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 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.

Ultrahuge cardinals

“…In fact: Theorem A cardinal κ is extendible if and only if it is jointly supercompact and κ‐superstrong if and only if it is jointly supercompact and superstrong. The previous theorem follows from [, Corollary 2.31] and its subsequent remarks; indeed, we furthermore showed in that such a characterization is also available for the C ( n ) ‐version of extendible cardinals, as this was introduced by Bagaria (cf. ).…”

“…The notation and terminology used in this article are (mostly) standard; we refer the reader to or for an account of all undefined set‐theoretic notions, as well as for a comprehensive presentation of the theory of large cardinals. Adopting the notation of , and for every natural number n , we let C ( n ) denote the closed and unbounded proper class of ordinals α which are ‐correct in , that is, ordinals α such that is a ‐elementary substructure of (denoted by ). Note that the statement “” is expressible by a ‐formula, for every .…”

“…The so‐called C ( n ) ‐cardinals were introduced in by Bagaria, who showed that such notions are closely related to the theme of reflection for the set‐theoretic universe; in particular, Bagaria obtained level‐by‐level correspondence between C ( n ) ‐extendible cardinals and Vopěnka's Principle (), with the latter being a sort of reflection principle which carries high consistency strength. Subsequently, the various C ( n ) ‐cardinals were further studied by the author in .…”

On the Symbiosis Between Model-Theoretic and Set-Theoretic Properties of Large Cardinals

“…We shall devote most of the rest of the paper to showing that the cardinals corresponding to items (1) − − (4) − above are precisely the first weakly-inaccessible, the first 2-weakly inaccessible, the first weakly Mahlo, and the first weakly compact.…”

“…If φ is in the extension of first order logic by the Härtig-quantifier I (see Section 4.1 for the definition), then Φ(x, y) can be taken to be Δ 1 (Cd ), that is, Δ 1 with respect to the predicate Cd (x) ⇐⇒ "x is a cardinal". This works also in the other direction: If a Φ(x, y) is given and it is Δ KP 1 , Δ 1 (Cd ) or Δ 2 , then there is a sentence φ in the respective logic such that (1) holds. This is indicative of a tight correspondence for properties of models between expressibility in an extension of first-order logic and definability in set theory.…”