翻訳と辞書 Words near each other ・ Second-generation wavelet transform ・ Second-Hand ・ Second-Hand Hearts ・ Second-hand shop ・ Second-hand Smoke ・ Second-harmonic generation ・ Second-harmonic imaging microscopy ・ Second-impact syndrome ・ Second-in-command ・ Second-language acquisition ・ Second-language acquisition classroom research ・ Second-language attrition ・ Second-language phonology ・ Second-level domain ・ Second-order ・ Second-order arithmetic ・ Second-order cellular automaton ・ Second-order co-occurrence pointwise mutual information ・ Second-order conditioning ・ Second-order cone programming ・ Second-order cybernetics ・ Second-order election ・ Second-order fluid ・ Second-order intercept point ・ Second-order logic ・ Second-order predicate ・ Second-order propositional logic ・ Second-order simulacra ・ Second-order stimulus ・ Second-person narrative Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Second-order arithmetic ： ウィキペディア英語版 Second-order arithmetic In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics. It was introduced by David Hilbert and Paul Bernays in their book Grundlagen der Mathematik. The standard axiomatization of second-order arithmetic is denoted Z2. Second-order arithmetic includes, but is significantly stronger than, its first-order counterpart Peano arithmetic. Unlike Peano arithmetic, second-order arithmetic allows quantification over sets of numbers as well as numbers themselves. Because real numbers can be represented as (infinite) sets of natural numbers in well-known ways, and because second order arithmetic allows quantification over such sets, it is possible to formalize the real numbers in second-order arithmetic. For this reason, second-order arithmetic is sometimes called “analysis”. Second-order arithmetic can also be seen as a weak version of set theory in which every element is either a natural number or a set of natural numbers. Although it is much weaker than Zermelo-Fraenkel set theory, second-order arithmetic can prove essentially all of the results of classical mathematics expressible in its language. A subsystem of second-order arithmetic is a theory in the language of second-order arithmetic each axiom of which is a theorem of full second-order arithmetic (Z2). Such subsystems are essential to reverse mathematics, a research program investigating how much of classical mathematics can be derived in certain weak subsystems of varying strength. Much of core mathematics can be formalized in these weak subsystems, some of which are defined below. Reverse mathematics also clarifies the extent and manner in which classical mathematics is nonconstructive. ==Definition== 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「Second-order 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.