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 $κ$ 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.