Archive for Mathematical Logic

, Volume 56, Issue 7–8, pp 911–934 | Cite as

The nonstationary ideal on \(P_\kappa (\lambda )\) for \(\lambda \) singular

  • Pierre MatetEmail author
  • Saharon Shelah


We give a new characterization of the nonstationary ideal on \(P_\kappa (\lambda )\) in the case when \(\kappa \) is a regular uncountable cardinal and \(\lambda \) a singular strong limit cardinal of cofinality at least \(\kappa \).


\(P_\kappa (\lambda )\) Nonstationary ideal Precipitous ideal 

Mathematics Subject Classification

03E05 03E55 


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The authors would like to express their gratitude to the referee for a number of helpful suggestions.


  1. 1.
    Abe, Y.: A hierarchy of filters smaller than \(CF_{\kappa \lambda }\). Arch. Math. Log. 36, 385–397 (1997)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Carr, D.M.: The minimal normal filter on \(P_\kappa \lambda \). Proc. Am. Math. Soc. 86, 316–320 (1982)Google Scholar
  3. 3.
    Cummings, J.: A model in which GCH holds at successors but fails at limits. Trans. Am. Math. Soc. 329, 1–39 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Foreman, M.: Potent axioms. Trans. Am. Math. Soc. 294, 1–28 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Galvin, F., Jech, T., Magidor, M.: An ideal game. J. Symb. Log. 43, 284–292 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gitik, M., Magidor, M.: The singular cardinal hypothesis revisited. In: Judah, H., Just, W., Woodin, W.H. (eds.) Set Theory of the Continuum Mathematical Sciences Research Institute Publication # 26, pp. 243–279. Springer, Berlin (1992)Google Scholar
  7. 7.
    Goldring, N.: The entire NS ideal on \(\cal{P}_\gamma \mu \) can be precipitous. J. Symb. Log. 62, 1161–1172 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Matet, P.: Large cardinals and covering numbers. Fundam. Math. 205, 45–75 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Matet, P.: Weak saturation of ideals on \(P_\kappa (\lambda )\). Math. Log. Q. 57, 149–165 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Matet, P., Péan, C., Shelah, S.: Cofinality of normal ideals on \(P_\kappa (\lambda )\)II. Isr. J. Math. 150, 253–283 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Matet, P., Péan, C., Shelah, S.: Cofinality of normal ideals on \(P_\kappa (\lambda )\)I. Arch. Math. Logic. 55, 799–834 (2016)Google Scholar
  12. 12.
    Matet, P., Rosłanowki, A., Shelah, S.: Cofinality of the nonstationary ideal. Trans. Am. Math. Soc. 357, 4813–4837 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Matsubara, Y., Shelah, S.: Nowhere precipitousness of the non-stationary ideal over \({\cal{P}}_\kappa \lambda \). J. Math. Log. 2, 81–89 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Matsubara, Y., Shioya, M.: Nowhere precipitousness of some ideals. J. Symb. Log. 63, 1003–1006 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Shelah, S.: Cardinal arithmetic. In: Oxford Logic Guides, vol. 29. Oxford University Press, Oxford (1994)Google Scholar
  16. 16.
    Shelah, S.: On the existence of large subsets of \([\lambda ]^{<\kappa }\) which contain no unbounded non-stationary subsets. Arch. Math. Log. 41, 207–213 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Laboratoire de MathématiquesUniversité de Caen - CNRSCaen CedexFrance
  2. 2.Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael
  3. 3.Department of MathematicsRutgers UniversityNew BrunswickUSA

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