Mathematical Sciences: Set Theory and Large Cardinals

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DMS 9626143 William Mitchell and Jean Larson, U. Florida Dr. Mitchell will investigate the core model, particularly near or at the first Woodin cardinal. This is the point at which the structure of the universe of sets seems to make a radical change. The aim of this project is better to understand this change. Dr. Larson's research concerns the combinatorial properties of small ordinals, specifically countable ordinals above the first countable ordinal of infinite exponent, and uncountable ordinals which are squares of cardinals. Ordinals are the backbone of the set theoretic universe, as well as the basis for many model-theoretic constructions. Additionally, they are used to assess complexity in proof theory. The sequence of ordinals (which are the numbers used to list possibly infinite sets) and the core model of set theory each provide part of the supporting structure of the universe of set theory. The funded projects seek a better understanding of these structures at critical points in their architecture. The aim in both instances is a keener understanding of the very fundamental building blocks of mathematics, namely numbers and sets.