Forcing a $\square (\kappa )$-like principle to hold at a weakly compact cardinal

Abstract

Hellsten \cite{MR2026390} proved that when $\kappa$ is ${\Pi }_n^1$-indescribable, the \emph{$n$-club} subsets of $\kappa$ provide a filter base for the ${\Pi }_n^1$-indescribability ideal, and hence can also be used to give a characterization of ${\Pi }_n^1$-indescribable sets which resembles the definition of stationarity: a set $S\subseteq \kappa$ is ${\Pi }_n^1$-indescribable if and only if $S\cap C\ne \varnothing$ for every $n$-club $C\subseteq \kappa$. By replacing clubs with $n$-clubs in the definition of $\square (\kappa )$, one obtains a $\square (\kappa )$-like principle ${\square }_n(\kappa )$, a version of which was first considered by Brickhill and Welch \cite{BrickhillWelch}. The principle ${\square }_n(\kappa )$ is consistent with the ${\Pi }_n^1$-indescribability of $\kappa$ but inconsistent with the ${\Pi }_{n+1}^1$-indescribability of $\kappa$. By generalizing the standard forcing to add a $\square (\kappa )$-sequence, we show that if $\kappa$ is ${\kappa }^{+}$-weakly compact and $\mathrm{GCH}$ holds then there is a cofinality-preserving forcing extension in which $\kappa$ remains ${\kappa }^{+}$-weakly compact and ${\square }_1(\kappa )$ holds. If $\kappa$ is ${\Pi }_2^1$-indescribable and $\mathrm{GCH}$ holds then there is a cofinality-preserving forcing extension in which $\kappa$ is ${\kappa }^{+}$-weakly compact, ${\square }_1(\kappa )$ holds and every weakly compact subset of $\kappa$ has a weakly compact proper initial segment. As an application, we prove that, relative to a ${\Pi }_2^1$-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.

Changed title and added citations to Brickhill-Welch