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Abstract

We continue the study of the virtual large cardinal hierarchy by analysing virtual versions of superstrong, Woodin, and Berkeley cardinals. Gitman and Schindler showed that virtualizations of strong and supercompact cardinals yield the same large cardinal notion. We provide various equivalent characterizations of virtually Woodin cardinals, including showing that $\mathsf{On}$ is virtually Woodin if and only if for every class $A$, there is a proper class of virtually $A$-extendible cardinals. We introduce the virtual Vopěnka principle for finite languages and show that it is not equivalent to the virtual Vopěnka principle (although the two principles are equiconsistent), but is equivalent to the assertion that $\mathsf{On}$ is virtually pre-Woodin, a weakening of virtually Woodin, which is equivalent to having, for every class $A$, a weakly virtually $A$-extendible cardinal. We show that if there are no virtually Berkeley cardinals, then $\mathsf{On}$ is virtually Woodin if and only if $\mathsf{On}$ is virtually pre-Woodin (if and only if the virtual Vopěnka principle for finite languages holds). In particular, if the virtual Vopěnka principle holds and $\mathsf{On}$ is not Mahlo, then $\mathsf{On}$ is not virtually Woodin, and hence there is a virtually Berkeley cardinal.