Transferring Compactness

TL;DR

This paper uses Radin forcing to transfer compactness properties from weakly inaccessible to strongly inaccessible cardinals, constructing models with unique stationary properties and answering open questions about stationarity.

Contribution

It introduces a novel application of Radin forcing to transfer compactness properties and constructs models with specific stationary characteristics.

Findings

Constructed a model with a cardinal that is n-d-stationary for all n but not weakly compact.

Demonstrated the consistency of a cardinal with universal n-stationarity for all λ ≥ κ.

Answered Sakai's question by showing the existence of a cardinal with n-stationarity properties across power sets.

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.