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== Controversy ==

== Controversy ==

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 name="willard">

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 name="willard" />

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, results provable with it may be verified with the other axioms. Any proof which uses the axiom of choice can be transformed into a proof that doesn't. Usually the resulting proof will be far more complicated, but the transformation is always possible, with proven reliability. <ref>http://web.unicam.it/matinf/aila/Scuola%20AILA/sulla%20varieta%20dei%20metodiAC_sem02.pdf</ref>  

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, results provable with it may be verified with the other axioms. Any proof which uses the axiom of choice can be transformed into a proof that doesn't. Usually the resulting proof will be far more complicated, but the transformation is always possible, with proven reliability. <ref>http://web.unicam.it/matinf/aila/Scuola%20AILA/sulla%20varieta%20dei%20metodiAC_sem02.pdf.gz</ref>  

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