Inner Model Theory and Descriptive Set Theory

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

Steel will work on the problem of constructing canonical inner models for large cardinal hypotheses, and on related problems in pure descriptive set theory. One very well known and central problem in this area is whether the Proper Forcing Axiom is equiconsistent with the existence of a supercompact cardinal. A solution to this problem is probably still far off, but Steel has recently made some progress using the powerful and flexible ``core model induction" method. He intends to pursue and develop this method further, as it applies to the Proper Forcing Axiom, and in several other contexts as well. This program leads naturally to a number of questions concerning the relationship between canonical inner models with Woodin cardinals and the derived models of the Axiom of Determinacy which 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?". Mathematicians prove things for a living; what should they take as their common assumptions in these proofs? In the period 1905-1927, Russell, Zermelo, Fraenkel, and Skolem isolated an elegant list of basic statements about sets, expressed in the language of set theory, and showed that from these axioms one could derive all of the mathematics of the time. This system of axioms is now known as Zermelo-Fraenkel Set Theory with Choice, or ZFC. While there is still no hint of a mathematical statement which cannot be expressed in the language of set theory, we have discovered that ZFC is incomplete in important ways. A surprising number of quite basic questions about sets in general are not decided by the axioms of ZFC; moreover, many of the more abstract questions of analysis, algebra, and topology are similarly left undecided. This leads to Godel's Program, first formulated by Kurt Godel in the late 1940's: Decide mathematically interesting questions which are independent of ZFC in well-justified extensions of ZFC. There have been great successes in this direction obtained by adding ``large cardinal hypotheses" to ZFC. Such hypotheses assert the existence of sets whose cardinality, or size, is inaccessible from below in ever stronger senses. Possibly our deepest understanding of large cardinal hypotheses comes from the inner model program. This program attempts to associate to each large cardinal hypothesis H a canonical minimal universe of sets (or ``inner model") in which H is true. The stronger H is, the largere this universe must be. The inner models we have so far constructed have internal structures which admit a systematic, detailed analysis, a ``fine structure theory" which gives us a very good idea as to what these universes look like. Inner models are crucial in several basic uses of large cardinal hypotheses to settle questions left undecided by ZFC. Steel will focus his efforts on furthering the inner model program.