A Mazurkiewicz set is a plane subset that intersects every straight line at exactly two points, and a Sierpiński–Zygmund function is a function from R into R that has as little of the standard continuity as possible. Building on the recent work of Kharazishvili, we construct a Mazurkiewicz set that contains a Sierpiński–Zygmund function in every direction and another one that contains none in any direction. Furthermore, we show that whether a Mazurkiewicz set can be expressed as a union of two Sierpiński–Zygmund functions is independent of Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC). Some open problems related to the containment of Hamel functions are stated.

Abstract This paper investigates histories in Branching Space–Time (BST) structures. We start by identifying necessary and sufficient conditions for the existence of free histories, and then we turn to the intangibility problem, and we show that the existence of histories in BST structures is equivalent to the axiom of choice, yielding the punchline “history gives us choice”.

The point-to-set principle [J. H. Lutz and N. Lutz, ACM Trans. Comput. Theory, 10(2), 7 (2018).] characterizes the Hausdorff dimension of a subset [Formula: see text] by the effective (or algorithmic) dimension of its individual points. This characterization has been used to prove several results in classical, i.e. without any computability requirements, analysis. Recent work has shown that algorithmic techniques can be fruitfully applied to Marstrand’s projection theorem, a fundamental result in fractal geometry. In this paper, we introduce the notion of optimal oracles for subsets [Formula: see text]. One of the primary motivations of this definition is that, if [Formula: see text] has optimal oracles, then the conclusion of Marstrand’s projection theorem holds for [Formula: see text]. We show that every analytic set has optimal oracles. We also prove that if the Hausdorff and packing dimensions of [Formula: see text] agree, then [Formula: see text] has optimal oracles. Moreover, we show that the existence of sufficiently nice outer measures on [Formula: see text] implies the existence of optimal Hausdorff oracles. In particular, the existence of exact gauge functions for a set [Formula: see text] is sufficient for the existence of optimal Hausdorff oracles, and is therefore sufficient for Marstrand’s theorem. Thus, the existence of optimal oracles extends the currently known sufficient conditions for Marstrand’s theorem to hold. Under certain assumptions, every set has optimal oracles. However, assuming the axiom of choice and the continuum hypothesis, we construct sets which do not have optimal oracles. This construction naturally leads to a generalization of Davies theorem on projections.

We investigate the set-theoretic strength of several maximality principles that play an important role in the study of modal and intuitionistic logics. We focus on well-known Fine’s and Esakia’s Maximality Principles, present two formulations of each, and show that the stronger formulations are equivalent to the Axiom of Choice (AC), while the weaker ones to the Boolean Prime Ideal Theorem (BPI).

Abstract Probabilism holds that rational credence functions are probability functions defined over some probability space $$(\Omega, \mathcal{F}, P)$$ . According to some recent philosophical arguments, in some situations, rational credence function must be total, i.e. $$\mathcal{F}=2^\Omega$$ , a view which I call credence totalism. Arguments for credence totalism are based on the premise that non-Lebesgue measurable subsets of $$\mathbb{R}$$ are epistemically significant, in the sense that an agent has reasons to assign probability to these sets. This paper argues that nonmeasurable sets are not epistemically significant in this sense. Consequently, the arguments for credence totalism are not successful. My argument is based on a careful consideration of the role of the Axiom of Choice in probabilistic practice. I also discuss some topics considered closely related, viz. the existence of total chance functions and the truth value of the Continuum Hypothesis. I argue that the role of nonmeasurability in epistemology does not shed light on these issues.

Gödel's first steps in set theory, from the summer of 1935 to the end of his stay in Princeton half a year later, are described in the light of his shorthand notebooks. The notes end with an English manuscript titled ‘The freedom from contradiction of the axiom of choice’ that is analyzed in detail. Gödel works out a logical hierarchical construction that systematically incorporates well-orderings, thereby affirming the title of his paper. He also sees an avenue to having his construction affirm the relative consistency of the Continuum Hypothesis, even to a key lemma about condensation along the hierarchy. However, he could not see a way to establishing the lemma until two years later.

Abstract The iterative conception (IC) is arguably the best worked out conception of set available. What is the status of the axiom of choice under this conception? Boolos argues that it is not justified by IC. We show that Boolos’s influential argument overgenerates. For, if cogent, it would imply that none of the axioms of ZFC which Boolos took to be justified by IC is so justified. We furthermore show that, to the extent that they are consequences of a plural formulation of stage theory, all those axioms are justified by IC — axiom of choice included.

