December 25th, 2013
(written by lawrence krubner, however indented passages are often quotes). You can contact lawrence at: email@example.com
Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin. In many cases such a selection can be made without invoking the axiom of choice; this is in particular the case if the number of bins is finite, or if a selection rule is available: a distinguishing property that happens to hold for exactly one object in each bin. To give an informal example, for any (even infinite) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection, but for an infinite collection of pairs of socks (assumed to have no distinguishing features), such a selection can be obtained only by invoking the axiom of choice.
Although originally controversial, the axiom of choice is now used without reservation by most mathematicians, and it is included in Zermelo–Fraenkel set theory with the axiom of choice (ZFC), the standard form of axiomatic set theory. One motivation for this use is that a number of generally accepted mathematical results, such as Tychonoff’s theorem, require the axiom of choice for their proofs. Contemporary set theorists also study axioms that are not compatible with the axiom of choice, such as the axiom of determinacy. The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced.