New consequences of PFA($T^{\ast }$)

Abstract

Let $T^{\ast }$ be an almost Suslin tree, that is, an Aronszajn tree with no stationary antichains. Krueger introduced a forcing axiom, $\mathrm{PFA}(T^{\ast })$, for the class of proper forcings that preserve that $T^{\ast }$ is almost Suslin. He showed that $\mathrm{PFA}(T^{\ast })$ implies several well-known consequences of the Proper Forcing Axiom ($\mathrm{PFA}$), including Suslin's Hypothesis and the P-ideal dichotomy. We extend this list by proving that $\mathrm{PFA}(T^{\ast })$ also implies the Mapping Reflection Principle ($\mathrm{MRP}$) and the Open Graph Axiom ($\mathrm{OGA}$). Additionally, we show that $\mathrm{PFA}(T^{\ast })$ implies that all special Aronszajn trees are club-isomorphic, but it does not imply that all almost Suslin trees are club-isomorphic.

v2: corrected citation to Lambie-Hanson's and Stejskalová's article