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Herbrand consistency of some arithmetical theories

Published online by Cambridge University Press:  12 March 2014

Saeed Salehi
Saeed Salehi*
Affiliation:
Department of Mathematics, University of Tabriz, P.O. BOX 51666-17766, Tabriz, Iran, E-mail: [email protected], URL: http://saeedsalehi.ir/
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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.

Keywords

Cut-free provabilityHerbrand provability, bounded arithmeticsweak arithmeticsGodel's Second Incompleteness Theorem
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 3 , September 2012 , pp. 807 - 827
Copyright
Copyright © Association for Symbolic Logic 2012

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References

REFERENCES

[1] Adamowicz, Zofia, On tableaux consistency in weak theories, preprint # 618, Institute of Mathematics, Polish Academy of Sciences, 34 pp., 2001.Google Scholar
[2] Adamowicz, Zofia, Herbrand consistency and bounded arithmetic, Fundamenta Mathematical vol. 171 (2002), no. 3, pp. 279–292.CrossRefGoogle Scholar
[3] Adamowicz, Zofia and Zbierski, Pawel, On Herbrand consistency in weak arithmetic, Archive for Mathematical Logic, vol. 40 (2001), no. 6, pp. 399–413.Google Scholar
[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. 191–216.CrossRefGoogle Scholar
[5] Boolos, George S. and Jeffrey, Richard C., Computability and Logic, Cambridge University Press, 2007.CrossRefGoogle Scholar
[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. 195–209.Google Scholar
[7] Hájek, Petr and Pudlák, Pavel, Metamathematics of first-order arithmetic, Springer-Verlag, 1998.Google Scholar
[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. 624–638.Google Scholar
[9] Krajíček, Jan, Bounded arithmetic, prepositional logic and complexity theory, Cambridge University Press, 1995.Google Scholar
[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. 237–248.Google Scholar
[11] Pudlák, Pavel, Cuts, consistency statements and interpretations, this Journal, vol. 50 (1985), no. 2, pp. 423–441.Google Scholar
[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. 265–274.Google Scholar
[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.Google Scholar
[14] Salehi, Saeed, Herbrand consistency of some finite fragments of bounded arithmetical theories, 14 pages, http://arxiv.org/abs/1110.1848, 2011.Google Scholar
[15] Salehi, Saeed, Separating bounded arithmetical theories by Herbrand consistency, Journal of Logic and Computation, vol. 22 (2012), no. 3, pp. 545–560.CrossRefGoogle Scholar
[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. 465–496.Google Scholar
[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. 124–149.CrossRefGoogle Scholar