August 28
Miha Habič,
Bard College at Simon’s Rock
Normal ultrapowers with many sets of ordinals
Any ultrapower $M$ of the universe by a normal measure on a cardinal $\kappa$ is quite far from $V$ in the sense that it computes $V_{\kappa+2}$ incorrectly. If GCH holds, this amounts to saying that $M$ is missing a subset of $\kappa^+$. Steel asked whether, even in the absence of GCH, normal ultrapowers at $\kappa$ must miss a subset of $\kappa^+$. In the early 90s Cummings gave a negative answer, building a model with a normal measure on $\kappa$ whose ultrapower captures the entire powerset of $\kappa^+$. I will present some joint work with Radek Honzík in which we improved Cummings’ result to get this capturing property to hold at the least measurable cardinal.