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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 Kleene's O ： ウィキペディア英語版 Kleene's O In set theory and computability theory, Kleene's $\backslash mathcal$ is a canonical subset of the natural numbers when regarded as ordinal notations. It contains ordinal notations for every recursive ordinal, that is, ordinals below Church–Kleene ordinal, $\backslash omega\_1^$. Since $\backslash omega\_1^$ is the first ordinal not representable in a computable system of ordinal notations the elements of $\backslash mathcal$ can be regarded as the canonical ordinal notations. Kleene (1938) described a system of notation for all recursive ordinals (those less than the Church–Kleene ordinal). It uses a subset of the natural numbers instead of finite strings of symbols. Unfortunately, there is in general no effective way to tell whether some natural number represents an ordinal, or whether two numbers represent the same ordinal. However, one can effectively find notations which represent the ordinal sum, product, and power (see ordinal arithmetic) of any two given notations in Kleene's $\backslash mathcal$; and given any notation for an ordinal, there is a recursively enumerable set of notations which contains one element for each smaller ordinal and is effectively ordered. == Kleene's $\backslash mathcal$ == The basic idea of Kleene's system of ordinal notations is to build up ordinals in an effective manner. For members $p$ of $\backslash mathcal$, the ordinal for which $p$ is a notation is $|\; p\; |$. The standard definition proceeds via transfinite induction and the ordering ) is defined simultaneously. * The natural number 0 belongs to Kleene's $\backslash mathcal$ and $|\; 0\; |\; =\; 0$. * If $i$ belongs to Kleene's $\backslash mathcal$ and $|i|\; =\; \backslash alpha$, then $2^i$ belongs to Kleene's $\backslash mathcal$ and $|2^i|=\; \backslash alpha\; +\; 1$ and . * Suppose $\backslash$ is the $e$-th partial recursive function. If $e$ is total, with range contained in $\backslash mathcal$, and for every natural number $n$, we have , then $3\backslash cdot\; 5^e$ belongs to Kleene's $\backslash mathcal$, for each $n$ and $|\; 3\; \backslash cdot\; 5^e\; |\; =\; \backslash lim\_k\; |\backslash \; (k)\; |$, i.e. $3\; \backslash cdot\; 5^e$ is a notation for the limit of the ordinals $\backslash gamma\_k$ where $|\; \backslash (k)|\; =\; \backslash gamma\_k$ for every natural number $k$. * and imply (this guarantees that is transitive.) This definition has the advantages that one can recursively enumerate the predecessors of a given ordinal (though not in the ordering) and that the notations are downward closed, i.e., if there is a notation for $\backslash gamma$ and then there is a notation for $\backslash alpha$. 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「Kleene's O」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.