Countable Choice Research Articles

Preservation of choice principles under realizability

AbstractEspecially nice models of intuitionistic set theories are realizability models $V({\mathcal A})$, where $\mathcal A$ is an applicative structure or partial combinatory algebra. This paper is concerned with the preservation of various choice principles in $V({\mathcal A})$ if assumed in the underlying universe $V$, adopting Constructive Zermelo–Fraenkel as background theory for all of these investigations. Examples of choice principles are the axiom schemes of countable choice, dependent choice, relativized dependent choice and the presentation axiom. It is shown that any of these axioms holds in $V(\mathcal{A})$ for every applicative structure $\mathcal A$ if it holds in the background universe.1 It is also shown that a weak form of the countable axiom of choice, $\textbf{AC}^{\boldsymbol{\omega , \omega }}$, is rendered true in any $V(\mathcal{A})$ regardless of whether it holds in the background universe. The paper extends work by McCarty (1984, Realizability and Recursive Mathematics, PhD Thesis) and Rathjen (2006, Realizability for constructive Zermelo–Fraenkel set theory. In Logic Colloquium 03, pp. 282–314).

On uniformly continuous functions between pseudometric spaces and the Axiom of Countable Choice

In this note we show that the Axiom of Countable Choice is equivalent to two statements from the theory of pseudometric spaces: the first of them is a well-known characterization of uniform continuity for functions between (pseudo)metric spaces, and the second declares that sequentially compact pseudometric spaces are $$\mathbf {UC}$$ —meaning that all real valued, continuous functions defined on these spaces are necessarily uniformly continuous.

On lightly and countably compact spaces in ZF

Given a topological space X = (X, T ), we show in the Zermelo-Fraenkel set theory ZF that:Every locally finite family of open sets of X is finite iff every pairwise disjoint, locally finite family of open sets is finite.Every locally finite family of subsets of X is finite iff every pairwise disjoint, locally finite family of subsets of X is finite iff every locally finite family of closed subsets of X is finite.The statement “every locally finite family of closed sets of X is finite” implies the proposition “every locally finite family of open sets of X is finite”. The converse holds true in case X is T4 and the countable axiom of choice holds true.We also show:It is relatively consistent with ZF the existence of a non countably compact T1 space such that every pairwise disjoint locally finite family of closed subsets is finite but some locally finite family of subsets is infinite.It is relatively consistent with ZF the existence of a countably compact T4 space including an infinite pairwise disjoint locally finite family of open (resp. closed) sets.

Some versions of second countability of metric spaces in ZF and their role to compactness

In the realm of metric spaces we show in ZF that: (i) A metric space is compact if and only if it is countably compact and for every $\varepsilon > 0$, every cover by open balls of radius $\varepsilon $ has a countable subcover. (ii) Every second countable metric space has a countable base consisting of open balls if and only if the axiom of countable choice restricted to subsets of $\mathbb{R}$ holds true. (iii) A countably compact metric space is separable if and only if it is second countable.

Two new equivalents of Lindelöf metric spaces

AbstractIn the realm of Lindelöf metric spaces the following results are obtained in : (i) If is a Lindelöf metric space then it is both densely Lindelöf and almost Lindelöf. In addition, under the countable axiom of choice , the three notions coincide. (ii) The statement “every separable metric space is almost Lindelöf” implies that every infinite subset of has a countably infinite subset). (iii) The statement “every almost Lindelöf metric space is quasi totally bounded implies . (iv) The proposition “every quasi totally bounded metric space is separable” lies, in the deductive hierarchy of choice principles, strictly between the countable union theorem and . Likewise, the statement “every pre‐Lindelöf (or Lindelöf) metric space is separable” lies strictly between and .

MINIMAL AXIOMATIC FRAMEWORKS FOR DEFINABLE HYPERREALS WITH TRANSFER

AbstractWe modify the definable ultrapower construction of Kanovei and Shelah (2004) to develop a ZF-definable extension of the continuum with transfer provable using countable choice only, with an additional mild hypothesis on well-ordering implying properness. Under the same assumptions, we also prove the existence of a definable, proper elementary extension of the standard superstructure over the reals.

