It is easy to prove in ZF− that a relation R satisfies the maximal condition if and only if its transitive hull R* does; equivalently: R is well-founded if and only if R* is. We will show in the following that, if the maximal condition is replaced by the chain condition, as is often the case in Algebra, the resulting statement is not provable in ZF− anymore . More precisely, we will prove that this statement is equivalent in ZF− to (...) the countable axiom of choice ACω. Moreover, applying this result we will prove that the axiom of dependent choices, restricted to partial orders as used in Algebra, already implies the general form for arbitrary relations as formulated first by Teichmüller and, independently, some time later by Bernays and Tarski. (shrink)

We show that the Axiom of Countable Choice is necessary and sufficient to prove that the existence of a Borel measure on a pseudometric space such that the measure of open balls is positive and finite implies separability of the space. In this way a negative answer to an open problem formulated in Górka (Am Math Mon 128:84–86, 2020) is given. Moreover, we study existence of maximal $$\delta $$ δ -separated sets in metric and pseudometric spaces from (...) the point of view the Axiom of Choice and its weaker forms. (shrink)

We show that the countable axiom of choice CAC strictly implies the statements “Lindelöf metric spaces are second countable” “Lindelöf metric spaces are separable”. We also show that CAC is equivalent to the statement: “If is a Lindelöf topological space with respect to the base ℬ, then is Lindelöf”.

We study statements about countable and well-ordered unions and their relation to each other and to countable and well-ordered forms of the axiom of choice. Using WO as an abbreviation for “well-orderable”, here are two typical results: The assertion that every WO family of countable sets has a WO union does not imply that every countable family of WO sets has a WO union; the axiom of choice for WO families of WO (...) sets does not imply that the countable union of countable sets is WO. (shrink)

In the realm of metric spaces the role of choice principles is investigated.

It is shown that for compact metric spaces the following statements are pairwise equivalent: “X is Loeb”, “X is separable”, “X has a we ordered dense subset”, “X is second countable”, and “X has a dense set G = ∪{Gn : n ∈ ω}, ∣Gn∣ < ω, with limn→∞ diam = 0”. Further, it is shown that the statement: “Compact metric spaces are weakly Loeb” is not provable in ZF0 , the Zermelo-Fraenkel set theory without the axiom of (...) regularity, and that the countable axiom of choice for families of finite sets CACfin does not imply the statement “Compact metric spaces are separable”. (shrink)

We show that some well known theorems in topology may not be true without the axiom of choice.

We work in set-theory without choice ZF. A set is Countable if it is finite or equipotent with ${\Bbb N}$ . Given a closed subset F of [0, 1] I which is a bounded subset of $\ell ^{1}(I)$ (resp. such that $F\subseteq c_{0}(I)$ ), we show that the countable axiom of choice for finite sets, (resp. the countable axiom of choice AC N ) implies that F is compact. This enhances previous results (...) where AC N (resp. the axiom of Dependent Choices) was required. If I is linearly orderable (for example $I={\Bbb R}$ ), then, in ZF, the closed unit ball of the Hilbert space $\ell ^{2}(I)$ is (Loeb-)compact in the weak topology. However, the weak compactness of the closed unit ball of $\ell ^{2}(\scr{P}({\Bbb R}))$ is not provable in ZF. (shrink)

This is a continuation of [2]. We study the Tychonoff Compactness Theorem for various definitions of compactness and for various types of spaces . We also study well ordered Tychonoff products and the effect that the multiple choice axiom has on such products.

