07/December/2012, 11:15–12:15

Fields institute,Room 230

Speaker: Sean Cox

Title: Antichain catching at $\omega_1$ versus antichain catching at $\omega_2$

Abstract: I’ll discuss a property of normal ideals, called

*projective antichain catching*, which lies (implication-wise) between saturation and precipitousness. For ideals on $\omega_1$, projective antichain catching is equivalent to precipitousness; in fact it gives a nice characterization of the statement “$NS_{\omega_1}$ is precipitous” in terms of Feng-Jech’s notion of*projective stationarity*(this is due essentially to Schindler). For ideals on $\omega_2$, however, projective antichain catching is strictly between saturation and precipitousness (and much stronger than precipitousness, in consistency strength). Proving that projective antichain catching does not imply saturation—in fact does not imply even strongness of the ideal—involves a modification of the Kunen–Magidor constructions of saturated ideals to work in the context of supercompact towers which are not almost huge. This is joint work with Martin Zeman.———————————————————————————————–

07/December/2012, 13:30–15:00

Fields institute,Room 230

Speaker: Bill Mitchell

Title: The Chang Model — Again

Abstract: A few years ago, I gave several talks — with varying degrees of tentativeness — describing a weak version of Woodin’s sharp for the Chang model. I will discuss what the optimal result, that is, the actual sharp, might look like, and how the picture I had of this is wrong in a major way. I will then discuss the proof of the result I did claim.