Archive for Mathematical Logic

, Volume 52, Issue 3–4, pp 317–333 | Cite as

Herbrand consistency of some finite fragments of bounded arithmetical theories

  • Saeed Salehi


We formalize the notion of Herbrand Consistency in an appropriate way for bounded arithmetics, and show the existence of a finite fragment of IΔ0 whose Herbrand Consistency is not provable in IΔ0. We also show the existence of an IΔ0-derivable Π 1-sentence such that IΔ0 cannot prove its Herbrand Consistency.


Herbrand consistency Bounded arithmetic Gödel’s Second Incompleteness Theorem 

Mathematics Subject Classification (2000)

03F40 03F25 03F30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adamowicz, Z.: On Tableaux consistency in weak theories. Preprint # 618, Institute of Mathematics, Polish Academy of Sciences (2001).
  2. 2.
    Adamowicz Z, Zbierski P: On Herbrand consistency in weak arithmetic. Arch. Math. Logic 40, 399–413 (2001). doi: 10.1007/s001530000072 MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Adamowicz, Z.: Herbrand consistency and bounded arithmetic. Fund. Math. 171, 279–292 (2002).
  4. 4.
    Adamowicz Z, Zdanowski K: Lower bounds for the unprovability of Herbrand consistency in weak arithmetics. Fund. Math. 212, 191–216 (2011). doi: 10.4064/fm212-3-1 MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Buss, S.R.: On Herbrand’s theorem. In: Maurice, D., Leivant, R. (eds.) Selected Papers from the International Workshop on Logic and Computational Complexity. Indianapolis, IN, USA, October 13–16, 1994, Lecture Notes in Computer Science, vol. 960, pp. 195–209. Springer, Berlin (1995).
  6. 6.
    Hájek, P., Pudlák, P.: Metamathematics of First-Order Arithmetic. Springer, Berlin (1998) (2nd printing).
  7. 7.
    Kołodziejczyk L.A.: On the Herbrand notion of consistency for finitely axiomatizable fragments of bounded arithmetic theories. J. Symb. Log. 71, 624–638 (2006). doi: 10.2178/jsl/1146620163 zbMATHCrossRefGoogle Scholar
  8. 8.
    Paris, J.B., Wilkie, A.J.: Δ0 sets and induction. In: Guzicki, W., Marek, W., Plec, A., Rauszer, C. (eds.) Proceedings of Open Days in Model Theory and Set Theory, Jadwisin, Poland 1981, pp. 237–248. Leeds University Press, Leeds (1981)Google Scholar
  9. 9.
    Salehi, S.: Unprovability of Herbrand consistency in weak arithmetics. In: Striegnitz, K. (ed.) Proceedings of the Sixth ESSLLI Student Session, European Summer School for Logic, Language, and Information, pp. 265–274 (2001).
  10. 10.
    Salehi, S.: Herbrand consistency in arithmetics with bounded induction. Ph.D. Dissertation, Institute of Mathematics, Polish Academy of Sciences (2002).
  11. 11.
    Salehi S.: Separating bounded arithmetical theories by Herbrand consistency. J. Log. Comput. 22, 545–560 (2012). doi: 10.1093/logcom/exr005 MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Salehi S.: Herbrand consistency of some arithmetical theories. J. Symb. Log. 77, 807–827 (2012). doi: 10.2178/jsl/1344862163 MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Willard D.E.: How to extend the semantic Tableaux and cut-free versions of the second incompleteness theorem almost to Robinson’s arithmetic Q. J. Symb. Log. 67, 465–496 (2002). doi: 10.2178/jsl/1190150055 MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Willard D.E.: Passive induction and a solution to a Paris–Wilkie open question. Ann. Pure Appl. Log. 146, 124–149 (2007). doi: 10.1016/j.apal.2007.01.003 MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TabrizTabrizIran
  2. 2.School of MathematicsInstitute for Research in Fundamental Sciences (IPM)TehranIran

Personalised recommendations