# Set Theory E-Seminar: Alejandro Poveda (Universitat de Barcelona) - Sigma-Prikry forcings and their iterations

Joint E-seminar of Bar-Ilan University and the Hebrew University

Title: Sigma-Prikry forcings and their iterations

Abstract:

In a joint project with A. Rinot and D. Sinapova we introduce a class of notions of forcing which we call $\Sigma$-Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are $\Sigma$-Prikry. Among these examples one may find Prikry forcing and its supercompact version, Gitik-Sharon forcing or the Extender Based Prikry forcing due to Gitik and Magidor.

Our first result shows that there is a functor $\mathbb{A}(\cdot,\cdot)$ which, given a $\Sigma$-Prikry poset $\mathbb P$ and a name for a non-reflecting stationary set $\dot{T}$, yields a $\Sigma$-Prikry poset $\mathbb{A}(\mathbb{P},\dot{T})$ that projects onto $\mathbb P$ and kills the stationarity of $T$. Afterwards, we develop a viable iteration scheme for $\Sigma$-Prikry posets.

In this talk I pretend to give an overview of this theory and, if time permits, present the very first application of the method: namely, the consistency of a failure of the SCH_\kappa with $Refl(<\omega,\kappa^+)$, where $\kappa$ is a strong limit singular cardinal of countable cofinality.

## Date:

Wed, 06/05/2020 - 11:00 to 13:00

## Location:

Zoom meeting ID 243-676-331