Skip to main content
×
×
Home

Herbrand consistency of some arithmetical theories

  • Saeed Salehi (a1)
Abstract

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 IΔ0 + Ωm 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 ⊇ IΔ0 + Ω2 in T itself.

In this paper, the above results are generalized for Δ0 + Ω1. 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 IΔ0 + Ω1 and IΔ0.

Copyright
References
Hide All
[1] Adamowicz, Zofia, On tableaux consistency in weak theories, preprint # 618, Institute of Mathematics, Polish Academy of Sciences, 34 pp., 2001.
[2] Adamowicz, Zofia, Herbrand consistency and bounded arithmetic, Fundamenta Mathematical vol. 171 (2002), no. 3, pp. 279292.
[3] Adamowicz, Zofia and Zbierski, Pawel, On Herbrand consistency in weak arithmetic, Archive for Mathematical Logic, vol. 40 (2001), no. 6, pp. 399413.
[4] Adamowicz, Zofia and Zdanowski, Konrad, Lower bounds for the unprovability of Herbrand consistency in weak arithmetics, Fundamenta Mathematicae, vol. 212 (2011), no. 3, pp. 191216.
[5] Boolos, George S. and Jeffrey, Richard C., Computability and Logic, Cambridge University Press, 2007.
[6] Buss, Samuel R., On Herbrand's theorem, Proceedings of the International Workshop on Logic and Computational Complexity, October 13-16, 1994 (Maurice, D. and Leivant, R., editors), Lecture Notes in Computer Science 960, Springer-Verlag, 1995, pp. 195209.
[7] Hájek, Petr and Pudlák, Pavel, Metamathematics of first-order arithmetic, Springer-Verlag, 1998.
[8] Kołodziejczyk, Leszek A., On the Herbrand notion of consistency for finitely axiomatizable fragments of bounded arithmetic theories, this Journal, vol. 71 (2006), no. 2, pp. 624638.
[9] Krajíček, Jan, Bounded arithmetic, prepositional logic and complexity theory, Cambridge University Press, 1995.
[10] Paris, Jeff B. and Wilkie, Alex J., Δ0 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. 237248.
[11] Pudlák, Pavel, Cuts, consistency statements and interpretations, this Journal, vol. 50 (1985), no. 2, pp. 423441.
[12] Salehi, Saeed, 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. 265274.
[13] Salehi, Saeed, 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 http://saeedsalehi.ir/pphd.html.
[14] Salehi, Saeed, Herbrand consistency of some finite fragments of bounded arithmetical theories, 14 pages, http://arxiv.org/abs/1110.1848, 2011.
[15] Salehi, Saeed, Separating bounded arithmetical theories by Herbrand consistency, Journal of Logic and Computation, vol. 22 (2012), no. 3, pp. 545560.
[16] Willard, Dan E., How to extend the semantic tableaux and cut-free versions of the second incompleteness theorem almost to Robinson's arithmetic Q, this Journal, vol. 67 (2002), no. 1, pp. 465496.
[17] Willard, Dan E., Passive induction and a solution to a Paris-Wilkie open question, Annals of Pure and Applied Logic, vol. 146 (2007), no. 2–3, pp. 124149.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed