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.
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 -conservativity.
See also
[edit]References
[edit]- 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.
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