Mathematical Sciences: Set Theory

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DMS 9625997 Sy D. Friedman Friedman intends to work in four directions, all aimed at an understanding of the structure of the set-theoretic universe through the techniques of fine structure and forcing. First, he intends to complete his book, entitled, "Fine Structure and Class Forcing". Second, Friedman will pursue a number of questions in the area of class forcing over L through the use of the coding method and his method of iterated class forcing. Third, Friedman intends to lay the foundation for an analogous theory relative to K=the core model for a strong cardinal, by first developing an improved fine structure theory for this model, and then adapting the coding method to this new context. This work will be carried out jointly with his collaborators Jensen, Koepke and Wylie. Lastly, Friedman will investigate a number of other forcing problems, related to 0# (with Velickovic), involving large cardinals (with Cummings) and in definability theory (with Bagaria). Early this century, mathematicians developed the system ZFC of axioms for set theory, adequate to express and prove the theorems of mathematics. However ZFC is incomplete: there are statements that can neither be proved nor refuted in this theory. An important task for the set theorist is to uncover new axioms to adjoin to ZFC so as to alleviate this incompleteness. This will provide powerful new tools for the working mathematician, which may in addition have exciting applications for science in general. Friedman's work is focused on the deepest possible understanding of the structure of models of ZFC, as a way of discovering the right axioms to add to this theory.