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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 beth number ： ウィキペディア英語版 beth number In mathematics, the infinite cardinal numbers are represented by the Hebrew letter $\backslash aleph$ (aleph) indexed with a subscript that runs over the ordinal numbers (see aleph number). The second Hebrew letter $\backslash beth$ (beth) is used in a related way, but does not necessarily index all of the numbers indexed by $\backslash aleph$. == Definition == To define the beth numbers, start by letting :$\backslash beth\_0=\backslash aleph\_0$ be the cardinality of any countably infinite set; for concreteness, take the set $\backslash mathbb$ of natural numbers to be a typical case. Denote by ''P''(''A'') the power set of ''A''; i.e., the set of all subsets of ''A''. Then define :$\backslash beth\_=2^$ is the cardinality of ''A''. Given this definition, :$\backslash beth\_0,\backslash \; \backslash beth\_1,\backslash \; \backslash beth\_2,\backslash \; \backslash beth\_3,\backslash \; \backslash dots$ are respectively the cardinalities of :$\backslash mathbb,\backslash \; P(\backslash mathbb),\backslash \; P(P(\backslash mathbb)),\backslash \; P(P(P(\backslash mathbb))),\backslash \; \backslash dots.$ so that the second beth number $\backslash beth\_1$ is equal to $\backslash mathfrak\; c$, the cardinality of the continuum, and the third beth number $\backslash beth\_2$ is the cardinality of the power set of the continuum. Because of Cantor's theorem each set in the preceding sequence has cardinality strictly greater than the one preceding it. For infinite limit ordinals λ the corresponding beth number is defined as the supremum of the beth numbers for all ordinals strictly smaller than λ: :$\backslash beth\_=\backslash sup\backslash .$ One can also show that the von Neumann universes $V\_\; \backslash !$ have cardinality $\backslash beth\_\; \backslash !$. 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「beth number」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.