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Superstrong and other large cardinals are never Laver indestructible

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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, 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.

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Authors and Affiliations

  1. ICREA and Departament de Lògica, Història i Filosofia de la Ciència, Universitat de Barcelona, Montalegre 6-8, 08001, Barcelona, Catalonia, Spain

    Joan Bagaria

  2. The Graduate Center, The City University of New York (CUNY), 365 Fifth Avenue, New York, NY, 10016, USA

    Joel David Hamkins

  3. Departament de Lògica, Història i Filosofia de la Ciència, Universitat de Barcelona, Montalegre 6-8, 08001, Barcelona, Catalonia, Spain

    Konstantinos Tsaprounis

  4. Organization of Advanced Science and Technology, Kobe University, Rokko-dai 1-1, Nada, Kobe, 657-8501, Japan

    Toshimichi Usuba

Authors
  1. Joan Bagaria
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  2. Joel David Hamkins
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  3. Konstantinos Tsaprounis
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  4. Toshimichi Usuba
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Corresponding author

Correspondence to Joan Bagaria.

Additional information

The research work of the first author was partially supported by the Spanish Ministry of Science and Innovation under Grant MTM2011-25229 and by the Generalitat de Catalunya (Catalan Government) under Grant 2009-SGR-187. The second author’s research has been supported in part by Simons Foundation Grant 209252, PSC-CUNY Grant 64732-00-42, and CUNY Collaborative Incentive grant 80209-06 20, and he is grateful for the support provided to him as a visitor at the University of Barcelona in December 2012, where much of this research took place in the days immediately following the dissertation defense of the third author, inspired by remarks made at the defense. The fourth author had subsequently proved essentially similar results independently, and so we joined forces, merged the papers and strengthened the results into the present account. Commentary concerning this article can be made at jdh.hamkins.org/superstrong-never-indestructible.

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Bagaria, J., Hamkins, J.D., Tsaprounis, K. et al. Superstrong and other large cardinals are never Laver indestructible. Arch. Math. Logic 55, 19–35 (2016). https://doi.org/10.1007/s00153-015-0458-3

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  • DOI: https://doi.org/10.1007/s00153-015-0458-3

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