This is a talk for the CUNY Set Theory Seminar on February 1, 2013, 10:00 am.
Abstract. Although the large cardinal indestructibility phenomenon, initiated with Laver’s seminal 1978 result that any supercompact cardinal $\kappa$ can be made indestructible by $\lt\kappa$-directed closed forcing and continued with the Gitik-Shelah treatment of strong cardinals, is by now nearly pervasive in set theory, nevertheless I shall show that no superstrong strong cardinal—and hence also no $1$-extendible cardinal, no almost huge cardinal and no rank-into-rank cardinal—can be made indestructible, even by comparatively mild forcing: all such cardinals $\kappa$ are destroyed by $\text{Add}(\kappa,1)$, by $\text{Add}(\kappa,\kappa^+)$, by $\text{Add}(\kappa^+,1)$ and by many other commonly considered forcing notions.
This is very recent joint work with Konstantinos Tsaprounis and Joan Bagaria.
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Do these results give as an automatic corollary that none of these cardinals carries an analog of the Laver Diamond? It seems that, at least for extendible cardinals, there is some sort of tension here since extendible cardinals are supercompact. Does this mean that a Laver diamond for an extendible can only preserve the supercompactness of the extendible? In a different direction, could there nevertheless be a forcing extension where one of these cardinals does carry an appropriate analog of the Laver Diamond (some kind of pre-preparation forcing)?
No, on the contrary, I think that there can be a Laver function for superstrongness. But the Laver preparation using such a function will not achieve indestructibility, and neither will the lottery preparation nor any other method of preparation. The main issue seems to be that the Laver preparation and all the other indestructibility argument exploit ultimately the fact that for the relevant embedding $j:V\to M$, one has a disconnect between $M$ and $V$ eventually between $\kappa$ and $j(\kappa)$. But with superstrongness, $M$ and $V$ agree all the way up to $j(\kappa)$, and so the Laver preparation argument breaks down. My argument shows that there is no way to repair it.
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