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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{{fact}}. 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{{fact}}. 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" /> However, this is another seemingly absurd consequence of the Axiom of Choice, that there are subsets of the real line which do not have a well-defined Lebesgue measure.

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" />

 

Another seemingly absurd consequence of the Axiom of Choice is that there are subsets of the real line which do not have a well-defined Lebesgue measure. However, it is also possible to construct such a subset without using the Axiom of Choice.<ref>[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 highly publicized proof of [[Fermat's Last Theorem]] relies on the Axiom of Choice, which was one of the points of criticism by [[Marilyn vos Savant]] in her column and book.<ref>Ask Marilyn ® by Marilyn vos Savant, Parade Magazine. November 21, 1993</ref><ref>''The World's Most Famous Math Problem: The Proof of Fermat's Last Theorem and Other Mathematical Mysteries'', Marilyn vos Savant. St. Martin's Griffin, 1993</ref>

The highly publicized proof of [[Fermat's Last Theorem]] relies on the Axiom of Choice, which was one of the points of criticism by [[Marilyn vos Savant]] in her column and book.<ref>Ask Marilyn ® by Marilyn vos Savant, Parade Magazine. November 21, 1993</ref><ref>''The World's Most Famous Math Problem: The Proof of Fermat's Last Theorem and Other Mathematical Mysteries'', Marilyn vos Savant. St. Martin's Griffin, 1993</ref>