Inner Model Theory and Descriptive Set Theory

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Steel will work in set theory, and in particular on the theory of canonical inner models for large cardinal hypotheses. His focus will be on extending the basics of this theory to stronger large cardinal hypotheses. The central open problem here is whether every countable elementary submodel of V is appropriately iterable. Steel will work on this question, its fragments, and related issues. This effort is closely connected to pure descriptive set theory, the general study of definability for sets of real numbers. Steel plans to work in this area too, and in particular on questions related to the determinacy of infinite games. One very nice question here is whether all closed (in the countable-initial-segment topology) games of length aleph-one with projectively definable payoff are determined. Many questions of mathematical interest are left undecided by ZFC, the commonly accepted system of axioms for mathematics. The most natural and successful way to extend ZFC so as to decide such questions is to add "large cardinal hypotheses" to ZFC. Such hypotheses play a central role in the foundations of mathematics, and because of this have been studied intensively over the last 30 to 40 years. One way to understand and use large cardinal hypotheses is by constructing canonical minimal universes, or "inner models", in which these hypotheses hold true. Inner model theory began with work of Kurt Godel in the 1930's, and from the late 1960's to the present the theory has undergone more or less continuous development. Set theorists have constructed and analyzed in detail canonical inner models for many large cardinal hypotheses, and these models have proved quite useful. Nevertheless, there are important large cardinal hypotheses which are beyond the reach of current inner model theory. Steel will attempt to extend the theory so that it applies to these hypotheses.