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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 Cardinal characteristic of the continuum ： ウィキペディア英語版 Cardinal characteristic of the continuum In the mathematical discipline of set theory, a cardinal characteristic of the continuum is an infinite cardinal number that may consistently lie strictly between $\backslash aleph\_0$ (the cardinality of the set of natural numbers), and the cardinality of the continuum, that is, the cardinality of the set $\backslash mathbb\; R$ of all real numbers. The latter cardinal is denoted $2^$ or $\backslash mathfrak\; c$. A variety of such cardinal characteristics arise naturally, and much work has been done in determining what relations between them are provable, and constructing models of set theory for various consistent configurations of them. == Background == Cantor's diagonal argument shows that $\backslash mathfrak\; c$ is strictly greater than $\backslash aleph\_0$, but it does not specify whether it is the ''least'' cardinal greater than $\backslash aleph\_0$ (that is, $\backslash aleph\_1$). Indeed the assumption that $\backslash mathfrak\; c=\backslash aleph\_1$ is the well-known Continuum Hypothesis, which was shown to be independent of the standard ZFC axioms for set theory by Paul Cohen. If the Continuum Hypothesis fails and so $\backslash mathfrak\; c$ is at least $\backslash aleph\_2$, natural questions arise about the cardinals strictly between $\backslash aleph\_0$ and $\backslash mathfrak\; c$, for example regarding Lebesgue measurability. By considering the least cardinal with some property, one may get a definition for an uncountable cardinal that is consistently less than $\backslash mathfrak\; c$. Generally one only considers definitions for cardinals that are provably greater than $\backslash aleph\_0$ and at most $\backslash mathfrak\; c$ as cardinal characteristics of the continuum, so if the Continuum Hypothesis holds they are all equal to $\backslash aleph\_1$. 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「Cardinal characteristic of the continuum」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.