Specializing trees and answer to a question of Williams

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

paper · doi:10.48550/arxiv.1708.02719

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.

Related

• More notions of forcing add a square
• Small forcings and Cohen reals
• Indestructibility of some compactness principles over models of PFA
• Specializing Trees with Small Approximations I
• Alternative Cichoń Diagrams and Forcing Axioms Compatible with CH