Squares and uncountably singularized cardinals
Volume 253 / 2021
Abstract
It is known that if $\kappa $ is inaccessible in $V$, and $W$ is an outer model of $V$ such that $(\kappa ^+)^V = (\kappa ^+)^W$, and $\mathop{\rm cf} ^W \!(\kappa ) = \omega $, then $\square _{\kappa ,\omega }$ holds in $W$. Many strengthenings of this theorem have been investigated as well. We show that this theorem does not generalize to uncountable cofinalities: There is a model $V$ in which $\kappa $ is inaccessible and there is a forcing extension $W$ of $V$ in which $(\kappa ^+)^V = (\kappa ^+)^W$, $\omega \lt \mathop{\rm cf} ^W \!(\kappa ) \lt \kappa $, and $\square _{\kappa ,\tau }$ fails in $W$ for all $\tau \lt \kappa $. We make use of Magidor’s forcing for singularizing an inaccessible $\kappa $ to have uncountable cofinality. Along the way, we analyze stationary reflection in this model, and we show that it is possible for $\square _{\kappa ,\mathop{\rm cf} (\kappa )}$ to hold in a forcing extension by Magidor’s poset if the ground model is prepared with a partial square sequence.
