Archive for Mathematical Logic

, Volume 51, Issue 5–6, pp 635–645 | Cite as

Fragility and indestructibility of the tree property



We prove various theorems about the preservation and destruction of the tree property at ω 2. Working in a model of Mitchell [9] where the tree property holds at ω 2, we prove that ω 2 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 ω 2 can be indestructible under ω 2-directed closed forcing.


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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