Skip to main content
Log in

C (n)-cardinals

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

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 : 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Bagaria, J., Casacuberta, C., Mathias, A.R.D., Rosický, J.: Definable orthogonality classes are small. Submitted for publication (2010)

  2. Barbanel J., Di Prisco C.A., Tan I.B.: Many times huge and superhuge cardinals. J. Symb. Log. 49, 112–122 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dimonte, V.: Non-Proper Elementary Embeddings beyond L(V λ+1). Doctoral dissertation. Universitá di Torino (2010)

  4. Jech T.: Set Theory. The Third Millenium Edition, Revised and Expanded. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg (2003)

    Google Scholar 

  5. Kanamori, A.: The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings. Perspectives in Mathematical Logic. Springer, Berlin, Heidelberg (1994)

  6. Kunen K.: Elementary embeddings and infinitary combinatorics. J. Symb. Log. 36, 407–413 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  7. Laver R.: Implications between strong large cardinal axioms. Ann. Pure Appl. Log. 90, 79–90 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Magidor M.: On the role of supercompact and extendible cardinals in logic. Israel J. Math. 10, 147–157 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  9. Martin D.A., Steel J.R.: A proof of Projective Determinacy. J. Am. Math. Soc. 2(1), 71–125 (1989)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Joan Bagaria.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bagaria, J. C (n)-cardinals. Arch. Math. Logic 51, 213–240 (2012). https://doi.org/10.1007/s00153-011-0261-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-011-0261-8

Keywords

Mathematics Subject Classification (2000)

Navigation