Stationary Reflection and the failure of SCH

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.

arXiv comment28 pages