Archive for Mathematical Logic

, Volume 55, Issue 1–2, pp 19–35 | Cite as

Superstrong and other large cardinals are never Laver indestructible

  • Joan Bagaria
  • Joel David Hamkins
  • Konstantinos Tsaprounis
  • Toshimichi Usuba
Article

Abstract

Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, superstrongly unfoldable cardinals, Σn-reflecting cardinals, Σn-correct cardinals and Σn-extendible cardinals (all for n ≥  3) are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if κ exhibits any of them, with corresponding target θ, then in any forcing extension arising from nontrivial strategically <κ-closed forcing \({\mathbb{Q} \in V_\theta}\), the cardinal κ will exhibit none of the large cardinal properties with target θ or larger.

Keywords

Large cardinals Forcing Indestructible cardinals 

Mathematics Subject Classification

03E55 03E40 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Apter A.W., Gitik M.: The least measurable can be strongly compact and indestructible. J. Symb. Log. 63(4), 1404–1412 (1998)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Apter A.W., Hamkins J.D.: Universal indestructibility. Kobe J. Math. 16(2), 119–130 (1999)MathSciNetMATHGoogle Scholar
  3. 3.
    Apter A.W.: Indestructibility and strong compactness. Proc. Log. Colloq. 2003 LNL 24, 27–37 (2006)MathSciNetGoogle Scholar
  4. 4.
    Apter A.W.: The least strongly compact can be the least strong and indestructible. Ann. Pure Appl. Log. 144, 33–42 (2006)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Brooke-Taylor A.D.: Indestructibility of Vopĕnka’s principle. Arch. Math. Log. 50(5–6), 515–529 (2011)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Fuchs, G., Hamkins, J.D., Reitz, J.: Set-theoretic geology. Ann. Pure Appl. Log. 166(4), 464–501 (2015)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Gitik M., Shelah S.: On certain indestructibility of strong cardinals and a question of Hajnal. Arch. Math. Log. 28(1), 35–42 (1989)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Hamkins J.: Fragile measurability. J. Symb. Log. 59(1), 262–282 (1994)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Hamkins, J.D.: Lifting and extending measures; fragile measurability. Ph.D. thesis, University of California, Berkeley, Department of Mathematics, May (1994)Google Scholar
  10. 10.
    Hamkins J.D.: Small forcing makes any cardinal superdestructible. J. Symb. Log. 63(1), 51–58 (1998)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Hamkins J.D.: Gap forcing: generalizing the Lévy–Solovay theorem. Bull. Symb. Log. 5(2), 264–272 (1999)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Hamkins J.D.: The lottery preparation. Ann. Pure Appl. Log. 101(2–3), 103–146 (2000)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Hamkins J.D.: Gap forcing. Israel J. Math. 125, 237–252 (2001)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Hamkins J.D.: Extensions with the approximation and cover properties have no new large cardinals. Fund. Math. 180(3), 257–277 (2003)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Hamkins J.D.: The Ground Axiom. Mathematisches Forschungsinstitut Oberwolfach Report 55, 3160–3162 (2005)Google Scholar
  16. 16.
    Hamkins, J.D.: (mathoverflow.net/users/1946). Can a model of set theory be realized as a Cohen-subset forcing extension in two different ways, with different grounds and different cardinals? MathOverflow, 2013. http://mathoverflow.net/questions/120546 (version: 2013-02-01)
  17. 17.
    Hamkins, J.D., Johnstone, T.: Strongly uplifting cardinals and the boldface resurrection axioms (2012, under review). http://arxiv.org/abs/1403.2788
  18. 18.
    Hamkins J.D., Johnstone T.A.: Indestructible strong unfoldability. Notre Dame J. Form. Log. 51(3), 291–321 (2010)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Hamkins, J.D., Johnstone, T.: Resurrection axioms and uplifting cardinals. Arch. Math. Log. 53(3–4), 463–485 (2014)Google Scholar
  20. 20.
    Hamkins J.D., Shelah S.: Superdestructibility: a dual to Laver’s indestructibility. J. Symb. Log. 63(2), 549–554 (1998)CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    Johnstone, T.A.: Strongly unfoldable cardinals made indestructible. Ph.D. thesis, The Graduate Center of the City University of New York, June (2007)Google Scholar
  22. 22.
    Johnstone T.A.: Strongly unfoldable cardinals made indestructible. J. Symb. Log. 73(4), 1215–1248 (2008)CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Laver R.: Making the supercompactness of κ indestructible under κ-directed closed forcing. Israel J. Math. 29, 385–388 (1978)CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    Laver R.: Certain very large cardinals are not created in small forcing extensions. Ann. Pure Appl. Log. 149(1–3), 1–6 (2007)CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    Mitchell W.: A note on Hamkins’ approximation lemma. Ann. Pure Appl. Log. 144, 126–129 (Conference in honor of James E. Baumgartner’s sixtieth birthday) (2006)CrossRefMATHGoogle Scholar
  26. 26.
    Reitz, J.: The ground axiom. Ph.D. thesis, The Graduate Center of the City University of New York, September (2006)Google Scholar
  27. 27.
    Reitz, J.: The ground axiom. J. Symb. Log. 72(4), 1299–1317 (2007)CrossRefMATHGoogle Scholar
  28. 28.
    Sargsyan G.: On indestructibility aspects of identity crises. Arch. Math. Log. 48, 493–513 (2009)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Joan Bagaria
    • 1
  • Joel David Hamkins
    • 2
  • Konstantinos Tsaprounis
    • 3
  • Toshimichi Usuba
    • 4
  1. 1.ICREA and Departament de Lògica, Història i Filosofia de la CiènciaUniversitat de BarcelonaBarcelonaSpain
  2. 2.The Graduate CenterThe City University of New York (CUNY)New YorkUSA
  3. 3.Departament de Lògica, Història i Filosofia de la CiènciaUniversitat de BarcelonaBarcelonaSpain
  4. 4.Organization of Advanced Science and TechnologyKobe UniversityKobeJapan

Personalised recommendations