Transferring Compactness
Fuente:
arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909182439981056 |
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| author | Benhamou, Tom Zhang, Jing |
| author_facet | Benhamou, Tom Zhang, Jing |
| contents | We demonstrate that the technology of Radin forcing can be used to transfer compactness properties at a weakly inaccessible but not strong limit cardinal to a strongly inaccessible cardinal.
As an application, relative to the existence of large cardinals, we construct a model of set theory in which there is a cardinal $κ$ that is $n$-$d$-stationary for all $n\in ω$ but not weakly compact. This is in sharp contrast to the situation in the constructible universe $L$, where $κ$ being $(n+1)$-$d$-stationary is equivalent to $κ$ being $\mathbfΠ^1_n$-indescribable. We also show that it is consistent that there is a cardinal $κ\leq 2^ω$ such that $P_κ(λ)$ is $n$-stationary for all $λ\geq κ$ and $n\in ω$, answering a question of Sakai. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_06910 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Transferring Compactness Benhamou, Tom Zhang, Jing Logic We demonstrate that the technology of Radin forcing can be used to transfer compactness properties at a weakly inaccessible but not strong limit cardinal to a strongly inaccessible cardinal. As an application, relative to the existence of large cardinals, we construct a model of set theory in which there is a cardinal $κ$ that is $n$-$d$-stationary for all $n\in ω$ but not weakly compact. This is in sharp contrast to the situation in the constructible universe $L$, where $κ$ being $(n+1)$-$d$-stationary is equivalent to $κ$ being $\mathbfΠ^1_n$-indescribable. We also show that it is consistent that there is a cardinal $κ\leq 2^ω$ such that $P_κ(λ)$ is $n$-stationary for all $λ\geq κ$ and $n\in ω$, answering a question of Sakai. |
| title | Transferring Compactness |
| topic | Logic |
| url | https://arxiv.org/abs/2307.06910 |