Cardinal Invariants and Descriptive Set Theory

University Of Florida, Gainesville FL

Investigators

Abstract

Abstract Award: DMS-0300201 Principal Investigator: Jindrich Zapletal The principal investigator plans to work on the connections between the theory of cardinal invariants and descriptive set theory, two subfields of set theory. The central idea in the field of cardinal invariants 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 measure differences between distinct Borel structures, and the principal investigator has developed a method for comparing many of these cardinal numbers. It turns out that the comparison of cardinal numbers frequently translates back to natural Borel questions about the Borel structures, without loss of information. Questions to be studied under this grant include the extension of the syntactically defined class of problems for which the technique sketched above works; the relevance of large cardinal axioms to the translation method; particular cases of Borel structures arising in dynamical systems or in the study of Borel equivalence relations, and a duality that relates these results to a method recently found by W. Hugh Woodin. Ordinary cardinal numbers are used for counting, i.e., for comparing the sizes of collections of objects. For more than one hundred years mathematicians have had a version of cardinal numbers for infinite sets, beginning with the notion of comparison: if two sets A and B can be put into a one-to-one correspondence then we say that A and B have the same cardinality. From this point of view the set of natural numbers {1, 2, 3, ...} and its subset of even natural numbers {2, 4, 6, ...} have the same cardinality (size) since multiplication by 2 gives a one-to-one correspondence from the first set to the second one. Both of these sets are infinite, i.e. larger than any finite set, and one of the key steps in the development of logic and set theory was the realization by Georg Cantor that the set of real numbers is definitely of a larger cardinality than the set of natural numbers. Modern set theory has developed notions and tools for working with sets of larger cardinality than the real line, and these tools are becoming useful in exploring constructions and properties that arise in dynamical systems and measure theory.