Forcing upper $Σ$-uniformization in the presence of lower $Π$-reduction or uniformization
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912693373370368 |
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| author | Hoffelner, Stefan |
| author_facet | Hoffelner, Stefan |
| contents | We present a method which allows the combination of forcing uniformization on the $Π$- and the $Σ$-side of the projective hierarchy to a certain extent. Using this method we construct a universe where $Π^1_3$-reduction holds, $Π^1_3$-uniformization fails, yet $Σ^1_n$ uniformization is true for $n \ge 4$. We also construct a universe where $Π^1_3$-uniformization holds and for every $n \ge 4, $ $ Σ^1_4$-uniformization holds, lowering best known upper bound for this statement from the existence of two Woodin cardinals to $Con(\ZFC)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_05081 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Forcing upper $Σ$-uniformization in the presence of lower $Π$-reduction or uniformization Hoffelner, Stefan Logic 03E15, 03E35, 03E45 We present a method which allows the combination of forcing uniformization on the $Π$- and the $Σ$-side of the projective hierarchy to a certain extent. Using this method we construct a universe where $Π^1_3$-reduction holds, $Π^1_3$-uniformization fails, yet $Σ^1_n$ uniformization is true for $n \ge 4$. We also construct a universe where $Π^1_3$-uniformization holds and for every $n \ge 4, $ $ Σ^1_4$-uniformization holds, lowering best known upper bound for this statement from the existence of two Woodin cardinals to $Con(\ZFC)$. |
| title | Forcing upper $Σ$-uniformization in the presence of lower $Π$-reduction or uniformization |
| topic | Logic 03E15, 03E35, 03E45 |
| url | https://arxiv.org/abs/2511.05081 |