New consequences of PFA($T^*$)

TL;DR

The paper investigates which consequences of the Proper Forcing Axiom carry over to the relativized axiom $\mathrm{PFA}(T^*)$ for almost Suslin trees. It constructs $T^*$-proper forcings to establish $\mathrm{MRP}$ and $\mathrm{OGA}$ under $\mathrm{PFA}(T^*)$, and uses the Abraham–Shelah forcing to show that $\,\mathrm{PFA}(T^*)$ implies that all normal special Aronszajn trees are club-isomorphic. It then exhibits a diamond-based construction showing that $\mathrm{PFA}(T^*)$ does not force all normal almost Suslin trees to be club-isomorphic, yielding a negative answer to a natural analogue of the PFA-club isomorphism phenomenon for almost Suslin trees. The results together strengthen the understanding of how relativized forcing axioms interact with tree combinatorics, including implications for weak Kurepa trees and $\omega_2$-Aronszajn trees.

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.

Theorems & Definitions (42)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Lemma 1.4
• Lemma 1.5
• Definition 1.6
• Definition 1.7
• Definition 1.8
• Theorem 1.9
• Definition 2.1