3:00–4:00 pm Eckhart 206
Title: Large cardinals, small cardinals, and the Ultrapower Axiom
Abstract:
Set theory begins with the transfinite cardinal numbers and the powerset operation. These concepts are notoriously intractable: the most basic questions about them (e.g., Cantor's continuum hypothesis) are left unanswered by the axioms of set theory. One approach to understanding the powerset of a cardinal is to analyze its space of ultrafilters (equivalently maximal ideals). This talk concerns the Ultrapower Axiom (UA), an additional set theoretic hypothesis that stratifies the ultrafilters into a well-ordered complexity hierarchy. The rigid structure on ultrafilters imposed by UA allows us to settle a number of classical problems on cardinals and powersets (e.g. instances of the generalized continuum hypothesis), but only in the realm of large cardinals. Finally, we discuss a recent theorem showing that UA extends to small cardinals (below the continuum) in the context of the Axiom of Determinacy.