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arXiv · 2506.15308

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

Jonathan Schilhan
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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.

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Jonathan Schilhan. 2025-06-18. $\Sigma^1_3$ sets in the Sacks model. https://arxiv.org/abs/2506.15308

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