (which $=\omega$) and should not be confused with ${\omega }_1$, the first uncountable ordinal). Binary encodings of finite binary sequences and binary sequences that allow infinite repetitions for larger ordinals than ω using ordinal notations for infinite ordinals (see [25]) are clearly possible using the same encoding function n (0 s represent natural number n for example), but are not the subject of this paper.

Definition 11 (Definition of a mathematical object).

A mathematical object is taken to be a well-founded set definable in the cumulative hierarchy of pure sets; see [1,26] 6.3. It is assumed in this paper that any mathematical object can be represented as a binary α-sequence for some ordinal α (which in general requires the Axiom of Choice).

3.2. Principles Relating to the Continuum Hypothesis

Property 1 (Continuum Hypothesis, CH).

We will start with CH itself written in logical notation:

CH: $(\forall X\subseteq \mathbb{R})((\left|X\right|\le {\aleph }_0)\vee (\left|X\right|=\left|\mathbb{R}\right|)$

CH can also be formulated as $2^{{\aleph }_0}={\aleph }_1$ because any uncountable set of real numbers, including any (set representing a) sequence of real numbers indexed by all countable ordinals (which has cardinality ${\aleph }_1$), must have the same cardinality as $\mathbb{R}$ (which has the cardinality $2^{{\aleph }_0}$).

Property 2 (Continuum Hypothesis, CH=).

CH can be expressed as follows as a statement about well-orderings of a set of real numbers (compare [16], which allows well-orderings of order type $<{\omega }_2$):

CH=: For all linear orderings of a set of real numbers X, there is a well-ordering of X of order type $\le {\omega }_1$.

Theorem 1.

CH= is equivalent to CH (uses Axiom of Choice).

Proof. 

Assume CH=. Since any set of real numbers can be well-ordered with order type ≤${\omega }_1$, the cardinality of X, $\left|X\right|$, is ≤${\aleph }_1$. The set of all real numbers, $\mathbb{R}$, has cardinality ${\aleph }_1$ because it is uncountable by Cantor’s theorem that $2^{{\aleph }_0}\ge {\aleph }_1$. It follows that every subset of the real numbers is countable or has the cardinality of $\mathbb{R}$, which is CH. Conversely, assume CH. Then take any set of real numbers, X. By CH, if X is countable then it has cardinality ≤${\aleph }_0$, and if X is uncountable it has cardinality ${\aleph }_1$; that is because by CH, there is only one uncountable cardinality of any set of real numbers; that must be ${\aleph }_1$. Well order X by applying AC, that is $x_{\alpha }:=f(X-{\bigcup }_{\beta <\alpha }\left\{x_{\beta }\right\})$, where $0\le$$\alpha <|X\parallel$ for choice function f that chooses a member of each set in a non-empty set of sets, where $|X\parallel =|X|$ for finite X, $|X\parallel =\gamma$ for some $\omega \le \gamma <{\omega }_1$ if X is countably infinite and $|X\parallel ={\omega }_1$ if X is uncountable. The linear ordering of X is not used but can be added as a premiss. Thus, CH follows. □

Theorem 2.

Each real number contains ≤ω bits of information, with almost all real numbers having exactly ω bits of information. All objects with a countable number of bits have ≤ω bits of information.

Proof. 

Each real number contains ≤$\omega$ bits of information, with some (in fact all but countably infinitely many) real numbers having exactly $\omega$ bits of information, as otherwise there would be a lossless compression, i.e. a one-to-one and onto function, from the set of all real numbers to the set of natural numbers (where each natural number contains <${\aleph }_0$ bits). In terms of the definition of number of bits in a set, we would have for the set of all real numbers $\mathbb{R}$, ${\aleph }_0\times 2^{{\aleph }_0}={\aleph }_0\times {\aleph }_0$, i.e., ${\aleph }_0=2^{{\aleph }_0}$, if all real numbers have finite information content as represented by the set of natural numbers, ($\mathbb{N}$, which contradicts Cantor’s theorem), which would violate Cantor’s theorem that $2^{{\aleph }_0}\ge {\aleph }_1>{\aleph }_0$. Let us assume that x is an object with a countable number of bits of information. Then either x has a finite number of bits of information or x has countably infinitely many bits of information. In the first case, x can be encoded as a natural number and therefore as a real number. In the second case, there is a lossless compression of a binary $\alpha$-sequence representation of x for $\omega \le \alpha <{\omega }_1$ to a binary $\omega$-sequence representation of x by definition of countability. Since a real number is identified with a binary $\omega$-sequence, x is a real number. Seen in this way, a real number can be identified with a mathematical object with a countable number of bits. □

Property 3.

We can generalize CH asCH*: For all linear orderings of a set of mathematical objects that contain a countable amount of information, X, there is a well-ordering of X of order type ≤${\omega }_1$.

