For each natural number n, let C (n) be the closed and unbounded proper class of ordinals α such that V α is a 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.

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We study the transfinite sequence of topologies on the ordinal numbers that is obtained through successive closure under Cantor's derivative operator on sets of ordinals, starting form the usual interval topology. We characterize the non-isolated points in the ξ-th topology as those ordinals that satisfy a strong iterated form of stationary reflection, which we call ξ-simultaneous-reflection. We prove some properties of the ideals of non-ξ-simultaneous-stationary sets and identify their tight connection with indescribable cardinals. We then introduce a new natural notion of Π 1 ξ-indescribability, for any ordinal ξ, which extends to the transfinite the usual notion of Π 1 n-indescribability, and prove that in the constructible universe L, a regular cardinal is (ξ + 1)-simultaneouslyreflecting if and only if it is Π 1 ξ-indescribable, a result that generalizes to all ordinals ξ previous results of Jensen [28] in the case ξ = 2, and Bagaria-Magidor-Sakai [5] in the case ξ = n. This yields a complete characterization in L of the non-discreteness of the ξ-topologies, both in terms of iterated stationary reflection and in terms of indescribability. 1. Introduction Cantor's derivative operator on a topological space (X, τ) is the map d τ that assigns to every subset A of X the set of its limit points. By declaring the sets d τ (A) to be open one generates a finer topology. Through successive applications of this process, and by taking unions at limit stages, one obtains a polytopological space (X, τ 0 , τ 1 ,. .. , τ ξ ,. . .), where τ 0 = τ and τ ζ ⊆ τ ξ whenever ζ < ξ. The study of such spaces has been of great interest in recent years, not so much in topology but, perhaps surprisingly, in proof theory and modal logic. When (X, τ) is scattered, the derived polytopological spaces (X, τ 0 , τ 1 ,. .. , τ ξ ,. . .) are known in the literature as GLP-spaces, for they provide a natural topological interpretation of the logic GLP (Japaridze [24]), namely the polymodal extension of the classical Gödel-Löb provability logic GL to infinitely-many modal operators [n],