Axiom Of Choice Research Articles

Infinite combinatorics revisited in the absence of Axiom of choice

AbstractWe 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: $$cf({\omega }_1)={\omega }_1$$ c f ( ω 1 ) = ω 1 , $${\omega }_1\rightarrow ({\omega }_1,{\omega }+1)^2$$ ω 1 → ( ω 1 , ω + 1 ) 2 , any family $$\mathcal {A}\subset [{On}]^{<{\omega }}$$ A ⊂ [ On ] < ω of size $${\omega }_1$$ ω 1 contains a $$\Delta $$ Δ -system of size $${\omega }_1$$ ω 1 . 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 $${{\omega }_2}\rightarrow ({\omega }_1,{\omega }+1)$$ ω 2 → ( ω 1 , ω + 1 ) , any family $$\mathcal {A}\subset [{On}]^{<{\omega }}$$ A ⊂ [ On ] < ω of size $${\omega }_2$$ ω 2 contains a $$\Delta $$ Δ -system of size $${\omega }_1$$ ω 1 . Some statements can be proven in ZF using purely combinatorial arguments, such as: given a set mapping $$F:{\omega }_1\rightarrow {[{\omega }_1]}^{<{\omega }}$$ F : ω 1 → [ ω 1 ] < ω , the set $${\omega }_1$$ ω 1 has a partition into $${\omega }$$ ω -many F-free sets. Other statements can be proven in ZF by employing certain methods of absoluteness, for example: given a set mapping $$F:{\omega }_1\rightarrow {[{\omega }_1]}^{<{\omega }}$$ F : ω 1 → [ ω 1 ] < ω , there is an F-free set of size $${\omega }_1$$ ω 1 , for each $$n\in {\omega }$$ n ∈ ω , every family $$\mathcal {A}\subset {[{\omega }_1]}^{{\omega }}$$ A ⊂ [ ω 1 ] ω with $$|A\cap B|\le n$$ | A ∩ B | ≤ n for $$\{A,B\}\in {[\mathcal {A}]}^{2}$$ { 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 + $$cf({\omega }_1)={\omega }_1$$ c f ( ω 1 ) = ω 1 : (6$$ ^*$$ ∗ ) every family $$\mathcal {A}\subset {[{\omega }_1]}^{{\omega }}$$ A ⊂ [ ω 1 ] ω with $$|A\cap B|\le 1$$ | A ∩ B | ≤ 1 for $$\{A,B\}\in {[\mathcal {A}]}^{2}$$ { A , B } ∈ [ A ] 2 is essentially disjoint. A function f is a uniform denumeration on$${\omega }_1$$ ω 1 iff $${\text {dom}}(f)={\omega }_1$$ dom ( f ) = ω 1 , and for every $$1\le {\alpha }<{\omega }_1$$ 1 ≤ α < ω 1 , $$f({\alpha })$$ f ( α ) is a function from $${\omega }$$ ω onto $${\alpha }$$ α . It is easy to see that the existence of a uniform denumeration of $${\omega }_1$$ ω 1 implies $$cf({\omega }_1)={\omega }_1$$ c f ( ω 1 ) = ω 1 . We prove that the failure of the reverse implication is equiconsistent with the existence of an inaccessible cardinal.

Philosophical Justifications for the Axiom of Choice

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.

The set of injections and the set of surjections on a set

AbstractIn this paper, we investigate the relationships between the cardinalities of the set of injections, the set of surjections, and the set of all functions on a set which is of cardinality , denoted by , and , respectively. Among our results, we show that “”, “” and “” are provable for an arbitrary infinite cardinal , and these are the best possible results, in the Zermelo‐Fraenkel set theory () without the Axiom of Choice. Also, we show that it is relatively consistent with that there exists an infinite cardinal such that where denotes the cardinality of the set of bijections on a set which is of cardinality .

