Investigations into Set Theory and Ergodic Theory

集合论和遍历理论的研究

• 批准号: 0400887
• 负责人: Matthew Foreman
• 金额: $ 20万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400887&HistoricalAwards=false
• 关键词: Investigations; into; Set; Theory; Ergodic

Foreman will investigate the descriptive set theoreticaspects of the classification problem for ergodic measure preservingsystems. He has partial results showing, for example, that they cannot beclassified by countable structures. He conjectures that the isomorphismproblem is provably complex, e.g. analytic but not Borel. The secondaspect of Foreman's proposal is to study ``canonical structure" in settheory. This is structure that requires the Axiom of Choice for itsexistence, but is independent of the choices made in it construction. Ofparticular interest are PCF objects and relations with squares anddiamonds.Foreman will investigate two areas of mathematics. The first has to dowith the evolution of complex systems in time. These systems appear toshow random behaviour, but of qualitatively different sorts. Foreman'sproject is to show that various behavior types are not distinguishable byconcrete observations. The second project has to do with the assumptionsthat the study of mathematics is based on. The usual assumptions are notstrong enough to be able to find the answers to all mathematicalquestions. For this reason it is necessary to investigate strengtheningsof the ordinary assumptions. Foreman proposes to do this by focussing on``canonical structure" in set theory.

工头将调查ergodic测度保护系统的分类问题的描述性集合。他的部分结果表明，例如，它们不能被可计数的结构进行分类。他猜想同构象征很复杂，例如分析但不是骨。工头提出的第二个方面是研究``set理论''``规范结构''。这是需要为ITSEXSISTINES的选择公理，但独立于IT构建中的选择。专门的兴趣是PCF对象，并且与Squares andDiamond的关系是与Squares andDiamond的关系。Foreman将随机研究数学上的两个系统。在定性的情况下，工头的措施表明，各种行为类型不是结束的观察，第二个项目与数学的研究相关。在集合理论中进行“``统一结构''。