This is a talk at the VCU MAMLS Conference in Richmond, Virginia, April 1, 2017.

The idea of considering *virtual* set theoretic assertions was introduced by Schindler, arising out of his work on remarkable cardinals. Suppose $\mathcal P$ is a set theoretic property asserting the existence of elementary embeddings between some first-order structures. We will say that $\mathcal P$ holds *virtually* if embeddings of structures from $V$ characterizing $\mathcal P$ exist in the generic multiverse of $V$ (in its set-forcing extensions). Large cardinals are primary candidates for virtualization. Recall, for instance, that a cardinal $\kappa$ is *extendible* if for every $\alpha>\kappa$, there is $j:V_\alpha\to V_\beta$ with $\text{crit}(j)=\kappa$ and $j(\kappa)>\alpha$. So we can say that $\kappa$ is *virtually extendible* if for every $\alpha>\kappa$ some set-forcing extension has an extendibility embedding $j:V_\alpha^V\to V_\beta^V$. We can do the same with an appropriately chosen characterization of supercompact cardinals based on the existence of embeddings of set-sized structures, as well as with several other large cardinals in the neighborhood of a supercompact. Other properties which seem to naturally lend themselves to virtualization are forcing axioms. Virtual versions of ${\rm PFA}$, ${\rm SCFA}$ (forcing axiom for subcomplete forcing) and resurrection axioms have been studied by Schindler and Fuchs [1], [2], [3]. Together with Bagaria and Schindler, we studied a virtual version of Vopěnka’s Principle [1].

We can even have (consistent) virtual versions of inconsistent set-theoretic assertions. Observe for example that there can be a virtual elementary embedding from the reals to the rationals. To achieve this we simply force to collapse the cardinality of $\mathbb R$ to become countable so that in the collapse extension $\mathbb R^V$ is a countable dense linear order without end-points and hence isomorphic to the rationals. Of course the reals of the forcing extension still cannot be embedded into $\mathbb Q$ but virtual properties are about $V$-structures de re and not de dicto. It also turns out that Kunen’s Inconsistency does not hold for virtual embeddings. In a set-forcing extension there can be elementary $j:V_\lambda^V\to V_\lambda^V$ with $\lambda$ much larger than the supremum of the critical sequence of $j$.

Schindler introduced remarkable cardinals when he discovered that a remarkable cardinal is equiconsistent with the assertion that the theory of $L(\mathbb R)$ cannot be changed by proper forcing [4]. He defined that $\kappa$ is remarkable if for every $\lambda>\kappa$, there is $\bar\lambda<\kappa$ such that in a set-forcing extension there is an elementary $j:V_{\bar\lambda}^V\to V_\lambda^V$ with $j(\text{crit}(j))=\kappa$. By a theorem of Magidor [5], a cardinal $\kappa$ is supercompact precisely when the embeddings $j$ as above exist in $V$ itself. So remarkable cardinals are *virtually supercompac*t. Although it was conjectured that absoluteness of the theory of $L(\mathbb R)$ by proper forcing would have strength in the neighborhood of a strong cardinal, Schindler showed that remarkable cardinals are consistent with $V=L$ [6].

Calling remarkable cardinals virtually supercompact can seem like cheating because we chose a very peculiar characterization of supercompact cardinals to virtualize. We recently observed with Schindler that equivalently $\kappa$ is remarkable if for every $\lambda>\kappa$, there is $\alpha>\lambda$ and a transitive $M$ with $M^\lambda\subseteq M$ such that in a set-forcing extension there is $j:V_\alpha^V\to M$ with $\text{crit}(j)=\kappa$ and $j(\kappa)>\lambda$. More surprising is another equivalent characterization that for every $\lambda>\kappa$, there is $\alpha>\lambda$ and a transitive $M$ with $V_\lambda\subseteq M$ such that in a set-forcing extension there is $j:V_\alpha^V\to M$ with $\text{crit}(j)=\kappa$ and $j(\kappa)>\lambda$, making remarkables also look like virtually strong cardinals. A deeper reason for this appears to be that closure (in $V$) of the target model does not calibrate the strength of virtual large cardinals. Only large cardinals with characterization involving $j:V_\alpha\to V_\beta$ have robust virtual versions [7]. So we have robust virtual versions of supercompact, $C^{(n)}$-extendible, and rank-into-rank cardinals. The $n$-huge cardinals do not appear to have a robust characterization for virtualizing, so we instead virtualized a related hierarchy of $n$-huge* cardinals, where $\kappa$ is $n$-*huge** if there is $\alpha>\kappa$ and $j:V_\alpha\to V_\beta$ with $\text{crit}(j)=\kappa$ and $j^n(\kappa)<\alpha$ [7]. Schindler and Wilson recently defined a *virtual Shelah for supercompactness* cardinal and showed that it is equiconsistent with the assertion that every universally Baire set has a perfect subset [8]. The hierarchy of virtual large cardinals mirrors that of their actual counterparts. If $0^{\#}$ exists, then the Silver indiscernibles have all the virtual large cardinal properties. The virtual large cardinals fit between 1-iterable and $\omega+1$-iterable cardinals and they are are downward absolute to $L$ [7].

