Adding a non-reflecting weakly compact set

Abstract

For $n<\omega$, we say that the ${\Pi }_n^1$-reflection principle holds at $\kappa$ and write ${\text{Refl}}_n(\kappa )$ 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 ${\text{Refl}}_n(\kappa )$ 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 $n=1$. 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.

Edited Sections 5 and 6 to match published version. This contains a correction to the previous version

References in corpus (1)

• Extensions with the approximation and cover properties have no new large cardinals