Superstrong and other large cardinals are never Laver indestructible

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, \Sigma_n-reflecting cardinals, \Sigma_n-correct cardinals and \Sigma_n-extendible cardinals (all for n>2) are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if \kappa\ exhibits any of them, with corresponding target \theta, then in any forcing extension arising from nontrivial strategically <\kappa-closed forcing Q in V_\theta, the cardinal \kappa\ will exhibit none of the large cardinal properties with target \theta\ or larger.Comment: 19 pages. Commentary concerning this article can be made at http://jdh.hamkins.org/superstrong-never-indestructible. Minor changes in v

Laver and set theory

In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip

Generically extendible cardinals

In this paper, we study generically extendible cardinal, which is a generic version of extendible cardinal. We prove that the generic extendibility of $\omega_1$ or $\omega_2$ has small consistency strength, but of a cardinal $>\omega_2$ is not. We also consider some results concerning with generic extendible cardinals, such as indestructibility, generic absoluteness of the reals, and Boolean valued second order logic