With Bagaria and Schindler we introduced, *Generic Vopěnka’s Principle*, a virtual version of Vopěnka’s Principle [1]. *Vopěnka’s Principle* asserts that every proper class of first-order structures has a pair of distinct structures that elementarily embed. Generic Vopěnka’s Principle asserts that the embedding exists in a set-forcing extension. Vopěnka’s Principle as well as its virtual version are second-order assertions formalizable in Godel-Bernays set theory. The first-order version of Vopěnka’s Principle which I will call here, *Vopěnka’s Scheme*, is the scheme of assertions ${\rm VP}(\Sigma_n)$ for every $n\in\omega$, which state that Vopěnka’s Principle holds for $\Sigma_n$-definable (with parameters) classes. *Generic Vopěnka’s Scheme* is the scheme of analogous assertions ${\rm gVP}(\Sigma_n)$. Bagaria showed that ${\rm VP}(\Sigma_2)$ holds precisely when there is a proper class of supercompact cardinals and ${\rm VP}(\Sigma_{n+2})$ holds precisely when there is a proper class of $C^{(n)}$-extendible cardinals [9]. Recall that $C^{(n)}$ is the class of all $\delta$ such that $V_\delta\prec_{\Sigma_n}V$. A cardinal $\kappa$ is $C^{(n)}$-*extendible* if for every $\alpha>\kappa$ there is an extendibility embedding $j:V_\alpha\to V_\beta$ with $j(\kappa)\in C^{(n)}$.

With Bagaria and Schindler we showed that ${\rm gVP}(\Sigma_2)$ is equiconsistent with a proper class of remarkable cardinals and ${\rm gVP}(\Sigma_{n+2})$ is equiconsistent with a proper class of virtually $C^{(n)}$-extendible cardinals [1]. If there is a proper class of remarkable or virtually $C^{(n)}$-extendible cardinals then ${\rm gVP}(\Sigma_2)$ or ${\rm gVP}(\Sigma_{n+2})$ respectively holds. If ${\rm gVP}(\Sigma_2)$ holds then there is a proper class of cardinals each of which is *either* remarkable or virtually rank-into-rank, and the analogous result holds for ${\rm gVP}(\Sigma_{n+2})$ with remarkable replaced by virtually $C^{(n)}$-extendible. In Bagaria’s argument you assumed that say there is no proper class of supercompacts and arrived at a contradiction by obtaining an embedding $j:V_{\lambda+2}\to V_{\lambda+2}$. But in the virtual case, such an embedding simply indicates the presence of a virtually rank-into-rank cardinal. Was it possible to eliminate the pesky case of a virtually rank-into-rank cardinal with a cleverer argument? I tried unsuccessfully for months. Then last summer with Joel Hamkins we showed that Kunen’s Inconsistency is fundamental to Bagaria’s proof. There is a model of Generic Vopěnka’s Scheme with no remarkable cardinals but a proper class of virtually rank-into-rank cardinals [10].

Slides to come!

[Bibtex]

```
@ARTICLE{BagariaGitmanSchindler:VopenkaPrinciple,
AUTHOR = {Bagaria, Joan and Gitman, Victoria and Schindler, Ralf},
TITLE = {Generic {V}op\v enka's {P}rinciple, remarkable cardinals, and the
weak {P}roper {F}orcing {A}xiom},
JOURNAL = {Arch. Math. Logic},
FJOURNAL = {Archive for Mathematical Logic},
VOLUME = {56},
YEAR = {2017},
NUMBER = {1-2},
PAGES = {1--20},
ISSN = {0933-5846},
MRCLASS = {03E35 (03E55 03E57)},
MRNUMBER = {3598793},
DOI = {10.1007/s00153-016-0511-x},
URL = {http://dx.doi.org/10.1007/s00153-016-0511-x},
pdf ={http://boolesrings.org/victoriagitman/files/2016/02/GenericVopenkaPrinciples.pdf},
}
```

[Bibtex]

```
@ARTICLE{Fuchs:HierarchiesVirtualResurrectionAxioms,
AUTHOR= {Gunter Fuchs},
TITLE= {Hierarchies of (virtual) resurrection axioms},
Note ={Preprint},
}
```

[Bibtex]

```
@ARTICLE{Fuchs:HierarchiesForcingAxiomsContinuumHypothesisSquarePrinciples,
AUTHOR= {Gunter Fuchs},
TITLE= {Hierarchies of forcing axioms, the continuum hypothesis and square principles},
Note ={Preprint},
}
```

