Trees and stationary reflection at double successors of regular cardinals

TL;DR

This paper presents new consistency results related to trees and stationary reflection principles at double successors of regular cardinals, extending classical constructions and exploring various combinations of these principles.

Contribution

It introduces novel models demonstrating the consistency of multiple stationary reflection and tree properties at cb2, including cases with large continuum and failure of cb2-strong hypotheses.

Findings

Models of cb2 with cb2-compact stationary reflection and tree properties.

Models with stationary reflection and weak tree property, with and without the approachability property.

Results hold even with arbitrarily large continuum, extending previous work.

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.