Superstrong and other large cardinals are never Laver indestructible

TL;DR

This paper proves that a wide class of large cardinals, including superstrong and extendible types, cannot be made indestructible by Laver-style forcing, and are instead superdestructible under certain forcing extensions.

Contribution

It establishes that many large cardinal properties are inherently superdestructible, showing they cannot be rendered Laver indestructible through forcing.

Findings

Superstrong and similar large cardinals are never Laver indestructible.

These cardinals are superdestructible under certain forcing extensions.

Large cardinal properties are inherently superdestructible.

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, \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.