$Σ^1_3$ sets in the Sacks model
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912437947596800 |
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| author | Schilhan, Jonathan |
| author_facet | Schilhan, Jonathan |
| contents | We show that in the iterated Sacks model over the constructible universe the Mansfield-Solovay Theorem holds for $Σ^1_3$ sets. In particular, every $\mathbfΣ^1_3$ set is Marczewski measurable and the optimal complexity for a Bernstein set is $Δ^1_4$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_15308 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $Σ^1_3$ sets in the Sacks model Schilhan, Jonathan Logic 03E15, 03E35, 03E45 We show that in the iterated Sacks model over the constructible universe the Mansfield-Solovay Theorem holds for $Σ^1_3$ sets. In particular, every $\mathbfΣ^1_3$ set is Marczewski measurable and the optimal complexity for a Bernstein set is $Δ^1_4$. 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. |
| title | $Σ^1_3$ sets in the Sacks model |
| topic | Logic 03E15, 03E35, 03E45 |
| url | https://arxiv.org/abs/2506.15308 |