Archive for Mathematical Logic

, Volume 46, Issue 2, pp 61–72 | Cite as

Lifting elementary embeddings j: V λV λ

  • Paul Corazza


We describe a fairly general procedure for preserving I3 embeddings j: V λV λ via λ-stage reverse Easton iterated forcings. We use this method to prove that, assuming the consistency of an I3 embedding, V =  HOD is consistent with the theory ZFC + WA where WA is an axiom schema in the language {∈, j} asserting a strong but not inconsistent form of “there is an elementary embedding VV”. This improves upon an earlier result in which consistency was established assuming an I1 embedding.


Large cardinal Forcing Wholeness axiom Elementary embedding I3 Liftings WA 

Mathematics Subject Classification (2000)

Primary 03E55 Secondary 03E40 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baumgartner J. (1983). Iterated forcing. In: Mathias, A.R.D. (eds) Surveys in Set Theory., pp 1–59. Cambridge University Press, Cambridge Google Scholar
  2. 2.
    Corazza, P.: Indestructibility of wholeness (in preparation)Google Scholar
  3. 3.
    Corazza P. (2006). The spectrum of elementary embeddings j: VV. Ann. Pure Appl. Logic 139: 327–399 CrossRefMathSciNetGoogle Scholar
  4. 4.
    Corazza P. (2000). Consistency of V = HOD with the wholeness axiom. Arch. Math. Logic 39: 219–226 CrossRefMathSciNetGoogle Scholar
  5. 5.
    Corazza P. (1999). Laver sequences for extendible and super-almost-huge cardinals. J. Symbolic Logic 64: 963–983 CrossRefMathSciNetGoogle Scholar
  6. 6.
    Corazza P. (2000). The wholeness axiom and Laver sequences. Ann. Pure Appl. Logic 105: 157–260 CrossRefMathSciNetGoogle Scholar
  7. 7.
    Friedman S. (2000). Fine Structure and Class Forcing. Walter de Gruyter, New York zbMATHGoogle Scholar
  8. 8.
    Hamkins J.D. (2001). The wholeness axioms and V = HOD. Arch. Math. Logic 40: 1–8 CrossRefMathSciNetGoogle Scholar
  9. 9.
    Hamkins J.D. (1994). Fragile measurability. J. Symbolic Logic 59: 262–282 CrossRefMathSciNetGoogle Scholar
  10. 10.
    Jech T. (1978). Set theory. Academic, New York Google Scholar
  11. 11.
    Kunen K. (1980). Set theory: an introduction to independence proofs. North-Holland Publishing Company, New York zbMATHGoogle Scholar
  12. 12.
    Kunen K. (1971). Elementary embeddings and infinitary combinatorics. J. Symbolic Logic 36: 407–413 CrossRefMathSciNetGoogle Scholar
  13. 13.
    Menas T. (1976). Consistency results concerning supercompactness. Trans. Am. Math. Soc. 223: 61–91 CrossRefMathSciNetGoogle Scholar
  14. 14.
    Roguski, S.: The theory of the class HOD. In: Set Theory and Hierarchy Theory, Lecture Notes in Mathematics, 619. Springer, Heidelberg (1977)Google Scholar
  15. 15.
    Zadrożny W. (1983). Iterating ordinal definability. Ann. Pure Appl. Logic 24: 263–310 CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceMaharishi University of ManagementFairfieldUSA

Personalised recommendations