Examples of using Set theory in English and their translations into Chinese
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The concept was first introduced by Georg Cantor in 1872 and he developed set theory in large part to study derived sets on the real line.
这个概念是格奥尔格·康托尔在1872年引入的，他开发集合论很大程度上就是为了研究在实直线上的导出集合。In 1908, Ernst Zermelo proposed the first axiomatic set theory, Zermelo set theory.
在1908年，ErnstZermelo提议了第一个公理化集合论，Zermelo集合论。In modern set theory, we usually use the Von Neumann cardinal assignment, which uses the theory of ordinal numbers and the full power of the axioms of choice and replacement.
在现代集合论中，我们通常使用冯·诺伊曼基数指派，它使用序数的理论与选择公理和替代公理的全部能力。Set theory, was founded by a single paper in 1874 by Georg Cantor:"On a Property of the Collection of All Real Algebraic Numbers".
然而，集理论，创立由一个单一的文件在1874年由乔治·康托尔：“在所有实代数数”的一个特征性质。For a further development of mathematics in NFU, with a comparison to the development of the same in ZFC, see implementation of mathematics in set theory.
关于数学在NFU中的进一步开发，和与在ZFC中相同的开发的比较，请参见数学的集合论实现（en:Implementationofmathematicsinsettheory）。The axiom of choice without replacement(ZC set theory) is not strong enough to show that Borel sets are determined; for this, replacement is required.
不带替代公理的选择公理（ZC集合论）不足以强到证明博雷爾集是确定（英语：Axiomofdeterminacy）的；为此你需要替代公理。Systems of constructive set theory, such as CST, CZF, and IZF, embed their set axioms in intuitionistic logic instead of first order logic.
建设性的一套理论系统，如CST，CZF，IZF，嵌入自己设定的一阶逻辑，直觉主义逻辑，而不是公理。Because RA can express Peano arithmetic and set theory, Gödel's incompleteness theorems apply to it; RA is incomplete, incompletable, and undecidable.
因为RA可以表达皮亚诺算术和集合论，哥德尔不完备性定理适用于它，RA是不完备的、不可完备的和不可判定性的。If κ is an inaccessible cardinal, then Vκ is a model of Zermelo-Fraenkel set theory(ZFC) itself, and Vκ+1 is a model of Morse- Kelley set theory.
如果κ是不可及基数（英语：Inaccessiblecardinal），则Vκ是Zermelo-Fraenkel集合论自身的模型，而Vκ+1是Morse-Kelley集合论的模型。Each chapter surveys some aspect of contemporary research in set theory.
每章集理论在当代研究调查的某些方面。Barkley Rosser proved that Quine's set theory was subject to the Burali-Forti paradox.
BarkleyRosser证明了蒯因的集合论遭受Burali-Forti悖论。The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment.
集理论的势头是这样的，辩论的矛盾并没有导致其放弃。De Morgan proposes two laws of set theory that are now known as"de Morgan's laws".
德摩根（DeMorgan）提出了两个集合论定律，被称为“德摩根律”。Therefore, no useful axiomatization of set theory can use unrestricted comprehension, at least not with classical logic.
所以，有用的集合论的公理化都不能包括无限制概括，至少不跟经典逻辑一同被使用。Russell discovers"Russell's paradox" which illustrates in a simple fashion the problems inherent in naive set theory.
罗素（Russell）发现了“罗素悖论”，用一种简单的方式说明了朴素集合论固有的问题。Therefore, no useful axiomatisation of set theory can use unrestricted comprehension, at least not with classical logic.
所以，有用的集合论的公理化都不能包括无限制概括，至少不跟经典逻辑一同被使用。(Since we are not trying to do set theory here, we just treat V as a unary predicate.).
由于我们不是要在这里讲述理论，我们只将V当作一元谓词。This utility of set theory led to the article"Mengenlehre" contributed in 1898 by Arthur Schoenflies to Klein's encyclopedia.
此实用程序集理论的的文章“Mengenlehre”贡献于1898年由阿瑟夫立克莱因的百科全书。This past July, Malliaris and Shelah were awarded the Hausdorff medal, one of the top prizes in set theory.
今年七月，马利亚里斯和希拉被授予豪斯多夫奖(HausdorffMedal)，集合论的最高奖项之一。Specifically, Alternative Set Theory(or AST) may refer to a particular set theory developed in the 1970s and 1980s by Petr Vopěnka and his students.
狭义地讲，代替集合论(theAlternativeSetTheory,AST)是指一种具体的集合论，它是在70年代至80年代之间由PetrVopěnka和他的学生所发展的。
