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

We present a method which forces the failure of ${\mathrm{\Pi }}_3^1$ and ${\mathrm{\Sigma }}_3^1$-separation, while $\mathsf{M}\mathsf{A}(\mathcal{I}$) holds, for $\mathcal{I}$ the family of indestructible ccc forcings. This shows that, in contrast to the assumption $\mathsf{B}\mathsf{P}\mathsf{F}\mathsf{A}$ and ${\mathrm{\aleph }}_1={\mathrm{\aleph }}_1^L$ which implies ${\mathrm{\Pi }}_3^1$-separation, that weaker forcing axioms do not decide separation on the third projective level.