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In mathematics, a set ''A'' is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset ''B'' of ''A'' is equinumerous to ''A''. Explicitly, this means that there is a bijective function from ''A'' onto some proper subset ''B'' of ''A''. A set is Dedekind-finite if it is not Dedekind-infinite. Proposed by Richard Dedekind in 1888, Dedekind-infiniteness was the first definition of "infinite" that did not rely on the definition of the natural numbers. Until the foundational crisis of mathematics showed the need for a more careful treatment of set theory most mathematicians assumed that a set is infinite if and only if it is Dedekind-infinite. In the early twentieth century Zermelo–Fraenkel set theory (ZF), today the most commonly used form of axiomatic set theory, was proposed as an axiomatic system to formulate a theory of sets without the paradoxes of naive set theory such as Russell's paradox. Using the axioms of ZF set theory with the originally highly controversial axiom of choice included (ZFC) one can show that a set is Dedekind-finite if and only if it is finite in the sense of having a finite number of elements. However, there exists a model of ZF in which there exists an infinite, Dedekind-finite set, showing that the axioms of ZF are not strong enough to prove that every set that is Dedekind-finite has a finite number of elements.〔 There are other definitions of finiteness and infiniteness of sets that do not depend on the axiom of choice. A vaguely related notion is that of a Dedekind-finite ring. A ring is said to be a Dedekind-finite ring if ''ab''=1 implies ''ba''=1 for any two ring elements ''a'' and ''b''. These rings have also been called directly finite rings. ==Comparison with the usual definition of infinite set== This definition of "infinite set" should be compared with the usual definition: a set ''A'' is infinite when it cannot be put in bijection with a finite ordinal, namely a set of the form for some natural number ''n'' – an infinite set is one that is literally "not finite", in the sense of bijection. During the latter half of the 19th century, most mathematicians simply assumed that a set is infinite if and only if it is Dedekind-infinite. However, this equivalence cannot be proved with the axioms of Zermelo–Fraenkel set theory without the axiom of choice (AC) (usually denoted "ZF"). The full strength of AC is not needed to prove the equivalence; in fact, the equivalence of the two definitions is strictly weaker than the axiom of countable choice (CC). (See the references below.) 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「Dedekind-infinite set」の詳細全文を読む スポンサード リンク
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