Adding a non-reflecting weakly compact set

Brent Cody

Published 2017-01-16Version 1

For $n<\omega$, we say that the $\Pi^1_n$-reflection principle holds at $\kappa$ and write $Refl_n(\kappa)$ if and only if every $\Pi^1_n$-indescribable subset of $\kappa$ has a $\Pi^1_n$-indescribable proper initial segment. The $\Pi^1_n$-reflection principle $Refl_n(\kappa)$ generalizes a certain stationary reflection principle and implies that $\kappa$ is $\Pi^1_n$-indescribable of order $\omega$. We prove that the converse of this implication is consistently false in the case $n=1$. Moreover, we prove that if $\kappa$ is $(\alpha+1)$-weakly compact 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.