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Line 1: − + − ::<math>\forall x\;(\forall y\;y\in x\Rightarrow\exists z\; z\in y)\;\exists S\;(\forall z\;z\in x\Rightarrow\exists w\;w\in z\;\wedge\;w\in S)\;\wedge\;(\forall v\;v\in z\;\wedge\;v\in S\Rightarrow v=w)</math> − and+ − ::<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.<ref>See, e.g., the [[Banach-Tarski Paradox]] discussed below.</ref> The Axiom of Choice holds that:<ref name="willard">Stephen Willard, "General Topology" 1.17, p. 9 (Dover 2004)</ref> − + − In layman's terms, the Axiom of Choice is "For every nonempty set there is a choice function." Or, "We can choose one element from every element of a nonempty set of disjoint sets and this process by which we choose these sets will set up a new set." More precisely, the Axiom of Choice states that:+ + + Line 14:
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trying desperately to make this article readable
The '''Axiom of Choice''' ('''AC''')<ref>There are numerous equivalent [[mathematical formula]]e which capture the idea of the Axiom of Choice. Here are two:
The '''Axiom of Choice''' ('''AC''') states that "For every nonempty set there is a choice function." It is a powerful but controversial idea frequently used by mathematicians who are attempting to prove otherwise unprovable problems, but which has also been used to prove statements that appear absurd. See, e.g., the [[Banach-Tarski Paradox]] discussed below.
See also [[Zermelo-Fraenkel]] [[set theory]].
::given any collection of sets, however large, we can pick one element from each set in the collection.
The Axiom of Choice holds that:<ref name="willard">Stephen Willard, "General Topology" 1.17, p. 9 (Dover 2004)</ref>
::given any collection of [[set]]s, however large, we can pick one element from each set in the collection.
More precisely, the Axiom of Choice states that:
::For every collection of nonempty sets S, there exists a function f such that f(S) is a member of S for every possible S.
::For every collection of nonempty sets S, there exists a function f such that f(S) is a member of S for every possible S.
::Given any set of mutually disjoint nonempty sets, there exists at least one set that contains exactly one element in common with each of the nonempty sets.
::Given any set of mutually disjoint nonempty sets, there exists at least one set that contains exactly one element in common with each of the nonempty sets.
Therefore, "We can choose one [[element]] from every element of a [[nonempty set]] of [[disjoint sets]] and this process by which we choose these sets will set up a new set."
Yet another helpful explanation of the Axiom of Choice is this:<ref name="kuro5hin">http://www.kuro5hin.org/story/2003/5/23/134430/275</ref>
Yet another helpful explanation of the Axiom of Choice is this:<ref name="kuro5hin">http://www.kuro5hin.org/story/2003/5/23/134430/275</ref>