Transferring Compactness

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 $\hat{I}º$ that is $n$-$d$-stationary for all $n\in \ddot{I}‰$ but not weakly compact. This is in sharp contrast to the situation in the constructible universe $L$, where $\hat{I}º$ being $(n+1)$-$d$-stationary is equivalent to $\hat{I}º$ being ${\hat{\mathbf{I}}}_n^1$-indescribable. We also show that it is consistent that there is a cardinal $\hat{I}º\le 2^{\ddot{I}}‰$ such that $P_{\hat{I}}º(\hat{I}»)$ is $n$-stationary for all $\hat{I}»\ge \hat{I}º$ and $n\in \ddot{I}‰$, answering a question of Sakai.

Accepted version