Adding a Nonreflecting Weakly Compact Set

Abstract

For $n<\omega$, we say that the${\Pi }_n^1$-reflection principle holds at $\kappa$ and write ${Refl}_n\left(\kappa \right)$ if and only if $\kappa$ is a ${\Pi }_n^1$-indescribable cardinal and every ${\Pi }_n^1$-indescribable subset of $\kappa$ has a ${\Pi }_n^1$-indescribable proper initial segment. The ${\Pi }_n^1$-reflection principle ${Refl}_n\left(\kappa \right)$ generalizes a certain stationary reflection principle and implies that $\kappa$ is ${\Pi }_n^1$-indescribable of order $\omega$. We define a forcing which shows that the converse of this implication can be false in the case $n=1$; that is, we show that $\kappa$ being ${\Pi }_1^1$-indescribable of order $\omega$ need not imply ${Refl}_1\left(\kappa \right)$. 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. We also formulate several open problems and highlight places in which standard arguments seem to break down.