On the Actual Potential of Virtual Large Cardinals

Question

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?

Answer 1

In the context of virtual large cardinals, it doesn't matter whether you consider arbitrary extensions or just forcing extensions.

The basic situation is that the virtual large cardinal properties are generally witnessed by the existence of an elementary embedding $j:M\to N$ between two structures $M$ and $N$ of the ground model, but where the embedding $j$ might exist only in some extension.

What I claim is that in any such situation, there is such an embedding $j$ in an outer model of $V$ if and only if there is such an embedding $j$ in the collapse forcing extension of $V$ in which $M$ becomes countable.

The reason has to do with the equivalence between the existence of $j$ and the existence of a winning strategy for player II in the game $G(M,N)$, where player I poses challenges with particular elements $x_i\in M$ and player II must play a response $y_i$, such that the type of $(x_0,x_1,\ldots,x_n)$ in $M$ is the same as $(y_0,y_1,\ldots,y_n)$ in $N$, for every $n$. This is an infinitary pebble game of the kind sometimes considered in model theory. The game is open for player I, since player II wins as long as no move has already lost at a finite stage. Thus, the game is determined in the ground model.

The main point is that the winner of an open game is absolute from a model to any outer model, since the assignment of the transfinite game values will proceed the same in any model with the same game tree, and the open player has a winning strategy just in case the initial position has an ordinal game value in that ranking.

Observe now that if there is some outer model in which there is an embedding $j:M\to N$, then player II can play in that model in accordance with that embedding $y_i=j(x_i)$, and thereby win the game there. So if there is a $j$ anywhere, then player II has a winning strategy in that model, and therefore has a winning strategy in the ground model, by the absoluteness we observed above. It follows that in the model collapsing $M$ to become countable, we can build a $j:M\to N$ simply by having player I systematically play all elements $x$ of $M$ and using the strategy to define $j(x)$.

Thus, if there is a $j$ anywhere, in any outer model, then there is a $j$ in the forcing extension collapsing $M$.

So it does not matter if you change the multiverse from forcing extensions to outer models for this notion; it suffices to look in the collapse forcing extensions.

It would seem to matter, however, if you restrict your potentialist concept to exclude cases where $M$ becomes countable, since the argument would break down in that case. Even in this case, however, many of the arguments with virtual large cardinals can simply proceed with the strategy, which is visible in the ground model, rather than the actual virtual embedding, which exists only in the collapse extensions. That is, the embedding is often used as a proxy for the strategy.