Adding a non-reflecting weakly compact set

Abstract

For $n<\omega$, we say that the $\Pi^1_n$-reflection principle holds at $\kappa$ and write $\text{Refl}_n(\kappa)$ if and only if $\kappa$ is a $\Pi^1_n$-indescribable cardinal and every $\Pi^1_n$-indescribable subset of $\kappa$ has a $\Pi^1_n$-indescribable proper initial segment. The $\Pi^1_n$-reflection principle $\text{Refl}_n(\kappa)$ generalizes a certain stationary reflection principle and implies that $\kappa$ is $\Pi^1_n$-indescribable of order $\omega$. We define a forcing which shows that the converse of this implication can be false in the case $n=1$. Moreover, we prove that if $\kappa$ is $(\alpha+1)$-weakly compact where $\alpha<\kappa^+$, then there is a forcing extension in which there is a weakly compact set $W\subseteq\kappa$ having no weakly compact proper initial segment, the class of weakly compact cardinals is preserved and $\kappa$ remains $(\alpha+1)$-weakly compact. Additionally, we prove a resurrection result for the $\Pi^1_1$-reflection principle.