On $\Sigma^1_3$ sets in the Sacks model

We show that in the iterated Sacks model over the constructible universe the Mansfield–Solovay theorem holds for $\Sigma^1_3$ sets. In particular, every $\mathbf{\Sigma}^1_3$ set is Marczewski measurable and the optimal complexity for a Bernstein set is $\Delta^1_4$. Based on a result by Kanovei, we also briefly show how to separate the Mansfield–Solovay theorem at non-trivial levels of the projective hierarchy.