Cardinal Invariants and Descriptive Set Theory

AbstractAward：DMS-0300201首席研究员：Jindrich Zapletal首席研究员计划研究基数不变量理论和描述性集合论（集合论的两个子领域）之间的联系。 基数不变量领域的中心思想是将基数分配给真实的直线或类似空间上的各种Borel结构（也就是说，可适当定义的结构）。这些基数的比较提供了一种方法来测量不同Borel结构之间的差异，并且主要研究者已经开发出一种方法来比较许多这些基数。 事实证明，基数的比较经常转换回关于Borel结构的自然Borel问题，而不会丢失信息。 根据这项资助，将研究的问题包括：上述技术适用的句法定义问题的扩展;大基数公理与平移方法的关系;动力系统中或Borel等价关系研究中出现的Borel结构的特殊情况;以及将这些结果与W.休·伍丁。普通基数用于计数，即，for comparing比较the sizes大小of collections集合of objects对象. 一百多年来，数学家们已经有了无穷集合基数的一个版本，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。 从这个角度来看，自然数的集合{1，2，3，...}它的偶数自然数子集{2，4，6，.} 具有相同的基数（大小），因为乘以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，即大于任何有限集合，逻辑和集合论发展的关键步骤之一是康托尔实现了实数集的基数肯定大于自然数集。 现代集合论已经发展了一些概念和工具，用于处理比真实的直线基数更大的集合，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.