Archive for Mathematical Logic

, Volume 51, Issue 3–4, pp 213–240 | Cite as

C(n)-cardinals

Article

Abstract

For each natural number n, let C(n) be the closed and unbounded proper class of ordinals α such that Vα is a Σn elementary substructure of V. We say that κ is a C(n)-cardinal if it is the critical point of an elementary embedding j : VM, 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 

Mathematics Subject Classification (2000)

03E55 03C55 

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References

  1. 1.
    Bagaria, J., Casacuberta, C., Mathias, A.R.D., Rosický, J.: Definable orthogonality classes are small. Submitted for publication (2010)Google Scholar
  2. 2.
    Barbanel J., Di Prisco C.A., Tan I.B.: Many times huge and superhuge cardinals. J. Symb. Log. 49, 112–122 (1984)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Dimonte, V.: Non-Proper Elementary Embeddings beyond L(V λ+1). Doctoral dissertation. Universitá di Torino (2010)Google Scholar
  4. 4.
    Jech T.: Set Theory. The Third Millenium Edition, Revised and Expanded. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg (2003)Google Scholar
  5. 5.
    Kanamori, A.: The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings. Perspectives in Mathematical Logic. Springer, Berlin, Heidelberg (1994)Google Scholar
  6. 6.
    Kunen K.: Elementary embeddings and infinitary combinatorics. J. Symb. Log. 36, 407–413 (1971)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Laver R.: Implications between strong large cardinal axioms. Ann. Pure Appl. Log. 90, 79–90 (1997)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Magidor M.: On the role of supercompact and extendible cardinals in logic. Israel J. Math. 10, 147–157 (1971)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Martin D.A., Steel J.R.: A proof of Projective Determinacy. J. Am. Math. Soc. 2(1), 71–125 (1989)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.ICREA (Institució Catalana de Recerca i Estudis Avançats) and Departament de Lògica, Història i Filosofia de la CiènciaUniversitat de BarcelonaBarcelonaSpain

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