This will be an invited talk at the ASL 2014 North American Annual Meeting (May 19-22, 2014) in the special session Set Theory in Honor of Rich Laver, organized by Bill Mitchell and Jean Larson.

Abstract.  The large cardinal indestructibility phenomenon, discovered by Richard Laver with his seminal result on supercompact cardinals, is by now often seen as pervasive in the large cardinal hierarchy. Nevertheless, a new never-indestrucible phenomenon has emerged.  Superstrong cardinals, for example, 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, $\Sigma_n$-reflecting cardinals, $\Sigma_n$-correct cardinals and $\Sigma_n$-extendible cardinals (all for $n\geq 3$) are never Laver indestructible.  The proof involves a detailed technical analysis of the complexity of the definition in Laver’s theorem on the definability of the ground model, thereby involving and extending results in set-theoretic geology.  This is joint work between myself and Joan Bagaria, Kostas Tasprounis and Toshimichi Usuba.

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