More Derived Models in PFA
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866912912781606912 |
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| author | Levinson, Derek Trang, Nam Wilson, Trevor |
| author_facet | Levinson, Derek Trang, Nam Wilson, Trevor |
| contents | This paper makes significant progress towards resolving a conjecture relating strong forcing axioms like $PFA$ and the derived model at a limit of Woodin cardinals $κ$. In particular, using a concept called Covering Matrices, we show that the $Θ$ of the derived model at $κ$ is strictly less than $κ^+$ under various circumstances; in particular, this shows that the conclusion holds under $PFA$ if $κ$ is a limit of Woodin cardinals of cofinality $ω$ and the derived model does not satisfy $LSA$. Assuming a form of mouse capturing, we show that the derived model satisfies $AD_{\mathbb{R}}$ under $PFA$ when $κ$ is a regular limit of Woodin cardinals. If $κ$ is an indestructibly $(κ,κ^+)$-weakly compact limit of Woodin cardinals, then the derived model outright satisfies $AD_{\mathbb{R}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_16854 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | More Derived Models in PFA Levinson, Derek Trang, Nam Wilson, Trevor Logic 03E45, 03E60 This paper makes significant progress towards resolving a conjecture relating strong forcing axioms like $PFA$ and the derived model at a limit of Woodin cardinals $κ$. In particular, using a concept called Covering Matrices, we show that the $Θ$ of the derived model at $κ$ is strictly less than $κ^+$ under various circumstances; in particular, this shows that the conclusion holds under $PFA$ if $κ$ is a limit of Woodin cardinals of cofinality $ω$ and the derived model does not satisfy $LSA$. Assuming a form of mouse capturing, we show that the derived model satisfies $AD_{\mathbb{R}}$ under $PFA$ when $κ$ is a regular limit of Woodin cardinals. If $κ$ is an indestructibly $(κ,κ^+)$-weakly compact limit of Woodin cardinals, then the derived model outright satisfies $AD_{\mathbb{R}}$. |
| title | More Derived Models in PFA |
| topic | Logic 03E45, 03E60 |
| url | https://arxiv.org/abs/2602.16854 |