On compactness of weak square at singulars of uncountable cofinality

Abstract

Cummings, Foreman, and Magidor proved that Jensen's square principle is non-compact at ${\aleph }_{\omega }$, meaning that it is consistent that ${\square }_{{\aleph }_n}$ holds for all $n<\omega$ while ${\square }_{{\aleph }_{\omega }}$ fails. We investigate the natural question of whether this phenomenon generalizes to singulars of uncountable cofinality. Surprisingly, we show that under some mild hypotheses, the weak square principle ${\square }_{\kappa }^{\ast }$ is in fact compact at singulars of uncountable cofinality, and that an even stronger version of these hypotheses is not enough for compactness of weak square at ${\aleph }_{\omega }$.

Partially revised

References in corpus (1)

• A Galvin-Hajnal theorem for generalized cardinal characteristics