Separating Subversion Forcing Axioms

TL;DR

This paper introduces variants of forcing axioms and develops techniques to demonstrate their non-implications, revealing that certain strong axioms do not imply others like Martin's Maximum.

Contribution

It presents a general method for proving non-implications among forcing axioms and applies it to show specific independence results.

Findings

SCFA does not imply MA^+(σ-closed)

SubPFA does not imply Martin's Maximum

Developed a technique for non-implication proofs

Abstract

We study a family of variants of Jensen's\emph{subcomplete forcing axiom}, $\mathsf{SCFA}$ and \emph{subproper forcing axiom}, $\mathsf{SubPFA}$. Using these we develop a general technique for proving non-implications of $\mathsf{SCFA}$, $\mathsf{SubPFA}$ and their relatives and give several applications. For instance we show that $\mathsf{SCFA}$ does not imply $\mathsf{MA}^+(\sigma$-closed$)$ and $\mathsf{SubPFA}$ does not imply Martin's Maximum.