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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 Cardinality of the continuum ： ウィキペディア英語版 Cardinality of the continuum In set theory, the cardinality of the continuum is the cardinality or “size” of the set of real numbers $\backslash mathbb\; R$, sometimes called the continuum. It is an infinite cardinal number and is denoted by $|\backslash mathbb\; R|$ or $\backslash mathfrak\; c$ (a lowercase fraktur script "c"). The real numbers $\backslash mathbb\; R$ are more numerous than the natural numbers $\backslash mathbb\; N$. Moreover, $\backslash mathbb\; R$ has the same number of elements as the power set of $\backslash mathbb\; N$. Symbolically, if the cardinality of $\backslash mathbb\; N$ is denoted as $\backslash aleph\_0$, the cardinality of the continuum is :$\backslash mathfrak\; c\; =\; 2^\; >\; \backslash aleph\_0\; \backslash ,.$ This was proven by Georg Cantor in his 1874 uncountability proof, part of his groundbreaking study of different infinities, and later more simply in his diagonal argument. Cantor defined cardinality in terms of bijective functions: two sets have the same cardinality if and only if there exists a bijective function between them. Between any two real numbers ''a'' < ''b'', no matter how close they are to each other, there are always infinitely many other real numbers, and Cantor showed that they are as many as those contained in the whole set of real numbers. In other words, the open interval (''a'',''b'') is equinumerous with $\backslash mathbb\; R.$ This is also true for several other infinite sets, such as any ''n''-dimensional Euclidean space $\backslash mathbb\; R^n$ (see space filling curve). That is, : $|(a,b)|\; =\; |\backslash mathbb\; R|\; =\; |\backslash mathbb\; R^n|.$ The smallest infinite cardinal number is $\backslash aleph\_0$ (aleph-naught). The second smallest is $\backslash aleph\_1$ (aleph-one). The continuum hypothesis, which asserts that there are no sets whose cardinality is strictly between $\backslash aleph\_0$ and $\backslash mathfrak\; c$, implies that $\backslash mathfrak\; c\; =\; \backslash aleph\_1$. ==Properties== 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「Cardinality 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.