Quotienting the delay monad by weak bisimilarity

The delay datatype was introduced by Capretta (Logical Methods in Computer Science, 1(2), article 1, 2005) as a means to deal with partial functions (as in computability theory) in Martin-Löf type theory. The delay datatype is a monad. It is often desirable to consider two delayed computations equal, if they terminate with equal values, whenever one of them terminates. The equivalence relation underlying this identification is called weak bisimilarity. In type theory, one commonly replaces quotients with setoids. In this approach, the delay datatype quotiented by weak bisimilarity is still a monad–a constructive alternative to the maybe monad. In this paper, we consider the alternative approach of Hofmann (Extensional Constructs in Intensional Type Theory, Springer, London, 1997) of extending type theory with inductive-like quotient types. In this setting, it is difficult to define the intended monad multiplication for the quotiented datatype. We give a solution where we postulate some principles, crucially proposition extensionality and the (semi-classical) axiom of countable choice. With the aid of these principles, we also prove that the quotiented delay datatype delivers free ω-complete pointed partial orders (ωcppos).Altenkirch et al. (Lecture Notes in Computer Science, vol. 10203, Springer, Heidelberg, 534–549, 2017) demonstrated that, in homotopy type theory, a certain higher inductive–inductive type is the free ωcppo on a type X essentially by definition; this allowed them to obtain a monad of free ωcppos without recourse to a choice principle. We notice that, by a similar construction, a simpler ordinary higher inductive type gives the free countably complete join semilattice on the unit type 1. This type suffices for constructing a monad, which is isomorphic to the one of Altenkirch et al. We have fully formalized our results in the Agda dependently typed programming language.

On fixed-point theorems in synthetic computability

Lawvere's fixed point theorem captures the essence of diagonalization arguments. Cantor's theorem, Gödel's incompleteness theorem, and Tarski's undefinability of truth are all instances of the contrapositive form of the theorem. It is harder to apply the theorem directly because non-trivial examples are not easily found, in fact, none exist if excluded middle holds. We study Lawvere's fixed-point theorem in synthetic computability, which is higher-order intuitionistic logic augmented with the Axiom of Countable Choice, Markov's principle, and the Enumeration axiom, which states that there are countably many countable subsets of $\mathbb{N}$. These extra-logical principles are valid in the effective topos, as well as in any realizability topos built over Turing machines with an oracle, and suffice for an abstract axiomatic development of a computability theory. We show that every countably generated $\omega$-chain complete pointed partial order ($\omega$cppo) is countable, and that countably generated $\omega$cppos are closed under countable products. Therefore, Lawvere's fixed-point theorem applies and we obtain fixed points of all endomaps on countably generated $\omega$cppos. Similarly, the Knaster-Tarski theorem guarantees existence of least fixed points of continuous endomaps. To get the best of both theorems, namely that all endomaps on domains ($\omega$cppos generated by a countable set of compact elements) have least fixed points, we prove a synthetic version of the Myhill-Shepherdson theorem: every map from an $\omega$cpo to a domain is continuous. The proof relies on a new fixed-point theorem, the synthetic Recursion Theorem, so called because it subsumes the classic Kleene-Rogers Recursion Theorem. The Recursion Theorem takes the form of Lawvere's fixed point theorem for multi-valued endomaps.

On the relation of three theorems of functional analysis to the axiom of choice

In what follows, essentially two things will be accomplished: Firstly, it will be proven that a version of the Arzel\`a--Ascoli theorem and the Fr\'echet--Kolmogorov theorem are equivalent to the axiom of countable choice for subsets of real numbers. Secondly, some progress is made towards determining the amount of axioms that have to be added to the Zermelo--Fraenkel system so that the uniform boundedness principle holds.

Bar recursion over finite partial functions

We introduce a new, demand-driven variant of Spector's bar recursion in the spirit of the Berardi–Bezem–Coquand functional of [4]. The recursion takes place over finite partial functions u, where the control parameter ω, used in Spector's bar recursion to terminate the computation at sequences s satisfying ω(sˆ)<|s|, now acts as a guide for deciding exactly where to make bar recursive updates, terminating the computation whenever ω(uˆ)∈dom(u). We begin by exploring theoretical aspects of this new form of recursion, then in the main part of the paper we show that demand-driven bar recursion can be directly used to give an alternative functional interpretation of classical countable choice. We provide a short case study as an illustration, in which we extract a new bar recursive program from the proof that there is no injection from N→N to N, and compare this with the program that would be obtained using Spector's original variant. We conclude by formally establishing that our new bar recursor is primitive recursively equivalent to the original Spector bar recursion, and thus defines the same class of functionals when added to Gödel's system T.

