$\mathsf{MA} (\mathcal{I}$) and a Failure of Separation on the third Level

TL;DR

The paper demonstrates that certain weaker forcing axioms like $ ext{MA}( ext{I})$ do not guarantee separation at the third projective level, contrasting with stronger axioms like $ ext{BPFA}$.

Contribution

It constructs a model where $ ext{MA}( ext{I})$ holds but $ ext{Pi}^1_3$ and $ ext{Sigma}^1_3$-separation fail, showing limits of weaker forcing axioms.

Findings

$ ext{MA}( ext{I})$ can coexist with failure of $ ext{Pi}^1_3$-separation.

Weaker forcing axioms do not imply separation at the third projective level.

Contrasts with results under $ ext{BPFA}$ and $ ext{aleph}_1= ext{aleph}_1^L$.

Abstract

We present a method which forces the failure of $\Pi^1_3$ and $\Sigma^1_3$-separation, while $\mathsf{MA} (\mathcal{I}$) holds, for $\mathcal{I}$ the family of indestructible ccc forcings. This shows that, in contrast to the assumption $\mathsf{BPFA}$ and $\aleph_1=\aleph_1^L$ which implies $\Pi^1_3$-separation, that weaker forcing axioms do not decide separation on the third projective level.