2008 Annual Meeting of the Association for Symbolic Logic

of 19 Annual Gödel Lecture W. HUGH WOODIN, The Continuum Hypothesis, the Ω Conjecture, and the inner model problem for one supercompact cardinal. Department of Mathematics, The University of California, 721 Evans Hall #3840, Berkeley, CA 94720-3840 US. E-mail: woodin@math.berkeley.edu. The Ω Conjecture is a key element of an argument that the Continuum Hypothesis is false, based on notions of simplicity. While this application is certainly debatable, there is a much stronger argument that if the Ω Conjecture is true then the Continuum Hypothesis must have an answer. In brief, the Ω Conjecture rules out amultiverse conception of truth in Set Theory which is based on the generic multiverse. The ΩConjecture is invariant under passing from the universe of sets to a generic extension of the universe of sets, and so it is reasonable to expect that if the Ω Conjecture is false then it must be refuted by some large cardinal hypothesis. The search for candidates for such a large cardinal hypothesis leads to the inner model problem since subject to fairly general criteria, if the inner model problem can be solved for a particular large cardinal hypothesis then that large cardinal hypothesis cannot refute the Ω Conjecture. Again subject to fairly general criteria for a successful inner model construction, the entire inner model program can now be reduced to the case of exactly one supercompact cardinal. If this one case can be solved then one obtains an ultimate generalization of Gödel’s, Axiom of Constructibility, and one also obtains that no (known) large cardinal hypothesis can refute the Ω Conjecture. I shall survey how the inner model problem for one supercompact cardinal has emerged as the critical case, the connections with the Ω Conjecture, and the connections to transfinite generalizations of the Axiom of Determinacy. My emphasis will be less on the technical details and more on what I think are the key foundational issues. Abstract of Special Lecture in honor of Paul Eklofof Special Lecture in honor of Paul Eklof SAHARON SHELAH, Incompactness in singular cardinals. Department of Mathematics, Hebrew University, Jerusalem, Israel 91905. E-mail: shelah@math.huji.ac.il. We prove the consistency of: is strong limit singular and for some properties of Abelian 42