Theorem 3.

CH* is equivalent to CH.

Proof. 

This follows from Theorems 1 and 2. □

Remark 7.

We hold CH* in reserve until the next section, but it will come in use when we examine arguments for CH. We can compare CH with AC. AC says that there is a choice function that for a set of non-empty sets chooses a member of each set and collects them in a set. As we have seen in Theorem 1, repeated application of AC will generate a well-ordering of a non-empty set (known as the well-ordering principle). In particular, we have (It should not be thought that the function that maps a linear ordering to a well-ordering will preserve the linear ordering. The choice function that operates on $b>a$ aftera is will in general select a different linear order of the set to form a well-ordering):

Property 4.

Consequence of AC: For all linear orderings of a set of real numbers X, there is a well-ordering of X.

Remark 8.

Both AC and the well-ordering principle as applied to real numbers are a consequence of CH because CH says that each member of a set of real numbers, X, can be indexed with a countable ordinal, which means that there is a method for choosing a member of any subset of X by taking the member with the least ordinal index (see [6]). At a minimum, we can say that CH is a choice principle that is as at least as strong as AC when applied to sets of real numbers. The fact that AC for sets in general follows from the Generalized Continuum Hypothesis is due to [27].

3.3. Main Results

CH as a Uniform Infinite Binary Search

Remark 9.

According to principle CH*, CH translates the countability of the number of bits in any member x of a linear ordering of a set of real numbers X to the countability of a (single) enumeration up to and including in a well-ordering of X. One problem is that this form of CH does not correspond in logical form to a standard search algorithm such as binary search, which is a very efficient way to decide membership of a set of real numbers (linear in the length of the real numbers and logarithmic in the size of the set of real numbers; see [28], for example, the finite case of binary search).

Definition 12

(Binary search).Binary search of a set of real numbers, X, is the process where, given a real number to be searched for, x, a linear ordering < of X is continually divided into two contiguous linear sub-orderings $A_{i+1}=\langle y\in X_i:y\le a_i\rangle$ and $B_{i+1}=\langle y\in X_i:y>a_i\rangle$ for some $a_i\in X_i$ where $in{f}_{x\in X_i}<a_i<su{p}_{x\in X_i}$ for greatest lower bound of $X_i$ along <, $in{f}_{x\in X_i}$, and least upper bound $X_i$ along <, $su{p}_{x\in X_i}$, where $X_0:=X$ and $X_{i+1}:=A_{i+1}$ if $x\le a_i$ and $X_{i+1}:=B_{i+1}$ if $x>a_i$ and i is a natural number index of the subdivision process. The process will stop if $x=a_i$ for some $i<\omega$; otherwise in the limit, we have $X_{\omega }=x$ if $x\in X$ and $X_{\omega }=\varnothing$ if $x\notin X$.

Theorem 4.

We can write binary search in the form ($\forall X\subseteq \mathbb{R})(\forall x\in X)(\exists F:{\omega }_1\to X)(\exists \beta <{\omega }_1)(F\left(\beta \right)=x)$ for R the set of all real numbers, and F is a function. CH is the stronger statement $(\forall X\subseteq \mathbb{R})(\exists F:{\omega }_1\to X)(\forall x\in X)(\exists \beta <{\omega }_1)(F\left(\beta \right)=x)$, as in CH F does not depend on x (The F used in binary search in the proof will have only one non-empty entry, while F used in CH is an enumeration of X of length $\le {\omega }_1$).

Proof. 

We can allow $x\in X$ to be represented by a binary sequence of countably infinite length ${\omega }_1>\beta \ge \omega$ rather than by a binary $\omega$-sequence. Then we can adapt the infinite binary search algorithm outlined in the definition of binary search above by recursively choosing $a_i\in X_i$ where $in{f}_{x\in X_i}<a_i<su{p}_{x\in X_i}$, setting $X_{i+1}:=A_{\gamma +1}$ if $x\le a_i$ and $X_{i+1}:=B_{i+1}$ for ordinal $i<\beta$, $X_{\lambda }={\bigcap }_{i<\lambda }{X}_i$ for limit ordinal $\lambda$, and setting $F\left(i\right)=a_i$ and $F\left(\gamma \right)=\varnothing$ for $i\ne \gamma <{\omega }_1$ if $i<\beta$ and $x=a_i$, otherwise setting $F\left(\beta \right)=x$ and $F\left(\gamma \right)=\varnothing$ for $\beta \ne \gamma <{\omega }_1$ (since $x\in X$, so the search will never fail). The statement of CH is a formalization of CH as a search algorithm. □

Corollary 1.

