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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 Creative and productive sets ： ウィキペディア英語版 Creative and productive sets In computability theory, productive sets and creative sets are types of sets of natural numbers that have important applications in mathematical logic. They are a standard topic in mathematical logic textbooks such as and . ==Definition and example== For the remainder of this article, assume that $\backslash varphi\_i$ is an acceptable numbering of the computable functions and ''W''''i'' the corresponding numbering of the recursively enumerable sets. A set ''A'' of natural numbers is called productive if there exists a total recursive (computable) function $f$ so that for all $i\; \backslash in\; \backslash mathbb$, if $W\_i\; \backslash subseteq\; A$ then $f(i)\; \backslash in\; A\; \backslash setminus\; W\_i.$ The function $f$ is called the productive function for $A.$ A set ''A'' of natural numbers is called creative if ''A'' is recursively enumerable and its complement $\backslash mathbb\backslash setminus\; A$ is productive. Not every productive set has a recursively enumerable complement, however, as illustrated below. The archetypal creative set is $K\; =\; \backslash$, the set representing the halting problem. Its complement $\backslash bar\; =\; \backslash$ is productive with productive function ''f''(''i'') = ''i'' (the identity function). To see this, we apply the definition of productivity function and show separately that $i\; \backslash in\; \backslash bar$ and $i\; \backslash not\; \backslash in\; W\_i$: * $i\; \backslash in\; \backslash bar$: suppose $i\; \backslash in\; K$, then $i\; \backslash in\; W\_i$, now given that $W\_i\; \backslash subseteq\; \backslash bar$ we have $i\; \backslash in\; \backslash bar$, this leads to a contradiction. So $i\; \backslash in\; \backslash bar$. * $i\; \backslash not\; \backslash in\; W\_i$: in fact if $i\; \backslash in\; W\_i$, then it would be true that $i\; \backslash in\; K$, but we have demonstrated the contrary in the previous point. So $i\; \backslash not\; \backslash in\; W\_i$. 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「Creative and productive sets」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.