On the Intermediate Models of Strongly Compact Prikry Forcing
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914548422803456 |
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| author | Benhamou, Tom Thei, Sebastiano Weltsch, Ben-Zion |
| author_facet | Benhamou, Tom Thei, Sebastiano Weltsch, Ben-Zion |
| contents | We analyze the intermediate models of the strongly compact Prikry forcing. We exhibit a simple combinatorial property which, for a given supercompact cardinal $κ$, characterize the projections of all projections of the strongly compact Prikry forcing using $κ$-complete fine measures. Considering level-by-level results, if $κ$ is $2^λ$-strongly compact, we characterize the forcings of size $\leqλ$ which are projections of that $λ$-strongly compact Prikry forcing. Our characterization generalizes several known results, including those of Benhamou-Hayut-Gitik and folklore results regarding the class of $κ$-distributive forcing notions which are embedded into the supercompact Prikry forcing. Fixing a $κ$-complete fine measure $\mathcal{U}$ on $P_κ(λ)$, we also provide Rudin-Keisler like critiria for the existence projections from the strongly compact Prikry forcing with $\mathcal{U}$. Finally, we prove that among all projections of the $λ$-strongly compact Prikry forcing, the class of forcings of cardinality $λ$ are exactly those for which there is a projection map which depends only on the stem of the Prikry condition. We also give partial results regarding projections of arbitrary cardinality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09161 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Intermediate Models of Strongly Compact Prikry Forcing Benhamou, Tom Thei, Sebastiano Weltsch, Ben-Zion Logic We analyze the intermediate models of the strongly compact Prikry forcing. We exhibit a simple combinatorial property which, for a given supercompact cardinal $κ$, characterize the projections of all projections of the strongly compact Prikry forcing using $κ$-complete fine measures. Considering level-by-level results, if $κ$ is $2^λ$-strongly compact, we characterize the forcings of size $\leqλ$ which are projections of that $λ$-strongly compact Prikry forcing. Our characterization generalizes several known results, including those of Benhamou-Hayut-Gitik and folklore results regarding the class of $κ$-distributive forcing notions which are embedded into the supercompact Prikry forcing. Fixing a $κ$-complete fine measure $\mathcal{U}$ on $P_κ(λ)$, we also provide Rudin-Keisler like critiria for the existence projections from the strongly compact Prikry forcing with $\mathcal{U}$. Finally, we prove that among all projections of the $λ$-strongly compact Prikry forcing, the class of forcings of cardinality $λ$ are exactly those for which there is a projection map which depends only on the stem of the Prikry condition. We also give partial results regarding projections of arbitrary cardinality. |
| title | On the Intermediate Models of Strongly Compact Prikry Forcing |
| topic | Logic |
| url | https://arxiv.org/abs/2605.09161 |