Transferring Compactness

DOI10.1112/JLMS.12940arXiv2307.06910OpenAlexW4399155831MaRDI QIDQ6443691

Author name not available (Why is that?)

Publication date: 13 July 2023

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 $ninomega$ 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 $mathbf{Pi}_n^1$-indescribable. We also show that it is consistent that there is a cardinal $kappaleq{2}^{o}mega$ such that $P_{k}appa(lambda)$ is $n$-stationary for all $lambdageqkappa$ and $ninomega$, answering a question of Sakai.

Full work available at URL: https://doi.org/10.1112/jlms.12940

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