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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 Paris–Harrington theorem ： ウィキペディア英語版 Paris–Harrington theorem In mathematical logic, the Paris–Harrington theorem states that a certain combinatorial principle in Ramsey theory, namely the strengthened finite Ramsey theorem, is true, but not provable in Peano arithmetic. This was the first "natural" example of a true statement about the integers that could be stated in the language of arithmetic, but not proved in Peano arithmetic; it was already known that such statements existed by Gödel's first incompleteness theorem. ==The strengthened finite Ramsey theorem== The strengthened finite Ramsey theorem is a statement about colorings and natural numbers and states that: *For any positive integers ''n'', ''k'', ''m'' we can find ''N'' with the following property: if we color each of the ''n''-element subsets of ''S'' = with one of ''k'' colors, then we can find a subset ''Y'' of ''S'' with at least ''m'' elements, such that all ''n'' element subsets of ''Y'' have the same color, and the number of elements of ''Y'' is at least the smallest element of ''Y''. Without the condition that the number of elements of ''Y'' is at least the smallest element of ''Y'', this is a corollary of the finite Ramsey theorem in $K\_$, with ''N'' given by: :$\backslash binom\; =\; |\backslash mathcal\_n(S)|\; \backslash ge\; R(\backslash ,\backslash underbrace\_k\backslash ,).$ Moreover, the strengthened finite Ramsey theorem can be deduced from the infinite Ramsey theorem in almost exactly the same way that the finite Ramsey theorem can be deduced from it, using a compactness argument (see the article on Ramsey's theorem for details). This proof can be carried out in second-order arithmetic. The Paris–Harrington theorem states that the strengthened finite Ramsey theorem is not provable in Peano arithmetic. 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「Paris–Harrington 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.