${\bigcap }_{i<{\omega }_1}{X}_i=\varnothing$ and ${\bigcap }_{i<\beta }{X}_i=\left\{x\right\}$ for any given countably infinite ordinal β and given $x\in X$ for uncountable set of real numbers X and $X_i$ and β are as defined in Theorem 4, while ${\bigcap }_{i<\omega }{G}_i=\varnothing$ and ${\bigcap }_{i<n}{G}_i=\left\{x\right\}$ for G a non-empty set of finite binary sequences such that $x\in G$ has length m and n ≥ m for some natural number n, $G_0=G$ and $G_{i+1}$ represents a subdivision of $G_i$ for $i<\omega$.

Proof. 

The first part is a restatement of Theorem 4, while G can be searched using finite binary search once it has been ordered lexicographically. The construction of $G_i$ follows $X_i$ for finite ordinals with $G_0=G$ and $G_{i\ge n}=\varnothing$. $n=m$ can be achieved if the midpoint, $mi{d}_i=(s_i+f_i)/2$ in binary, of $G_i:=[s_i,f_i]$, the set of all finite binary sequences between and including $s_i$ and $f_i$, is chosen as $a_i$ (which always exists for sets of binary sequences), where $mi{d}_i$ does not have to be member of G. Note that all binary sequences start with ordinal index 0. □

Remark 10.

Theorem 4 and Corollary 1 work on a set of real numbers, X, that can be thought of as a binary tree of binary ω-sequences, the idea being that by continually subdividing the tree by choosing a member in the interior of the subtree $X_{i<\omega }$, $a_i,$ a given $x\in X$ will be found in ≤ω steps. The same approach works for a set of natural numbers with <ω steps. It is noted that CH is a uniform version of (infinite) binary search, as the choice function does not depend on x. In computer science, binary search for a given finite sequence x on finite sets that have already been ordered is a logarithmic run time algorithm (see [28]).

3.4. A Choice-Based Argument for CH

Definition 13 (Definition of an interleaved enumeration).

An enumeration of a non-empty set X is interleaved one-to-one with an enumeration of a non-empty set Y if one member of X is chosen followed by a member of Y and the operation repeated until all members of X and Y have been chosen; the enumeration is no longer alternating if $\left|X\right|<\left|Y\right|$ or $\left|Y\right|<\left|X\right|$.

Theorem 5.

CH is equivalent to the statement that any cardinal enumeration of any $X\subseteq R$ to find any $x\in R$ when interleaved one-to-one with an enumeration of $R-X$ takes $<{\omega }_1$ steps.

Proof. 

If CH, then since X and $\mathbb{R}-X$ can be linearly ordered by the lexicographical ordering, they can be enumerated in ≤${\omega }_1$ steps as a single cardinal enumeration of $\mathbb{R}$. Since either $x\in X$ or $x\in \mathbb{R}-X$, x will be enumerated in <${\omega }_1$ steps. Conversely, if $x\in X$, then x can be found by an enumeration of X in <${\omega }_1$ steps by assumption, and if $x\in \mathbb{R}-X$, then x can be found by an enumeration of $\mathbb{R}-X$ in <${\omega }_1$ steps by assumption. If X is empty, $\mathbb{R}-X$ can be enumerated in ≤${\omega }_1$ steps since all $x\in \mathbb{R}$ can be enumerated in <${\omega }_1$ steps by assumption; if $\mathbb{R}-X$ is empty, X can be enumerated in ≤${\omega }_1$ steps by the same argument; otherwise, both X and $\mathbb{R}-X$ can be enumerated in ≤${\omega }_1$ steps again by the same argument. Since X is an arbitrary set of real numbers that is assumed to be linearly ordered, we have shown that X is well-ordered (by enumeration) with an order type ≤${\omega }_1$. CH follows. □

Remark 11.

Interleaved enumeration is an exhaustive technique, particularly in the case of a set and its complement, as it will be possible to decide whether $x\in X$ or not. But the only (strict upper) bound on the number of steps is the minimum of the cardinality of X and the cardinality of $\mathbb{R}-X$ in the case of Theorem 5 or the cardinality of X in general if $x\in X$ is being tested.

Property 5.

CH*-: For all linear orderings of a set of mathematical objects that contains a strictly increasing countable number of bits of information, X, there is awell-ordering of X of order type $\le {\omega }_1$.

4. Discussion

Theorem 3 shows that CH is really a choice principle that is at least as strong as AC applied to a set of real numbers. This supports thesis (ii) in the introduction.

