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

Abstract

The weakly compact reflection principle$${\mathrm{Refl}}_{{\mathrm{wc}}}(\kappa )$$ 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.