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 strongly inaccessible 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 ${\Pi }_n^1$-indescribable. We also show that it is consistent that there is a cardinal $\kappa ⩽2^{\omega }$ such that $P_{\kappa }\left(\lambda \right)$ is $n$-stationary for all $\lambda ⩾\kappa$ and $n\in \omega$, answering a question of Sakai.