From Theorem 4, CH can be thought of as a uniformization condition on individual countable binary searches for $x\in X\subseteq \mathbb{R}$ to produce a single countable search that can find every $x\in X$. Combined with the observation that CH is at least as strong as AC for sets of real numbers, we regard CH as a uniform choice principle. This supports thesis (ii) in the introduction. Without this countable uniformization condition, it is possible that an enumeration of a set C of size $2^{{\aleph }_0}$ is such that every countable ordinal label for a real number $2^{{\aleph }_0}$ is re-used times or otherwise that almost all ordinal labels used for members of a well-ordering are ≥${\aleph }_1$. There is nothing in Zermelo Fraenkel set theory that prevents a mapping $2^{{\aleph }_0}\to {\aleph }_1$ from being many-to-one. In fact, Cohen (1963) showed that it is possible to force a mapping from $2^{{\aleph }_0}\to {\aleph }_1$ to be many-to-one in a countably infinite model of set theory by adding a countably infinite set of computably decidable conditions using knowledge-based semantics (see [29] for an accessible treatment and [30] for a comprehensive account of ways of understanding forcing and [17] for reference). The knowledge-based semantics was set out in [31] from [3] and is based on the view that knowledge forms a tree (a partial ordering) of cumulative conditions where a condition and its negation will appear in separate branches.

Corollary 1 shows that there is an analogy between the cardinality of the set of all finite binary sequences, which is ${\aleph }_0$, and the finiteness of individual binary sequences and the cardinality of an uncountable set of real numbers and the countability of each real number. A similar analogy appears in Theorem 5. This supports thesis (i) in the introduction.

Theorem 5 shows that there is an analogy between $\omega$ and ${\omega }_1$. That is, we know that to decide whether $x\in X$ for $X\subseteq \mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers, requires no more than finitely many steps, and to find x by enumeration of X when interleaved one-to-one with an enumeration of the complement of X, $N-X$, requires <$\omega$ steps. Replacing $\mathbb{N}$ by $\mathbb{R}$, $\omega$ by ${\omega }_1$ and “finitely” by “countably many” we get: to decide whether $x\in X$ for $X\subseteq \mathbb{R}$ requires no more than countably many steps and to find x by enumeration of X when interleaved one-to-one with an enumeration of $\mathbb{R}-X$ requires <${\omega }_1$ steps. In order to translate “requires” into mathematical language, we restrict enumerations to cardinal enumerations. This supports thesis (i) in the introduction.

While an analogy is a weak argument in general, there is a reason why the analogy between “finitely” by “countably many” may hold. That is, if CH is false, then it follows that there would be no uniform method (such as enumeration) for deciding whether any $x\in X\subseteq \mathbb{R}$ in countably many steps. Thus CH being false would imply that a countable set of decision computations of $x\in X$ could not be enumerated in a way that any member computation is accessible in countably many steps, or, more starkly, that almost all countable decision computations of x ∈ X require uncountably many steps to complete if the set of decision computations is well-ordered. But the assertion that almost all membership decisions that require countably many bits of information are decided by enumeration in uncountably many steps is counter-intuitive. This argument supports thesis (i) and thesis (ii) in the introduction.

Given that CH is a choice function and CH* applies to all mathematical objects with a countable number of bits of information, it is plausible to believe that a choice function could be selected to minimize the total number of steps in the uniform method (i.e., to ≤${\omega }_1$ steps) and to avoid having almost all countable content being decided in uncountably many steps.

Principle CH*- is a consequence of AC since any strictly increasing linear order with a countable infinity of information can be mapped to a strictly increasing linear order of countable ordinals. CH* is stronger than Principle CH*- because it is possible that at an uncountable limit ordinal the increasing number of bits becomes uncountable. Thus with CH*, there is no natural way to map the countable content of real numbers to ordinals. The choice function in AC may provide such a mapping. In fact, we can say that CH* is really a claim that CH* is the same as CH*-, not in the sense of logical equivalence; rather that CH* not only has a choice function to well-order any set of mathematical objects with a countably infinite number of bits of information, but also that same choice function can also re-order the objects in an increasing countable number of bits. Thus CH is a natural strong choice principle (based on Principles CH* and CH*-) that is maximally efficient at deciding whether a real number is in a set of real numbers in the sense that it is uniform in deciding membership for every real number in a countable number of steps. This supports thesis (i) in the introduction.

5. Conclusions

This paper concludes that the following two theses can be supported: (i) that the real numbers are as similar to the natural numbers as possible in the sense that the relationship between any general method of deciding membership of a set of real numbers and the cardinality of the set should be a natural generalization of the case of the same relationship in the case of a set of natural numbers; and (ii) that CH is a very strong choice principle that is maximally efficient as a principle for deciding whether a real number is in a set of real numbers, in the sense that it is uniform in deciding membership for every real number in a countable number of steps. Moreover, if CH is false, it follows that almost all membership decisions that require countably many bits of information are decided by enumeration in uncountably many steps. That this last assertion is counter-intuitive leads to CH being a reasonable principle to adopt. This is supported by the fact that CH reflects the analogy between the finiteness of computation of the number of steps to check membership of a set of natural numbers by (cardinal) enumeration with the countability of computation of the number of steps to check membership of a set of real numbers by (cardinal) enumeration.