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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 Arithmetical set ： ウィキペディア英語版 Arithmetical set In mathematical logic, an arithmetical set (or arithmetic set) is a set of natural numbers that can be defined by a formula of first-order Peano arithmetic. The arithmetical sets are classified by the arithmetical hierarchy. The definition can be extended to an arbitrary countable set ''A'' (e.g. the set of n-tuples of integers, the set of rational numbers, the set of formulas in some formal language, etc.) by using Gödel numbers to represent elements of the set and declaring a subset of ''A'' to be arithmetical if the set of corresponding Gödel numbers is arithmetical. A function $f:\backslash subseteq\; \backslash mathbb^k\; \backslash to\; \backslash mathbb$ is called arithmetically definable if the graph of $f$ is an arithmetical set. A real number is called arithmetical if the set of all smaller rational numbers is arithmetical. A complex number is called arithmetical if its real and imaginary parts are both arithmetical. == Formal definition == A set ''X'' of natural numbers is arithmetical or arithmetically definable if there is a formula φ(''n'') in the language of Peano arithmetic such that each number ''n'' is in ''X'' if and only if φ(''n'') holds in the standard model of arithmetic. Similarly, a ''k''-ary relation $R(n\_1,\backslash ldots,n\_k)$ is arithmetical if there is a formula $\backslash psi(n\_1,\backslash ldots,n\_k)$ such that $R(n\_1,\backslash ldots,n\_k)\; \backslash Leftrightarrow\; \backslash psi(n\_1,\backslash ldots,n\_k)$ holds for all ''k''-tuples $(n\_1,\backslash ldots,n\_k)$ of natural numbers. A finitary function on the natural numbers is called arithmetical if its graph is an arithmetical binary relation. A set ''A'' is said to be arithmetical in a set ''B'' if ''A'' is definable by an arithmetical formula which has ''B'' as a set parameter. 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「Arithmetical set」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.