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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 Goodstein's theorem ： ウィキペディア英語版 Goodstein's theorem In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every ''Goodstein sequence'' eventually terminates at 0. Kirby and Paris showed that it is unprovable in Peano arithmetic (but it can be proven in stronger systems, such as second order arithmetic). This was the third example of a true statement that is unprovable in Peano arithmetic, after Gödel's incompleteness theorem and Gerhard Gentzen's 1943 direct proof of the unprovability of ε0-induction in Peano arithmetic. The Paris–Harrington theorem was a later example. Laurence Kirby and Jeff Paris introduced a graph theoretic hydra game with behavior similar to that of Goodstein sequences: the "Hydra" is a rooted tree, and a move consists of cutting off one of its "heads" (a branch of the tree), to which the hydra responds by growing a finite number of new heads according to certain rules. Kirby and Paris proved that the Hydra will eventually be killed, regardless of the strategy that Hercules uses to chop off its heads, though this may take a very long time.〔 == Hereditary base-''n'' notation == Goodstein sequences are defined in terms of a concept called "hereditary base-''n'' notation". This notation is very similar to usual base-''n'' positional notation, but the usual notation does not suffice for the purposes of Goodstein's theorem. In ordinary base-''n'' notation, where ''n'' is a natural number greater than 1, an arbitrary natural number ''m'' is written as a sum of multiples of powers of ''n'': :$m\; =\; a\_k\; n^k\; +\; a\_\; n^\; +\; \backslash cdots\; +\; a\_0,$ where each coefficient ''ai'' satisfies , and . For example, in base 2, :$35\; =\; 32\; +\; 2\; +\; 1\; =\; 2^5\; +\; 2^1\; +\; 2^0.$ Thus the base 2 representation of 35 is 100011, which means . Similarly, 100 represented in base 3 is 10201: :$100\; =\; 81\; +\; 18\; +\; 1\; =\; 3^4\; +\; 2\backslash cdot\; 3^2\; +\; 3^0.$ Note that the exponents themselves are not written in base-''n'' notation. For example, the expressions above include 25 and 34. To convert a base-''n'' representation to hereditary base ''n'' notation, first rewrite all of the exponents in base-''n'' notation. Then rewrite any exponents inside the exponents, and continue in this way until every digit appearing in the expression is ''n'' or less. For example, while 35 in ordinary base-2 notation is , it is written in hereditary base-2 notation as :$35\; =\; 2^+2+1,$ using the fact that Similarly, 100 in hereditary base 3 notation is :$100\; =\; 3^\; +\; 2\backslash cdot\; 3^2\; +\; 1.$ 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「Goodstein's 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.