The weakly compact reflection principle need not imply a high order of weak compactness

TL;DR

This paper investigates the limits of the weakly compact reflection principle, showing it does not imply higher levels of weak compactness and exploring its preservation under forcing extensions.

Contribution

It demonstrates that the weakly compact reflection principle does not imply -weakly compactness and constructs models where -weakly compactness is minimal, extending known ideal preservation results.

Findings

The weakly compact reflection principle does not imply -weakly compactness.

Existence of forcing extensions where -weakly compactness is minimal.

Generalization of ideal preservation under -c.c. forcing to weakly compact ideals.

Abstract

The weakly compact reflection principle $\text{Refl}_{\text{wc}}(\kappa)$ states that $\kappa$ is a weakly compact cardinal and every weakly compact subset of $\kappa$ has a weakly compact proper initial segment. The weakly compact reflection principle at $\kappa$ implies that $\kappa$ is an $\omega$-weakly compact cardinal. In this article we show that the weakly compact reflection principle does not imply that $\kappa$ is $(\omega+1)$-weakly compact. Moreover, we show that if the weakly compact reflection principle holds at $\kappa$ then there is a forcing extension preserving this in which $\kappa$ is the least $\omega$-weakly compact cardinal. Along the way we generalize the well-known result which states that if $\kappa$ is a regular cardinal then in any forcing extension by $\kappa$-c.c. forcing the nonstationary ideal equals the ideal generated by the ground model nonstationary ideal; our generalization states that if $\kappa$ is a weakly compact cardinal then after forcing with a `typical' Easton-support iteration of length $\kappa$ the weakly compact ideal equals the ideal generated by the ground model weakly compact ideal.