# Sh:1120

• Golshani, M., & Shelah, S. (2021). Specializing trees and answer to a question of Williams. J. Math. Log., 21(1), 2050023, 20.
• 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}
}