Cointerpretability

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 (1993), who also proved that, for theories of Peano arithmetic and any stronger theories with computable axiomatizations, cointerpretability is equivalent to

Σ

1

{\displaystyle \Sigma _{1}}

-conservativity.

See also

Cotolerance

Interpretability logic

Tolerance (in logic)

References

Japaridze, Giorgi (1993), "A generalized notion of weak interpretability and the corresponding modal logic", Annals of Pure and Applied Logic, 61 (1–2): 113–160, doi:10.1016/0168-0072(93)90201-N, MR 1218658.

Japaridze, Giorgi; de Jongh, Dick (1998), "The logic of provability", in Buss, Samuel R. (ed.), Handbook of Proof Theory, Studies in Logic and the Foundations of Mathematics, vol. 137, Amsterdam: North-Holland, pp. 475–546, doi:10.1016/S0049-237X(98)80022-0, MR 1640331.