Fri Oct 4, 2019
60 Evans Hall, 4:10–6 PM
Gabriel Goldberg (NSF Postdoctoral Fellow, UC Berkeley)
Comparison principles and very large cardinals
In 1947, Gödel proposed a program to solve the Continuum Problem, as well as many other unsolvable problems of set theory, by supplementing the traditional ZFC axioms with large cardinal axioms. Although this program has been remarkably successful, serious limitations have since been discovered: for example, it turns out that large cardinal axioms do not help resolve the Continuum Problem itself. This talk is about new set theoretic principles that serve to amplify traditional large cardinals by endowing them with a powerful structure theory analogous to both the Comparison Lemma of inner model theory and the Wadge order of descriptive set theory. The first part of this talk deals with one of these principles, the Ultrapower Axiom (UA) and its relationship with the long-standing problem of building inner models with supercompact cardinals. The second part of the talk explores more recent work on generalizations of UA, especially the refutation of the Extender Power Axiom. The talk concludes with some speculation about the prospect of solving the Continuum Problem using UA.