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::If you have a collection of sets C (which may potentially contain an uncountably large number of sets), then there exists a set H, called the choice set, which contains precisely one element from each (non-empty) set in C. H is called the "choice set" because you are essentially going through each set in C and choosing one element from it. One feature of the Axiom of Choice is that H is simply assumed to exist; there is no algorithm given which might tell you how to construct an example of H.

::If you have a collection of sets C (which may potentially contain an uncountably large number of sets), then there exists a set H, called the choice set, which contains precisely one element from each (non-empty) set in C. H is called the "choice set" because you are essentially going through each set in C and choosing one element from it. One feature of the Axiom of Choice is that H is simply assumed to exist; there is no algorithm given which might tell you how to construct an example of H.

The Axiom of Choice has many equivalent statements, such as the [[Tychonoff theorem]], the [[Well-Ordering Theorem]], the existence of [[cardinal number]]s, the existence of a basis for every vector space, and the existence of subsets of the real line which do not have a well-defined [[Lebesgue measure]]. In [[algebra]] it is common to use [[Zorn's Lemma]] (also equivalent to the Axiom of Choice) 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]], the existence of [[cardinal number]]s, the existence of a basis for every vector space, and the existence of subsets of the real line which do not have a well-defined [[Lebesgue measure]]. In [[algebra]] it is common to use [[Zorn's Lemma]] (also equivalent to the Axiom of Choice) to study [[ideal]]s in infinite [[Noetherian ring]]s.

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>

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>http://www.kuro5hin.org/story/2003/5/23/134430/275</ref>

 

The highly publicized proof of [[Fermat's Last Theorem]] relies on the Axiom of Choice.<ref>http://www.chronon.org/articles/fermat_undecidable.html</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 show that physical spheres are not [[Lebesgue measurable]].<ref>http://www.kuro5hin.org/story/2003/5/23/134430/275</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>

== Use of the Axiom of Choice ==

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 highly publicized proof of [[Fermat's Last Theorem]] relies on the Axiom of Choice. {{fact}}

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

Press, 1982. p. 595.</ref>

== Sources ==

== Sources ==