The Second Incompleteness Theorem and Bounded Interpretations Authors Albert Visser Department of Philosophy Utrecht University Article

First Online: 09 February 2012 DOI :
10.1007/s11225-012-9385-z

Cite this article as: Visser, A. Stud Logica (2012) 100: 399. doi:10.1007/s11225-012-9385-z
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 Dedicated to the memory of Leo Esakia

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References 1.

Berarducci A., Verbrugge R.: ‘On the provability logic of bounded arithmetic’. Annals of Pure and Applied Logic

61 , 75–93 (1993)

CrossRef 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)

CrossRef 3.

Feferman S.: ‘Arithmetization of metamathematics in a general setting’. Fundamenta Mathematicae 49 , 35–92 (1960)

4.

Ferrante, J., and C. W. Rackoff, The computational complexity of logical theories , volume 718 of Lecture Notes in Mathematics . Springer, Berlin, 1979.

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.

6.

Gerhardy, P., ‘The Role of Quantifier Alternations in Cut Elimination’. Notre Dame Journal of Formal Logic , 46, no. 2:165–171, 2005.

7.

Hájek P., Pudlák P.: Metamathematics of First-Order Arithmetic. Perspectives in Mathematical Logic. Springer, Berlin (1991)

8.

Paris, J. B., and C. Dimitracopoulos, ‘Truth definitions and Δ_{0} formulae’. In Logic and algorithmic , Monographie de L’Enseignement Mathematique 30, Geneve, 1982, pp. 317–329.

9.

Pudlák P.: ‘Cuts, consistency statements and interpretations’. The Journal of Symbolic Logic

50 , 423–441 (1985)

CrossRef 10.

Vaught R. A.: ‘Axiomatizability by a schema’. The Journal of Symbolic Logic

32 (4), 473–479 (1967)

CrossRef 11.

Visser A.: ‘An inside view of

EXP ’. The Journal of Symbolic Logic

57 , 131–165 (1992)

CrossRef 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.

13.

Visser A.: ‘Pairs, sets and sequences in first order theories’. Archive for Mathematical Logic,

47 (4), 299–326 (2008)

CrossRef 14.

Visser, A., ‘Can we make the Second Incompleteness Theorem coordinate free’.

Journal of Logic and Computation , 2009. doi:

10.1093/logcom/exp048 .

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.

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)

CrossRef © Springer Science+Business Media B.V. 2012