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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2510.21374 |
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| _version_ | 1866911229734289408 |
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| author | Tornquist, Asger Schrittesser, David |
| author_facet | Tornquist, Asger Schrittesser, David |
| contents | Let $x$ denote a Laver real over $L$. We prove that in $L[x]$ there is a $Π^1_1$ infinite mad family. Since $Π^1_1$ and $Σ^1_2$ sets are Laver measurable in $L[x]$, this shows that there are examples of well-behaved classical pointclasses $Γ$, namely $Γ=Π^1_1$ and $Γ=Σ^1_2$, where $Γ$-uniformization and ``all sets in $Γ$ are Laver measurable'' hold, but there is a mad family in $Γ$. This result stands in contrast to that for reasonable pointclasses, the $Γ$-Ramsey property together with uniformization implies that there are no mad families in $Γ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_21374 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The happy coexistence of mad families and Laver measurability Tornquist, Asger Schrittesser, David Logic 03E15, 05D10 Let $x$ denote a Laver real over $L$. We prove that in $L[x]$ there is a $Π^1_1$ infinite mad family. Since $Π^1_1$ and $Σ^1_2$ sets are Laver measurable in $L[x]$, this shows that there are examples of well-behaved classical pointclasses $Γ$, namely $Γ=Π^1_1$ and $Γ=Σ^1_2$, where $Γ$-uniformization and ``all sets in $Γ$ are Laver measurable'' hold, but there is a mad family in $Γ$. This result stands in contrast to that for reasonable pointclasses, the $Γ$-Ramsey property together with uniformization implies that there are no mad families in $Γ$. |
| title | The happy coexistence of mad families and Laver measurability |
| topic | Logic 03E15, 05D10 |
| url | https://arxiv.org/abs/2510.21374 |