Specializing trees and answer to a question of Williams

TL;DR

This paper proves that under certain set-theoretic assumptions, all non-trivial -closed forcing notions of size  are equivalent to Cohen forcing, answering a question from 1978 and extending results about forcing and cardinal collapse.

Contribution

It establishes the uniqueness of -closed forcing notions of a given size under specific conditions and constructs models where these notions collapse cardinals, addressing longstanding questions.

Findings

-closed forcing notions of size  are equivalent to Cohen forcing under certain conditions.

Existence of models where all forcing notions adding subsets of  collapse  or .

Answers to Scott Williams' 1978 question about forcing equivalences.

Abstract

We show that if $cf(2^{\aleph_0})=\aleph_1,$ then any non-trivial $\aleph_1$-closed forcing notion of size $\leq 2^{\aleph_0}$ 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 $\leq 2^{\aleph_0}$ 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 $\aleph_3.$