Specializing trees and answer to a question of Williams

2017/08/09 by Mohammad Golshani, Saharon Shelah, Golshani, Mohammad +1
Mathematics · #FOS: Mathematics #Logic (math.LO) #math.LO

paper · pdf · doi:10.48550/arxiv.1708.02719

published as J. Math. Log. 21 No. 1 (2021) 2050023, 20

arxiv created 2020/03/10 · arxiv updated 2020/03/11

Abstract

We show that if cf(2^ℵ₀)=ℵ₁, then any non-trivial ℵ₁-closed forcing notion of size ≤ 2^ℵ₀ is forcing equivalent to Add(ℵ₁, 1), the Cohen forcing for adding a new Cohen subset of ω₁. We also produce, relative to the existence of suitable large cardinals, a model of ZFC in which 2^ℵ₀=ℵ₂ and all ℵ₁-closed forcing notion of size ≤ 2^ℵ₀ collapse ℵ₂, and hence are forcing equivalent to Add(ℵ₁, 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 ℵ₂, collapses ℵ₂ or ℵ₃.

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