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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 Continuum (set theory) ： ウィキペディア英語版 Continuum (set theory) In the mathematical field of set theory, the continuum means the real numbers, or the corresponding (infinite) cardinal number, $\backslash mathfrak$. Georg Cantor proved that the cardinality $\backslash mathfrak$ is larger than the smallest infinity, namely, $\backslash aleph\_0$. He also proved that $\backslash mathfrak$ equals $2^$, the cardinality of the power set of the natural numbers. The ''cardinality of the continuum'' is the size of the set of real numbers. The continuum hypothesis is sometimes stated by saying that no cardinality lies between that of the continuum and that of the natural numbers, $\backslash aleph\_0$. == Linear continuum == (詳細はRaymond Wilder (1965) there are four axioms that make a set ''C'' and the relation < into a linear continuum: * ''C'' is simply ordered with respect to <. * If () is a cut of ''C'', then either ''A'' has a last element or ''B'' has a first element. (compare Dedekind cut) * There exists a non-empty, countable subset ''S'' of ''C'' such that, if ''x,y'' ∈ ''C'' such that ''x'' < ''y'', then there exists ''z'' ∈ ''S'' such that ''x'' < ''z'' < ''y''. (separability axiom) * ''C'' has no first element and no last element. (Unboundedness axiom) These axioms characterize the order type of the real number line. == See also == * Suslin's problem 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「Continuum (set theory)」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.