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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 Lévy hierarchy ： ウィキペディア英語版 Lévy hierarchy In set theory and mathematical logic, the Lévy hierarchy, introduced by Azriel Lévy in 1965, is a hierarchy of formulas in the formal language of the Zermelo–Fraenkel set theory, which is typically called just the language of set theory. This is analogous to the arithmetical hierarchy which provides the classifications but for sentences of the language of arithmetic. ==Definitions== In the language of set theory, atomic formulas are of the form x = y or x ∈ y, standing for equality and respectively set membership predicates. The first level of the Levy hierarchy is defined as containing only formulas with no unbounded quantifiers, and is denoted by $\backslash Delta\; \_0=\backslash Sigma\_0=\backslash Pi\_0$.〔Walicki, Michal (2012). ''Mathematical Logic'', p. 225. World Scientific Publishing Co. Pte. Ltd. ISBN 9789814343862〕 The next levels are given by finding an equivalent formula in Prenex normal form, and counting the number of changes of quantifiers: In the theory ZFC, a formula $A$ is called:〔 $\backslash Sigma\; \_$ if $A$ is equivalent to $\backslash exists\; x\; \_1\; ...\; \backslash exists\; x\; \_n\; B$ in ZFC, where $B$ is $\backslash Pi\; \_i$ $\backslash Pi\; \_$ if $A$ is equivalent to $\backslash forall\; x\; \_1\; ...\; \backslash forall\; x\; \_n\; B$ in ZFC, where $B$ is $\backslash Sigma\; \_i$ If a formula is both $\backslash Sigma\; \_i$ and $\backslash Pi\; \_i$, it is called $\backslash Delta\; \_i$. As a formula might have several different equivalent formulas in Prenex normal form, it might belong to several different levels of the hierarchy. In this case, the lowest possible level is the level of the formula. The Lévy hierarchy is sometimes defined for other theories ''S''. In this case $\backslash Sigma\; \_i$ and $\backslash Pi\; \_i$ by themselves refer only to formulas that start with a sequence of quantifiers with at most ''i''−1 alternations, and $\backslash Sigma\; \_i^S$ and $\backslash Pi\; \_i^S$ refer to formulas equivalent to $\backslash Sigma\; \_i$ and $\backslash Pi\; \_i$ formulas in the theory ''S''. So strictly speaking the levels $\backslash Sigma\; \_i$ and $\backslash Pi\; \_i$ of the Lévy hierarchy for ZFC defined above should be denoted by $\backslash Sigma^\; \_i$ and $\backslash Pi^\; \_i$. 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「Lévy hierarchy」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.