Errata: multi-level nonstandard analysis and the axiom of choice

Errata: multi-level nonstandard analysis and the axiom of choice

Multi-level nonstandard analysis and the axiom of choice

Model-theoretic frameworks for Nonstandard Analysis depend on the existence of nonprincipal ultrafilters, a strong form of the Axiom of Choice (AC). Hrbacek and Katz, APAL Volume 72 formulate axiomatic nonstandard set theories SPOT and SCOT that are conservative extensions of respectively ZF and ZF + ADC (the Axiom of Dependent Choice), and in which a significant part of Nonstandard Analysis can be developed.The present paper extends these theories to theories with many levels of standardness, called respectively SPOTS and SCOTS. It shows that Jin's recent nonstandard proof of Szemeredi's Theorem can be carried out in SPOTS and that SCOTS is a conservative extension of ZF + ADC.

A generalisation of Läuchli's lemma

AbstractLäuchli showed in the absence of the Axiom of Choice () that and, consequently, for all infinite cardinals , where and are the cardinalities of the set of finite subsets and the power set, respectively, of a set which is of cardinality . In this article, we give a generalisation of a simple form of Läuchli's lemma from which several results can be obtained. That is, in the latter equation can be replaced by other cardinals which are equal to in but not in , for example, and , the cardinalities of the set of permutations and the set of partitions, respectively, of a set which is of cardinality .

Non‐accessible localizations

AbstractIn a 2005 paper, Casacuberta, Scevenels, and Smith construct a homotopy idempotent functor on the category of simplicial sets with the property that whether it can be expressed as localization with respect to a map is independent of the ZFC axioms. We show that this construction can be carried out in homotopy type theory. More precisely, we give a general method of associating to a suitable (possibly large) family of maps, a reflective subuniverse of any universe . When specialized to an appropriate family, this produces a localization which when interpreted in the ‐topos of spaces agrees with the localization corresponding to . Our approach generalizes the approach of Casacuberta et al. (Adv. Math. 197 (2005), no. 1, 120–139) in two ways. First, by working in homotopy type theory, our construction can be interpreted in any ‐topos. Second, while the local objects produced by Casacuberta et al. are always 1‐types, our construction can produce ‐types, for any . This is new, even in the ‐topos of spaces. In addition, by making use of universes, our proof is very direct. Along the way, we prove many results about “small” types that are of independent interest. As an application, we give a new proof that separated localizations exist. We also give results that say when a localization with respect to a family of maps can be presented as localization with respect to a single map, and show that the simplicial model satisfies a strong form of the axiom of choice that implies that sets cover and that the law of excluded middle holds.

Separablilty of metric measure spaces and choice axioms

Separablilty of metric measure spaces and choice axioms

DOES IMPLY , UNIFORMLY?

Abstract 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 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.

Reifying dynamical algebra: Maximal ideals in countable rings, constructively

The existence of a maximal ideal in a general nontrivial commutative ring is tied together with the axiom of choice. Following Berardi, Valentini and thus Krivine but using the relative interpretation of negation (that is, as “implies 0 = 1”) we show, in constructive set theory with minimal logic, how for countable rings one can do without any kind of choice and without the usual decidability assumption that the ring is strongly discrete (membership in finitely generated ideals is decidable). By a functional recursive definition we obtain a maximal ideal in the sense that the quotient ring is a residue field (every noninvertible element is zero), and with strong discreteness even a geometric field (every element is either invertible or else zero). Krull’s lemma for the related notion of prime ideal follows by passing to rings of fractions. By employing a construction variant of set-theoretic forcing due to Joyal and Tierney, we expand our treatment to arbitrary rings and establish a connection with dynamical algebra: We recover the dynamical approach to maximal ideals as a parametrized version of the celebrated double negation translation. This connection allows us to give formal a priori criteria elucidating the scope of the dynamical method. Along the way we do a case study for proofs in algebra with minimal logic, and generalize the construction to arbitrary inconsistency predicates. A partial Agda formalization is available at an accompanying repository.11 See https://github.com/iblech/constructive-maximal-ideals/. This text is a revised and extended version of the conference paper (In Revolutions and Revelations in Computability. 18th Conference on Computability in Europe (2022) Springer). The conference paper only briefly sketched the connection with dynamical algebra; did not compare this connection with other flavors of set-theoretic forcing; and sticked to the case of commutative algebra only, passing on the generalization to inconsistency predicates and well-orders.

