$Σ_1$ gaps as derived models and correctness of mice
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866910939231551488 |
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| author | Schlutzenberg, Farmer Steel, John |
| author_facet | Schlutzenberg, Farmer Steel, John |
| contents | Assume ZF + AD + V=L(R). Let $[α,β]$ be a $Σ_1$ gap with $J_α(R)$ admissible. We analyze $J_β(R)$ as a natural form of "derived model" of a premouse $P$, where $P$ is found in a generic extension of $V$. In particular, we will have $\mathcal{P}(R)\cap J_β(R)=\mathcal{P}(R)\cap D$, and if $J_β(R)\models$ "$Θ$ exists", then $J_β(R)$ and $D$ in fact have the same universe. This analysis will be employed in further work, yet to appear, toward a resolution of a conjecture of Rudominer and Steel on the nature of $(L(R))^M$, for $ω$-small mice $M$. We also establish some preliminary work toward this conjecture in the present paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_08856 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $Σ_1$ gaps as derived models and correctness of mice Schlutzenberg, Farmer Steel, John Logic 03E45, 03E55, 03E60 Assume ZF + AD + V=L(R). Let $[α,β]$ be a $Σ_1$ gap with $J_α(R)$ admissible. We analyze $J_β(R)$ as a natural form of "derived model" of a premouse $P$, where $P$ is found in a generic extension of $V$. In particular, we will have $\mathcal{P}(R)\cap J_β(R)=\mathcal{P}(R)\cap D$, and if $J_β(R)\models$ "$Θ$ exists", then $J_β(R)$ and $D$ in fact have the same universe. This analysis will be employed in further work, yet to appear, toward a resolution of a conjecture of Rudominer and Steel on the nature of $(L(R))^M$, for $ω$-small mice $M$. We also establish some preliminary work toward this conjecture in the present paper. |
| title | $Σ_1$ gaps as derived models and correctness of mice |
| topic | Logic 03E45, 03E55, 03E60 |
| url | https://arxiv.org/abs/2307.08856 |