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== Controversy ==

== Controversy ==

 

Despite its usefulness, many mathematicians prefer to avoid the Axiom of Choice when possible. 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]]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 avoidance 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