This is a talk at the Harvard Logic Colloquium, Cambridge, November 8, 2017.

**Abstract**:

Given a set-theoretic property $\mathcal P$ characterized by the existence of elementary embeddings between some first-order structures, let’s say that $\mathcal P$ holds *virtually* if the embeddings between structures from $V$ characterizing $\mathcal P$ exist somewhere in the generic multiverse. We showed with Schindler that virtual versions of supercompact, $C^{(n)}$-extendible, $n$-huge and rank-into-rank cardinals form a large cardinal hierarchy consistent with $V=L$. Included in the hierarchy are virtual versions of inconsistent large cardinal notions such as the existence of an elementary embedding $j:V_\lambda\to V_\lambda$ for $\lambda$ much larger than the supremum of the critical sequence. The Silver indiscernibles, under $0^\sharp$, which have a number of large cardinal properties in $L$, are also natural examples of virtual large cardinals. Virtual versions of forcing axioms, including ${\rm PFA}$, ${\rm SCFA}$, and resurrection axioms, have been studied by Schindler and Fuchs, who showed that they are equiconsistent with virtual large cardinals. We showed with Bagaria and Schindler that the virtual version of Vopěnka’s Principle is consistent with $V=L$. Bagaria had showed that Vopěnka’s Principle holds if and only if the universe has a proper class of $C^{(n)}$-extendible cardinals for every $n\in\omega$. We almost generalized his result by showing that the virtual version is equiconsistent with the existence, for every $n\in\omega$, of a proper class of virtually $C^{(n)}$-extendible cardinals. With Hamkins we showed that Bagaria’s result cannot generalize by constructing a model of virtual Vopěnka’s Principle in which there are no virtually extendible cardinals. The difference arises from the failure of Kunen’s Inconsistency in the virtual setting. In the talk, I will discuss a mixture of results about the virtual large cardinal hierarchy and virtual Vopěnka’s Principle.

Super interesting Vika! Very quick question:

What do you think these new uses of the virtual setting tell us about how essential the virtual world is for understanding set theory properly?

In particular, the failure of the Kunen Inconsistency in the virtual setting (well, strictly speaking it’s not the failure of the Kunen Inconsistency, but the fact that you can have an embedding between an inner model (i.e. V) and some different inner model M when viewing V as a ground of V[G]), leads me to think that really this external perspective is essential to understanding V in all its glory. This holds, I think, even if you think there’s just one universe of sets (indeed the problem is even more pressing for the universe theorist, as these seem like legitimate tools for studying *the* universe).

I don’t know if there are any specific results that you think are especially interesting from this point of view?

Hi Neil, glad you are following my posts!

So first, let me clarify that because virtual properties are about embeddings between set structures, by the “failure of Kunen’s Inconsistency” I mean the existence of embeddings $j:V_\lambda\to V_\lambda$ with $\lambda\gg\text{sup}\langle k_n\mid n< \omega\rangle$.

On the one hand, the virtual large cardinals have definitely proved their importance because we need them to measure consistency strength of natural set theoretic assertions (for example, Schindler showed that absoluteness of $L(\mathbb R)$ by proper forcing is equiconsistent with a remarkable cardinal). On the other hand, virtual properties have equivalent characterizations (for example, in terms of the existence of winning strategies in certain determined games) that have nothing to do with forcing, although one can argue that they are nowhere near as natural.