October 20, 2025Open Access

Σ¹₃ sets in the Sacks model

Key Points

Every $Σ^1_3$ set is Marczewski measurable, indicating significant properties of these sets.
The optimal complexity of a Bernstein set is identified as $Δ^1_4$, which has important implications in set theory.
The study demonstrates the applicability of the Mansfield-Solovay Theorem in the Sacks model, enriching the understanding of related sets.
A method for separating the Mansfield-Solovay Theorem at non-trivial levels of the projective hierarchy is also presented.

Abstract

We show that in the iterated Sacks model over the constructible universe the Mansfield-Solovay Theorem holds for Σ¹₃ sets. In particular, every Σ¹₃ set is Marczewski measurable and the optimal complexity for a Bernstein set is Δ¹₄. 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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Cite This Study

Jonathan Schilhan (Wed,) studied this question.

synapsesocial.com/papers/68f6379bb481a140a36cf74c https://doi.org/https://doi.org/10.48550/arxiv.2506.15308

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