2022/08/04 by Radek Honzík, Chris Lambie‐Hanson, Honzik, Radek +3
Computer Science · Mathematics · #03E35 #03E55 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.2208.02684
openalex publication_date 2022/08/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that PFA (Proper Forcing Axiom) implies that adding any number of Cohen subsets of ω will not add an ω2-Aronszajn tree or a weak ω1-Kurepa tree, and moreover no σ-centered forcing can add a weak ω1-Kurepa tree (a tree of height and size ω1 with at least ω2 cofinal branches). This partially answers an open problem whether ccc forcings can add ω2-Aronszajn or ω1-Kurepa trees. We actually prove more: We show that a consequence of PFA, namely the guessing model principle, GMP, which is equivalent to the ineffable slender tree property, ISP, is preserved by adding any number of Cohen subsets of ω. And moreover, GMP implies that no σ-centered forcing can add a weak ω1-Kurepa tree. For more generality, we study the principle GMP at an arbitrary regular cardinal κ= κ<κ (we denote this principle GMPκ++), and as an application we show that there is a model in which there are no weak ℵω+1-Kurepa trees and no ℵω+2-Aronszajn trees.