The Second Incompleteness Theorem and Bounded Interpretations
 Albert Visser
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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 Vsentences are Uprovably equivalent to sentences of complexity less than n. We call a sequential sentence with consistency power over T a proconsistency statement for T. We study proconsistency statements. We provide an example of a proconsistency 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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 Title
 The Second Incompleteness Theorem and Bounded Interpretations
 Journal

Studia Logica
Volume 100, Issue 12 , pp 399418
 Cover Date
 20120401
 DOI
 10.1007/s112250129385z
 Print ISSN
 00393215
 Online ISSN
 15728730
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Second Incompleteness Theorem
 interpretability
 Authors

 Albert Visser ^{(1)}
 Author Affiliations

 1. Department of Philosophy, Utrecht University, Janskerkhof 13A, 3512BL, Utrecht, The Netherlands