Superstrong and other large cardinals are never Laver indestructible

Joan Bagaria, Joel David Hamkins, Konstantinos Tsaprounis, Toshimichi Usuba

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

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 (Formula presented.), the cardinal κ will exhibit none of the large cardinal properties with target θ or larger.

Original languageEnglish
Pages (from-to)19-35
Number of pages17
JournalArchive for Mathematical Logic
Volume55
Issue number1-2
DOIs
Publication statusPublished - 2016 Feb 1

Keywords

  • Forcing
  • Indestructible cardinals
  • Large cardinals

ASJC Scopus subject areas

  • Philosophy
  • Logic

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