|
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''the Cartesian product of a collection of non-empty sets is non-empty''. It states that for every indexed family of nonempty sets there exists an indexed family of elements such that for every . The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.〔.〕 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,〔Jech, 1977, p. 348''ff''; Martin-Löf 2008, p. 210. According to : : The status of the Axiom of Choice has become less controversial in recent years. To most mathematicians it seems quite plausible and it has so many important applications in practically all branches of mathematics that not to accept it would seem to be a wilful hobbling of the practicing mathematician.〕 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. ==Statement== A choice function is a function ''f'', defined on a collection ''X'' of nonempty sets, such that for every set ''A'' in ''X'', ''f''(''A'') is an element of ''A''. With this concept, the axiom can be stated: :For any set ''X'' of nonempty sets, there exists a choice function ''f'' defined on ''X''. Formally, this may be expressed as follows: : Thus the negation of the axiom of choice states, there exists a collection of nonempty sets that has no choice function. Each choice function on a collection ''X'' of nonempty sets is an element of the Cartesian product of the sets in ''X''. This is not the most general situation of a Cartesian product of a family of sets, where a given set can occur more than once as a factor; however, one can focus on elements of such a product that select the same element every time a given set appears as factor, and such elements correspond to an element of the Cartesian product of all ''distinct'' sets in the family. The axiom of choice asserts the existence of such elements; it is therefore equivalent to: :Given any family of nonempty sets, their Cartesian product is a nonempty set. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Axiom of choice」の詳細全文を読む スポンサード リンク
|