Skip to main content
SpringerLink
Log in

Gentzen’s consistency proof without heightlines

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

Abstract

This paper gives a Gentzen-style proof of the consistency of Heyting arithmetic in an intuitionistic sequent calculus with explicit rules of weakening, contraction and cut. The reductions of the proof, which transform derivations of a contradiction into less complex derivations, are based on a method for direct cut-elimination without the use of multicut. This method treats contractions by tracing up from contracted cut formulas to the places in the derivation where each occurrence was first introduced. Thereby, Gentzen’s heightline argument, which introduces additional cuts on contracted compound cut formulas, is avoided. To show termination of the reduction procedure an ordinal assignment based on techniques of Howard for Gödel’s T is used.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Borisavljević M., Dosen K., Petrić Z.: On permuting cut with contraction. Math. Struct. Comput. Sci. 10(2), 99–136 (2000)

    Article  MATH  Google Scholar 

  2. Buss S.: Handbook Proof Theory. Elsevier, Amsterdam (1998)

    MATH  Google Scholar 

  3. Gentzen G.: The Collected Papers of Gerhard Gentzen. North-Holland, Amsterdam (1969)

    MATH  Google Scholar 

  4. Howard, W.A.: Assignment of ordinals to terms for primitive recursive functionals of finite type. In: Myhill, J., Kino, A., Vesley, R. (eds.) Intuitionism and Proof Theory. North-Holland, Amsterdam (1970)

  5. Kanckos A.: Consistency of Heyting arithmetic in natural deduction. Math. Log. Q. 56(6), 611–624 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. von Plato J.: A proof of Gentzen’s Hauptsatz without multicut. Arch. Math. Log. 40, 9–18 (2001)

    Article  MATH  Google Scholar 

  7. von Plato J.: Normal form and existence property for derivations in heyting arithmetic. Acta Philos. Fennica 78, 159–163 (2006)

    Google Scholar 

  8. Schütte K.: Proof Theory. Springer, Berlin (1977)

    Book  MATH  Google Scholar 

  9. Siders, A.: A direct Gentzen-style consistency proof for Heyting arithmetic (to appear)

  10. Takeuti G.: Proof Theory. North-Holland, Amsterdam (1987)

    MATH  Google Scholar 

  11. Troelstra A.S., Schwichtenberg H.: Basic Proof Theory. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  12. Troelstra A.S., van Dalen D.: Constructivism in Mathematics vol. 1. North-Holland, Amsterdam (1988)

    Google Scholar 

  13. Wilken G., Weiermann A.: Derivation lengths classification of Goedel’s T extending Howard’s assignment. Log. Methods Comput. Sci. 8(1), 19–144 (2012)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Philosophy, University of Helsinki, P.O. Box 24 (UNIONINKATU 40 A), 00014, Helsinki, Finland

    Annika Siders

Authors
  1. Annika Siders
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Annika Siders.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Siders, A. Gentzen’s consistency proof without heightlines. Arch. Math. Logic 52, 449–468 (2013). https://doi.org/10.1007/s00153-013-0324-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-013-0324-0

Keywords

Mathematics Subject Classification (2000)

Search

Navigation