Friedman intends to investigate four areas within set theory in addition to beginning work on a book on the method of L-coding. (The latter should help make the many significant applications of this method accessible to a wider audience of set-theorists.) (1) With his former student, Dorshka Wylie (now an NSF fellow and an instructor at M.I.T.), he is engaged in a radically new approach to the core model K for a strong cardinal, which should provide a powerful new tool for establishing combinatorial principles in K. (2) A second goal is to enable coding methods to lift smoothly to K, permitting extension of results such as his proof of the Pi-1-2- Singleton Conjecture to the core model context. (3) A long- standing open question in the theory of iterated forcing concerns the problem of extending the method of countable support iteration beyond omega-2 without collapsing cardinals. A number of important consistency problems could be solved by such a technique. With his student, Dimitiros Tzimas, Friedman will attempt to develop such a method through the use of morasses. (4) Friedman will also explore questions in descriptive set theory as to whether the full structure theory for projective sets can be developed directly from the existence of large cardinals, without the use of infinite game theory. Set theory provides the popular way to lay the foundations for all of mathematics, the best known systematic attempt to do this being Russell and Whitehead's Principia Mathematica, dating from the early part of the 20th Century. Important foundational questions concern the independence and the consistency of the axioms used to establish set theory. Zermelo and Fraenkel's axioms (ZF for short) are one of the most convenient sets of basic axioms. However, it has been known since the work of Kurt Goedel in the 1930's that no axioms for set theory can be complete as well as consistent. This means that no set of axioms can be powerful enough to prove every possible proposition or else its negation, but not both. Upon this startling theorem has been erected a rich theory, treating possible propositions P such that either P or Not P can be added to ZF without resulting in a contradiction. Any such proposition P is said to be independent of ZF and can be taken as an additional axiom of set theory. The principal technique for finding such independent propositions is Paul J. Cohen's so-called method of forcing and its offspring. This is the circle of ideas involved in and motivating the investigator's research.
