This is a talk at the Young Set Theory 2016 Conference in Copenhagen, Denmark, June 13-17, 2016.

Since this talk is similar to the one I gave a few weeks ago at the Rutgers Logic Seminar, I will just include here the abstract and links to other posts related to this work.

**Abstract**: *Vopěnka’s Principle*, introduced by Petr Vopěnka in the 1970’s, is the second-order assertion that for every proper class $\mathcal C$ of first-order structures in the same language, there are $B\neq A$, both in $\mathcal C$, such that $B$ elementarily embeds into $A$. In ${\rm ZFC}$, we can consider *first-order* Vopěnka’s Principle, which is the scheme of assertions ${\rm VP}(\Sigma_n)$, for $n\in\omega$, stating that Vopěnka’s Principle holds for $\Sigma_n$-definable (with parameters) classes. The principle ${\rm VP}(\Sigma_1)$ is a theorem of ${\rm ZFC}$; Bagaria showed that the principle ${\rm VP}(\Sigma_2)$ holds if and only if there is a proper class of supercompact cardinals, and for $n\geq 1$, ${\rm VP}(\Sigma_{n+2})$ holds if and only if there is a proper class of $C^{(n)}$-extendible cardinals, where $\kappa$ is $C^{(n)}$-*extendible* if for every $\alpha>\kappa$, there is an extendibility $j:V_\alpha\to V_\beta$ with $V_{j(\kappa)}\prec_{\Sigma_n} V$. We introduce *Generic Vopěnka’s Principle*, which asserts that the embeddings of Vopěnka’s Principle exist in some set-forcing extension. First-order Generic Vopěnka’s Principle is the scheme of assertions ${\rm gVP}(\Sigma_n)$ for $\Sigma_n$-definable classes of structures. The consistency strength of Generic Vopěnka’s Principle is measured by *virtual* large cardinals. Given a very large cardinal property $\mathcal A$, such as supercompact, $C^{(n)}$-extendible, or rank-into-rank, characterized by the existence of suitable set-sized embeddings, we say that a cardinal $\kappa$ is *virtually* $\mathcal A$ if the embeddings of $V$-structures characterizing $\mathcal A$ exist in some set-forcing extension. Unlike the similar sounding generic large cardinals, virtual large cardinals are actual large cardinals that fit between ineffables and $0^{\sharp}$ in the hierarchy. Remarkable cardinals introduced by Schindler turned out to be virtually supercompact. We show that ${\rm gVP}(\Sigma_2)$ is equiconsistent with a proper class of remarkable cardinals and for $n\geq 1$, ${\rm gVP}(\Sigma_{n+2})$ is equiconsistent with a proper class of virtually $C^{(n)}$-extendible cardinals. We conjecture that the equiconsistency results can be improved to get an equivalence. This is joint work with Joan Bagaria and Ralf Schindler.

Here are links to other posts related to this work:

Looking forward to your talk in Kopenhagen!

I am very much looking forward to the conference and meeting all my colleagues again :).