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

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.