A forcing axiom for a non-special Aronszajn tree

TL;DR

This paper introduces a new forcing axiom PFA(T*) tailored for a specific class of Aronszajn trees, demonstrating it implies many strong set-theoretic consequences similar to PFA while also exhibiting unique properties.

Contribution

It defines a forcing axiom PFA(T*) for proper forcings preserving a non-special Aronszajn tree and explores its implications, bridging properties of PFA and diamond principles.

Findings

PFA(T*)) implies the failure of very weak club guessing.

All continuum cardinal characteristics are greater than ω₁ under PFA(T*).

PFA(T*)) entails the P-ideal dichotomy.

Abstract

Suppose that $T^*$ is an $\omega_1$-Aronszajn tree with no stationary antichain. We introduce a forcing axiom PFA($T^*$) for proper forcings which preserve these properties of $T^*$. We prove that PFA($T^*$) implies many of the strong consequences of PFA, such as the failure of very weak club guessing, that all of the cardinal characteristics of the continuum are greater than $\omega_1$, and the $P$-ideal dichotomy. On the other hand, PFA($T^*$) implies some of the consequences of diamond principles, such as the existence of Knaster forcings which are not stationarily Knaster.