Total Boundedness and the Axiom of Choice

A metric space is Totally Bounded (also called preCompact) if it has a finite e-net for every e > 0 and it is preLindelof if it has a countable e-net for every e > 0. Using the Axiom of Countable Choice (CC), one can prove that a metric space is topologically equivalent to a Totally Bounded metric space if and only if it is a preLindelof space if and only if it is a Lindelof space. In the absence of CC, it is not clear anymore what should the definition of preLindelofness be. There are two distinguished options. One says that a metric space X is: (a) preLindelof if, for every e > 0, there is a countable cover of X by open balls of radius ?? (Keremedis, Math. Log. Quart. 49, 179–186 2003); (b) Quasi Totally Bounded if, for every e > 0, there is a countable subset A of X such that the open balls with centers in A and radius e cover X.

On metric spaces where continuous real valued functions are uniformly continuous in ZF

We show that the negation of each one of the following statements is consistent with ZF:(i) Every sequentially compact metric space X=(X,d) is normal, i.e., the distance of any two disjoint non-empty closed subsets of X is strictly positive.(ii) If (X,d) is a sequentially compact metric space then X is a UC space, i.e., every continuous real valued on X is uniformly continuous.(iii) If (X,d) is a UC metric space then X is Lebesgue, i.e., every open cover of X has a Lebesgue number.(iv) If (X,d) is a metric space such that every countable open cover of X has a Lebesgue number then X is Lebesgue.We also show:(v) For every metric space X, the following are equivalent:(1) Every sequence in X admits a Cauchy subsequence;(2) For every sequence (xn)n∈N of X, for each ε>0 there is an infinite Nε⊆N such that d(xn,xm)<ε whenever n,m∈Nε;(3) For every sequence (xn)n∈N of X, for each ε>0 and for each n0∈N there exist n,m∈N, n,m≥n0, n≠m such that d(xn,xm)<ε.(vi) The axiom of countable choice CAC implies that for every metric space X the following statements are equivalent:(1) X is Lebesgue;(2) Every countable open cover of X has a Lebesgue number;(3) X is UC.

Solution, extensions and applications of the Schauder's 54th problem in Scottish book

This paper presents the Axiom of Infinite Choice: Given any set P, there exist at least countable choice functions or there exist at least finite choice functions. The author continues herein with the further study of two papers of the Axiom of Choice in order by E. Zermelo [Neuer Beweis für die Möglichkeit einer Wohlordung, Math. Annalen, 65 (1908), 107-128; translated in van Heijenoort 1967, 183-198], and by M. Taskovic [The axiom of choice, fixed point theorems, and inductive ordered sets, Proc. Amer. Math. Soc., 116 (1992), 897-904]. Fredholm and Leray-Schauder alternatives are two direct consequences of the Axiom of Infinite Choice! This paper presents applications of the Axiom of Infinite Choice to the Fredholm and Leray-Schauder theory. In this sense, I give a solution and some extensions of Schauder's problem (in Scottish book, problem 54). This paper presents some new mathematical n-person games. In the theory of n-person games, there have been some further developments in the direction of transversal games and mathematical alternative theory.

On the metric reflection of a pseudometric space in ZF

We show: (i) The countable axiom of choice $\mathbf{CAC}$ is equivalent to each one of the statements: (a) a pseudometric space is sequentially compact iff its metric reflection is sequentially compact, (b) a pseudometric space is complete iff its metric reflection is complete. (ii) The countable multiple choice axiom $\mathbf{CMC}$ is equivalent to the statement: (a) a pseudometric space is Weierstrass-compact iff its metric reflection is Weierstrass-compact. (iii) The axiom of choice $\mathbf{AC}$ is equivalent to each one of the statements: (a) a pseudometric space is Alexandroff-Urysohn compact iff its metric reflection is Alexandroff-Urysohn compact, (b) a pseudometric space $\mathbf{X}$ is Alexandroff-Urysohn compact iff its metric reflection is ultrafilter compact. (iv) We show that the statement “The preimage of an ultrafilter extends to an ultrafilter” is not a theorem of $\mathbf{ZFA}$.

