Set Theory Seminar: Yair Hayut (HUJI)

Wed, 20/10/202114:00-16:00
Ross 7

Stationary Reflection and the Successors of Singular cardinals (Part 1)

 In this series of talks I'm going to present a few old and new results concerning the consistency of special assertions at successors of singular cardinals (i.e. $\aleph_{\omega + 1}$) - the reflection principles. The so called "reflection principle" are properties of the form: 
Let X be a subset of $\lambda$. such that X has some property. Then there is some $M$ subset of $\lambda$ of small cardinality, such that X \cap M has the same property.
We will start with a very gentle introduction to Prikry forcing, showing its basic properties. Then, we will focus on stationary reflection and prove Magidor's theorem on the consistency of stationary reflection at $\aleph_{\omega+1}$, starting with supercompact cardinals. Then, we will show how to get stationary reflection except one bad set, using Prikry forcing. 
After that, we will work towards the stronger result, getting full stationary reflection at $\aleph_{\omega+1}$, using a variant of the Prikry forcing (Unger and H.). We will introduce the Extender Based Prikry forcing, and prove the consistency of stationary reflection with the negation of SCH, using a partial supercompact cardinal (Ben-Neria, Unger and H.).
In the last part, I will talk about a recent project with Magidor, improving the upper bound of the Delta reflection.       

The first lecture is going to be just an introduction and will require only familiarity with forcing.