# Sh:1120

- Golshani, M., & Shelah, S. (2021).
*Specializing trees and answer to a question of Williams*. J. Math. Log.,**21**(1), 2050023, 20. arXiv: 1708.02719 DOI: 10.1142/S0219061320500233 MR: 4194557 -
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. - Version 2020-06-03 (23p) published version (20p)

Bib entry

@article{Sh:1120, author = {Golshani, Mohammad and Shelah, Saharon}, title = {{Specializing trees and answer to a question of Williams}}, journal = {J. Math. Log.}, fjournal = {Journal of Mathematical Logic}, volume = {21}, number = {1}, year = {2021}, pages = {2050023, 20}, issn = {0219-0613}, mrnumber = {4194557}, mrclass = {03E35 (03E05 03E55)}, doi = {10.1142/S0219061320500233}, note = {\href{https://arxiv.org/abs/1708.02719}{arXiv: 1708.02719}}, arxiv_number = {1708.02719} }