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 is virtually $A$ if the embeddings characterizing $A$ exist in some set-forcing extensions. We call such a reformulation of a large cardinal axiom a virtualization of $A$.

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 to observe how the large cardinal strength of the 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?

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 is virtually $A$ if the embeddings characterizing $A$ exist in some set-forcing extensions. We call such a reformulation of a large cardinal axiom a virtualization of $A$.

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?