We show that the axiom of choice , a basic yet controversial postulate of set theory, together with one of the pillars of our best physical theories, namely the no-signalling principle , may point to a limitation in modelling physical problems using standard mathematics. While it is well known that probabilistic no-signalling resources (such as quantum non-locality) are stronger than deterministic ones, we show—by invoking the axiom of choice—the opposite: functional (deterministic) no-signalling resources can be stronger than probabilistic ones (e.g. stronger than quantum entanglement or non-locality in general). To prove this, we devise a Bell-like game that shows a systematic advantage of functional no-signalling with respect to any probabilistic no-signalling resource.

This essay deals with one of the most basic questions that concern the philosophy of mathematics, which has come to be known as the Continuum Hypothesis. First put forth by Georg Cantor in 1878, the Continuum Hypothesis is a postulate on whether there exists an infinite set of real numbers whose cardinality lies strictly between that of the natural numbers and that of the real numbers themselves. The independence of Continuum Hypothesis from the standard axiomatic system of Zermelo-Fraenkel set theory with the Axiom of Choice was shown by Kurt Gödel and Paul Cohen; the independence has given rise to much interesting philosophical debate about the nature of mathematical truth. This essay argues from a Platonist perspective, maintaining that Continuum Hypothesis must have a determinate truth value, independent of the limitations of formal systems. The essay contrasts this view with formalism, which sees mathematical truths as dependent on the choice of axioms. By drawing historical analogies and examining both Platonist and formalist viewpoints, the paper advocates for the pursuit of new axioms and alternative frameworks—such as large cardinal and forcing axioms—that might ultimately resolve the Continuum Hypothesis. The discussion highlights the broader implications of Continuum Hypothesis for understanding the nature of infinity, the completeness of mathematical systems, and the foundations of mathematics itself.

Abstract This article concerns the Herrlich-Chew theorem stating that a Hausdorff zero-dimensional space is ℕ-compact if and only if every clopen ultrafilter with the countable intersection property in this space is fixed. It also concerns Hewitt’s theorem stating that a Tychonoff space is realcompact if and only if every z-ultrafilter with the countable intersection property in this space is fixed. The axiom of choice was involved in the original proofs of these theorems. The aim of this article is to show that the Herrlich-Chew theorem is valid in ZF, but it is an open problem if Hewitt’s theorem can be false in a model of ZF. It is proved that Hewitt’s theorem is true in every model of ZF in which the countable axiom of multiple choice is satisfied. A modification of Hewitt’s theorem is given and proved true in ZF. Several applications of the results obtained are shown.

The logic of cardinality comparison without the axiom of choice

For a topological space is the ring of all continuous real functions f on X such that, for every real number ϵ > 0, there exists a countable clopen cover of X such that the oscillation of f on each member of is less than ϵ. For a zero-dimensional T 1-space X, the ring and its subring of bounded functions from are applied to necessary and sufficient conditions on X to admit the Banaschewski compactification in the absence of the Axiom of Choice. For a zero-dimensional T 1-space X and a Tychonoff space Y, the problem of when the ring can be isomorphic to or to the ring of all (bounded) continuous real functions on Y is investigated. Several new equivalences of the Boolean Prime Ideal Theorem are deduced. Some results about are obtained under the Principle of Countable Multiple Choices.

ABSTRACTAbout 95% of criminal convictions in the United States are obtained through plea decisions, a growing global practice. Courts justify these convictions based on defendant choice—defendants, as rational agents, can freely choose to plead guilty or go to trial. However, a fundamental axiom of rational choice—descriptive invariance—has never been effectively tested for plea decisions. To test this axiom, we manipulated gain–loss framing of plea options. The shadow‐of‐trial model, the leading theory of plea decision‐making, is predicated on expected utility theory which is in turn predicated on the invariance axiom; if the axiom is falsified, the entire structure collapses. Thus, framing effects are important as a test of fundamental assumptions undergirding practice and as an empirical phenomenon revealing effects of context. We tested framing effects in students and community members including those with criminal involvement for whom plea bargaining has personal relevance. Varying subtle changes in wording of outcomes, we produced pronounced differences in choices to accept a plea rather than proceed to trial. These framing effects were robust to age, sex, educational attainment, risk propensity (DOSPERT and sensation seeking), and loss aversion. Perceived fairness of the legal system increased acceptance and risk propensity decreased it (each about 32%). However, controlling for those effects, loss (compared to gain) framing increased the odds of going to trial by 664%. Criminal involvement did not account for additional variance. These results are consistent with prospect theory and fuzzy‐trace theory, but they challenge the legal theory of bargaining in “the shadow of trial.”

