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C (n)-cardinals

C (n)-cardinals 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Mathematical Logic Springer Journals

C (n)-cardinals

Bagaria, Joan
Archive for Mathematical Logic , Volume 51 (4) – Dec 23, 2011
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References (6)

Publisher
Springer Journals
Copyright
Copyright © 2011 by Springer-Verlag
Subject
Mathematics; Mathematics, general; Algebra; Mathematical Logic and Foundations
ISSN
0933-5846
eISSN
1432-0665
DOI
10.1007/s00153-011-0261-8
Publisher site
See Article on Publisher Site

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.

Journal

Archive for Mathematical Logic – Springer Journals

Published: Dec 23, 2011

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