Archive for Mathematical Logic

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

Fragility and indestructibility of the tree property

Article

Abstract

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.

Keywords

Tree property Indestructibility Fragility Large cardinals Forcing 

Mathematics Subject Classification

03E35 03E55 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abraham U.: Aronszajn trees on \({\aleph_2}\) and \({\aleph_3}\) . Ann. Pure Appl. Log. 24(3), 213–230 (1983)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Cummings J., Foreman M.: The tree property. Adv. Math. 133(1), 1–32 (1998)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Devlin K.J.: Constructibility. Springer, Berlin (1984)MATHGoogle Scholar
  4. 4.
    Foreman, M., Magidor, M.: UnpublishedGoogle Scholar
  5. 5.
    Jech T.J.: Set Theory, 3rd edn. Springer, Berlin (2003)MATHGoogle Scholar
  6. 6.
    König D.: Sur les correspondence multivoques des ensembles. Fundam. Math. 8, 114–134 (1926)MATHGoogle Scholar
  7. 7.
    Knig B., Yoshinobu Y.: Fragments of martin’s maximum in generic extensions. Math. Log. Q. 50(3), 297–302 (2004)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Magidor M., Shelah S.: The tree property at successors of singular cardinals. Arch. Math. Log. 35(5–6), 385–404 (1996) doi: 10.1007/s001530050052 MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Mitchell W.: Aronszajn trees and the independence of the transfer property. Ann. Pure Appl. Log. 5, 21–46 (1972)MATHGoogle Scholar
  10. 10.
    Neeman, I.: The tree property up to \({\aleph_{\omega+1}}\) (to appear)Google Scholar
  11. 11.
    Sinpova, D.: The tree property and the failure of the singular cardinal hypothesis at \({\aleph_{\omega^2}}\) (to appear)Google Scholar
  12. 12.
    Specker E.: Sur un problème de Sikorski. Colloquium Mathematicum 2, 9–12 (1949)MathSciNetMATHGoogle Scholar
  13. 13.
    Viale M., Wei C.: On the consistency strength of the proper forcing axiom. Adv. Math. 228(5), 2672–2687 (2011)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations