Inner Model Theory and Descriptive Set Theory

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

Steel is working in the theory of canonical inner models for large cardinal hypotheses, and in descriptive set theory. In inner model theory, he has focussed on questions related to the fundamental iterability problem, and on questions concerning how to construct inner models in various situations so as to obtain consistency-strength lower bounds. He is particularly interested in obtaining large cardinal strength from the Proper Forcing Axiom, from the failure of Jensen's square principle at a singular cardinal, and from the failure of the Unique Branches Hypothesis. In descriptive set theory, Steel is working on various questions related to the structure of models satisfying strong forms of determinacy. Strong axioms of infinity, or as they are more often called, large cardinal hypotheses, have been a focal point of work in set theory and the foundations of mathematics for thirty or forty years, for at least two reasons. First, large cardinal hypotheses can be used to decide in a natural way many questions which cannot be decided on the basis of the commonly accepted system of axioms for mathematics, and second, large cardinal hypotheses provide a way of organizing and surveying all possible natural extensions of this commonly accepted system. One important way to study large cardinal hypotheses is to construct canonical minimal inner models in which these hypotheses are true. Such models admit a systematic, detailed analysis of their internal structure which makes them an invaluable technical tool in both sorts of application of large cardinal hypotheses. At the moment, we have a good theory of canonical inner models satisfying ``There is a Woodin cardinal", and even slightly stronger large cardinal hypotheses. A fundamental open problem, one of the most important open problems in set theory, is to extend this theory to models satisfying ``There is a supercompact cardinal". Steel is working in this direction.