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.