Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Primitive recursive analogues of regular cardinals based on ordinal representation systems for KPi and KPM

  • 50 Accesses


In this paper, we develop primitive recursive analogues of regular cardinals by using ordinal representation systems for KPi and KPM. We also define primitive recursive analogues of inaccessible and hyperinaccessible cardinals. Moreover, we characterize the primitive recursive analogue of the least (uncountable) regular cardinal.

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


  1. 1.

    Buchholz, W.: A note on the ordinal analysis of KPM, Proceedings Logic Colloquium 90 (Edited by J. Väänänen), pp. 1–9 (1992)

  2. 2.

    Buchholz, W.: A simplified version of local predicativity, Proof Theory (Edited by P. Aczel, H. Simmons and S. Wainer), pp. 117–147 (1993)

  3. 3.

    Drake, F.: Set Theory: An introduction to large cardinals, (North Holland, (1974)

  4. 4.

    Kanamori, A.: The higher infinite, Springer-Verlag, (1997)

  5. 5.

    Pohlers, W.: Subsystems of set theory and second order number theory, Handbook of Proof Theory (Edited by S.R. Buss), pp. 209–335 (1998)

  6. 6.

    Rathjen, M.: Ordinal notations based on a weakly Mahlo cardinal. Archive for Mathematical Logic 29, 249–263 (1990)

  7. 7.

    Rathjen, M.: Proof-theoretic analysis of KPM. Archive for Mathematical Logic 30, 377–403 (1991)

  8. 8.

    Rathjen, M.: The higher infinite in proof theory. Logic Colloquium ‘95, Lecture Notes in Logic 11, 275–304 (1998)

  9. 9.

    Rathjen, M.: The realm of ordinal analysis. Sets and Proofs, Cambridge University Press, 219–279 (1999)

  10. 10.

    Takaki, O.: Primitive recursive analogues of the least regular cardinal and the least weakly inaccessible cardinal. Scientiae Mathematicae Japonicae 57–1, 103–111 (2003): e7 157–167

  11. 11.

    Takaki, O.: Elementary ordinal representation systems for KP theories, preprint

Download references

Author information

Correspondence to Osamu Takaki.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Takaki, O. Primitive recursive analogues of regular cardinals based on ordinal representation systems for KPi and KPM. Arch. Math. Logic 44, 689–709 (2005). https://doi.org/10.1007/s00153-005-0308-9

Download citation

Key words or phrases

  • Proof-theoretic ordinal
  • Ordinal representation system
  • Inaccessible cardinal