Squares and uncountably singularized cardinals
Volume 253 / 2021
Abstract
It is known that if 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.