Large cardinals and small sets: The AD+ Duality Program

Speaker: Hugh Woodin

W. Hugh Woodin is a Distinguished Professor Emeritus of Mathematics at the University of California, Berkeley and a Professor of Philosophy and of Mathematics at Harvard University. Woodin is a leading set theorist. He has made many notable contributions to the theory of inner models and determinacy, to functional analysis, and recursion theory. A type of large cardinal, the Woodin cardinal, bears his name. He is a managing editor of the Journal of Mathematical Logic. In 1989 he received the Carol Karp Prize from the Association for Symbolic Logic.

Abstract

The determinacy axiom, AD, was introduced by Mycielski and Steinhaus over 60 years ago as an alternative to the Axiom of Choice for the study of arbitrary sets of real numbers. The modern view is that determinacy axioms concern generalizations of the borel sets, and deep connections with large cardinal axioms have emerged.

The study of determinacy axioms has led to a specific technical refinement of AD, this is the axiom AD+. The further connections with large axioms have in turn implicitly led to a duality program, this is the AD+ Duality Program.

The main open problems here are intertwined with those of the Inner Model Program, which is the central program in the study of large cardinal axioms.

This has now all been distilled into a series of specific conjectures.