A $\square(κ)$-like principle consistent with weak compactness

Brent Cody, Victoria Gitman, Chris Lambie-Hanson

Published 2019-02-11Version 1

Sun proved that when $\kappa$ is weakly compact, the \emph{$1$-club} subsets of $\kappa$ provide a filter base for the weakly compact ideal, and hence can also be used to give a characterization of weakly compact sets which resembles the definition of stationarity: a set $S\subseteq\kappa$ is weakly compact (or equivalently $\Pi^1_1$-indescribable) if and only if $S\cap C\neq\emptyset$ for every $1$-club $C\subseteq\kappa$. By replacing clubs with $1$-clubs in the definition of $\Box(\kappa)$, we obtain a $\Box(\kappa)$-like principle we call $\Box_1(\kappa)$ that is consistent with the weak compactness of $\kappa$ but inconsistent with the $\Pi^1_2$-indescribability of $\kappa$. By generalizing the standard forcing to add a $\Box(\kappa)$-sequence, we show that if $\kappa$ is $\kappa^+$-weakly compact and GCH holds then there is a cofinality-preserving forcing extension in which $\kappa$ remains $\kappa^+$-weakly compact and $\Box_1(\kappa)$ holds. If $\kappa$ is $\Pi^1_2$-indescribable and GCH holds then there is a cofinality-preserving forcing extension in which $\kappa$ is $\kappa^+$-weakly compact, $\Box_1(\kappa)$ holds and every weakly compact subset of $\kappa$ has a weakly compact proper initial segment. As an application, we prove that, relative to a $\Pi^1_2$-indescribable cardinal, it is consistent that $\kappa$ is $\kappa^+$-weakly compact, every weakly compact subset of $\kappa$ has a weakly compact proper initial segment, and there exist two weakly compact subsets $S^0$ and $S^1$ of $\kappa$ such that there is no $\beta<\kappa$ for which both $S^0\cap\beta$ and $S^1\cap\beta$ are weakly compact.