The Second Incompleteness Theorem and Bounded Interpretations

Studia Logica 100 (1-2):399-418 (2012)
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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 S21{{\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

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Citations of this work

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References found in this work

Arithmetization of Metamathematics in a General Setting.Solomon Feferman - 1960 - Journal of Symbolic Logic 31 (2):269-270.
On the scheme of induction for bounded arithmetic formulas.A. J. Wilkie & J. B. Paris - 1987 - Annals of Pure and Applied Logic 35 (C):261-302.
Cuts, consistency statements and interpretations.Pavel Pudlák - 1985 - Journal of Symbolic Logic 50 (2):423-441.
The computational complexity of logical theories.Jeanne Ferrante - 1979 - New York: Springer Verlag. Edited by Charles W. Rackoff.

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