Mathematical Sciences: Topics in Logic and Category Theory

数学科学：逻辑和范畴论主题

• 批准号: 9204276
• 负责人: Andreas Blass
• 金额: $ 12.18万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-08-15 至 1996-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9204276&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topics; Logic; Category

The research of Andreas Blass is in three areas: (1) combinatorial set theory, particularly the study of cardinal characteristics of the continuum; (2) logic of topoi, particularly connections between geometric morphisms of topoi, the internal higher-order logic, and domains for denotational semantics; and (3) game semantics for linear logic and related systems. Set theory provides the popular way to lay the foundations for all of mathematics, the best known systematic attempt to do this being Russell and Whitehead's Principia Mathematica, dating from the early days of the 20th Century. Important foundational questions concern the independence and the consistency of the axioms used to establish set theory. Zermelo and Fraenkel's axioms (ZF for short) are one of the most convenient sets of basic axioms. However, it has been known since the work of Kurt Goedel in the 1930's that no axioms for set theory can be complete as well as consistent. This means that no set of axioms can be powerful enough to prove every possible proposition or else its negation, but not both. Upon this startling theorem has been erected a rich theory, treating possible propositions P such that either P or Not P can be added to ZF without resulting in a contradiction. Any such proposition P is said to be independent of ZF and can be taken as an additional axiom of set theory. The principal technique for finding such independent propositions is Paul J. Cohen's so-called method of forcing and its offspring. This is the circle of ideas involved in and motivating a large part of the investigator's research.

Andreas Blass的研究主要集中在三个方面：(1)组合集论，特别是研究连续体的基数特征；(2)拓扑逻辑，特别是拓扑图的几何态射、内部高阶逻辑和指称语义域之间的联系；(3)线性逻辑及其相关系统的博弈语义学。集合论提供了为所有数学奠定基础的流行方法，最著名的系统尝试是罗素和怀特海的《数学原理》，可以追溯到20世纪初。重要的基础性问题涉及用于建立集合论的公理的独立性和一致性。Zermelo和Fraenkel公理(简称ZF)是最方便的基本公理之一。然而，自从哥德尔在20世纪30年代的S的工作以来，人们就知道，集合论的任何公理都不可能既完整又一致。这意味着，没有一组公理能够强大到足以证明每一个可能的命题或其否定，但不能两者兼而有之。在这个惊人的定理的基础上，建立了一个丰富的理论，处理可能的命题P，使P或不P可以加到ZF上，而不会导致矛盾。任何这样的命题P都可以说是独立于ZF的，并且可以作为集合论的一个附加公理。寻找这种独立命题的主要技术是保罗·J·科恩的所谓强迫方法及其产物。这就是研究者研究的一大部分涉及和激励的思想圈子。