Title: The weakly compact reflection principle need not imply a high order of weak compactness,
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.