In this paper 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.Comment: 23 page