Superstrong and other large cardinals are never Laver indestructible

Journal article

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.

Authors: Bagaria, J; Hamkins, J; Tsaprounis, K; Usuba, T

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer
• Journal: Archive for Mathematical Logic
• Volume: 55
• Issue: 1-2
• Pages: 19-35
• Publication date: 2015-12-23
• Acceptance date: 2014-03-31
• DOI: 10.1007/s00153-015-0458-3
• EISSN: 1432-0665
• ISSN: 0933-5846
• Language: English
• Keywords: SBTMR; Large cardinals; Journal Article; Forcing; Indestructible cardinals