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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 Vaught conjecture ： ウィキペディア英語版 Vaught conjecture The Vaught conjecture is a conjecture in the mathematical field of model theory originally proposed by Robert Lawson Vaught in 1961. It states that the number of countable models of a first-order complete theory in a countable language is finite or ℵ0 or 2. Morley showed that number of countable models is finite or ℵ0 or ℵ1 or 2, which solves the conjecture except for the case of ℵ1 models when the continuum hypothesis fails. For this remaining case, has announced a counterexample to the Vaught conjecture and the topological Vaught conjecture. ==Statement of the conjecture== Let $T$ be a first-order, countable, complete theory with infinite models. Let $I(T,\; \backslash alpha)$ denote the number of models of T of cardinality $\backslash alpha$ up to isomorphism, the spectrum of the theory $T$. Morley proved that if I(T,ℵ0) is infinite then it must be ℵ0 or ℵ1 or the cardinality of the continuum. The Vaught conjecture is the statement that it is not possible for . The conjecture is a trivial consequence of the continuum hypothesis; so this axiom is often excluded in work on the conjecture. Alternatively there is a sharper form of the conjecture which states that any countable complete T with uncountably many countable models will have a perfect set of uncountable models (as pointed out by John Steel, in On Vaught's conjecture. Cabal Seminar 76—77 (Proc. Caltech-UCLA Logic Sem., 1976—77), pp. 193–208, Lecture Notes in Math., 689, Springer, Berlin, 1978, this form of the Vaught conjecture is equiprovable with the original). 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「Vaught conjecture」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.