On Singular Stationarity II (tight stationarity and extenders-based methods)

TL;DR

This paper explores tightly stationary sets and demonstrates their consistency in certain models using variations of short-extenders forcing, advancing understanding of stationary set properties in set theory.

Contribution

It provides new consistency results showing that sequences of stationary sets can be tightly stationary in generic extensions, using extender-based forcing methods.

Findings

Sequences of stationary sets can be tightly stationary in generic extensions.

Variations of short-extenders forcing can establish these consistency results.

The work extends the understanding of stationary set behavior under forcing.

Abstract

We study the notion of tightly stationary sets which was introduced by Foreman and Magidor in \cite{ForMag-MS}. We obtain two consistency results which show that it is possible for a sequence of regular cardinals $( \kappa_n )_{n < \omega}$ to have the property that for every sequence $\vec{S}$, of some fixed-cofinality stationary sets $S_n \subseteq \kappa_n$, $\vec{S}$ is tightly stationary in a generic extension. The results are obtained using variations of the short-extenders forcing method.