Easton's theorem for the tree property below aleph_omega

2019/07/08 by Šárka Stejskalová, Sarka Stejskalova, Stejskalova, Sarka
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Limits and Structures in Graph Theory #math.LO

paper · pdf · doi:10.48550/arxiv.1907.03737

24 pages, submitted

arxiv created 2019/07/08 · arxiv updated 2019/07/09

Abstract

Starting with infinitely many supercompact cardinals, we show that the tree property at every cardinal ℵ_n, 1 < n <ω, is consistent with an arbitrary continuum function below ℵ_ω which satisfies 2^ℵ_n > ℵ_n+1, n<ω. Thus the tree property has no provable effect on the continuum function below ℵ_ω except for the restriction that the tree property at κ⁺⁺ implies 2^κ>κ⁺ for every infinite κ.

Related

• The Tree Property up to ℵ_ω²
• Tree Properties at Successors of Singulars of Many Cofinalities
• The strong tree property and weak square
• Strong Tree Properties for Small Cardinals
• Strong Tree Properties at Successors of Singular Cardinals