THE DEFINABILITY OF THE EXTENDER SEQUENCE FROM IN

AbstractLet M be a short extender mouse. We prove that if $E\in M$ and $M\models $ “E is a countably complete short extender whose support is a cardinal $\theta $ and $\mathcal {H}_\theta \subseteq \mathrm {Ult}(V,E)$ ”, then E is in the extender sequence $\mathbb {E}^M$ of M. We also prove other related facts, and use them to establish that if $\kappa $ is an uncountable cardinal of M and $\kappa ^{+M}$ exists in M then $(\mathcal {H}_{\kappa ^+})^M$ satisfies the Axiom of Global Choice. We prove that if M satisfies the Power Set Axiom then $\mathbb {E}^M$ is definable over the universe of M from the parameter $X=\mathbb {E}^M\!\upharpoonright \!\aleph _1^M$ , and M satisfies “Every set is $\mathrm {OD}_{\{X\}}$ ”. We also prove various local versions of this fact in which M has a largest cardinal, and a version for generic extensions of M. As a consequence, for example, the minimal proper class mouse with a Woodin limit of Woodin cardinals models “ $V=\mathrm {HOD}$ ”. This adapts to many other similar examples. We also describe a simplified approach to Mitchell–Steel fine structure, which does away with the parameters $u_n$ .

The equivalency of the Banach-Alaoglu theorem and the axiom of choice

The Axiom of Choice can be used to prove the Banach-Alaoglu theorem, while a weakened form of the Axiom of Choice is required to prove the full strength of the Hahn-Banach theorem, which is equivalent to the Banach-Alaoglu theorem. Although the Banach-Alaoglu theorem and the full- strength Axiom of Choice are not exactly equivalent, they are closely related.
 The Banach-Alaoglu theorem is a compactness theorem whose proof mainly depends on Tychonoff's theorem. In this article, Banach-Alaoglu theorem is equivalent to the axiom of choice, has been proved.

CANTOR’S THEOREM MAY FAIL FOR FINITARY PARTITIONS

Abstract A partition is finitary if all its members are finite. For a set A, $\mathscr {B}(A)$ denotes the set of all finitary partitions of A. It is shown consistent with $\mathsf {ZF}$ (without the axiom of choice) that there exist an infinite set A and a surjection from A onto $\mathscr {B}(A)$ . On the other hand, we prove in $\mathsf {ZF}$ some theorems concerning $\mathscr {B}(A)$ for infinite sets A, among which are the following: (1) If there is a finitary partition of A without singleton blocks, then there are no surjections from A onto $\mathscr {B}(A)$ and no finite-to-one functions from $\mathscr {B}(A)$ to A. (2) For all $n\in \omega $ , $|A^n|<|\mathscr {B}(A)|$ . (3) $|\mathscr {B}(A)|\neq |\mathrm {seq}(A)|$ , where $\mathrm {seq}(A)$ is the set of all finite sequences of elements of A.

