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section on instruction, formatting
and
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::<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>
::<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>
::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.
It has been shown that neither the Axiom of Choice nor its negation will contradict the other axioms of set theory. Thus assuming the Axiom of Choice and assuming its negation will each result in differingly consistent set theories. It is the choice of the mathematician whether to believe the axiom or believe its negation; many people choose to not take a position on this issue at all because the majority of mathematics can be proven without invoking the axiom altogether.
It has been shown that neither the Axiom of Choice nor its negation will contradict the other axioms of set theory. Thus assuming the Axiom of Choice and assuming its negation will each result in differingly consistent set theories. It is the choice of the mathematician whether to believe the axiom or believe its negation; many people choose to not take a position on this issue at all because the majority of mathematics can be proven without invoking the axiom altogether.
== Instruction ==
In many undergraduate mathematics curricula, the Axiom of Choice is introduced early as a tool for solving problems. It often goes under one of its other guises, such as [[Zorn's Lemma]] or the [[Well-Ordering Theorem]]. Students are taught it as a proof method, but often with little mention of the questionable nature of the axiom. Often, a full discussion of the [[paradox]]es that accompany the axiom is left to a higher (graduate) level special topics course in [[logic]], despite the fact that mathematicians in numerous fields may be tempted to use it and thus should be aware of its controversy. Undergraduate courses also neglect to adequately mention when the axiom is being invoked, giving a false sense that use of the axiom is incidental, rather than documenting its widespread nature in recent mathematical proofs.
== Use of the Axiom of Choice ==
== Use of the Axiom of Choice ==
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" />
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" />
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"]. It is worth noting that the Hahn-Banach theorem is another consequence of the Axiom of Choice.</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>
== Controversy ==
== Controversy ==
Despite its usefulness, many mathematicians reject the Axiom of Choice. Stephen Willard, author of the textbook ''"General Topology"'', wrote the following after describing the axiom of choice:
Despite its usefulness, many mathematicians reject the Axiom of Choice. Stephen Willard, author of the textbook ''General Topology'', wrote the following after describing the Axiom of Choice:
<blockquote>It is left to the reader to decide that these two statements both say the same thing. What they say is: given any collection of sets, however large, we can pick one element from each set in the collection. It bothers some people because it asserts the existence of a set (...) without giving enough information to determine that set uniquely (by applying a finite number of rules), and it is the ''only'' formal set-theoretic axiom which does this. For this reason it is customary to mention the axiom of choice whenever it is used. It need not be used if the number of sets is finite.<ref>[http://books.google.com/books?id=-o8xJQ7Ag2cC&printsec=frontcover#PPA9,M1 Page 9 of "General Topology"]</ref></blockquote>
<blockquote>It is left to the reader to decide that these two statements both say the same thing. What they say is: given any collection of sets, however large, we can pick one element from each set in the collection. It bothers some people because it asserts the existence of a set (...) without giving enough information to determine that set uniquely (by applying a finite number of rules), and it is the ''only'' formal set-theoretic axiom which does this. For this reason it is customary to mention the axiom of choice whenever it is used. It need not be used if the number of sets is finite.<ref>[http://books.google.com/books?id=-o8xJQ7Ag2cC&printsec=frontcover#PPA9,M1 Page 9 of "General Topology"]</ref></blockquote>
The rejection of the Axiom of Choice reflects a preference for constructive [[Proof|mathematical proofs]]. AC, by its very nature, is nonconstructive, since it merely asserts that a choice function exists, but does not give an explicit method for its construction. Since the Axiom of Choice is independent of the other axioms, assuming this axiom results in a more constrained system than otherwise. Thus results provable with it might not be provable or might not even be true, without it.
The rejection of the Axiom of Choice reflects a preference for [[constructive proof]]s. AC, by its very nature, is nonconstructive, since it merely asserts that a choice function exists, but does not give an explicit method for its construction. Since the Axiom of Choice is independent of the other axioms, assuming this axiom results in a more constrained system than otherwise. Thus results provable with it might not be provable or might not even be true, without it.
The formulation of a constructive Axiom of Choice is one of three major problems which challenge 21st century logicians.<ref>Edna Ernestine Kramer, "The Nature and Growth of Modern Mathematics", Princeton University
The formulation of a constructive Axiom of Choice is one of three major problems which challenge 21st century logicians.<ref>Edna Ernestine Kramer, "The Nature and Growth of Modern Mathematics", Princeton University