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

TL;DR

This paper proves that in the iterated Sacks model over the constructible universe, the Mansfield-Solovay Theorem applies to -sets, establishing their measurability and complexity bounds for Bernstein sets.

Contribution

It extends the Mansfield-Solovay Theorem to -sets within the iterated Sacks model and discusses separations at higher projective levels.

Findings

-sets are Marczewski measurable in the Sacks model

Optimal complexity for Bernstein sets is 

The theorem can be separated at higher projective levels

Abstract

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.