New consequences of PFA($T^*$)

TL;DR

This paper explores the consequences of the forcing axiom PFA(T*) for almost Suslin trees, showing it implies several important set-theoretic principles and properties of trees.

Contribution

It extends known implications of PFA(T*) by proving it implies MRP, OGA, and certain isomorphism properties of Aronszajn trees, revealing new consequences.

Findings

PFA(T*) implies the Mapping Reflection Principle (MRP).

PFA(T*) implies the Open Graph Axiom (OGA).

All special Aronszajn trees are club-isomorphic under PFA(T*).

Abstract

Let $T^*$ be an almost Suslin tree, that is, an Aronszajn tree with no stationary antichains. Krueger introduced a forcing axiom, $\mathrm{PFA}(T^*)$, for the class of proper forcings that preserve that $T^*$ is almost Suslin. He showed that $\mathrm{PFA}(T^*)$ 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 $\mathrm{PFA}(T^*)$ also implies the Mapping Reflection Principle ($\mathrm{MRP}$) and the Open Graph Axiom ($\mathrm{OGA}$). Additionally, we show that $\mathrm{PFA}(T^*)$ implies that all special Aronszajn trees are club-isomorphic, but it does not imply that all almost Suslin trees are club-isomorphic.