C ⁽ⁿ⁾-cardinals

Abstract

For each natural number n, let C ⁽ⁿ⁾ be the closed and unbounded proper class of ordinals α such that V _α is a Σ _n elementary substructure of V. We say that κ is a C ⁽ⁿ⁾ -cardinal if it is the critical point of an elementary embedding j : V → M, M transitive, with j(κ) 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 Bagaria et al. (2010), we give new characterizations of Vopeňka’s Principle in terms of C ⁽ⁿ⁾-extendible cardinals.