Σ¹₃ sets in the Sacks model

2025/06/18 by Jonathan Schilhan, Schilhan, Jonathan
Mathematics · #03E15 #03E35 #03E45 #Advanced Topology and Set Theory #FOS: Mathematics #Limits and Structures in Graph Theory #Logic (math.LO) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2506.15308

openalex publication_date 2025/06/18 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28

Abstract

We show that in the iterated Sacks model over the constructible universe the Mansfield-Solovay Theorem holds for Σ¹₃ sets. In particular, every \mathbfΣ¹₃ 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.

Citations

• Constructibility real degrees in the side-by-side Sacks model
• Tree forcing and definable maximal independent sets in hypergraphs
• Solovay-type characterizations for forcing-algebras

Related

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• Menger and consonant sets in the Sacks model
• Constructibility real degrees in the side-by-side Sacks model
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• The Consistency of the \bfΣ¹₃-Separation Property