Absoluteness and Choice

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Larson plans to study the extent to which the absoluteness results and detailed analysis of inner models of determinacy in the context of large cardinals can be lifted to larger models satisfying larger fragments of the Axiom of Choice. One topic in this class of problems concerns the problem of producing a model of Choice with the same ordinal cardinalities as some inner model of determinacy. Another concerns the question of whether the minimal model containing the reals, the ordinals and one ultrafilter on the the integers collapses any non-ordinal cardinals in the minimal inner model containing the reals and the ordinals.Related problems concern forcing-absoluteness for models of Choice, determinacy for games of uncountable length, and properties of the nonstationary ideal on the first uncountable cardinal.Set theory provides a foundation for mathematics, in that the axioms of set theory aim to describe the universe of mathematical concepts. Many set theorists study questions which are not resolved by the standard axioms for set theory, in hope of finding the right extension of these axioms. Developments in the study of such possible extensions have had a dramatic foundational impact in the last twenty years, affecting many areas of mathematics, and even philosophy. The PI works on the border between some of the more technical, inward-directed areas of set theory and more classical areas with connections to other fields. Much of his work consists of finding applications of these more technical areas, and exposing them to a wider audience.