Israel Journal of Mathematics

, Volume 52, Issue 4, pp 273–292 | Cite as

Some results on consecutive large cardinals II: Applications of radin forcing

  • Arthur W. Apter


Letκ be a 3 huge cardinal in a countable modelV of ZFC, and letA andB be subsets of the successor ordinals <κ so thatAB={α<κ:α is a successor ordinal}. Using techniques of Gitik, we construct a choiceless modelN A of ZF of heightκ so thatN A ╞“ZF+⌍AC ω+ForαA, ℵa is a Ramsey cardinal+ForβB, ℵβ is a singular Rowbottom cardinal which carries a Rowbottom filter+Forγ a limit ordinal, ℵy is a Jonsson cardinal which carries a Jonsson filter”.


Partial Ordering Normal Measure Closure Property Regular Cardinal Measurable Cardinal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Apter,Some results on consecutive large cardinals, Ann. Pure Appl. Logic25 (1983), 1–17.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    E. Bull,Successive large cardinals, Ann. Math. Logic15 (1978), 161–191.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    M. Foreman and H. Woodin,The GCH can fail everywhere, to appear.Google Scholar
  4. 4.
    M. Gitik,Regular cardinals in models of ZF, Trans. Am. Math. Soc., to appear.Google Scholar
  5. 5.
    A. Kanamori and M. Magidor,The evolution of large cardinal axioms in set theory, Lecture Notes in Mathematics685, Springer-Verlag, Berlin, 1979.Google Scholar
  6. 6.
    E. Kleinberg,Infinitary Combinatorics and the Axiom of Determinateness, Lecture Notes in Mathematics612, Springer-Verlag, Berlin, 1977.MATHGoogle Scholar
  7. 7.
    R. Laver,Making the supercompactness of κ indestructible under κ directed closed forcing, Israel J. Math.29 (1978), 383–388.CrossRefMathSciNetGoogle Scholar
  8. 8.
    A. Lévy and R. Solovay,Measurable cardinals and the Continuum Hypothesis, Israel J. Math.5 (1967), 234–248.MATHMathSciNetGoogle Scholar
  9. 9.
    T. Menas, A combinatorial property ofP K(λ), J. Symbolic Logic41 (1975), 225–233.MathSciNetGoogle Scholar
  10. 10.
    T. Menas,On strong compactness and supercompactness, Ann. Math. Logic7 (1975), 327–359.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    W. Mitchell,How weak is a closed unbounded filter?, Logic Colloq. ’80 (Van Dalen, Lascar and Smiley, eds.), North-Holland, 1982.Google Scholar
  12. 12.
    K. Prikry,Changing measurable into accessible cardinals, Dissertationes Math. (Rozprany Mathematyczne)68 (1970), 5–52.MathSciNetGoogle Scholar
  13. 13.
    L. Radin,Adding closed cofinal sequences to large cardinals, Ann. Math. Logic23 (1982), 263–283.MathSciNetGoogle Scholar
  14. 14.
    F. Rowbottom,Some strong axioms of infinity incompatible with the axiom of constructibility, Ann. Math. Logic3 (1971), 1–44.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    H. Woodin, Handwritten notes on the closed unbounded filter.Google Scholar
  16. 16.
    H. Woodin, Handwritten notes on Radin forcing and the Prikry property.Google Scholar

Copyright information

© The Weizmann Science Press of Israel 1985

Authors and Affiliations

  • Arthur W. Apter
    • 1
  1. 1.Department of MathematicsRutgers UniversityNewarkUSA

Personalised recommendations