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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 Löb's theorem ： ウィキペディア英語版 Löb's theorem In mathematical logic, Löb's theorem states that in a theory with Peano arithmetic, for any formula ''P'', if it is provable that "if ''P'' is provable then ''P'' is true", then ''P'' is provable. I.e. :$\backslash mathrm\backslash \; PA\; \backslash vdash\; (Bew(\backslash \#\; P)\; \backslash rightarrow\; P)\backslash mathrm\backslash \; PA\; \backslash vdash\; P$ where ''Bew''(#''P'') means that the formula ''P'' with Gödel number #''P'' is provable (from the German "beweisbar"). Löb's theorem is named for Martin Hugo Löb, who formulated it in 1955. ==Löb's theorem in provability logic== Provability logic abstracts away from the details of encodings used in Gödel's incompleteness theorems by expressing the provability of $\backslash phi$ in the given system in the language of modal logic, by means of the modality . Then we can formalize Löb's theorem by the axiom :$\backslash Box(\backslash Box\; P\backslash rightarrow\; P)\backslash rightarrow\; \backslash Box\; P,$ known as axiom GL, for Gödel-Löb. This is sometimes formalised by means of an inference rule that infers :$P$ from :$\backslash Box\; P\backslash rightarrow\; P.$ The provability logic GL that results from taking the modal logic ''K4'' (or ''K'', since the axiom schema 4, $\backslash Box\; A\backslash rightarrow\backslash Box\backslash Box\; A$, then becomes redundant) and adding the above axiom GL is the most intensely investigated system in provability logic. 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「Löb's theorem」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.