Models of computation for partial functions on the reals

We compare models of computation for partial functions f:R⇀R. We consider four models: two concrete (Grzegorczyk–Lacombe and tracking computability), one abstract (approximability by a While program with “countable choice”) and a new hybrid model: multipolynomial approximability. We show that these four models are equivalent, under the two assumptions:(1)the domain of f is the union of an effective exhaustion, i.e. a sequence of “stages”, each of which is a finite union of disjoint rational open intervals, and(2)f is effectively locally uniformly continuous w.r.t. this exhaustion. These assumptions seem to hold for all unary elementary functions of real analysis, many of which are, of course, partial. We make a conjecture with regard to this.

Step-Indexed Relational Reasoning for Countable Nondeterminism

Programming languages with countable nondeterministic choice are computationally interesting since countable nondeterminism arises when modeling fairness for concurrent systems. Because countable choice introduces non-continuous behaviour, it is well-known that developing semantic models for programming languages with countable nondeterminism is challenging. We present a step-indexed logical relations model of a higher-order functional programming language with countable nondeterminism and demonstrate how it can be used to reason about contextually defined may- and must-equivalence. In earlier step-indexed models, the indices have been drawn from {\omega}. Here the step-indexed relations for must-equivalence are indexed over an ordinal greater than {\omega}.

What do Freyd’s Toposes Classify?

We describe a method for presenting (a topos closely related to) either of Freyd’s topos-theoretic models for the independence of the axiom of choice as the classifying topos for a geometric theory. As an application, we show that no such topos can admit a geometric morphism from a two-valued topos satisfying countable dependent choice.

A generalization of Nash's theorem with higher-order functionals

The recent theory of sequential games and selection functions by Escardó & Oliva is extended to games in which players move simultaneously. The Nash existence theorem for mixed-strategy equilibria of finite games is generalized to games defined by selection functions. A normal form construction is given, which generalizes the game-theoretic normal form, and its soundness is proved. Minimax strategies also generalize to the new class of games, and are computed by the Berardi–Bezem–Coquand functional, studied in proof theory as an interpretation of the axiom of countable choice.

Lifschitz realizability for intuitionistic Zermelo–Fraenkel set theory

A variant of realizability for Heyting arithmetic which validates Church's thesis with uniqueness condition, but not the general form of Church's thesis, was introduced by Lifschitz (Proc Am Math Soc 73:101---106, 1979). A Lifschitz counterpart to Kleene's realizability for functions (in Baire space) was developed by van Oosten (J Symb Log 55:805---821, 1990). In that paper he also extended Lifschitz' realizability to second order arithmetic. The objective here is to extend it to full intuitionistic Zermelo---Fraenkel set theory, IZF. The machinery would also work for extensions of IZF with large set axioms. In addition to separating Church's thesis with uniqueness condition from its general form in intuitionistic set theory, we also obtain several interesting corollaries. The interpretation repudiates a weak form of countable choice, AC ?,? , asserting that a countable family of inhabited sets of natural numbers has a choice function. AC ?,? is validated by ordinary Kleene realizability and is of course provable in ZF. On the other hand, a pivotal consequence of AC ?,? , namely that the sets of Cauchy reals and Dedekind reals are isomorphic, remains valid in this interpretation. Another interesting aspect of this realizability is that it validates the lesser limited principle of omniscience.

Some axioms for constructive analysis

This note explores the common core of constructive, intuitionistic, recursive and classical analysis from an axiomatic standpoint. In addition to clarifying the relation between Kleene’s and Troelstra’s minimal formal theories of numbers and number-theoretic sequences, we propose some modified choice principles and other function existence axioms which may be of use in reverse constructive analysis. Specifically, we consider the function comprehension principles assumed by the two minimal theories EL and M, introduce an axiom schema CFd asserting that every decidable property of numbers has a characteristic function, and use it to describe a precise relationship between the minimal theories. We show that the axiom schema AC00 of countable choice can be decomposed into a monotone choice schema AC 00 m (which guarantees that every Cauchy sequence has a modulus) and a bounded choice schema BC00. We relate various (classically correct) axiom schemas of continuous choice to versions of the bar and fan theorems, suggest a constructive choice schema AC1/2,0 (which incidentally guarantees that every continuous function has a modulus of continuity), and observe a constructive equivalence between restricted versions of the fan theorem and correspondingly restricted bounding axioms $${AB_{1/2,0}^{2^{\mathbb{N}}}}$$ . We also introduce a version WKL!! of Weak König’s Lemma with uniqueness which is intermediate in strength between WKL and the decidable fan theorem FTd.