Takeuti introduced an infinitary proof system for determinate logic and showed that for transitive models of Zermelo-Fraenkel set theory with the Axiom of Dependent Choice that contain all reals, the cut-elimination theorem is equivalent to the Axiom of Determinacy, and in particular contradicts the Axiom of Choice. We consider variants of Takeuti's theorem without assuming the failure of the Axiom of Choice. For instance, we show that if one removes atomic formulae of infinite (...) arity from the language of Takeuti's proof system, then cut elimination is equivalent to a determinacy hypothesis provable, e.g., in ZFC + “there are infinitely many Woodin cardinals.” A slight extension of the proof system admits cut elimination for countable sequents under the same assumptions. A simple modification of the proof system yields analogs of the results above for the Axiom of Real Determinacy and uncountably many Woodin cardinals. (shrink)

We study within the framework of Zermelo-Fraenkel set theory ZF the role that the axiom of choice plays in the theory of Lindelöf metric spaces. We show that in ZF the weak choice principles: Every Lindelöf metric space is separable and Every Lindelöf metric space is second countable are equivalent. We also prove that a Lindelöf metric space is hereditarily separable iff it is hereditarily Lindelöf iff it hold as well the axiom of choice (...) restricted to countable sets and to topologies of Lindelöf metric spaces as the countable union theorem restricted to Lindelöf metric spaces. (shrink)

We study theorems from Functional Analysis with regard to their relationship with various weak choice principles and prove several results about them: “Every infinite‐dimensional Banach space has a well‐orderable Hamel basis” is equivalent to ; “ can be well‐ordered” implies “no infinite‐dimensional Banach space has a Hamel basis of cardinality ”, thus the latter statement is true in every Fraenkel‐Mostowski model of ; “No infinite‐dimensional Banach space has a Hamel basis of cardinality ” is not provable in ; “No (...) infinite‐dimensional Banach space has a well‐orderable Hamel basis of cardinality ” is provable in ; (the Axiom of Choice for denumerable families of non‐empty finite sets) is equivalent to “no infinite‐dimensional Banach space has a Hamel basis which can be written as a denumerable union of finite sets”; Mazur's Lemma (“If X is an infinite‐dimensional Banach space, Y is a finite‐dimensional vector subspace of X, and, then there is a unit vector such that for all and all scalars α”) is provable in ; “A real normed vector space X is finite‐dimensional if and only if its closed unit ball is compact” is provable in ; (Principle of Dependent Choices) + “ can be well‐ordered” does not imply the Hahn‐Banach Theorem () in ; and “no infinite‐dimensional Banach space has a Hamel basis of cardinality ” are independent from each other in ; “No infinite‐dimensional Banach space can be written as a denumerable union of finite‐dimensional subspaces” lies in strength between (the Axiom of Countable Choice) and ; implies “No infinite‐dimensional Banach space can be written as a denumerable union of closed proper subspaces” which in turn implies ; “Every infinite‐dimensional Banach space has a denumerable linearly independent subset” is a theorem of, but not a theorem of ; and “Every infinite‐dimensional Banach space has a linearly independent subset of cardinality ” implies “every Dedekind‐finite set is finite”. (shrink)

Let ZF denote Zermelo-Fraenkel set theory (without the axiom of choice), and let $M$ be a countable transitive model of ZF. The method of forcing extends $M$ to another model $M\lbrack G\rbrack$ of ZF (a "generic extension"). If the axiom of choice holds in $M$ it also holds in $M\lbrack G\rbrack$, that is, the axiom of choice is preserved by generic extensions. We show that this is not true for many weak forms of (...) the axiom of choice, and we derive an application to Boolean toposes. (shrink)

We present a possible computational content of the negative translation of classical analysis with the Axiom of (countable) Choice. Interestingly, this interpretation uses a refinement of the realizability semantics of the absurdity proposition, which is not interpreted as the empty type here. We also show how to compute witnesses from proofs in classical analysis of ∃-statements and how to extract algorithms from proofs of ∀∃-statements. Our interpretation seems computationally more direct than the one based on Godel's Dialectica (...) interpretation. (shrink)

We show that the Axiom of Dependent Choice,, holds in countably iterable, passive premice constructed over their reals which satisfy the Axiom of Determinacy,, in a background universe. This generalizes an argument of Kechris for using Steel's analysis of scales in mice. In particular, we show that for any and any countable set of reals A so that and, we have that.

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 metric spaces, and the second declares that sequentially compact pseudometric spaces are \—meaning that all real valued, continuous functions defined on these spaces are necessarily uniformly continuous.

We study the relationship between the countable axiom of choice and the Tychonoff product theorem for countable families of topological spaces.

