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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 Implementation of mathematics in set theory ： ウィキペディア英語版 Implementation of mathematics in set theory This article examines the implementation of mathematical concepts in set theory. The implementation of a number of basic mathematical concepts is carried out in parallel in ZFC (the dominant set theory) and in NFU, the version of Quine's New Foundations shown to be consistent by R. B. Jensen in 1969 (here understood to include at least axioms of Infinity and Choice). What is said here applies also to two families of set theories: on the one hand, a range of theories including Zermelo set theory near the lower end of the scale and going up to ZFC extended with large cardinal hypotheses such as "there is a measurable cardinal"; and on the other hand a hierarchy of extensions of NFU which is surveyed in the New Foundations article. These correspond to different general views of what the set-theoretical universe is like, and it is the approaches to implementation of mathematical concepts under these two general views that are being compared and contrasted. It is not the primary aim of this article to say anything about the relative merits of these theories as foundations for mathematics. The reason for the use of two different set theories is to illustrate that multiple approaches to the implementation of mathematics are feasible. Precisely because of this approach, this article is not a source of "official" definitions for any mathematical concept. ==Preliminaries== The following sections carry out certain constructions in the two theories ZFC and NFU and compare the resulting implementations of certain mathematical structures (such as the natural numbers). Mathematical theories prove theorems (and nothing else). So saying that a theory allows the construction of a certain object means that it is a theorem of that theory that that object exists. This is a statement about a definition of the form "the x such that $\backslash phi$ exists", where $\backslash phi$ is a formula of our language: the theory proves the existence of "the x such that $\backslash phi$" just in case it is a theorem that "there is one and only one x such that $\backslash phi$". (See Bertrand Russell's theory of descriptions.) Loosely, the theory "defines" or "constructs" this object in this case. If the statement is not a theorem, the theory cannot show that the object exists; if the statement is provably false in the theory, it proves that the object cannot exist; loosely, the object cannot be constructed. ZFC and NFU share the language of set theory, so the same formal definitions "the x such that $\backslash phi$" can be contemplated in the two theories. A specific form of definition in the language of set theory is set-builder notation: $\backslash$ means "the set A such that for all x, $x\; \backslash in\; A\; \backslash leftrightarrow\; \backslash phi$" (A cannot be free in $\backslash phi$). This notation admits certain conventional extensions: $\backslash$ is synonymous with $\backslash$; $\backslash$ is defined as $\backslash$, where $f(x\_1,\backslash ldots,x\_n)$ is an expression already defined. Expressions definable in set-builder notation make sense in both ZFC and NFU: it may be that both theories prove that a given definition succeeds, or that neither do (the expression $\backslash$ fails to refer to anything in ''any'' set theory with classical logic; in class theories like NBG this notation does refer to a class, but it is defined differently), or that one does and the other doesn't. Further, an object defined in the same way in ZFC and NFU may turn out to have different properties in the two theories (or there may be a difference in what can be proved where there is no provable difference between their properties). Further, set theory imports concepts from other branches of mathematics (in intention, ''all'' branches of mathematics). In some cases, there are different ways to import the concepts into ZFC and NFU. For example, the usual definition of the first infinite ordinal $\backslash omega$ in ZFC is not suitable for NFU because the object (defined in purely set theoretical language as the set of all finite von Neumann ordinals) cannot be shown to exist in NFU. The usual definition of $\backslash omega$ in NFU is (in purely set theoretical language) the set of all infinite well-orderings all of whose proper initial segments are finite, an object which can be shown not to exist in ZFC. In the case of such imported objects, there may be different definitions, one for use in ZFC and related theories, and one for use in NFU and related theories. For such "implementations" of imported mathematical concepts to make sense, it is necessary to be able to show that the two parallel interpretations have the expected properties: for example, the implementations of the natural numbers in ZFC and NFU are different, but both are implementations of the same mathematical structure, because both include definitions for all the primitives of Peano arithmetic and satisfy (the translations of) the Peano axioms. It is then possible to compare what happens in the two theories as when only set theoretical language is in use, as long as the definitions appropriate to ZFC are understood to be used in the ZFC context and the definitions appropriate to NFU are understood to be used in the NFU context. Whatever is proven to exist in a theory clearly provably exists in any extension of that theory; moreover, analysis of the proof that an object exists in a given theory may show that it exists in weaker versions of that theory (one may consider Zermelo set theory instead of ZFC for much of what is done in this article, for example). 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「Implementation of mathematics in set theory」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.