$\hat{I}{\pounds }_3^1$ sets in the Sacks model

Abstract

We show that in the iterated Sacks model over the constructible universe the Mansfield-Solovay Theorem holds for $\hat{I}{\pounds }_3^1$ sets. In particular, every $\hat{\mathbf{I}}{\pounds }_3^1$ set is Marczewski measurable and the optimal complexity for a Bernstein set is $\hat{I}{”}_4^1$. 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.

16 pages