Mathematical Sciences: Set Theory

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Set theory underlies all of mathematics. The two main avenues to laying the foundations of set theory rely either on axioms postulating large cardinal numbers or on various axioms about the determinacy of so-called set theoretic games. Woodin will investigate the relationship between large cardinals and determinacy, descriptive set theory in the context of determinacy, and the possibilities for the definability of the well-ordering of the reals in the presence of very large cardinals. He also intends to investigate the global structure of the Turing degrees and of more general degree structures. The basic question concerns rigidity.