Abstract In $\mathsf {ZF}$ (i.e., Zermelo–Fraenkel set theory minus the axiom of choice ( $\mathsf {AC}$ )), we investigate the open problem of the deductive strength of the principle UFwob(ω): “There exists a free ultrafilter on ω with a well-orderable base”, which was introduced by Herzberg, Kanovei, Katz, and Lyubetsky [(2018), Journal of Symbolic Logic, 83(1), 385–391]. Typical results are: (1) “ $\aleph _{1}\leq 2^{\aleph _{0}}$ ” is strictly weaker than $\mathsf {UF_{wob}}(\omega )$ in $\mathsf {ZF}$ . (2) “There exists a free ultrafilter on $\omega $ ” does not imply “ $\aleph _{1}\leq 2^{\aleph _{0}}$ ” in $\mathsf {ZF}$ , and thus (by (1)) neither does it imply $\mathsf {UF_{wob}}(\omega )$ in $\mathsf {ZF}$ . This fills the gap in information in Howard and Rubin [Mathematical Surveys and Monographs, American Mathematical Society, 1998], as well as in Herzberg et al. (2018). (3) Martin’s Axiom ( $\mathsf {MA}$ ) implies “no free ultrafilter on $\omega $ has a well-orderable base of cardinality $<2^{\aleph _{0}}$ ”, and the latter principle is not implied by $\aleph _{0}$ -Martin’s Axiom ( $\mathsf {MA}(\aleph _{0})$ ) in $\mathsf {ZF}$ . (4) $\mathsf {MA} + \mathsf {UF_{wob}}(\omega )$ implies $\mathsf {AC}(\mathbb {R})$ (the axiom of choice for non-empty sets of reals), which in turn implies $\mathsf {UF_{wob}}(\omega )$ . Furthermore, $\mathsf {MA}$ and $\mathsf {UF_{wob}}(\omega )$ are mutually independent in $\mathsf {ZF}$ . (5) For any infinite linearly orderable set X, each of “every filter base on X can be well ordered” and “every filter on X has a well-orderable base” is equivalent to “ $\wp (X)$ can be well ordered”. This yields novel characterizations of the principle “every linearly ordered set can be well ordered” in $\mathsf {ZFA}$ (i.e., Zermelo–Fraenkel set theory with atoms), and of $\mathsf {AC}$ in $\mathsf {ZF}$ . (6) “Every filter on $\mathbb {R}$ has a well-orderable base” implies “every filter on $\omega $ has a well-orderable base”, which in turn implies $\mathsf {UF_{wob}}(\omega )$ , and none of these implications are reversible in $\mathsf {ZF}$ . (7) “Every filter on $\omega $ can be extended to an ultrafilter with a well-orderable base” is equivalent to $\mathsf {AC}(\mathbb {R}),$ and thus is strictly stronger than $\mathsf {UF_{wob}}(\omega )$ in $\mathsf {ZF}$ . (8) “Every filter on $\omega $ can be extended to an ultrafilter” implies “there exists a free ultrafilter on $\omega $ which has no well-orderable base of cardinality ${<2^{\aleph _{0}}}$ ”. The former principle does not imply “there exists a free ultrafilter on $\omega $ which has no well-orderable base” in $\mathsf {ZF}$ , and the latter principle is true in the Basic Cohen Model.

On two theorems of Arens–Dugundji and Scott–Watson–Förster without the Axiom of Choice

In many branches of mathematics, especially set theory, algebra, and topology, the Axiom of Choice (AC) is crucial and helps to enable the existence of choice functions for any arbitrary collection of non-empty sets without explicit construction. Examining the arguments both for and against the Axiom of Choice, this work addresses the paradoxes and challenges it presents, including the Banach-Tarski dilemma and problems in measure theory, so furthering mathematical theory. Although AC is commended for its theoretical contributions—especially in terms of enabling work with abstract and infinite sets—it is attacked by constructivist mathematicians who stress the need of specific approaches of proof. The study comes to the conclusion that, despite its non-constructive character and the disputes it causes, the Axiom of Choice stays a vital instrument in modern mathematics, so extending the limits of theoretical investigation and application. Its application should, however, be carefully considered in order to balance the needs for mathematical rigour and practical relevance with the advantages of abstraction.

