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

Abstract

We present a method which forces the failure of ${\Pi }_3^1$ and ${\Sigma }_3^1$-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 ${\Pi }_3^1$-separation, that weaker forcing axioms do not decide separation on the third projective level.

arXiv admin note: text overlap with arXiv:2506.21778