Zachary Faubion: Obtaining stationary reflecion at small singulars cardinal via Prikry type forcings I

Monday, April 16, 2012, 4:00 PM
RH 440R

Speaker: Zachary Faubion (UCI)

Title: Obtaining stationary reflecion at small singulars cardinal via Prikry type forcings I

Abstract:

Given a regular cardinal $\kappa$, an uncountably cofinal ordinal $\nu<\kappa$ is a reflection point of the stationry set $S\subseteq\kappa$ just in the case where $S\cap\alpha$ is stationary in $\alpha$. Starting from ininitely many supercompact cardinals, Magidor constructed a model of set theory where every stationary $S\subseteq\aleph_{\omega+1}$ has a reflection point. In this series of talks we present a construction of a model of set theory where we obtain a large amount of stationary reflection (although not full) using a significantly weaker large cardinal hypothesis. We start from a quasicompact (quasicompactness is a large cardinal hypothesis significantly weaker than any nontrivial variant of supercompactness) cardinal $\kappa$ and use modified Prikry forcing to turn $\kappa$ into $\aleph_{\omega+1}$. We then show that in the resulting model every stationray $S\subseteq\aleph_{\omega+1}$ not concentrating on ordinals of ground model cofinality $\kappa$ has a reflection point.

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