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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 Ordinal analysis ： ウィキペディア英語版 Ordinal analysis In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. The field was formed when Gerhard Gentzen in 1934 used cut elimination to prove, in modern terms, that the proof theoretic ordinal of Peano arithmetic is ε0. ==Definition== Ordinal analysis concerns true, effective (recursive) theories that can interpret a sufficient portion of arithmetic to make statements about ordinal notations. The proof theoretic ordinal of such a theory $T$ is the smallest recursive ordinal that the theory cannot prove is well founded — the supremum of all ordinals $\backslash alpha$ for which there exists a notation $o$ in Kleene's sense such that $T$ proves that $o$ is an ordinal notation. Equivalently, it is the supremum of all ordinals $\backslash alpha$ such that there exists a recursive relation $R$ on $\backslash omega$ (the set of natural numbers) that well-orders it with ordinal $\backslash alpha$ and such that $T$ proves transfinite induction of arithmetical statements for $R$. The existence of any recursive ordinal that the theory fails to prove is well ordered follows from the $\backslash Sigma^1\_1$ bounding theorem, as the set of natural numbers that an effective theory proves to be ordinal notations is a $\backslash Sigma^0\_1$ set (see Hyperarithmetical theory). Thus the proof theoretic ordinal of a theory will always be a countable ordinal less than the Church-Kleene ordinal $\backslash omega\_1^\{\backslash mathrm\{CK\}\}$. In practice, the proof theoretic ordinal of a theory is a good measure of the strength of a theory. If theories have the same proof theoretic ordinal they are often equiconsistent, and if one theory has a larger proof theoretic ordinal than another it can often prove the consistency of the second theory. 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「Ordinal analysis」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.