This essay deals with one of the most basic questions that concern the philosophy of mathematics, which has come to be known as the Continuum Hypothesis. First put forth by Georg Cantor in 1878, the Continuum Hypothesis is a postulate on whether there exists an infinite set of real numbers whose cardinality lies strictly between that of the natural numbers and that of the real numbers themselves. The independence of Continuum Hypothesis from the standard axiomatic system of Zermelo-Fraenkel set theory with the Axiom of Choice was shown by Kurt Gödel and Paul Cohen; the independence has given rise to much interesting philosophical debate about the nature of mathematical truth. This essay argues from a Platonist perspective, maintaining that Continuum Hypothesis must have a determinate truth value, independent of the limitations of formal systems. The essay contrasts this view with formalism, which sees mathematical truths as dependent on the choice of axioms. By drawing historical analogies and examining both Platonist and formalist viewpoints, the paper advocates for the pursuit of new axioms and alternative frameworks—such as large cardinal and forcing axioms—that might ultimately resolve the Continuum Hypothesis. The discussion highlights the broader implications of Continuum Hypothesis for understanding the nature of infinity, the completeness of mathematical systems, and the foundations of mathematics itself.

We investigate whether classical combinatorial theorems are provable in ZF. Some statements are not provable in ZF, but they are equivalent within ZF. For example, the following statements (i)–(iii) are equivalent: cf(ω1)=ω1,ω1→(ω1,ω+1)2,any family A⊂[On]<ω of size ω1 contains a Δ-system of size ω1. Some classical results cannot be proven in ZF alone; however, we can establish weaker versions of these statements within the framework of ZF, such as ω2→(ω1,ω+1),any family A⊂[On]<ω of size ω2 contains a Δ-system of size ω1. Some statements can be proven in ZF using purely combinatorial arguments, such as: given a set mapping F:ω1→[ω1]<ω, the set ω1 has a partition into ω-many F-free sets. Other statements can be proven in ZF by employing certain methods of absoluteness, for example: given a set mapping F:ω1→[ω1]<ω, there is an F-free set of size ω1,for each n∈ω, every family A⊂[ω1]ω with |A∩B|≤n for {A,B}∈[A]2 has property B. In contrast to statement (5), we show that the following ZFC theorem of Komjáth is not provable from ZF + cf(ω1)=ω1: (6∗)every family A⊂[ω1]ω with |A∩B|≤1 for {A,B}∈[A]2 is essentially disjoint. A function f is a uniform denumeration onω1 iff dom(f)=ω1, and for every 1≤α<ω1, f(α) is a function from ω onto α. It is easy to see that the existence of a uniform denumeration of ω1 implies cf(ω1)=ω1. We prove that the failure of the reverse implication is equiconsistent with the existence of an inaccessible cardinal.

This paper investigates two kinds of philosophical justifications—intrinsic and extrinsic justifications—for the Axiom of Choice. A standard intrinsic justification for the Axiom of Choice is based on the iterative conception of set. However, this form of justification implicitly adopts the maximality principle of sets, which is incoherent. The extrinsic justification turns out to be more effective due to the theoretical merits of the Axiom of Choice, despite the paradoxes caused by AC.

Abstract Denoting by the free group over a two‐element alphabet, we show in set‐theory without the axiom of choice that the existence of a (2, 2)‐paradoxical decomposition of free ‐sets follows from the conjunction of a weakened consequence of the Hahn‐Banach axiom and a weakened consequence of the axiom of choice for pairs. The existence in of a paradoxical decomposition with 4 pieces of the sphere in the 3‐dimensional euclidean space follows from the same two statements restricted to the set of real numbers. Our result is linked to the ‐paradoxical decompositions of free ‐sets previously obtained by Pawlikowski (, cf. [11]) and then by Sato and Shioya ( and , cf. [13]) with the sole Hahn‐Banach axiom.