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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.