Parameter free induction and reflection

  • Lev D. Beklemishev
Contributed Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1289)


We give a precise characterization of parameter free Σn and IIn induction schemata, n and I II n , in terms of reflection principles. This allows us to show that I II n+1 is conservative over n w.r.t. boolean combinations of Σn+1 sentences, for n ≥ 1. In particular, we give a positive answer to a question by R. Kaye, whether the provably recursive functions of I II 2 are exactly the primitive recursive ones.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Z. Adamovicz and T. Bigorajska. Functions provably total in I Σ 1. Fundamenta mathematicae, 132:189–194, 1989.Google Scholar
  2. 2.
    Z. Adamovicz and R. Kossak. A note on n and an intermediate induction schema. Zeitschrift f. math. Logik and Grundlagen d. Math., 34:261–264, 1988.Google Scholar
  3. 3.
    L.D. Beklemishev. Notes on local reflection principles. Logic Group Preprint Series 133, University of Utrecht, 1995.Google Scholar
  4. 4.
    L.D. Beklemishev. Induction rules, reflection principles, and provably recursive functions. Logic Group Preprint Series 168, University of Utrecht, 1996. To appear in Annals of Pure and Applied Logic, 1997.Google Scholar
  5. 5.
    L.D. Beklemishev. A proof-theoretic analysis of collection. To appear in Archive for Mathematical Logic.Google Scholar
  6. 6.
    T. Bigorajska.On Σ1-definable functions provably total in I II 1. Mathematical Logic Quarterly, 41:135–137, 1995.Google Scholar
  7. 7.
    G. Boolos.The Logic of Provability. Cambridge University Press, Cambridge, 1993.Google Scholar
  8. 8.
    S. Goryachev. On interpretability of some extensions of arithmetic. Mat. Zametki, 40:561–572, 1986. In Russian.Google Scholar
  9. 9.
    P. Hájek and P. Pudlák. Metamathematics of First Order Arithmetic. Springer-Verlag, Berlin, Heidelberg, New-York, 1993.Google Scholar
  10. 10.
    R. Kaye. Parameter free universal induction. Zeitschrift f. math. Logik und Grundlagen d. Math., 35(5):443–456, 1989.Google Scholar
  11. 11.
    R. Kaye, J. Paris, and C. Dimitracopoulos. On parameter free induction schemas. Journal of Symbolic Logic, 53(4):1082–1097, 1988.Google Scholar
  12. 12.
    G. Kreisel and A. Lévy. Reflection principles and their use for establishing the complexity of axiomatic systems. Zeitschrift f. math. Logik und Grundlagen d. Math., 14:97–142, 1968.Google Scholar
  13. 13.
    Z. Ratajczyk. Functions provably total in I Σ n. Fundamenta Mathematicae, 133:81–95, 1989.Google Scholar
  14. 14.
    C. Smorynski. The incompleteness theorems. In J. Barwise, editor, Handbook of Mathematical Logic, pages 821–865. North Holland, Amsterdam, 1977.Google Scholar
  15. 15.
    C. Smorynski. Self-Reference and Modal Logic. Springer-Verlag, Berlin-Heidelberg-New York, 1985.Google Scholar
  16. 16.
    A. Wilkie and J. Paris. On the scheme of induction for bounded arithmetic formulas. Annals of Pure and Applied Logic, 35:261–302, 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Lev D. Beklemishev
    • 1
  1. 1.Steklov Mathematical InstituteMoscowRussia

Personalised recommendations