Virtual large cardinals belong to a relatively new breed of strong axioms of infinity. They often appear as statements of the following form:

Definition.Suppose $A$ is a large cardinal property characterized by the existence of suitable embeddings. A cardinal isvirtually$A$ if the embeddings characterizing $A$ exist insome set-forcing extensions. We call such a reformulation of a large cardinal axiom avirtualizationof $A$.

Generic Vopěnka's principle and Schindler's remarkable cardinals are two iconic and well-studied examples of virtual large cardinal axioms. The former is the virtualization of Vopěnka's principle/scheme while the latter could be considered a virtualization of both supercompact (in Magidor's characterization) and strong cardinals (Gitman and Schindler's result).

So virtual large cardinals are mainly produced by replacing the *actual existence* of certain elementary embeddings with the *possible existence* of such embeddings in a set-forcing generic extension.

On the other hand, from a potentialist perspective, set forcing generic extensions aren't the only nice interpretations of possible worlds in the set-theoretic multiverse. As stated in Joel's answer to my previous post, outer models, class forcing extensions, inaccessible levels of the von Neumann hierarchy, transitive models of $ZFC$, etc., may serve as pretty good alternatives of a possible world as well.

Thus, one may virtualize a given large cardinal axiom in many different ways depending on the way they interpret the possibility modality in the multiverse. For instance, an *outer virtual supercompact cardinal* could be defined just like a remarkable cardinal (that is set-forcing virtual supercompact) with the slight difference that the embeddings are required to exist in some *outer model* rather than a set-forcing generic extension.

Set-forcing virtual large cardinals lie between $1$-iterable and $\omega+1$-iterable cardinals in the consistency strength order and are known to be downward absolute to $L$ (and so are consistent with $V=L$). Here the following natural questions arise:

Question 1.What is the large cardinal strength of virtual large cardinals of different potentialist interpretations? Can any such virtual cardinal be inconsistent with $V=L$? Precisely, is there any large cardinal axiom $A$ and a potetialist interpretation $P$ for a possible world such that the consistency strength of $P$-virtual $A$-large cardinal is greater than or equal to that of $0^{\sharp}$?

Another question is about how the large cardinal strength of the different virtualizations of a fixed large cardinal axiom $A$ varies by changing our potentialist interpretation $P$. I am particularly interested in the case of virtualization of supercompact cardinals (and Vopěnka's principle). To be more specific:

Question 2.Is there a potentialist interpretation $P$ of the possible worlds on the set-theoretic multiverse such that the corresponding $P$-virtual supercompact cardinal is strictly stronger than remarkable cardinals (and so Weak Proper Forcing Axiom $WPFA$) in the consistency strength order?