Specializing trees and answer to a question of Williams

Abstract

We show that if $cf(2^{{\aleph }_0})={\aleph }_1,$ then any non-trivial ${\aleph }_1$-closed forcing notion of size $\le 2^{{\aleph }_0}$ is forcing equivalent to $Add({\aleph }_1,1),$ the Cohen forcing for adding a new Cohen subset of ${\omega }_1.$ We also produce, relative to the existence of suitable large cardinals, a model of $ZFC$ in which $2^{{\aleph }_0}={\aleph }_2$ and all ${\aleph }_1$-closed forcing notion of size $\le 2^{{\aleph }_0}$ collapse ${\aleph }_2,$ and hence are forcing equivalent to $Add({\aleph }_1,1).$ These results answer a question of Scott Williams from 1978. We also extend a result of Todorcevic and Foreman-Magidor-Shelah by showing that it is consistent that every partial order which adds a new subset of ${\aleph }_2,$ collapses ${\aleph }_2$ or ${\aleph }_3.$