Stationary Reflection and the Failure OF SCH at $\aleph_{\omega_1}$

TL;DR

This paper demonstrates the consistency of stationary reflection at l_{41} with the failure of the Singular Cardinal Hypothesis, using large cardinal assumptions to combine these seemingly contradictory properties.

Contribution

It proves, from uncountably many supercompact cardinals, that l_{41} can be a strong limit with a large power set, while stationary sets reflect, answering a longstanding open question.

Findings

l_{41} can be strong limit with _{41}+1 power set.

Stationary reflection can coexist with the failure of SCH at l_{41}.

Consistency results based on large cardinal assumptions.

Abstract

Combining stationary reflection (a compactness property) with the failure of SCH (an instance of non-compactness) has been a long-standing theme. We obtain this at $\aleph_{\omega_1}$, answering a question of Ben-Neria, Hayut, and Unger: We prove from the existence of uncountably many supercompact cardinals the consistency of $\aleph_{\omega_1}$ is strong limit together with $2^{\aleph_{\omega_1}}>\aleph_{\omega_1+1}$ and every stationary set of $\aleph_{\omega_1+1}$ reflects.