Stationary Reflection and the failure of SCH
We prove that from large cardinals it is consistent that there is a singular strong limit cardinal ν such that the singular cardinal hypothesis fails at ν and every collection of fewer than cf(ν) stationary subsets of ν+ reflects simultaneously. For uncountable cofinality, 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 cf(ν)=ω to below a single partially supercompact cardinal. The previous upper bound of infinitely many supercompact cardinals was due to Sharon. Joint work with Omer Ben-Neria and Yair Hayut.