I'm currently reading through Larson's book on the stationary tower and a point confused me.

Let $\delta$ be Woodin and let $j:V\to M\subseteq V[G]$ be an elementary embedding associated with the (full) stationary tower $\mathbb P_{<\delta}$. Then $V[G]\models {^{<\delta}}M\subseteq M$. This seems close to $\operatorname{crit}j$ being $\theta$-supercompact for every $\theta<\delta$, in $V[G]$, with the exception that the generic extension might contain more subsets of $\operatorname{crit}j$, so that the measure on $\operatorname{crit}j$ isn't total.

Now say there is a proper class of Woodins, and let $j:V\to V[G]$ be an elementary embedding associated with the stationary tower $\mathbb P_\infty$. Again, is $\operatorname{crit}j$ now 'close' to being supercompact in $V[G]$?

Is this the same thing as $\operatorname{crit}j$ being generically supercompact? Since we can choose $\operatorname{crit}j$ to be any inaccessible, does this then mean that a proper class of Woodins implies a proper class of generically supercompacts?

  • $\begingroup$ This is probably weaker than it sounds. By the same arguments, one Woodin cardinal is enough in order to make $\omega_1$ "generically almost-huge". It is not as useful as it sounds, since the forcing that adds this embedding is very wild. $\endgroup$
    – Yair Hayut
    Nov 20 '16 at 15:20
  • $\begingroup$ @Yair: So the generic multiverse of a model with a proper class of Woodins is "Where the Wild Things Are"? $\endgroup$
    – Asaf Karagila
    Nov 20 '16 at 21:40
  • $\begingroup$ I would call the $\mathbb P_\infty$ embedding "virtually Reinhardt"! $\endgroup$ Apr 5 '18 at 11:53

Apparently this has a name. To any large cardinal notion, there is a virtual variant, in which the elementary embeddings in question lie in some generic extension. Reformulating my scenario, I pointed out that a Woodin $\delta$ consistency-wise implies a virtually $\theta$-supercompact for every $\theta<\delta$. Using the countable tower instead, as Yair pointed out, we also get the result that a Woodin consistency-wise implies that $\omega_1$ is virtually almost huge.

But as is pointed out in Gitman's slides, all virtual large cardinals lie consistency-wise below $0^\sharp$, so this observation (that Woodins lie above virtuals) is pretty moot.


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