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Despite its usefulness, many mathematicians reject the Axiom of Choice. "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-theoretical 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>Willard, ''supra'', 1.17, p. 9.</ref>

Despite its usefulness, many mathematicians reject the Axiom of Choice. "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-theoretical 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>Willard, ''supra'', 1.17, p. 9.</ref>

The rejection of the Axiom of Choice reflects the preference that all [[mathematical proofs]] should be constructive. 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. Suppose we consider a set consisting of many pairs of shoes. The Axiom of Choice says that we can create a new set by selecting just the left shoe from each pair. It is obvious that we can do this for shoes, since left and right shoes are distinguishable. What if the pairs of objects in question are <i>not</i> distinguishable and it is not acceptable to say 'just pick any one of them'?

The rejection of the Axiom of Choice reflects a preference for constructive [[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, results provable with it may be verified with the other axioms.

 

Put another way, it may be objectionable to insist that a function exists based on the Axiom of Choice, and then to derive results based on the existence of that function, when in fact there may be no known way to construct such a function. This objection to use of the Axiom of Choice is analogous to the objection to the use of non-[[falsifiable]] claims in science.  Since the Axiom of Choice is independent of the other axioms, results provable with it may be verified with the other axioms.

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>

== Use of the Axiom of Choice ==

== Use of the Axiom of Choice ==

Although not publicized or widely recognized even among mathematicians, the highly publicized proof of [[Fermat's Last Theorem]] relies on the Axiom of Choice.<ref>http://jimcaprioli.blogspot.com/2005/04/fermats-last-theorem.html</ref>

The highly publicized proof of [[Fermat's Last Theorem]] relies on the Axiom of Choice. <ref>http://jimcaprioli.blogspot.com/2005/04/fermats-last-theorem.html</ref>