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On $\Sigma^1_3$ sets in the Sacks model
(2026) Schilhan, Jonathan
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.

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