$C^{(n)}$-Cardinals

For each natural number $n$, let $C^{(n)}$ be the closed and unbounded proper class of ordinals $\alpha$ such that $V_\alpha$ is a $\Sigma_n$ elementary substructure of $V$. We say that $\kappa$ is a \emph{$C^{(n)}$-cardinal} if it is the critical point of an elementary embedding $j:V\to M$, $M$ transitive, with $j(\kappa)$ in $C^{(n)}$. By analyzing the notion of $C^{(n)}$-cardinal at various levels of the usual hierarchy of large cardinal principles we show that, starting at the level of superstrong cardinals and up to the level of rank-into-rank embeddings, $C^{(n)}$-cardinals form a much finer hierarchy. The naturalness of the notion of $C^{(n)}$-cardinal is exemplified by showing that the existence of $C^{(n)}$-extendible cardinals is equivalent to simple reflection principles for classes of structures, which generalize the notions of supercompact and extendible cardinals. Moreover, building on results of \cite{BCMR}, we give new characterizations of Vope\v{n}ka's Principle in terms of $C^{(n)}$-extendible cardinals.