Trees and stationary reflection at double successors of regular cardinals

Abstract

We obtain an array of consistency results concerning trees and stationary reflection at double successors of regular cardinals $\kappa$, updating some classical constructions in the process. This includes models of $\mathsf{CSR}({\kappa }^{++})\wedge \mathsf{TP}({\kappa }^{++})$ (both with and without $\mathsf{AP}({\kappa }^{++})$) and models of the conjunctions $\mathsf{SR}({\kappa }^{++})\wedge \mathsf{wTP}({\kappa }^{++})\wedge \mathsf{AP}({\kappa }^{++})$ and $\neg \mathsf{AP}({\kappa }^{++})\wedge \mathsf{SR}({\kappa }^{++})$ (the latter was originally obtained in joint work by Krueger and the first author \cite{GilKru:8fold}, and is here given using different methods). Analogs of these results with the failure of $\mathsf{SH}({\kappa }^{++})$ are given as well. Finally, we obtain all of our results with an arbitrarily large $2^{\kappa }$, applying recent joint work by Honzik and the third author.