Ladder mice
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916281770311680 |
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| author | Schlutzenberg, Farmer |
| author_facet | Schlutzenberg, Farmer |
| contents | Assume ZF + AD + $V=L(\mathbb{R})$. We prove some "mouse set" theorems, for definability over $J_α(\mathbb{R})$ where $[α,α]$ is a projective-like gap (of $L(\mathbb{R})$) and $α$ is either a successor ordinal or has countable cofinality, but $α\neqβ+1$ where $β$ ends a strong gap. For such ordinals $α$ and integers $n\geq 1$, we show that there is a mouse $M$ with $\mathbb{R}\cap M=\mathrm{OD}_{αn}$.
The proof involves an analysis of ladder mice and their generalizations to $J_α(\mathbb{R})$. This analysis is related to earlier work of Rudominer, Woodin and Steel on ladder mice. However, it also yields a new proof of the mouse set theorem even at the least point where ladder mice arise -- one which avoids the stationary tower. The analysis also yields a corresponding "anti-correctness" result on a cone, generalizing facts familiar in the projective hierarchy; for example, that $(Π^1_3)^V\upharpoonright M_1$ truth is $(Σ^1_3)^{M_1}$-definable and $(Σ^1_3)^{M_1}$ truth is $(Π^1_3)^V\upharpoonright M_1$-definable.
We also define and study versions of ladder mice on a cone at the end of weak gap, and at the successor of the end of a strong gap, and an anti-correctness result on a cone there. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_06289 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ladder mice Schlutzenberg, Farmer Logic 03E45, 03E55, 03E15 Assume ZF + AD + $V=L(\mathbb{R})$. We prove some "mouse set" theorems, for definability over $J_α(\mathbb{R})$ where $[α,α]$ is a projective-like gap (of $L(\mathbb{R})$) and $α$ is either a successor ordinal or has countable cofinality, but $α\neqβ+1$ where $β$ ends a strong gap. For such ordinals $α$ and integers $n\geq 1$, we show that there is a mouse $M$ with $\mathbb{R}\cap M=\mathrm{OD}_{αn}$. The proof involves an analysis of ladder mice and their generalizations to $J_α(\mathbb{R})$. This analysis is related to earlier work of Rudominer, Woodin and Steel on ladder mice. However, it also yields a new proof of the mouse set theorem even at the least point where ladder mice arise -- one which avoids the stationary tower. The analysis also yields a corresponding "anti-correctness" result on a cone, generalizing facts familiar in the projective hierarchy; for example, that $(Π^1_3)^V\upharpoonright M_1$ truth is $(Σ^1_3)^{M_1}$-definable and $(Σ^1_3)^{M_1}$ truth is $(Π^1_3)^V\upharpoonright M_1$-definable. We also define and study versions of ladder mice on a cone at the end of weak gap, and at the successor of the end of a strong gap, and an anti-correctness result on a cone there. |
| title | Ladder mice |
| topic | Logic 03E45, 03E55, 03E15 |
| url | https://arxiv.org/abs/2406.06289 |