Journal of Symbolic Logic

Herbrand consistency of some arithmetical theories

Saeed Salehi

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Gödel's second incompleteness theorem is proved for Herbrand consistency of some arithmetical theories with bounded induction, by using a technique of logarithmic shrinking the witnesses of bounded formulas, due to Z. Adamowicz [Herbrand consistency and bounded arithmetic, Fundamenta Mathematicae, vol. 171 (2002), pp. 279—292]. In that paper, it was shown that one cannot always shrink the witness of a bounded formula logarithmically, but in the presence of Herbrand consistency, for theories IΔ0m with m ≥ 2, any witness for any bounded formula can be shortened logarithmically. This immediately implies the unprovability of Herbrand consistency of a theory T⊇ IΔ02 in T itself.

In this paper, the above results are generalized for IΔ01. Also after tailoring the definition of Herbrand consistency for IΔ0 we prove the corresponding theorems for IΔ0. Thus the Herbrand version of Gödel's second incompleteness theorem follows for the theories IΔ01 and IΔ0.

Article information

J. Symbolic Logic, Volume 77, Issue 3 (2012), 807-827.

First available in Project Euclid: 13 August 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: Primary 03F40, 03F30; Secondary 03F05, 03H15

Cut-free provability Herbrand provability bounded arithmetics, weak arithmetics Gödel's Second Incompleteness Theorem


Salehi, Saeed. Herbrand consistency of some arithmetical theories. J. Symbolic Logic 77 (2012), no. 3, 807--827. doi:10.2178/jsl/1344862163.

Export citation


  • Zofia Adamowicz On tableaux consistency in weak theories, preprint # 618, Institute of Mathematics, Polish Academy of Sciences, 34 pp.,2001.
  • –––– Herbrand consistency and bounded arithmetic, Fundamenta Mathematicae, vol. 171(2002), no. 3, pp. 279–292.
  • Zofia Adamowicz and Paweł Zbierski On Herbrand consistency in weak arithmetic, Archive for Mathematical Logic, vol. 40(2001), no. 6, pp. 399–413.
  • Zofia Adamowicz and Konrad Zdanowski Lower bounds for the unprovability of Herbrand consistency in weak arithmetics, Fundamenta Mathematicae, vol. 212(2011), no. 3, pp. 191–216.
  • George S. Boolos and Richard C. Jeffrey Computability and Logic, Cambridge University Press,2007.
  • Samuel R. Buss On Herbrand's theorem, Proceedings of the International Workshop on Logic and Computational Complexity, October 13–16, 1994 (D. Maurice and R. Leivant, editors), Lecture Notes in Computer Science 960, Springer-Verlag,1995, pp. 195–209.
  • Petr Hájek and Pavel Pudlák Metamathematics of first-order arithmetic, Springer-Verlag,1998.
  • Leszek A. Kołodziejczyk On the Herbrand notion of consistency for finitely axiomatizable fragments of bounded arithmetic theories, Journal of Symbolic Logic, vol. 71(2006), no. 2, pp. 624–638.
  • Jan Krajíček Bounded arithmetic, propositional logic and complexity theory, Cambridge University Press,1995.
  • Jeff B. Paris and Alex J. Wilkie $\Delta_0$ sets and induction, Proceedings of Open Days in Model Theory and Set Theory (Guzicki W., Marek W., Plec A., and Rauszer C., editors), Leeds University Press,1981, pp. 237–248.
  • Pavel Pudlák Cuts, consistency statements and interpretations, Journal of Symbolic Logic, vol. 50(1985), no. 2, pp. 423–441.
  • Saeed Salehi Unprovability of Herbrand consistency in weak arithmetics, Proceedings of the sixth ESSLLI student session, European Summer School for Logic, Language, and Information (Striegnitz K., editor),2001, pp. 265–274.
  • –––– Herbrand consistency in arithmetics with bounded induction, Ph.D. Dissertation, Institute of Mathematics of the Polish Academy of Sciences,2002, 84 pages, available on the net at family
  • –––– Herbrand consistency of some finite fragments of bounded arithmetical theories, 14 pages, family,2011.
  • –––– Separating bounded arithmetical theories by Herbrand consistency, Journal of Logic and Computation, vol. 22(2012), no. 3, pp. 545–560.
  • Dan E. Willard How to extend the semantic tableaux and cut-free versions of the second incompleteness theorem almost to Robinson's arithmetic Q, Journal of Symbolic Logic, vol. 67(2002), no. 1, pp. 465–496.
  • –––– Passive induction and a solution to a Paris–Wilkie open question, Annals of Pure and Applied Logic, vol. 146(2007), no. 2–3, pp. 124–149.