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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 Disjunction and existence properties ： ウィキペディア英語版 Disjunction and existence properties In mathematical logic, the disjunction and existence properties are the "hallmarks" of constructive theories such as Heyting arithmetic and constructive set theories (Rathjen 2005). The disjunction property is satisfied by a theory if, whenever a sentence ''A ∨ B'' is a theorem, then either ''A'' is a theorem, or ''B'' is a theorem. The existence property or witness property is satisfied by a theory if, whenever a sentence is a theorem, where ''A''(''x'') has no other free variables, then there is some term ''t'' such that the theory proves . == Related properties == Rathjen (2005) lists five properties that a theory may possess. These include the disjunction property (DP), the existence property (EP), and three additional properties: * The numerical existence property (NEP) states that if the theory proves $(\backslash exists\; x\; \backslash in\; \backslash mathbb)\backslash varphi(x)$, where φ has no other free variables, then the theory proves $\backslash varphi(\backslash bar)$ for some $n\; \backslash in\; \backslash mathbb$. Here $\backslash bar$ is a term in $T$ representing the number ''n''. * Church's rule (CR) states that if the theory proves $(\backslash forall\; x\; \backslash in\; \backslash mathbb)(\backslash exists\; y\; \backslash in\; \backslash mathbb)\backslash varphi(x,y)$ then there is a natural number ''e'' such that, letting $f\_e$ be the computable function with index ''e'', the theory proves $(\backslash forall\; x)\backslash varphi(x,f\_e(x))$. * A variant of Church's rule, CR1, states that if the theory proves $(\backslash exists\; f\; \backslash colon\; \backslash mathbb\backslash to\backslash mathbb)\; \backslash psi(f)$ then there is a natural number ''e'' such that the theory proves $f\_e$ is total and proves $\backslash psi(f\_e)$. These properties can only be directly expressed for theories that have the ability to quantify over natural numbers and, for CR1, quantify over functions from $\backslash mathbb$ to $\backslash mathbb$. In practice, one may say that a theory has one of these properties if a definitional extension of the theory has the property stated above (Rathjen 2005). 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「Disjunction and existence properties」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.