# Sh:1120

• Golshani, M., & Shelah, S. Specializing trees and answer to a question of Williams. arXiv: 1708.02719
• 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 the existence of some 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). Our 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, collapes \aleph_2 or \aleph_3.
@article{Sh:1120,
}