Toward the Consistency Strength of Stationary Set Reflection on Small Cardinals

OF THE DISSERTATION Toward the Consistency Strength of Stationary Set Reflection on Small Cardinals By Cynthia Northrup Doctor of Philosophy in Mathematics University of California, Irvine, 2015 Professor Dr. Martin Zeman, Chair A set S ⊆ ω3 is stationary in ω3 if it intersects all closed and unbounded subsets in ω3. We say that S reflects below ω3 if there is some α < ω3 for which S ∩ α is stationary in α. In my thesis I obtain a model M for which every subset of ω3 focusing on cofinality ω reflects at a point of cofinality ω1. That is, M |= Refl(ω3, ω, ω1). To do this, I adjust the methods of Gitik in order to get set reflection. This argument may help show the way to getting the consistency strength. Due to existing bounds, our strategy begins with κ of Mitchell order κ which concentrates on α-supercompact cardinals α. We collapse κ to ω3, change the cofinality of many measurables between ω2 and ω3, and shoot clubs through the non-reflecting stationary sets. For any nonreflecting stationary set S, we use the forcing to add a club C for which C ∩ S = ∅, thus rendering this S no longer stationary. We take care that any more stationary sets we add in this process, have a club shot through them and so are no longer stationary in the final generic extension. Additionally, S ⊆ S3 ω sets can only have a club shot through them under certain circumstances. In our situation, we use the fact that the complement of these nonreflecting stationary sets will be fat, i.e. for any club C ⊆ α, E ∩ C contains closed sets of ordinals of arbitrarily large order-type below α, to allow us to shoot a club without collapsing κ. vii