Israel Journal of Mathematics

, Volume 70, Issue 3, pp 381–394 | Cite as

Large cardinals imply that every reasonably definable set of reals is lebesgue measurable

  • Saharon Shelah
  • Hugh Woodin
Article

Abstract

We prove that if there is a supercompact cardinal or much smaller large cardinals, then every set of reals from L(R) is Lebesgue measurable, and similar results. We also introduce some large cardinals.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [F]
    M. Foreman,Potent axioms, Trans. Am. Math. Soc.294 (1986), 1–28.MATHCrossRefMathSciNetGoogle Scholar
  2. [FMS1]
    M. Foreman, M. Magidor and S. Shelah,Martin maximum, saturated ideals and non-regular ultrafilters I, Ann. of Math.127 (1988), 1–47.CrossRefMathSciNetGoogle Scholar
  3. [FMS2]
    M. Foreman, M. Magidor and S. Shelah,Martin maximum, saturated ideals and non-regular ultrafilters II, Ann. of Math.127 (1988), 521–545.CrossRefMathSciNetGoogle Scholar
  4. [L]
    R. Laver,Making the supercompactness of κ indestructible under κ-directed closed forcing, Isr. J. Math.29 (1978), 385–388.MATHCrossRefMathSciNetGoogle Scholar
  5. [Mg]
    M. Magidor,Precipitous ideal Σsets, Isr. J. Math.35 (1980), 109–134.MATHCrossRefMathSciNetGoogle Scholar
  6. [M1]
    Y. Moschovakis,Descriptive Set Theory, North-Holland, Amsterdam, 1980.MATHGoogle Scholar
  7. [MS]
    D. A. Martin and J. R. Steel,Determinacy in L(R)follows from large cardinals, Proc. Natl. Acad. Sci. U.S.A., submitted.Google Scholar
  8. [Mt]
    A. D. R. Mathias,On sequences generic in the sense of Prikry, J. Aust. Math. Soc.15 (1973), 409–416.MATHMathSciNetCrossRefGoogle Scholar
  9. [Sh1]
    S. Shelah,Proper Forcing, Lecture Notes in Math.940, Springer-Verlag, Berlin, 1982.MATHGoogle Scholar
  10. [Sh2]
    S. Shelah,Notes, part (A): On normal ideals and Boolean algebras, inAround Classification Theory, Lecture Notes in Math.1182, Springer-Verlag, Berlin, 1986.Google Scholar
  11. [Sh3]
    S. Shelah,Iterated forcing and normal ideals on θ 1, Isr. J. Math.60 (1987), 345–380.MATHCrossRefGoogle Scholar
  12. [Sh4]
    S. Shelah, Notes 3/85, to appear in [Sh 1], new edition.Google Scholar
  13. [So1]
    R. M. Solovay,The cardinality Σsets of reals, inFoundation of Mathematics, Symposium paper commemorating the sixtieth birthday of Kurt Godel, Springer-Verlag, Berlin, 1969, pp. 59–73.Google Scholar
  14. [So2]
    R. M. Solovay,A model of set theory in which every set of reals is Lebesgue measurable, Ann. of Math.92 (1970), 1–56.CrossRefMathSciNetGoogle Scholar
  15. [W1]
    H. Woodin, Embedding j:\(L\left( {V_{\lambda + 1} } \right) \to L\left( {V_{\lambda + 1} } \right)\) and determinacy, Lectures L.A., Cabal seminar, Spring 1984.Google Scholar
  16. [W2]
    H. Woodin,Every set of L(R)has a weakly homogeneous tree resp. from a supercompact, L.A., Cabal seminar, Spring 1985.Google Scholar
  17. [W3]
    H. Woodin, Σabsoluteness and supercompact cardinals, Notes, May 1985.Google Scholar

Copyright information

© The Weizmann Science Press of Israel 1990

Authors and Affiliations

  • Saharon Shelah
    • 1
    • 3
  • Hugh Woodin
    • 2
    • 3
  1. 1.Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael
  2. 2.Department of MathematicsCalifornia Institute of TechnologyPasadenaUSA
  3. 3.Department of MathematicsRutgers UniversityNew BrunswickUSA

Personalised recommendations