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::<math>\forall x\neq\varnothing\;\exists S\;(\forall z\in x\;\exists_1w\in z\cap S).</math></ref> is an [[axiom]] of [[Zermelo-Fraenkel]] set theory holding that:<ref name="willard">Stephen Willard, "General Topology" 1.17, p. 9 (Dover 2004)</ref>

::<math>\forall x\neq\varnothing\;\exists S\;(\forall z\in x\;\exists_1w\in z\cap S).</math></ref> is a powerful [[axiom]] of [[Zermelo-Fraenkel]] set theory that is useful for "proving" otherwise unprovable problems, but which has also been used to "prove" statements that appear absurd.  The '''Axiom of Choice''' holds that:<ref name="willard">Stephen Willard, "General Topology" 1.17, p. 9 (Dover 2004)</ref>

::given any collection of sets, however large, we can pick one element from each set in the collection.

::given any collection of sets, however large, we can pick one element from each set in the collection.

::If you have a collection of sets C (which may potentially contain an uncountably large number of sets), then there exists a set H, called the choice set, which contains precisely one element from each (non-empty) set in C. H is called the "choice set" because you are essentially going through each set in C and choosing one element from it. One feature of the Axiom of Choice is that H is simply assumed to exist; there is no algorithm given which might tell you how to construct an example of H.

::If you have a collection of sets C (which may potentially contain an uncountably large number of sets), then there exists a set H, called the choice set, which contains precisely one element from each (non-empty) set in C. H is called the "choice set" because you are essentially going through each set in C and choosing one element from it. One feature of the Axiom of Choice is that H is simply assumed to exist; there is no algorithm given which might tell you how to construct an example of H.

It has been shown that the Axiom of Choice is consistent with and independent of the other axioms of set theory.  Thus assuming the Axiom of Choice and assuming its negation each result in consistent set theories, differing from eachother.

It has been shown that the Axiom of Choice is consistent with and independent of the other axioms of set theory.  Thus assuming the Axiom of Choice and assuming its negation each result in consistent set theories, differing from each other.

== Use of the Axiom of Choice ==

== Use of the Axiom of Choice ==