Studia Logica

, Volume 100, Issue 1–2, pp 399–418

The Second Incompleteness Theorem and Bounded Interpretations

Article

Abstract

In this paper we formulate a version of Second Incompleteness Theorem. The idea is that a sequential sentence has ‘consistency power’ over a theory if it enables us to construct a bounded interpretation of that theory. An interpretation of V in U is bounded if, for some n, all translations of V-sentences are U-provably equivalent to sentences of complexity less than n. We call a sequential sentence with consistency power over T a pro-consistency statement for T. We study pro-consistency statements. We provide an example of a pro-consistency statement for a sequential sentence A that is weaker than an ordinary consistency statement for A. We show that, if A is \({{\sf S}^{1}_{2}}\) , this sentence has some further appealing properties, specifically that it is an Orey sentence for EA.

The basic ideas of the paper essentially involve sequential theories. We have a brief look at the wider environment of the results, to wit the case of theories with pairing.

Keywords

Second Incompleteness Theorem interpretability 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Berarducci A., Verbrugge R.: ‘On the provability logic of bounded arithmetic’. Annals of Pure and Applied Logic 61, 75–93 (1993)CrossRefGoogle Scholar
  2. 2.
    Cégielski P., Richard D.: ‘Decidability of the natural integers equipped with the Cantor pairing function and successor’. Theoretical Computer Science, 257(1-2), 51–77 (2001)CrossRefGoogle Scholar
  3. 3.
    Feferman S.: ‘Arithmetization of metamathematics in a general setting’. Fundamenta Mathematicae 49, 35–92 (1960)Google Scholar
  4. 4.
    Ferrante, J., and C. W. Rackoff, The computational complexity of logical theories, volume 718 of Lecture Notes in Mathematics. Springer, Berlin, 1979.Google Scholar
  5. 5.
    Gerhardy, P., ‘Refined Complexity Analysis of Cut Elimination’. In Matthias Baaz and Johann Makovsky, (eds.), Proceedings of the 17th International Workshop CSL 2003, volume 2803 of LNCS, pp. 212–225. Springer-Verlag, Berlin, 2003.Google Scholar
  6. 6.
    Gerhardy, P., ‘The Role of Quantifier Alternations in Cut Elimination’. Notre Dame Journal of Formal Logic, 46, no. 2:165–171, 2005.Google Scholar
  7. 7.
    Hájek P., Pudlák P.: Metamathematics of First-Order Arithmetic. Perspectives in Mathematical Logic. Springer, Berlin (1991)Google Scholar
  8. 8.
    Paris, J. B., and C. Dimitracopoulos, ‘Truth definitions and Δ0 formulae’. In Logic and algorithmic, Monographie de L’Enseignement Mathematique 30, Geneve, 1982, pp. 317–329.Google Scholar
  9. 9.
    Pudlák P.: ‘Cuts, consistency statements and interpretations’. The Journal of Symbolic Logic 50, 423–441 (1985)CrossRefGoogle Scholar
  10. 10.
    Vaught R. A.: ‘Axiomatizability by a schema’. The Journal of Symbolic Logic 32(4), 473–479 (1967)CrossRefGoogle Scholar
  11. 11.
    Visser A.: ‘An inside view of EXP’. The Journal of Symbolic Logic 57, 131–165 (1992)CrossRefGoogle Scholar
  12. 12.
    Visser, A., ‘Categories of Theories and Interpretations’. In Ali Enayat, Iraj Kalantari, and Mojtaba Moniri (eds.), Logic in Tehran. Proceedings of the workshop and conference on Logic, Algebra and Arithmetic, held October 18–22, 2003, volume 26 of Lecture Notes in Logic. ASL, A.K. Peters, Ltd., Wellesley, Mass., 2006, pp. 284–341.Google Scholar
  13. 13.
    Visser A.: ‘Pairs, sets and sequences in first order theories’. Archive for Mathematical Logic, 47(4), 299–326 (2008)CrossRefGoogle Scholar
  14. 14.
    Visser, A., ‘Can we make the Second Incompleteness Theorem coordinate free’. Journal of Logic and Computation, 2009. doi:10.1093/logcom/exp048.
  15. 15.
    Wilkie, A. J., ‘On sentences interpretable in systems of arithmetic’. In Logic Colloquium ’84, volume 120 of Studies in Logic and the Foundations of Mathematics. Elsevier, 1986, pp. 329–342.Google Scholar
  16. 16.
    Wilkie A., Paris J. B.: ‘On the scheme of of induction for bounded arithmetic formulas’. Annals of Pure and Applied Logic, 35, 261–302 (1987)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department of PhilosophyUtrecht UniversityUtrechtThe Netherlands

Personalised recommendations