On the consistency of ZF with an elementary embedding from Vλ+2 into Vλ+2

According to a theorem due to Kenneth Kunen, under ZFC, there is no ordinal [Formula: see text] and nontrivial elementary embedding [Formula: see text]. His proof relied on the Axiom of Choice (AC), and no proof from ZF alone is has been discovered. [Formula: see text] is the assertion, introduced by Hugh Woodin, that [Formula: see text] is an ordinal and there is an elementary embedding [Formula: see text] with critical point [Formula: see text]. And [Formula: see text] asserts that [Formula: see text] holds for some [Formula: see text]. The axiom [Formula: see text] is one of the strongest large cardinals not known to be inconsistent with AC. It is usually studied assuming ZFC in the full universe [Formula: see text] (in which case [Formula: see text] must be a limit ordinal), but we assume only ZF. We prove, assuming ZF [Formula: see text] [Formula: see text] [Formula: see text] “[Formula: see text] is an even ordinal”, that there is a proper class transitive inner model [Formula: see text] containing [Formula: see text] and satisfying ZF [Formula: see text] [Formula: see text] [Formula: see text] “there is an elementary embedding [Formula: see text]”; in fact we will have [Formula: see text] ⊆[Formula: see text], where [Formula: see text] witnesses [Formula: see text] in [Formula: see text]. This result was first proved by the author under the added assumption that [Formula: see text] exists; Gabe Goldberg noticed that this extra assumption was unnecessary. If also [Formula: see text] is a limit ordinal and [Formula: see text]-DC holds in [Formula: see text], then the model [Formula: see text] will also satisfy [Formula: see text]-DC. We show that ZFC [Formula: see text] “[Formula: see text] is even” [Formula: see text] [Formula: see text] implies [Formula: see text] exists for every [Formula: see text], but if consistent, this theory does not imply [Formula: see text] exists.

Nonstandard proof methods in toposes

We determine sufficient structure for an elementary topos to emulate Nelson's Internal Set Theory in its internal language, and show that any topos satisfying the internal axiom of choice occurs as a universe of standard objects and maps. This development allows one to employ the proof methods of nonstandard analysis (transfer, standardisation, and idealisation) in new environments such as toposes of G-sets and Boolean étendues.

The finite sequences and the partitions whose members are finite of a set

Abstract In this paper, we investigate relationships between $|\text{{seq}}(A)|$ and $|\text{{Part}}_{\text{{fin}}}(A)|$ in the absence of the Axiom of Choice, where $\text{{seq}}(A)$ is the set of finite sequences of elements in a set $A$ and $\text{{Part}}_{\text{{fin}}}(A)$ is the set of partitions of $A$ whose members are finite. We show that $|\text{{seq}}(A)|<|\text{{Part}}_{\text{{fin}}}(A)|$ if $A$ is Dedekind-infinite and the condition cannot be removed. Moreover, this relationship holds for an arbitrary infinite set $A$ if we restrict $\text{{seq}}(A)$ to the set of finite sequences with a bounded length.

A note on Łoś theorem without the Axiom of Choice

A note on Łoś theorem without the Axiom of Choice

The Axiom of Choice and Its Influence on LLM Hallucinations: An Exploration

The Axiom of Choice and Its Influence on LLM Hallucinations: An Exploration

The axiom of choice and its applications: An exploration of the '100 prisoners problem'

The Axiom of Choice (AC), a cardinal principle in set theory, postulates that for any assortment of disjoint non-empty sets, its possible to construct a new set by selecting one element from each set in the collection. Using choice functions, this idea suggests that every collection of nonempty sets can be associated with a choice function. In the mathematical landscape, ACs significance is accentuated by its extensive application in a myriad of mathematical deductions, marking it as a cornerstone among mathematical axioms. This study delves into the practical implications and applications of AC, employing rigorous analytical methods to investigate its vast and multifaceted influence on the broader mathematical domain. Key findings indicate that AC has facilitated the proof of various theorems, some of which, on the surface, appear unrelated. While its inception sparked considerable debates due to concerns over its intuitive validity, its impact on contemporary mathematics is profound. This research underscores ACs central role in advancing mathematical thought, highlighting its contributions to both foundational theories and intricate proofs.

Quantum Theory without the Axiom of Choice, and Lefschetz Quantum Physics

In this paper, we discuss quantum formalisms that do not use the axiom of choice. We also consider the fundamental problem that addresses the (in)correctness of having the complex numbers as the base field for Hilbert spaces in the København interpretation of quantum theory, and propose a new approach to this problem (based on the Lefschetz principle). Rather than a theorem–proof paper, this paper describes two new research programs on the foundational level, and focuses on basic open questions that arise in these programs.