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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 Conservativity theorem ： ウィキペディア英語版 Conservativity theorem In mathematical logic, the conservativity theorem states the following: Suppose that a ''closed'' formula :$\backslash exists\; x\_1\backslash ldots\backslash exists\; x\_m\backslash ,\backslash varphi(x\_1,\backslash ldots,x\_m)$ is a theorem of a first-order theory $T$. Let $T\_1$ be a theory obtained from $T$ by extending its language with new constants :$a\_1,\backslash ldots,a\_m$ and adding a new axiom :$\backslash varphi(a\_1,\backslash ldots,a\_m)$. Then $T\_1$ is a conservative extension of $T$, which means that the theory $T\_1$ has the same set of theorems in the original language (i.e., without constants $a\_i\backslash ,\backslash !$) as the theory $T$. In a more general setting, the conservativity theorem is formulated for extensions of a first-order theory by introducing a new functional symbol: :Suppose that a ''closed'' formula $\backslash forall\; \backslash vec\backslash ,\backslash exists\; x\backslash ,\backslash !\backslash ,\backslash varphi(x,\backslash vec)$ is a theorem of a first-order theory $T$, where we denote $\backslash vec:=(y\_1,\backslash ldots,y\_n)$. Let $T\_1$ be a theory obtained from $T$ by extending its language with new functional symbol $f\backslash ,\backslash !$ (of arity $n$) and adding a new axiom $\backslash forall\; \backslash vec\backslash ,\backslash varphi(f(\backslash vec),\backslash vec)$. Then $T\_1$ is a conservative extension of $T$, i.e. the theories $T$ and $T\_1$ prove the same theorems not involving the functional symbol $f\backslash ,\backslash !$). ==References== * Elliott Mendelson (1997). ''Introduction to Mathematical Logic'' (4th ed.) Chapman & Hall. * J.R. Shoenfield (1967). ''Mathematical Logic''. Addison-Wesley Publishing Company. 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「Conservativity 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.