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][1] and Schindler's [remarkable cardinals][2] are two iconic and well-studied examples of virtual large cardinal axioms. The former is the virtualization of [Vopěnka's principle/scheme][3] 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][4], 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][5], 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][6] 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?  


  [1]: http://jdh.hamkins.org/a-model-of-the-generic-vopenka-principle-in-which-the-ordinals-are-not-delta_2-mahlo/
  [2]: https://en.wikipedia.org/wiki/Remarkable_cardinal
  [3]: https://en.wikipedia.org/wiki/Vop%C4%9Bnka%27s_principle
  [4]: http://jdh.hamkins.org/set-theoretic-potentialism/
  [5]: https://mathoverflow.net/questions/306319/what-is-the-modal-logic-of-outer-multiverse
  [6]: https://en.wikipedia.org/wiki/Iterable_cardinal