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Squares and uncountably singularized cardinals

Volume 253 / 2021

Maxwell Levine, Dima Sinapova Fundamenta Mathematicae 253 (2021), 277-296 MSC: Primary 03E35; Secondary 03E55. DOI: 10.4064/fm955-9-2020 Published online: 25 November 2020


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.


  • Maxwell LevineInstitute of Mathematics
    Albert Ludwig University of Freiburg
    Ernst-Zermelo-Str. 1
    79104 Freiburg im Breisgau, Germany
  • Dima SinapovaDepartment of Mathematics,
    Statistics, and Computer Science
    University of Illinois at Chicago
    851 S. Morgan St.
    Chicago, IL 60607, U.S.A.

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