References & Citations
Mathematics > Logic
Title: Transferring Compactness
(Submitted on 13 Jul 2023 (v1), last revised 25 Apr 2024 (this version, v3))
Abstract: 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 $\kappa$ that is $n$-$d$-stationary for all $n\in \omega$ but not weakly compact. This is in sharp contrast to the situation in the constructible universe $L$, where $\kappa$ being $(n+1)$-$d$-stationary is equivalent to $\kappa$ being $\mathbf{\Pi}^1_n$-indescribable. We also show that it is consistent that there is a cardinal $\kappa\leq 2^\omega$ such that $P_\kappa(\lambda)$ is $n$-stationary for all $\lambda\geq \kappa$ and $n\in \omega$, answering a question of Sakai.
Submission history
From: Tom Benhamou [view email][v1] Thu, 13 Jul 2023 17:15:39 GMT (71kb)
[v2] Fri, 14 Jul 2023 07:10:00 GMT (71kb)
[v3] Thu, 25 Apr 2024 19:30:10 GMT (59kb)
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