In
mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
, cointerpretability is a
binary relation on
formal theories
Formal, formality, informal or informality imply the complying with, or not complying with, some set of requirements ( forms, in Ancient Greek). They may refer to:
Dress code and events
* Formal wear, attire for formal events
* Semi-formal atti ...
: 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
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
of ''T''. The "translation" here is required to preserve the logical structure of formulas.
This concept, in a sense dual to
interpretability In mathematical logic, interpretability is a relation between formal theories that expresses the possibility of interpreting or translating one into the other.
Informal definition
Assume ''T'' and ''S'' are formal theories. Slightly simplified, '' ...
, was introduced by , who also proved that, for theories of
Peano arithmetic and any stronger theories with effective
axiomatizations, cointerpretability is equivalent to
-conservativity.
See also
*
Cotolerance
*
Interpretability logic.
*
Tolerance (in logic)
References
*.
*.
Mathematical relations
Mathematical logic
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