Virtual Set Theory and Generic Vopěnka's Principle

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 supercompact. 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 [bibcite key=GitmanSchindler:virtualCardinals]. 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].

References

  1. J. Bagaria, V. Gitman, and R. Schindler, “Generic Vopěnka’s Principle, remarkable cardinals, and the weak Proper Forcing Axiom,” Arch. Math. Logic, vol. 56, no. 1-2, pp. 1–20, 2017.
  2. G. Fuchs, “Hierarchies of (virtual) resurrection axioms,” J. Symb. Log., vol. 83, no. 1, pp. 283–325, 2018.
  3. G. Fuchs, “Hierarchies of forcing axioms, the continuum hypothesis and square principles,” J. Symb. Log., vol. 83, no. 1, pp. 256–282, 2018.
  4. R.-D. Schindler, “Proper forcing and remarkable cardinals. II,” J. Symbolic Logic, vol. 66, no. 3, pp. 1481–1492, 2001.
  5. M. Magidor, “On the role of supercompact and extendible cardinals in logic,” Israel J. Math., vol. 10, pp. 147–157, 1971.
  6. R.-D. Schindler, “Proper forcing and remarkable cardinals,” Bull. Symbolic Logic, vol. 6, no. 2, pp. 176–184, 2000.
  7. V. Gitman and R. Schindler, “Virtual large cardinals,” Ann. Pure Appl. Logic, vol. 169, no. 12, pp. 1317–1334, 2018.
  8. R. Schindler and T. M. Wilson, “The consistency strength of the perfect set property for universally Baire sets of reals,” J. Symb. Log., vol. 87, no. 2, pp. 508–526, 2022. Available at: https://doi.org/10.1017/jsl.2019.63
  9. J. Bagaria, “$C^{(n)}$-cardinals,” Arch. Math. Logic, vol. 51, no. 3-4, pp. 213–240, 2012.
  10. V. Gitman and J. D. Hamkins, “A model of the generic Vopěnka principle in which the ordinals are not Mahlo,” Arch. Math. Logic, vol. 58, no. 1-2, pp. 245–265, 2019.