Archive for Mathematical Logic

, Volume 55, Issue 5–6, pp 785–798 | Cite as

The definable tree property for successors of cardinals



Strengthening a result of Leshem (J Symb Logic 65(3):1204–1214, 2000), we prove that the consistency strength of \(\textit{GCH}\) together with the definable tree property for all successors of regular cardinals is precisely equal to the consistency strength of existence of proper class many \(\varPi ^{1}_1\)-reflecting cardinals. Moreover it is proved that if \(\kappa \) is a supercompact cardinal and \(\lambda > \kappa \) is measurable, then there is a generic extension of the universe in which \(\kappa \) is a strong limit singular cardinal of cofinality \(\omega , ~ \lambda =\kappa ^+,\) and the definable tree property holds at \(\kappa ^+\). Additionally we can have \(2^\kappa > \kappa ^+,\) so that \(\textit{SCH}\) fails at \(\kappa \).


Aronszajn tree Definable tree property \(\varPi ^{1}_1\)-Reflecting cardinal Easton reverse iteration Extender based Prikry forcing 

Mathematics Subject Classification

03E35 03E55 


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  1. 1.
    Abraham, U.: Aronszajn trees on \(\aleph _2\) and \(\aleph _3\). Ann. Pure Appl. Logic 24, 213–230 (1983)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Cummings, J.: Iterated Forcing and Elementary Embeddings. Handbook of Set Theory, vol. 2. Springer, Dordrecht (2010)MATHGoogle Scholar
  3. 3.
    Cummings, J., Foremann, M.: The tree property. Adv. Math. 133(1), 1–32 (1998)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dobrinen, N., Friedman, S.D.: Homogeneous iteration and measure one covering relative to HOD. Arch. Math. Logic 47, 711–718 (2008)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Foreman, M., Magidor, M., Schindler, R.: The consistency strength of successive cardinals with the tree property. J. Symb. Logic 66(4), 1837–1847 (2001)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Jech, T.: Multiple Forcing. Cambridge University Press, London (1986)MATHGoogle Scholar
  7. 7.
    Jech, T.: Set Theory: The Third Millennium Edition. Revised and Expanded, Springer Monographs in Mathematics. Springer, Berlin (2003)Google Scholar
  8. 8.
    Leshem, A.: On the consistency of the definable tree property on \(\aleph _1\). J. Symb. Logic 65(3), 1204–1214 (2000)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Mitchell, W.: Aronszajn trees and the independence of the transfer property. Ann. Math. Logic 5, 473–478 (1973)MathSciNetGoogle Scholar
  10. 10.
    Magidor, M., Shelah, S.: The tree property at successors of singular cardinals. Arch. Math. Logic 35(5–6), 385–404 (1996)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Merimovich, C.: Supercompact extender based Prikry forcing. Arch. Math. Logic 50(5–6), 591–602 (2011)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Neeman, I.: Aronszajn trees and failure of the singular cardinal hypothesis. J. Math. Logic 9(1), 139–157 (2009)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Sinapova, D.: The tree property at \(\aleph _{\omega +1}\). J. Symb. Logic 77(1), 279–290 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Schindler, R., Steel, J.: The self-Iterability of \(L[E]\). J. Symb. Logic 74(3), 751–779 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceAmirkabir University of TechnologyTehranIran

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