Under the axiom of choice, every first countable space is a Fréchet-Urysohn space. Although, in its absence even ℝ may fail to be a sequential space.Our goal in this paper is to discuss under which set-theoretic conditions some topological classes, such as the first countable spaces, the metric spaces, or the subspaces of ℝ, are classes of Fréchet-Urysohn or sequential spaces.In this context, it is seen that there are metric spaces which are not sequential spaces. This (...) fact raises the question of knowing if the completion of a metric space exists and it is unique. The answer depends on the definition of completion.Among other results it is shown that: every first countable space is a sequential space if and only if the axiom of countable choice holds, the sequential closure is idempotent in ℝ if and only if the axiom of countable choice holds for families of subsets of ℝ, and every metric space has a unique equation image-completion. (shrink)

Especially nice models of intuitionistic set theories are realizability models $V$, where $\mathcal A$ is an applicative structure or partial combinatory algebra. This paper is concerned with the preservation of various choice principles in $V$ 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 $V$ 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 $V$ regardless of whether it holds in the background universe. The paper extends work by McCarty and Rathjen. (shrink)

Let {: i ∈I } be a family of compact spaces and let X be their Tychonoff product. [MATHEMATICAL SCRIPT CAPITAL C] denotes the family of all basic non-trivial closed subsets of X and [MATHEMATICAL SCRIPT CAPITAL C]R denotes the family of all closed subsets H = V × Πmath imageXi of X, where V is a non-trivial closed subset of Πmath imageXi and QH is a finite non-empty subset of I. We show: Every filterbase ℋ ⊂ [MATHEMATICAL SCRIPT CAPITAL (...) C]R extends to a [MATHEMATICAL SCRIPT CAPITAL C]R-ultrafilter ℱ if and only if every family H ⊂ [MATHEMATICAL SCRIPT CAPITAL C] with the finite intersection property extends to a maximal [MATHEMATICAL SCRIPT CAPITAL C] family F with the fip. The proposition “if every filterbase ℋ ⊂ [MATHEMATICAL SCRIPT CAPITAL C]R extends to a [MATHEMATICAL SCRIPT CAPITAL C]R-ultrafilter ℱ, then X is compact” is not provable in ZF. The statement “for every family {: i ∈ I } of compact spaces, every filterbase ℋ ⊂ [MATHEMATICAL SCRIPT CAPITAL C]R, Y = Πi ∈IYi, extends to a [MATHEMATICAL SCRIPT CAPITAL C]R-ultrafilter ℱ” is equivalent to Tychonoff's compactness theorem. The statement “for every family {: i ∈ ω } of compact spaces, every countable filterbase ℋ ⊂ [MATHEMATICAL SCRIPT CAPITAL C]R, X = Πi ∈ωXi, extends to a [MATHEMATICAL SCRIPT CAPITAL C]R-ultrafilter ℱ” is equivalent to Tychonoff's compactness theorem restricted to countable families. The countable Axiom of Choice is equivalent to the proposition “for every family {: i ∈ ω } of compact topological spaces, every countable family ℋ ⊂ [MATHEMATICAL SCRIPT CAPITAL C] with the fip extends to a maximal [MATHEMATICAL SCRIPT CAPITAL C] family ℱ with the fip”. (shrink)

In set theory without the Axiom of Choice we prove that the assertion “For every metric space (_X_, _d_) with a Borel measure \(\mu \) such that the measure of every open ball is positive and finite, (_X_, _d_) is separable.’ is implied by the axiom of choice for countable collections of sets and implies the axiom of choice for countable collections of finite sets. We also show that neither implication is reversible (...) in Zermelo–Fraenkel set theory weakend to permit the existence of atoms and that the second implication is not reversible in Zermelo–Fraenkel set theory. This gives an answer to a question of Dybowski and Górka (Arch Math Logic 62:735–749, 2023. https://doi.org/10.1007/s00153-023-00868-4 ). (shrink)

We study the role that the axiom of choice plays in Tychonoff's product theorem restricted to countable families of compact, as well as, Lindelöf metric spaces, and in disjoint topological unions of countably many such spaces.

