Cardinal Invariants and Descriptive Set Theory

项目负责人：Jindrich zapletal主要研究集合论的两个分支——基数不变量理论和描述集合理论之间的联系。基数不变量领域的中心思想是为实线上或类似空间上的各种Borel（即适当可定义的）结构分配基数。这些基数的比较提供了一种测量不同Borel结构之间差异的方法，并且主要研究者已经开发了一种比较许多这些基数的方法。事实证明，基数的比较经常会转换回关于Borel结构的自然Borel问题，而不会丢失信息。在这项拨款下研究的问题包括扩展上述技术所适用的语法定义的问题类别；大基数公理与翻译方法的关系在动力系统中出现的Borel结构的特殊情况或在Borel等价关系的研究中，以及将这些结果与W. Hugh Woodin最近发现的方法联系起来的对偶性。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，数学家们已经有了无限集基数的一个版本，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{1,2,3，…}及其偶自然数子集{2,4,6，…}具有相同的基数（大小），因为乘以2得到从第一个集合到第二个集合的一对一对应。这两个集合都是无限的，也就是说，比任何有限集都大，逻辑和集合论发展的关键步骤之一是乔治·康托尔认识到实数集合肯定比自然数集合具有更大的基数。现代集合理论已经发展了一些概念和工具，用于处理比实线基数更大的集合，这些工具在探索动力系统和测量理论中出现的结构和性质方面变得非常有用。

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.