Cointerpretability
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In mathematical logic, cointerpretability is a binary relation on formal theories: a formal theory T is cointerpretable in another such theory S, when the language of S can be translated into the language of T in such a way that S proves every formula whose translation is a theorem of T. The "translation" here is required to preserve the logical structure of formulas.
This concept, in a sense dual to interpretability, was introduced by Japaridze (http://www.csc.villanova.edu/~japaridz/) in 1993, who also proved that, for theories Peano arithmetic and any stronger theories with effective axiomatizations, cointerpretability is equivalent to <math>\Sigma_1<math>-conservativity.
See also: tolerance, cotolerance, interpretability logic.
References
- G.Japaridze (http://www.csc.villanova.edu/~japaridz/), A generalized notion of weak interpretability and the corresponding logic. Annals of Pure and Applied Logic 61 (1993), pp. 113-160.
- G.Japaridze (http://www.csc.villanova.edu/~japaridz/study.html) and D. de Jongh, The logic of provability. Handbook of Proof Theory. S.Buss, ed. Elsevier, 1998, pp. 476-546.