Axiom of choice

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In mathematics, the axiom of choice, or AC, is an axiom of set theory. 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, even if there are infinitely many bins and there is no "rule" for which object to pick from each. The axiom of choice is not required if the number of bins is finite or if such a selection "rule" is available. For example, given an infinite collection of pairs of socks, one needs AC to pick one sock out of each pair; but given an infinite collection of pairs of shoes, one shoe out of each pair can be specified even without AC, by choosing the left one.

The axiom of choice was formulated in 1904 by Ernst Zermelo.[1] Although originally controversial, it is now used without reservation by most mathematicians.[2] One motivation for this use is that a number of important 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. Unlike the axiom of choice, these alternatives are not ordinarily proposed as axioms for mathematics, but only as principles in set theory with interesting consequences.

Contents

Statement

A choice function is a function f, defined on a collection X of nonempty sets, such that for every set s in X, f(s) is an element of s. With this concept, the axiom can be stated:

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