The Second Incompleteness Theorem and Bounded Interpretations 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
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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 \({{\sf S}^{1}_{2}}\) , 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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Authors and Affiliations 1. Department of Philosophy Utrecht University Utrecht The Netherlands