Singular cardinals and square properties

Abstract:

We analyze the effect of singularizing cardinals on square properties. By work of Džamonja-Shelah and of Gitik, if you singularize an inaccessible cardinal to countable cofinality while preserving its successor, then $\square _{\kappa , \omega }$ holds in the bigger model. We extend this to the situation where every regular cardinal in an interval $[\kappa ,\nu ]$ is singularized, for some regular cardinal $\nu$. More precisely, we show that if $V\subset W$, $\kappa <\nu$ are cardinals, where $\nu$ is regular in $V$, $\kappa$ is a singular cardinal in $W$ of countable cofinality, $\mathrm {cf}^W(\tau )=\omega$ for all $V$-regular $\kappa \leq \tau \leq \nu$, and $(\nu ^+)^V=(\kappa ^+)^W$, then $W\models \square _{\kappa ,\omega }$.