We investigate in ZF conditions that are necessary and sufficient for countable products ∏m∈ℕXm of finite Hausdorff spaces Xm resp. Hausdorff spaces Xm with at most n points to be compact resp. Baire. Typica results: Countable products of finite Hausdorff spaces are compact if and only if countable products of non-empty finite sets are non-empty. Countable products of discrete spaces with at most n + 1 points are compact if and only if countable products of (...) non-empty sets with at most n points are non-empty. (shrink)

For a centerless group G, we can define its automorphism tower. We define G α : G 0 = G, G α+1 = Aut(G α ) and for limit ordinals ${G^{\delta}=\bigcup_{\alpha<\delta}G^{\alpha}}$ . Let τ G be the ordinal when the sequence stabilizes. Thomas’ celebrated theorem says ${\tau_{G}<(2^{|G|})^{+}}$ and more. If we consider Thomas’ proof too set theoretical (using Fodor’s lemma), we have here a more direct proof with little set theory. However, set theoretically we get a parallel theorem without the (...) Axiom of Choice. Moreover, we give a descriptive set theoretic approach for calculating an upper bound for τ G for all countable groups G (better than the one an analysis of Thomas’ proof gives). We attach to every element in G α , the αth member of the automorphism tower of G, a unique quantifier free type over G (which is a set of words from ${G*\langle x\rangle}$ ). This situation is generalized by defining “(G, A) is a special pair”. (shrink)

Should weak forms of the axiom of choice really be accepted within constructive mathematics? A critical view of the Brouwer-Heyting-Kolmogorov interpretation, accompanied by the intention to include nondeterministic algorithms, leads us to subscribe to Richman's appeal for dropping countable choice. As an alternative interpretation of intuitionistic logic, we propose to renew dialogue semantics.

We show that there is a model of ZF in which the Borel hierarchy on the reals has length ω2. This implies that ω1 has countable cofinality, so the axiom of choice fails very badly in our model. A similar argument produces models of ZF in which the Borel hierarchy has exactly λ + 1 levels for any given limit ordinal λ less than ω2. We also show that assuming a large cardinal hypothesis there are models of (...) ZF in which the Borel hierarchy is arbitrarily long. (shrink)

In this paper we prove three theorems about the theory of Borel sets in models of ZF without any form of the axiom of choice. We prove that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${B\subseteq 2^\omega}$$\end{document} is a Gδσ-set then either B is countable or B contains a perfect subset. Second, we prove that if 2ω is the countable union of countable sets, then there exists an Fσδ set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} (...) \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C\subseteq 2^\omega}$$\end{document} such that C is uncountable but contains no perfect subset. Finally, we construct a model of ZF in which we have an infinite Dedekind finite \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D\subseteq 2^\omega}$$\end{document} which is Fσδ. (shrink)

We establish the following results: 1. In ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC), for every set I and for every ordinal number α ≥ ω, the following statements are equivalent: (a) The Tychonoff product of| α| many non-empty finite discrete subsets of I is compact. (b) The union of| α| many non-empty finite subsets of I is well orderable. 2. The statement: For every infinite set I, every closed subset of the Tychonoff product (...) [0, 1] I which consists of functions with finite support is compact, is not provable in ZF set theory. 3. The statement: For every set I, the principle of dependent choices relativised to I implies the Tychonoff product of countably many non-empty finite discrete subsets of I is compact, is not provable in ZF⁰ (i.e., ZF minus the Axiom of Regularity). 4. The statement: For every set I, every ℵ₀-sized family of non-empty finite subsets of I has a choice function implies the Tychonoff product of ℵ₀ many non-empty finite discrete subsets of I is compact, is not provable in ZF⁰. (shrink)

Aim of the paper is to revise Boolos’ reinterpretation of second-order monadic logic in terms of plural quantification ([4], [5]) and expand it to full second order logic. Introducing the idealization of plural acts of choice, performed by a suitable team of agents, we will develop a notion of plural reference . Plural quantification will be then explained in terms of plural reference. As an application, we will sketch a structuralist reconstruction of second-order arithmetic based on the axiom (...) of infinite à la Dedekind, as the unique non-logical axiom. We will also sketch a virtual interpretation of the classical continuum involving no other infinite than a countable plurality of individuals. (shrink)

