Changes

::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.

It has been shown that the Axiom of Choice is consistent with and independent of the other axioms of set theory. Thus assuming the Axiom of Choice and assuming its negation each result in consistent set theories, differing from each other.

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.

== Use of the Axiom of Choice ==

== Use of the Axiom of Choice ==

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 ==