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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク axiom of infinity ： ウィキペディア英語版 axiom of infinity In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908.〔Zermelo: ''Untersuchungen über die Grundlagen der Mengenlehre'', 1907, in: Mathematische Annalen 65 (1908), 261-281; Axiom des Unendlichen p. 266f.〕 == Formal statement == In the formal language of the Zermelo–Fraenkel axioms, the axiom reads: :$\backslash exists\; \backslash mathbf\; \backslash ,\; (\; \backslash empty\; \backslash in\; \backslash mathbf\; \backslash ,\; \backslash and\; \backslash ,\; \backslash forall\; x\; \backslash in\; \backslash mathbf\; \backslash ,\; (\; \backslash ,\; (\; x\; \backslash cup\; \backslash \; )\; \backslash in\; \backslash mathbf\; )\; )\; .$ In words, there is a set I (the set which is postulated to be infinite), such that the empty set is in I and such that whenever any ''x'' is a member of I, the set formed by taking the union of ''x'' with its singleton is also a member of I. Such a set is sometimes called an inductive 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 infinity」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.