Superstrong and other large cardinals are never Laver indestructible

2013/07/12 by Bagaria, Joan, Hamkins, Joel David, Tsaprounis, Konstantinos +1 · 1 citation
#03E40 #03E55 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.1307.3486

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>2) 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

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