More notions of forcing add a square

Abstract

Foreman and Magidor showed that the continuum hypothesis implies the existence of a countably-closed ${\aleph }_2$-cc forcing notion $\mathbb{P}$ for adding ${\square }_{{\aleph }_1}$. Here, we show that $\mathbb{P}$ may consistently be realized as an ${\aleph }_2$-Souslin tree. More generally, we prove that ${\square }_{\hat{I}}»$ may be added by a $\hat{I}{»}^{+}$-Souslin tree, providing the first analog of the Foreman--Magidor forcing at the level of successors of singular cardinals. Our construction is uniform and extends to inaccessible cardinals as well.