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arXiv · 1307.3486

Superstrong and other large cardinals are never Laver indestructible

Joan Bagaria·Joel David Hamkins·Konstantinos Tsaprounis·Toshimichi Usuba
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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.

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Joan Bagaria, Joel David Hamkins, Konstantinos Tsaprounis, Toshimichi Usuba. 2013-07-12. Superstrong and other large cardinals are never Laver indestructible. https://arxiv.org/abs/1307.3486

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