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.
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Dedicated to the memory of Leo Esakia
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Visser, A. The Second Incompleteness Theorem and Bounded Interpretations. Stud Logica 100, 399–418 (2012). https://doi.org/10.1007/s11225-012-9385-z
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DOI: https://doi.org/10.1007/s11225-012-9385-z