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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 True arithmetic ： ウィキペディア英語版 True arithmetic In mathematical logic, true arithmetic is the set of all true statements about the arithmetic of natural numbers (Boolos, Burgess, and Jeffrey 2002:295). This is the theory associated with the standard model of the Peano axioms in the language of the first-order Peano axioms. True arithmetic is occasionally called Skolem arithmetic, though this term usually refers to the different theory of natural numbers with multiplication. == Definition == The signature of Peano arithmetic includes the addition, multiplication, and successor function symbols, the equality and less-than relation symbols, and a constant symbol for 0. The (well-formed) formulas of the language of first-order arithmetic are built up from these symbols together with the logical symbols in the usual manner of first-order logic. The structure $\backslash mathcal$ is defined to be a model of Peano arithmetic as follows. * The domain of discourse is the set $\backslash mathbb$ of natural numbers. * The symbol 0 is interpreted as the number 0. * The function symbols are interpreted as the usual arithmetical operations on $\backslash mathbb$ * The equality and less-than relation symbols are interpreted as the usual equality and order relation on $\backslash mathbb$ This structure is known as the standard model or intended interpretation of first-order arithmetic. A sentence in the language of first-order arithmetic is said to be true in $\backslash mathcal$ if it is true in the structure just defined. The notation $\backslash mathcal\; \backslash models\; \backslash varphi$ is used to indicate that the sentence φ is true in $\backslash mathcal.$ True arithmetic is defined to be the set of all sentences in the language of first-order arithmetic that are true in $\backslash mathcal$, written . This set is, equivalently, the (complete) theory of the structure $\backslash mathcal$ (see theories associated with a structure). 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「True arithmetic」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.