Journal of Symbolic Logic

On the available partial respects in which an axiomatization for real valued arithmetic can recognize its consistency

Dan E. Willard

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

Abstract

Gödel’s Second Incompleteness Theorem states axiom systems of sufficient strength are unable to verify their own consistency. We will show that axiomatizations for a computer’s floating point arithmetic can recognize their cut-free consistency in a stronger respect than is feasible under integer arithmetics. This paper will include both new generalizations of the Second Incompleteness Theorem and techniques for evading it.

Article information

Source
J. Symbolic Logic, Volume 71, Issue 4 (2006), 1189-1199.

Dates
First available in Project Euclid: 20 November 2006

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1164060451

Digital Object Identifier
doi:10.2178/jsl/1164060451

Mathematical Reviews number (MathSciNet)
MR2275855

Zentralblatt MATH identifier
1109.03068

Citation

Willard, Dan E. On the available partial respects in which an axiomatization for real valued arithmetic can recognize its consistency. J. Symbolic Logic 71 (2006), no. 4, 1189--1199. doi:10.2178/jsl/1164060451. https://projecteuclid.org/euclid.jsl/1164060451


Export citation

References

  • Z. Adamowicz, Herbrand consistency and bounded arithmetic, Fundamenta Mathematicae, vol. 171 (2002), pp. 279--292.
  • Z. Adamowicz and P. Zbierski, On Herbrand consistency in weak theories, Archive for Mathematical Logic, vol. 40 (2001), pp. 399--413.
  • K. Atkinson, Elementary numerical analysis, Wiley Press, 1993.
  • J. Bennett, On spectra, Ph.D. thesis, Princeton University, 1962.
  • A. Bezboruah and J. C. Shepherdson, Gödel's second incompleteness theorem for \mboxQ, Journal of Symbolic Logic, vol. 41 (1976), pp. 503--512.
  • R. Burden and J. Faires, Numerical methods, Brookes-Cole, 2003.
  • K. Gödel, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173--198.
  • P. Hájek and P. Pudlák, Metamathematics of first order arithmetic, Springer Verlag, 1991.
  • S. C. Kleene, On the notation of ordinal numbers, Journal of Symbolic Logic, vol. 3 (1938), pp. 150--156.
  • E. Nelson, Predicative arithmetic, Princeton Math Notes Press, 1986.
  • J. B. Paris and C. Dimitracopoulos, Truth definitions for $\Delta_0$ formulae, Logic and algorithmic, Monographie de L'Enseignement Mathematique, vol. 30, 1982, pp. 317--329.
  • --------, A note on the undefinability of cuts, Journal of Symbolic Logic, vol. 48 (1983), pp. 564--569.
  • P. Pudlák, Cuts, consistency statements and interpretations, Journal of Symbolic Logic, vol. 50 (1985), pp. 423--442.
  • --------, On the lengths of proofs of consistency, Collegium Logicum: Annals of the Kurt Goedel Society, vol. 2 (1996), pp. 65--86, Springer-Wien.
  • R. M. Solovay Several telephone conversations (during 1994) discussing how to modify Theorem 2.3 from Pudlák's article [Pu85?] with the methods of Nelson, and Wilki-Paris [Ne86,WP87?]. (The Appendix A of [ww1?] offers a 4-page interpretation of the underlying idea behind Solovay's unpublished observation.).,
  • A. J. Wilkie and J. B. Paris, On the scheme of induction for bounded arithmetic, Annals on Pure and Applied Logic, vol. 35 (1987), pp. 261--302.
  • D. Willard, Self-verifying systems, the incompleteness theorem and the tangibility reflection principle, Journal of Symbolic Logic, vol. 66 (2001), pp. 536--596.
  • --------, How to extend the semantic tableaux and cut-free versions of the second incompleteness theorem almost to Robinson's arithmetic \mboxQ, Journal of Symbolic Logic, vol. 67 (2002), pp. 465--496.
  • --------, A version of the second incompleteness theorem for axiom systems that recognize addition as a total function, First order logic revisited (V. Hendricks et. al., editor), Logos Verlag, 2004, pp. 337--368.
  • --------, An exploration of the partial respects in which an axiom system recognizing solely addition as a total function can verify its own consistency, Journal of Symbolic Logic, vol. 70 (2005), pp. 1171--1209.
  • --------, A new variant of Hilbert styled generalization of the second incompleteness theorem and some exceptions to it, Annals on Pure and Applied Logic, vol. 141 (2006), pp. 472--496.
  • C. Wrathall, Rudimentary predicates and relative computation, Siam Journal on Computing, vol. 7 (1978), pp. 194--209.