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The Axiom of Choice (AC)<ref>There are numerous equivalent mathematical descriptions of the Axiom of Choice.  Here are two:

The '''Axiom of Choice''' ('''AC''')<ref>There are numerous equivalent mathematical descriptions of the Axiom of Choice.  Here are two:

::<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>

::<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>

Use of the Axiom of Choice has led to some seemingly absurd results.  In the [[Banach-Tarski Paradox]], the Axiom of Choice is used to prove that a solid sphere of infinitely divisible parts may be chopped up and reconstructed as two new spheres of identical size, thereby creating 2 out of only 1.  This paradox is proven only through use of the Axiom of Choice, and the authors of this proof did so to criticize this Axiom.  One attempt to resolve this apparent contradiction is to say that physical spheres are not [[Lebesgue measurable]].<ref>http://www.kuro5hin.org/story/2003/5/23/134430/275</ref>

Use of the Axiom of Choice has led to some seemingly absurd results.  In the [[Banach-Tarski Paradox]], the Axiom of Choice is used to prove that a solid sphere of infinitely divisible parts may be chopped up and reconstructed as two new spheres of identical size, thereby creating 2 out of only 1.  This paradox is proven only through use of the Axiom of Choice, and the authors of this proof did so to criticize this Axiom.  One attempt to resolve this apparent contradiction is to say that physical spheres are not [[Lebesgue measurable]].<ref>http://www.kuro5hin.org/story/2003/5/23/134430/275</ref>

The formulation of a constructive Axiom of Choice is one of three major problems which challenge 21st century logicians. {{fact}}

The formulation of a constructive Axiom of Choice is one of three major problems which challenge 21st century logicians. {{fact-math}}

== Use of the Axiom of Choice ==

== Use of the Axiom of Choice ==