Israel Journal of Mathematics

, Volume 35, Issue 1–2, pp 61–88 | Cite as

All uncountable cardinals can be singular

  • M. Gitik


Assuming the consistency of the existence of arbitrarily large strongly compact cardinals, we prove the consistency with ZF of the statement that every infinite set is a countable union of sets of smaller cardinality. Some other statements related to this one are investigated too.


Force Condition Finite Subset Generic Subset Countable Union Regular 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.
    K. Devlin and R. Jensen,Marginalia to a theorem of Silver, in Proc. Internat. Summer Institute and Logic Colloq., Lecture Notes in Math.499, Springer-Verlag, Berlin, 1975, pp. 115–143.Google Scholar
  2. 2.
    J. Dodd and R. Jensen,The core model, handwritten notes.Google Scholar
  3. 3.
    U. Felgner,Comparison of the axioms of local and universal choice, Fund. Math.71 (1971), 73–62.MathSciNetGoogle Scholar
  4. 4.
    T. J. Jech,Lectures in set theory with particular emphasis on the method of forcing, Lecture Notes in Mathematics217, Springer-Verlag, Berlin, 1971, p. 137.zbMATHGoogle Scholar
  5. 5.
    T. J. Jech,The Axiom of Choice, North-Holland, Amsterdam, 1977.Google Scholar
  6. 6.
    A. Levy,Independence results in set theory by Cohen's Method IV (abstract), Notices Amer. Math. Soc.10 (1963), 592–593.Google Scholar
  7. 7.
    A. Levy,Definability in Axiomatic Set Theory II, inMath Logic and Formulations of Set Theory, Proc. Internat. Colloq. (J. Bar-Hillel, ed.), North-Holland, Amsterdam, 1970, pp. 129–145.Google Scholar
  8. 8.
    K. Prikry,Changing measurable, Dissertationes Math.68 (1970), 5–52.MathSciNetGoogle Scholar
  9. 9.
    J. R. Shoenfield,Unramified forcing in ‘Axiomatic Set Theory’, Proc. Symposia in Pure Math.13 (D. Scott, ed.), Providence, Rhode Island, 1971, pp. 351–382.Google Scholar
  10. 10.
    E. Speker,Zur Axiomatik der Mengenlehre, Z. Math. Logik Grundlagen Math.3 (1957), 173–210.CrossRefMathSciNetGoogle Scholar

Copyright information

© The Weizmann Science Press of Israel 1980

Authors and Affiliations

  • M. Gitik
    • 1
  1. 1.Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael

Personalised recommendations