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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 Extension by definitions ： ウィキペディア英語版 Extension by definitions In mathematical logic, more specifically in the proof theory of first-order theories, extensions by definitions formalize the introduction of new symbols by means of a definition. For example, it is common in naive set theory to introduce a symbol $\backslash emptyset$ for the set which has no member. In the formal setting of first-order theories, this can be done by adding to the theory a new constant $\backslash emptyset$ and the new axiom $\backslash forall\; x(x\backslash notin\backslash emptyset)$, meaning 'for all ''x'', ''x'' is not a member of $\backslash emptyset$'. It can then be proved that doing so adds essentially nothing to the old theory, as should be expected from a definition. More precisely, the new theory is a conservative extension of the old one. ==Definition of relation symbols== ''Let'' $T$ be a first-order theory and $\backslash phi(x\_1,\backslash dots,x\_n)$ a formula of $T$ such that $x\_1$, ..., $x\_n$ are distinct and include the variables free in $\backslash phi(x\_1,\backslash dots,x\_n)$. Form a new first-order theory $T\text{'}\backslash ,$ from $T$ by adding a new $n$-ary relation symbol $R$, the logical axioms featuring the symbol $R$ and the new axiom :$\backslash forall\; x\_1\backslash dots\backslash forall\; x\_n(R(x\_1,\backslash dots,x\_n)\backslash leftrightarrow\backslash phi(x\_1,\backslash dots,x\_n))$, called the ''defining axiom'' of $R$. If $\backslash psi$ is a formula of $T\text{'}\backslash ,$, let $\backslash psi^\backslash ast$ be the formula of $T$ obtained from $\backslash psi$ by replacing any occurrence of $R(t\_1,\backslash dots,t\_n)$ by $\backslash phi(t\_1,\backslash dots,t\_n)$ (changing the bound variables in $\backslash phi$ if necessary so that the variables occurring in the $t\_i$ are not bound in $\backslash phi(t\_1,\backslash dots,t\_n)$). Then the following hold: # $\backslash psi\backslash leftrightarrow\backslash psi^\backslash ast$ is provable in $T\text{'}\backslash ,$, and # $T\text{'}\backslash ,$ is a conservative extension of $T$. The fact that $T\text{'}\backslash ,$ is a conservative extension of $T$ shows that the defining axiom of $R$ cannot be used to prove new theorems. The formula $\backslash psi^\backslash ast$ is called a ''translation'' of $\backslash psi$ into $T$. Semantically, the formula $\backslash psi^\backslash ast$ has the same meaning as $\backslash psi$, but the defined symbol $R$ has been eliminated. 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「Extension by definitions」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.