Core Model Theory

Carnegie Mellon University, Pittsburgh PA

Investigators

Abstract

Abstract: Core models are generalizations of Godel's Constructible Universe. Jensen's pioneering work on the fine structure of core models and on techniques for constructing core models with large cardinals has led to many applications in diverse parts of set theory. More recently, Martin, Mitchell, Steel, and others have extended earlier work on core models to the level of Woodin cardinals. The investigator's questions concern 1) the combinatorial structure of core models, 2) the application of core model techniques in determining the large cardinal consistency strength of various set theoretic statements, 3) the application of core model techniques in descriptive set theory, 4) problems in infinitary combinatorics, forcing, and model theory that were in part inspired by his earlier work on core models. The most widely accepted theory (system of axioms) of sets is known as ZFC. Kurt Godel is famous for having shown that if ZFC is consistent, then ZFC is incomplete in the sense that there are statements which are neither provable nor refutable in ZFC. This is particularly striking since all of mathematics can be expressed in the language of sets and most mathematicians work exclusively within ZFC. There are important open questions that are not decided by ZFC. The most famous example is Georg Cantor's Continuum Problem, which Godel showed cannot be answered negatively in ZFC and Paul Cohen showed cannot be answered positively in ZFC. (Cohen received a Fields Medal for this work.) There are many interesting extensions of ZFC and the relationships between them are quite complicated. The investigator's project has to do with a natural hierarchy of axioms known as the ``large cardinals''. Many beautiful results, including some longstanding conjectures, were proved by a number of researchers using ZFC and large cardinals in the 1980's and 90's; further success is anticipated. And, also, details of how the various theories of sets are related have been discovered by comparing these theories with the large cardinal hierarchy.