Small u(kappa) at singular kappa with compactness at kappa++

Abstract

We show that the tree property, stationary reflection and the failure of approachability at ${\kappa }^{++}$ are consistent with $\mathfrak{u}(\kappa )={\kappa }^{+}<2^{\kappa }$, where $\kappa$ is a singular strong limit cardinal with the countable or uncountable cofinality. As a by-product, we show that if $\lambda$ is a regular cardinal, then stationary reflection at ${\lambda }^{+}$ is indestructible under all $\lambda$-cc forcings (out of general interest, we also state a related result for the preservation of club stationary reflection).

20 pages, submitted