Adjoin, to a countable standard model M of Zermelo-Fraenkel set theory (ZF), a countable set A of independent Cohen generic reals. If one attempts to construct the model generated over M by these reals (not necessarily containing A as an element) as the intersection of all standard models that include M ∪ A, the resulting model fails to satisfy the power set axiom, although it does satisfy all the other ZF axioms. Thus, there is no smallest ZF (...) model including M ∪ A, but there are minimal such models. These are classified by their sets of reals, and there is one minimal model whose set of reals is the smallest possible. We give several characterizations of this model, we determine which weak axioms of choice it satisfies, and we show that some better known models are forcing extensions of it. (shrink)

The axiom of dependent choice ( $\mathsf {DC}$ ) and the axiom of countable choice ( ${\mathsf {AC}}_\omega $ ) are two weak forms of the axiom of choice that can be stated for a specific set: $\mathsf {DC} ( X )$ asserts that any total binary relation on X has an infinite chain, while ${\mathsf {AC}}_\omega ( X )$ asserts that any countable collection of nonempty subsets of X has a (...) class='Hi'>choice function. It is well-known that $\mathsf {DC} \Rightarrow {\mathsf {AC}}_\omega $. We study for which sets and under which hypotheses $\mathsf {DC} ( X ) \Rightarrow {\mathsf {AC}}_\omega ( X )$, and then we show it is consistent with $\mathsf {ZF}$ that there is a set $A \subseteq \mathbb {R}$ for which $\mathsf {DC} ( A )$ holds, but ${\mathsf {AC}}_\omega ( A )$ fails. (shrink)

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.

We construct peculiar Hilbert spaces from counterexamples to the axiom of choice. We identify the intrinsically effective Hamiltonians with those observables of quantum theory which may coexist with such spaces. Here a self adjoint operator is intrinsically effective if and only if the Schrödinger equation of its generated semigroup is soluble by means of eigenfunction series expansions.

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, ACω,ω, asserting that a countable family of inhabited sets of natural numbers has a choice function. ACω,ω is validated by ordinary Kleene realizability and is of course provable in ZF. On the other hand, a pivotal consequence of ACω,ω, 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. (shrink)

A weak form of intuitionistic set theory WST lacking the axiom of extensionality is introduced. While WST is too weak to support the derivation of the law of excluded middle from the axiom of choice, we show that bee.ng up WST with moderate extensionality principles or quotient sets enables the derivation to go through.

There seems to be a view that intuitionists not only take the Axiom of Choice (AC) to be true, but also believe it a consequence of their fundamental posits. Widespread or not, this view is largely mistaken. This article offers a brief, yet comprehensive, overview of the status of AC in various intuitionistic and constructivist systems. The survey makes it clear that the Axiom of Choice fails to be a theorem in most contexts and is even (...) outright false in some important contexts. Of the systems surveyed, only intensional type theory renders AC a theorem, but the extent of AC in that theory does not include, for instance, real analysis. Only a small amount of extensionality is required in order for the obvious proof an intuitionist might offer for AC to break down. (shrink)

The present article deals with the power of the axiom of choice within the second-order predicate logic. We investigate the relationship between several variants of AC and some other statements, known as equivalent to AC within the set theory of Zermelo and Fraenkel with atoms, in Henkin models of the one-sorted second-order predicate logic with identity without operation variables. The construction of models follows the ideas of Fraenkel and Mostowski. It is e. g. shown that the well-ordering theorem (...) for unary predicates is independent from AC for binary predicates and from the trichotomy law for unary predicates. Moreover, we show that the AC for binary predicates follows neither from the trichotomy law for unary predicates nor from Zorn's lemma for unary predicates nor from the formalization of the axiom of choice for disjoint families of sets for binary predicates, and that the trichotomy law for unary predicates does not follow from AC for binary predicates. (shrink)

