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

DOI10.1016/J.APAL.2021.102960arXiv1902.04146MaRDI QIDQ6313936

Brent Cody, Chris Lambie-Hanson, Victoria Gitman

Publication date: 11 February 2019

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

Mathematics Subject Classification ID

Consistency and independence results (03E35) Large cardinals (03E55) Other combinatorial set theory (03E05) Other aspects of forcing and Boolean-valued models (03E40) Ordered sets and their cofinalities; pcf theory (03E04)

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