Archive for Mathematical Logic

, Volume 51, Issue 5–6, pp 635–645

Fragility and indestructibility of the tree property

Article

Abstract

We prove various theorems about the preservation and destruction of the tree property at ω2. Working in a model of Mitchell [9] where the tree property holds at ω2, we prove that ω2 still has the tree property after ccc forcing of size \({\aleph_1}\) or adding an arbitrary number of Cohen reals. We show that there is a relatively mild forcing in this same model which destroys the tree property. Finally we prove from a supercompact cardinal that the tree property at ω2 can be indestructible under ω2-directed closed forcing.

Keywords

Tree property Indestructibility Fragility Large cardinals Forcing 

Mathematics Subject Classification

03E35 03E55 

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA

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