We show that there is a [Formula: see text]-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a [Formula: see text]-assertion, confirming a conjecture of Stephen Simpson. We obtain as a corollary that the Reflection Principle, stating that every formula reflects to a transitive set, can fail in models of [Formula: see text]. This work is a rediscovery by the first two authors of a result obtained by the third author (...) in [V. G. Kanovei, On descriptive forms of the countable axiom of choice, Investigations on nonclassical logics and set theory, Work Collect., Moscow, 3-136 ]. (shrink)

The work of Roshdi Rashed has set a landmark in many senses, but perhaps the most striking one is his inexhaustible thrive to open new paths for the study of conceptual links between science and philosophy deeply rooted in the interaction of historic with systematic perspectives. In the present talk I will focus on how a framework that has its source in philosophy of logic, interacts with some new results on the foundations of mathematics. More precisely, the main objective of (...) my brief remarks is to discuss some claims of the late Hintikka who brought forward the idea that a game-theoretical interpretation of the Axiom of Choice yields its meaning “evident”. More precisely I will show that if we develop Per Martin-Löf’s demonstration of the axiom within a dialogical setting, the claim of Hintikka can be upheld. However, the dialogical demonstration, shows that, contrary to the expectations of Hintikka, the meaning that the game-theoretical setting provides to the Axiom is compatible with constructivist rather than with classical tenets. (shrink)

It is shown that in ZF set theory the axiom of choice holds iff every non empty topological space has a maximal closed filter.

The principle of set theory known as the Axiom of Choice has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid's axiom of parallels which was introduced more than two thousand years ago” (Fraenkel, Bar-Hillel & Levy 1973, §II.4). The fulsomeness of this description might lead those unfamiliar with the axiom to expect it to be as startling as, say, the (...) Principle of the Constancy of the Velocity of Light or the Heisenberg Uncertainty Principle. But in fact the Axiom of Choice as it is usually stated appears humdrum, even self-evident. For it amounts to nothing more than the claim that, given any collection of mutually disjoint nonempty sets, it is possible to assemble a new set — a transversal or choice set — containing exactly one element from each member of the given collection. Nevertheless, this seemingly innocuous principle has far-reaching mathematical consequences — many indispensable, some startling — and has come to figure prominently in discussions on the foundations of mathematics. It (or its equivalents) have been employed in countless mathematical papers, and a number of monographs have been exclusively devoted to it. (shrink)

In this paper we study some statements similar to the Partition Principle and the Trichotomy. We prove some relationships between these statements, the Axiom of Choice, and the Generalized Continuum Hypothesis. We also prove some independence results. MSC: 03E25, 03E50, 04A25, 04A50.

In 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. (shrink)

From 1908 to 1916, articles supporting the axiom of choice were scant. The situation changed in 1916, when Wacław Sierpiński published a series of articles reviving the debate. The posterity of the axiom of choice as we know it would be unimaginable without Sierpiński’s efforts.

The method of morphisms is a well-known application of Dialectica categories to set theory. In a previous work, Valeria de Paiva and the author have asked how much of the Axiom of Choice is needed in order to carry out the referred applications of such method. In this paper, we show that, when considered in their full generality, those applications of Dialectica categories give rise to equivalents of either the Axiom of Choice or Partition Principle —which (...) is a consequence of $\textbf{AC}$ whose precise status of its relationship with$\textbf{AC}$ itself is an open problem for more than a hundred years. (shrink)

We study conditions for a topological space to be metrizable, properties of metrizable spaces, and the role the axiom of choice plays in these matters.

The principle of set theory known as the Axiom of Choice (AC) has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid’s axiom of parallels which was introduced more than two thousand years ago”1 It has been employed in countless mathematical papers, a number of monographs have been exclusively devoted to it, and it has long played a prominently role in discussions (...) on the foundations of mathematics. (shrink)

We combine a variety of constructive methods (including forcing, realizability, asymmetric interpretation), to obtain consistency results concerning combinatory logic with extensionality and (forms of) the axiom of choice.