In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to …

选择公理（Axiom of Choice，缩写AC）是数学中的一条集合论公理，以下用一个较简单的描述： 选择公理 设C为一个由非空集合所组成的集合。 那么，我们可以从每一个在C中的集合中，都 …

Jan 8, 2008 · As the debate concerning the Axiom of Choice rumbled on, it became apparent that the proofs of a number of significant mathematical theorems made essential use of it, thereby …

The axiom of choice is an axiom in set theory with wide-reaching and sometimes counterintuitive consequences. It states that for any collection of sets, one can construct a new set containing …

Axiom of choice, statement in the language of set theory that makes it possible to form sets by choosing an element simultaneously from each member of an infinite collection of sets even …

Since f is surjective, it follows from the Axiom of Choice that there exists a map g : Y → X such that g(y) ∈ f−1(y) ⊂ A for every y ∈ Y . This is the required injective map from Y to X.

The Axiom of Choice (AC) was formulated about a century ago, and it was controversial for a few of decades after that; it might be considered the last great controversy of mathematics.

The Axiom of Choice is a fairly natural statement: given a collection of non-empty sets, you can simultaneously pick one element from each one, but it does lead to some perhaps unintuitive …

Apr 27, 2025 · Although it seems intuitively obvious ("surely you can just pick an element?"), when it comes to infinite sets of sets this axiom leads to non-intuitive results, notably the …

Dec 4, 2017 · The axiom of choice does not contradict the other axioms of set theory (e.g. the system ZF) and cannot be logically deduced from them if they are non-contradictory. The …