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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 New Foundations ： ウィキペディア英語版 New Foundations In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of ''Principia Mathematica''. Quine first proposed NF in a 1937 article titled "New Foundations for Mathematical Logic"; hence the name. Much of this entry discusses NFU, an important variant of NF due to Jensen (1969) and exposited in Holmes (1998).〔Holmes, Randall, 1998. ''(Elementary Set Theory with a Universal Set )''. Academia-Bruylant.〕 In 1940 and 1951 Quine introduced an extension of NF sometimes called "Mathematical Logic" or "ML", that included classes as well as sets. New Foundations has a universal set, so it is a non well founded set theory.〔(Quine's New Foundations ) - Stanford Encyclopedia of Philosophy〕 That is to say, it is a logical theory that allows infinite descending chains of membership such as …xn ∈ xn-1 ∈ …x3 ∈ x2 ∈ x1. It avoids Russell's paradox by only allowing stratifiable formulae in the axiom of comprehension. For instance x ∈ y is a stratifiable formula, but x ∈ x is not (for details of how this works see below). ==The type theory TST== The primitive predicates of Russellian unramified typed set theory (TST), a streamlined version of the theory of types, are equality ($=$) and membership ($\backslash in$). TST has a linear hierarchy of types: type 0 consists of individuals otherwise undescribed. For each (meta-) natural number ''n'', type ''n''+1 objects are sets of type ''n'' objects; sets of type ''n'' have members of type ''n''-1. Objects connected by identity must have the same type. The following two atomic formulas succinctly describe the typing rules: $x^\; =\; y^\backslash !$ and $x^\; \backslash in\; y^$. (Quinean set theory seeks to eliminate the need for such superscripts.) The axioms of TST are: *Extensionality: sets of the same (positive) type with the same members are equal; *An axiom schema of comprehension, namely: :If $\backslash phi(x^n)\backslash !$ is a formula, then the set $\backslash ^\backslash !$ exists. :In other words, given any formula $\backslash phi(x^n)\backslash !$, the formula $\backslash exists\; A^\; \backslash forall\; x^n\; (x^n\; \backslash in\; A^\; \backslash leftrightarrow\; \backslash phi(x^n))$ is an axiom where $A^\backslash !$ represents the set $\backslash ^\backslash !$ and is not free in $\backslash phi(x^n)$. This type theory is much less complicated than the one first set out in the ''Principia Mathematica'', which included types for relations whose arguments were not necessarily all of the same type. In 1914, Norbert Wiener showed how to code the ordered pair as a set of sets, making it possible to eliminate relation types in favor of the linear hierarchy of sets described here. 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「New Foundations」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.