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The Axiom of Choice has many equivalent statements, such as the [[Tychonoff theorem]], the Well-Ordering Theorem<ref name="mathworld" />, the existence of [[cardinal number]]s and the existence of a basis for every vector space.<ref>http://marauder.millersville.edu/~bikenaga/linearalgebra/basis/basis.html</ref> In [[algebra]] it is common to use Zorn's Lemma (also equivalent to the Axiom of Choice<ref>http://mathworld.wolfram.com/ZornsLemma.html</ref>) to study [[ideal]]s in infinite [[Noetherian ring]]s.

The Axiom of Choice has many equivalent statements, such as the [[Tychonoff theorem]], the Well-Ordering Theorem<ref name="mathworld" />, the existence of [[cardinal number]]s and the existence of a basis for every vector space.<ref>http://marauder.millersville.edu/~bikenaga/linearalgebra/basis/basis.html</ref> In [[algebra]] it is common to use Zorn's Lemma (also equivalent to the Axiom of Choice<ref>http://mathworld.wolfram.com/ZornsLemma.html</ref>) to study [[ideal]]s in infinite [[Noetherian ring]]s.

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 show that physical spheres are not [[Lebesgue measurable]].<ref name="kuro5hin" />

Use of the Axiom of Choice leads 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 observe that the "pieces" constructed in the course of the paradox are not Lebesgue measure, so there is no way that this process could ever be carried out with a physical object.

The Axiom of Choice can also be used to prove the apparently absurd existence of subsets of the real line which do not have a well-defined Lebesgue measure.<ref>However, it is also possible to construct such a subset without using the Axiom of Choice. See [http://matwbn.icm.edu.pl/ksiazki/fm/fm138/fm13812.pdf "The Hahn-Banach theorem implies the existence of a non-Lebesgue measurable set"]</ref>

The Axiom of Choice can also be used to prove the apparently absurd existence of subsets of the real line which do not have a well-defined Lebesgue measure.<ref>However, it is also possible to construct such a subset without using the Axiom of Choice. See [http://matwbn.icm.edu.pl/ksiazki/fm/fm138/fm13812.pdf "The Hahn-Banach theorem implies the existence of a non-Lebesgue measurable set"]</ref>