Cardinal Invariants and Descriptive Set Theory

基数不变量和描述集合论

• 批准号: 0300201
• 负责人: Jindrich Zapletal
• 金额: --
• 依托单位: University of Florida
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2008-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0300201&HistoricalAwards=false
• 关键词: Cardinal; Invariants; Descriptive; Set; Theory

AbstractAward: DMS-0300201Principal Investigator: Jindrich ZapletalThe principal investigator plans to work on the connections betweenthe theory of cardinal invariants and descriptive set theory, twosubfields of set theory. The central idea in the field of cardinalinvariants is to assign cardinal numbers to various Borel (that is,suitably definable) structures on the real line or similar spaces.The comparison of these cardinal numbers provides a way to measuredifferences between distinct Borel structures, and the principalinvestigator has developed a method for comparing many of thesecardinal numbers. It turns out that the comparison of cardinalnumbers frequently translates back to natural Borel questions about theBorel structures, without loss of information. Questions to be studiedunder this grant include the extension of the syntactically definedclass of problems for which the technique sketched above works; therelevance of large cardinal axioms to the translation method; particularcases of Borel structures arising in dynamical systems or in the studyof Borel equivalence relations, and a duality that relates these resultsto a method recently found by W. Hugh Woodin.Ordinary cardinal numbers are used for counting, i.e., for comparingthe sizes of collections of objects. For more than one hundred yearsmathematicians have had a version of cardinal numbers for infinitesets, beginning with the notion of comparison: if two sets A and B canbe put into a one-to-one correspondence then we say that A and B havethe same cardinality. From this point of view the set of naturalnumbers {1, 2, 3, ...} and its subset of even natural numbers {2, 4,6, ...} have the same cardinality (size) since multiplication by 2gives a one-to-one correspondence from the first set to the secondone. Both of these sets are infinite, i.e. larger than any finiteset, and one of the key steps in the development of logic and settheory was the realization by Georg Cantor that the set of realnumbers is definitely of a larger cardinality than the set of naturalnumbers. Modern set theory has developed notions and tools forworking with sets of larger cardinality than the real line, andthese tools are becoming useful in exploring constructions andproperties that arise in dynamical systems and measure theory.

摘要奖：DMS-0300201首席研究员：金德里奇·扎普莱特首席研究员计划研究集合论的两个子领域--基数不变量理论和描述集合论之间的联系。基数不变量领域的核心思想是将基数赋给实直线或相似空间上的各种Borel结构(即，可适当定义的)。这些基数的比较提供了一种度量不同Borel结构之间差异的方法，而PrintalInvestigator已经开发了一种方法来比较许多这些基数。事实证明，基数的比较经常被翻译回关于Borel结构的自然Borel问题，而不会丢失信息。在这项资助下要研究的问题包括：上述技术适用的句法定义问题的扩展；大型基数公理与翻译方法的关系；在动力系统或Borel等价关系的研究中出现的Borel结构的特殊情况；以及将这些结果与W.Hugh Woodin最近发现的方法相关联的对偶性。普通基数用于计数，即用于比较对象集合的大小。一百多年来，数学家们一直有一种关于无穷集合的基数版本，从比较的概念开始：如果两个集合A和B可以一一对应，那么我们说A和B具有相同的基数。从这个观点来看，自然数集合{1，2，3，…}及其偶数子集{2，4，6，…}具有相同的基数(大小)，因为乘2给出了从第一个集合到第二个集合的一一对应。这两个集合都是无限的，即比任何有限集合都大，而逻辑和集合论发展的关键步骤之一是乔治·坎托认识到实数集肯定比自然数集具有更大的基数。现代集合论发展了一些概念和工具，用于处理基数比实数大的集合，这些工具在探索动力系统和测度论中出现的结构和性质方面正变得有用。