Archive for Mathematical Logic 51 (3-4):213-240 (2012)

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
Keywords C (n)-cardinals  Supercompact cardinals  Extendible cardinals  Vopenka’s Principle  Reflection
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DOI 10.1007/s00153-011-0261-8
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References found in this work BETA

Elementary Embeddings and Infinitary Combinatorics.Kenneth Kunen - 1971 - Journal of Symbolic Logic 36 (3):407-413.
Implications Between Strong Large Cardinal Axioms.Richard Laver - 1997 - Annals of Pure and Applied Logic 90 (1-3):79-90.

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Citations of this work BETA

Elementary Chains and C (N)-Cardinals.Konstantinos Tsaprounis - 2014 - Archive for Mathematical Logic 53 (1-2):89-118.
On Resurrection Axioms.Konstantinos Tsaprounis - 2015 - Journal of Symbolic Logic 80 (2):587-608.
On Colimits and Elementary Embeddings.Joan Bagaria & Andrew Brooke-Taylor - 2013 - Journal of Symbolic Logic 78 (2):562-578.
The Large Cardinals Between Supercompact and Almost-Huge.Norman Lewis Perlmutter - 2015 - Archive for Mathematical Logic 54 (3-4):257-289.

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