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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 Markov's principle ： ウィキペディア英語版 Markov's principle Markov's principle, named after Andrey Markov Jr, is a specific statement in computability theory that is obvious true classically (i.e. it is a tautology), but must be proved when using constructive mathematics. There are many equivalent formulations of Markov's principle. == Statements of the principle == In the language of computability theory, Markov's principle is a formal expression of the claim that if it is impossible that an algorithm does not terminate, then it does terminate. This is equivalent to the claim that if a set and its complement are both computably enumerable, then the set is decidable. In predicate logic, if ''P'' is a predicate over the natural numbers, it is expressed as: :$(\backslash forall\; n\; (P(n)\; \backslash vee\; \backslash neg\; P(n))\; \backslash wedge\; (\backslash neg\; \backslash forall\; n\; \backslash neg\; P(n)))\; \backslash rightarrow\; (\backslash exists\; n\backslash ;P(n)).$ That is, if ''P'' is decidable, and it cannot be false for every natural number ''n'', then it is true for some ''n''. (In general, a predicate ''P'' over some domain is called ''decidable'' if for every ''x'' in the domain, either ''P''(''x'') is true, or ''P''(''x'') is not true, which is not always the case constructively.) It is equivalent in the language of arithmetic to: :$\backslash neg\; \backslash neg\; \backslash exists\; n\backslash ;f(n)=0\; \backslash rightarrow\; \backslash exists\; n\backslash ;f(n)=0,$ for $f$ a total recursive function on the natural numbers. It is equivalent, in the language of real analysis, to the following principles: * For each real number ''x'', if it is contradictory that ''x'' is equal to 0, then there exists ''y'' ∈ Q such that 0 < ''y'' < |''x''|, often expressed by saying that ''x'' is apart from, or constructively unequal to, 0. * For each real number ''x'', if it is contradictory that ''x'' is equal to 0, then there exists ''y'' ∈ R such that ''xy'' = 1. 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「Markov's principle」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.