[Bibtex]

```
@article {schindler:remarkable2,
AUTHOR = {Schindler, Ralf-Dieter},
TITLE = {Proper forcing and remarkable cardinals. {II}},
JOURNAL = {J. Symbolic Logic},
FJOURNAL = {The Journal of Symbolic Logic},
VOLUME = {66},
YEAR = {2001},
NUMBER = {3},
PAGES = {1481--1492},
ISSN = {0022-4812},
CODEN = {JSYLA6},
MRCLASS = {03E55 (03E15 03E35)},
MRNUMBER = {1856755 (2002g:03111)},
MRREVIEWER = {A. Kanamori},
DOI = {10.2307/2695120},
URL = {http://dx.doi.org.ezproxy.gc.cuny.edu/10.2307/2695120},
}
```

[Bibtex]

```
@article {magidor:supercompact,
AUTHOR = {Magidor, M.},
TITLE = {On the role of supercompact and extendible cardinals in logic},
JOURNAL = {Israel J. Math.},
FJOURNAL = {Israel Journal of Mathematics},
VOLUME = {10},
YEAR = {1971},
PAGES = {147--157},
ISSN = {0021-2172},
MRCLASS = {02K35},
MRNUMBER = {0295904 (45 \#4966)},
MRREVIEWER = {J. L. Bell},
}
```

[Bibtex]

```
@article {schindler:remarkable1,
AUTHOR = {Schindler, Ralf-Dieter},
TITLE = {Proper forcing and remarkable cardinals},
JOURNAL = {Bull. Symbolic Logic},
FJOURNAL = {The Bulletin of Symbolic Logic},
VOLUME = {6},
YEAR = {2000},
NUMBER = {2},
PAGES = {176--184},
ISSN = {1079-8986},
MRCLASS = {03E40 (03E45 03E55)},
MRNUMBER = {1765054 (2001h:03096)},
MRREVIEWER = {A. Kanamori},
DOI = {10.2307/421205},
URL = {http://dx.doi.org.ezproxy.gc.cuny.edu/10.2307/421205},
}
```

```
@ARTICLE{GitmanSchindler:virtualCardinals,
AUTHOR= {Gitman, Victoria and Schindler, Ralf},
TITLE= {Virtual large cardinals},
Note ={Submitted},
pdf={https://boolesrings.org/victoriagitman/files/2017/03/virtualLargeCardinals.pdf},
}
```

[Bibtex]

```
@ARTICLE{SchindlerWilson:UniversallyBaireSetsOfRealsPerfectSetProperty,
AUTHOR= {Ralf Schindler and Trevor Wilson},
TITLE= {Universally {B}aire sets of reals and the perfect set property},
Note ={In preparation},
}
```

[Bibtex]

```
@article {Bagaria:CnCardinals,
AUTHOR = {Bagaria, Joan},
TITLE = {{$C^{(n)}$}-cardinals},
JOURNAL = {Arch. Math. Logic},
FJOURNAL = {Archive for Mathematical Logic},
VOLUME = {51},
YEAR = {2012},
NUMBER = {3-4},
PAGES = {213--240},
ISSN = {0933-5846},
CODEN = {AMLOEH},
MRCLASS = {03E55 (03C55)},
MRNUMBER = {2899689},
MRREVIEWER = {Bernhard A. K{\"o}nig},
DOI = {10.1007/s00153-011-0261-8},
URL = {http://dx.doi.org.ezproxy.gc.cuny.edu/10.1007/s00153-011-0261-8},
}
```

[Bibtex]

```
@ARTICLE{GitmanHamkins:GVP,
AUTHOR= {Victoria Gitman and Joel David Hamkins},
TITLE= {A model of the generic Vop\v enka principle in which the ordinals are not $\Delta_2$-Mahlo},
PDF={https://boolesrings.org/victoriagitman/files/2017/06/Generic-Vopenka-with-Ord-not-Mahlo.pdf},
Note ={Submitted},
EPRINT ={1706.00843},
}
```

All this virtual stuff is really awesome. I guess it’s also virtually awesome, but it’s proper awesome. Well, maybe improperly awesome, because you keep collapsing everything to be countable…

I hope you’ll be posting slides, and do you know if there will be a live feed (or video for later)? I’d love to watch along as I did last year on Joel’s 50th.

Good to hear from you, Asaf! I doubt they will have a live feed, but I will ask. I will post the slides after I weed out all the typos :).

Nice post! I remembered you mentioning in Copenhagen that you didn’t know whether Kunen’s incosistency was crucial in Bagaria’s results. I would be interested in seeing your and Joel’s result. Will you upload any preprint soon?

Thanks, Stamatis! I am starting to write up the results this week. If things go smoothly, I might have something by the end of the month. I came back from YST and met with Joel, who immediately had a brilliant strategy for obtaining a counterexample, which worked out :).