Archive for Mathematical Logic

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

Superstrong and other large cardinals are never Laver indestructible

  • Joan BagariaEmail author
  • Joel David Hamkins
  • Konstantinos Tsaprounis
  • Toshimichi Usuba


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, Σ n -reflecting cardinals, Σ n -correct cardinals and Σ n -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.


Large cardinals Forcing Indestructible cardinals 

Mathematics Subject Classification

03E55 03E40 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Joan Bagaria
    • 1
    Email author
  • 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

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