Archive for Mathematical Logic

, Volume 54, Issue 5–6, pp 491–510 | Cite as

The least weakly compact cardinal can be unfoldable, weakly measurable and nearly \({\theta}\)-supercompact

  • Brent Cody
  • Moti Gitik
  • Joel David Hamkins
  • Jason A. Schanker


We prove from suitable large cardinal hypotheses that the least weakly compact cardinal can be unfoldable, weakly measurable and even nearly \({\theta}\)-supercompact, for any desired \({\theta}\). In addition, we prove several global results showing how the entire class of weakly compactcardinals, a proper class, can be made to coincide with the class of unfoldable cardinals, with the class of weakly measurable cardinals or with the class of nearly \({\theta_\kappa}\)-supercompact cardinals \({\kappa}\), for nearly any desired function \({\kappa\mapsto\theta_\kappa}\). These results answer several questions that had been open in the literature and extend to these large cardinals the identity-crises phenomenon, first identified by Magidor with the strongly compact cardinals.


Weakly compact Unfoldable Weakly measurable Nearly supercompact Identity crisis 

Mathematics Subject Classification

Primary 03E55 03E35 


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  1. 1.
    Dz̆amonja, M., Hamkins, J.D.: Diamond (on the regulars) can fail at any strongly unfoldable cardinal. Ann. Pure Appl. Log., 144(1–3), 83–95. Conference in honor of sixtieth birthday of James E. Baumgartner (2006)Google Scholar
  2. 2.
    Gitman, V., Hamkins, J.D., Johnstone, T.A.: What is the theory ZFC without Powerset? (submitted)Google Scholar
  3. 3.
    Gitman V., Welch P.D.: Ramsey-like cardinals II. J. Symb. Log. 76(2), 541–560 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Hamkins J.D.: The lottery preparation. Ann. Pure Appl. Log. 101(2–3), 103–146 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hamkins J.D.: Unfoldable cardinals and the GCH. J. Symb. Log. 66(3), 1186–1198 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hamkins J.D., Johnstone T.A.: Indestructible strong unfoldability. Notre Dame J. Form. Log. 51(3), 291–321 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Johnstone T.A.: Strongly unfoldable cardinals made indestructible. J. Symb. Log. 73(4), 1215–1248 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kunen K.: Saturated ideals. J. Symb. Log. 43(1), 65–76 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Magidor M.: How large is the first strongly compact cardinal? Or a study on identity crises. Ann. Math. Log. 10(1), 33–57 (1976)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Reitz, J.: The Ground Axiom. In: PhD thesis, The Graduate Center of the City University of New York (2006)Google Scholar
  11. 11.
    Schanker J.A.: Weakly measurable cardinals. MLQ Math. Log. Q. 57(3), 266–280 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Schanker, J.A.: Weakly measurable cardinals and partial near supercompactness. In: PhD thesis, CUNY Graduate Center (2011)Google Scholar
  13. 13.
    Schanker J.A.: Partial near supercompactness. Ann. Pure Appl. Log. 164(2), 67–85 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Villaveces A.: Chains of end elementary extensions of models of set theory. J. Symb. Log. 63(3), 1116–1136 (1998)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Brent Cody
    • 1
  • Moti Gitik
    • 2
  • Joel David Hamkins
    • 3
    • 4
  • Jason A. Schanker
    • 5
  1. 1.Department of Mathematics and Applied MathematicsVirginia Commonwealth UniversityRichmondUSA
  2. 2.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  3. 3.MathematicsThe Graduate Center of The City University of New YorkNew YorkUSA
  4. 4.Program in MathematicsCollege of Staten Island of CUNYStaten IslandUSA
  5. 5.Mathematics and Computer StudiesMolloy CollegeRockville CentreUSA

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