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and

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 unproven 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>

::<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 unproven 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>

::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.