Stationary Reflection and the failure of SCH

TL;DR

This paper demonstrates the consistency of a singular strong limit cardinal where the SCH fails at that cardinal, yet all small collections of stationary subsets of its successor reflect simultaneously, using large cardinal assumptions.

Contribution

It establishes the first known consistency of SCH failure at a singular strong limit with simultaneous reflection for uncountable cofinality, and reduces the consistency strength for countable cofinality cases.

Findings

SCH fails at a singular strong limit cardinal under large cardinal assumptions.

Simultaneous reflection of stationary sets occurs despite SCH failure.

Consistency strength is lowered to below a single partially supercompact cardinal for countable cofinality.

Abstract

In this paper we prove that from large cardinals it is consistent that there is a singular strong limit cardinal $\nu$ such that the singular cardinal hypothesis fails at $\nu$ and every collection of fewer than $\mathrm{cf}(\nu)$ stationary subsets of $\nu^+$ reflects simultaneously. For $\mathrm{cf}(\nu) > \omega$, this situation was not previously known to be consistent. Using different methods, we reduce the upper bound on the consistency strength of this situation for $\mathrm{cf}(\nu) = \omega$ to below a single partially supercompact cardinal. The previous upper bound of infinitely many supercompact cardinals was due to Sharon.