Inner Model Theory and Descriptive Set Theory

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Steel will work on the problem of constructing canonical inner modelsfor large cardinal hypotheses, and on related problems in pure descriptiveset theory. One very well known and central problem in this area iswhether the Proper Forcing Axiom is equiconsistent with the existence of asupercompact cardinal. A solution to this problem is probably still faroff, but Steel has recently made some progress using the powerful andflexible ``core model induction" method. He intends to pursue and developthis method further, as it applies to the Proper Forcing Axiom, and inseveral other contexts as well. This program leads naturally to a numberof questions concerning the relationship between canonical inner modelswith Woodin cardinals and the derived models of the Axiom of Determinacywhich are associated to them. Steel has some results on this connection,and plans to investigate the area further. Much of set theory is motivated by the simple question``What are the proper axioms for mathematics?".Mathematiciansprove things for a living; what should they take as theircommon assumptions in these proofs? In the period 1905-1927, Russell,Zermelo, Fraenkel, and Skolemisolated an elegant list of basic statements about sets, expressedin the language of set theory, and showed that from these axiomsone could derive all of the mathematics of the time. This system ofaxioms is now known as Zermelo-Fraenkel Set Theory with Choice,or ZFC. While there is still no hint of a mathematical statement whichcannot be expressed in the language of set theory, we havediscovered that ZFC is incomplete in important ways. Asurprising number of quite basic questions about sets in generalare not decided by the axioms of ZFC; moreover, many of themore abstract questions of analysis, algebra, and topology aresimilarly left undecided. This leads to Godel's Program, first formulatedby Kurt Godel in the late 1940's:Decidemathematically interesting questionswhich are independent of ZFC in well-justified extensions ofZFC. There have been great successes in this direction obtainedby adding ``large cardinal hypotheses" to ZFC.Such hypothesesassert the existence of sets whose cardinality, or size, isinaccessible from below in ever stronger senses. Possibly our deepestunderstanding of large cardinal hypothesescomes from the inner model program. This program attempts toassociate to each large cardinal hypothesis H a canonical minimaluniverse of sets (or ``inner model") in which H is true. The strongerH is, the largere this universe must be.The inner modelswe have so far constructed have internal structures whichadmit a systematic, detailed analysis, a ``fine structure theory"which gives us a very good idea as to what these universeslook like. Inner models are crucial in severalbasic uses of large cardinal hypotheses to settle questionsleft undecided by ZFC. Steel will focus his efforts on furtheringthe inner model program.