# THE AXIOM OF CHOICE

@inproceedings{Russell2003THEAO, title={THE AXIOM OF CHOICE}, author={Bertrand Russell}, year={2003} }

We propose that failures of the axiom of choice, that is, surjective functions admitting no sections, can be reasonably classified by means of invariants borrowed from algebraic topology. We show that cohomology, when defined so that its usual exactness properties hold even in the absence of the axiom of choice, is adequate for detecting failures of this axiom in the following sense. If a set X, viewed as a discrete space, has trivial first cohomology for all coefficient groups, then every J…

## 262 Citations

Notions of compactness for special subsets of ℝ I and some weak forms of the axiom of choice

- Mathematics, Computer ScienceThe Journal of Symbolic Logic
- 2010

It is shown that the countable axiom of choice of choice for finite sets implies that F is compact, and the weak compactness of the closed unit ball of the Hilbert space ℓ 2 (I) is (Loeb-)compact in the weak topology.

The Axiom of Choice and Related Topics

- 2020

In this paper I will be discussing the Axiom of Choice and its equivalent statements. The Axiom of Choice is an axiom of Zermelo-Fraenkel set theory that states that given a collection of non-empty…

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

- Mathematics, Computer ScienceJ. Log. Anal.
- 2017

In the following, we prove by extending methods of Rhineghost, that both the Arzel\`a--Ascoli theorem and the Fr\'echet--Kolmogorov theorem are equivalent to the axiom of countable choice for real…

Weak axioms of choice for metric spaces

- Mathematics
- 2005

In the framework of ZF, i.e., Zermelo-Fraenkel set theory without the axiom of choice AC, we show that if the family of all non-empty, closed subsets of a metric space (X, d) has a choice function,…

Every set has at least three choice functions

- Mathematics
- 2009

This paper continues the study of the Axiom of Choice by E. Z e r m e l o [Neuer Beweis für die Möglichkeit einer Wohlordung, Math. Annalen, 65 (1908), 107–128; translated in van Heijenoort 1967,…

Partition models, Permutations of infinite sets without fixed points, Variants of CAC, and weak forms of AC

- Mathematics
- 2021

We study new relations of the following statements with weak choice principles in ZF (ZermeloFraenkel set theory without the Axiom of Choice (AC)) and ZFA (ZF with the axiom of extensionality…

On the Leibniz–Mycielski axiom in set theory

- Mathematics
- 2004

Motivated by Leibniz’s thesis on the identity of indiscernibles, Mycielski introduced a set-theoretic axiom, here dubbed the Leibniz–Mycielski axiom LM, which asserts that for each pair of distinct…

Products of some special compact spaces and restricted forms of AC

- Mathematics, Computer ScienceThe Journal of Symbolic Logic
- 2010

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: The Tychonoff product of many non-empty finite discrete subsets of I is compact.

Iterated failures of choice

- Mathematics
- 2019

We make an important observation regarding iterated symmetric extensions which allows us to easily iterate certain counterexamples to the axiom of choice. We use this to improve previous results and…

On Sequentially Compact Subspaces of ℝ without the Axiom of Choice

- Mathematics, Computer ScienceNotre Dame J. Formal Log.
- 2003

It is shown that forms 152 and 214 of Howard and Rubin are equivalent and that every non-well-orderable set is the union of a pairwise disjoint well- orderable family of denumerable sets in ZF.

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