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2024-07-25T17:54:40Z

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In [[mathematical logic]], '''cointerpretability''' is a [[binary relation]] on [[theory (mathematical logic)|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 {{harvtxt|Japaridze|1993}}, who also proved that, for theories of [[Peano arithmetic]] and any stronger theories with effective [[axiomatization]]s, cointerpretability is equivalent to <math>\Sigma_1</math>-conservativity.

==See also==
* [[Cotolerance]]
* [[Interpretability logic]]
* [[Tolerance (in logic)]]

==References==
*{{citation
| last = Japaridze| first = Giorgi | authorlink = Giorgi Japaridze
| doi = 10.1016/0168-0072(93)90201-N
| issue = 1–2
| journal = Annals of Pure and Applied Logic
| mr = 1218658
| pages = 113–160
| title = A generalized notion of weak interpretability and the corresponding modal logic
| volume = 61
| year = 1993| doi-access =
}}.
*{{citation
| last1 = Japaridze | first1 = Giorgi | author1-link = Giorgi Japaridze
| last2 = de Jongh | first2 = Dick | author2-link = Dick de Jongh
| editor-last = Buss | editor-first = Samuel R. | editor-link = Samuel Buss
| contribution = The logic of provability
| doi = 10.1016/S0049-237X(98)80022-0
| location = Amsterdam
| mr = 1640331
| pages = 475–546
| publisher = North-Holland
| series = Studies in Logic and the Foundations of Mathematics
| title = Handbook of Proof Theory
| volume = 137
| year = 1998| doi-access = free
}}.

[[Category:Mathematical relations]]
[[Category:Mathematical logic]]

{{logic-stub}}

Cointerpretability - Revision history

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