Throughout this post, theory means first-order theory. In fact, we are concerned with theories that are recursively presented, though the abstract framework applies more generally. Thanks to Fredrik Engström Ellborg for suggesting in Google+ the reference Kaye-Wong, and to Ali Enayat for additional references and many useful conversations on this topic.
Informally, to say that a theory interprets a theory means that there is a procedure for associating structures in the language of to structures in the language of in such a way that if is a model of , then is a model of .
Let us be a bit more precise, and do this syntactically to reduce the requirements of the metatheory. The original notion is due to Tarski, see
Alfred Tarski. Undecidable theories. In collaboration with Andrzej Mostowski and Raphael M. Robinson. Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Company, Amsterdam, 1953. MR0058532 (15,384h).
I follow here the modern reference on interpretations,
Albert Visser, Categories of theories and interpretations, in Logic in Tehran, Lecture Notes in Logic, vol. 26, Association for Symbolic Logic, La Jolla, CA, 2006, pp. 284–341. MR2262326 (2007j:03083).
One can take “the theory interprets the theory ” to mean that there are
- A map that assigns formulas in the language of to the symbols of the language of , and
- A formula in the language of ,
with the following properties: We can extend to all -formulas recursively: , etc, and . It then holds that proves
- , and
- for each axiom of (including the axioms of first-order logic).
Here, are taken to be recursive, and so is .
If the above happens, then we can see as a strong witness to the fact that the consistency of implies the consistency of .
Two theories are mutually interpretable iff each one interprets the other. By the above, this is a strong version of the statement that they are equiconsistent.
Two theories are bi-interpretable iff they are mutually interpretable, and in fact, the interpretations from is and from in can be taken to be “inverses” of each other, in the sense that proves that and are equivalent for each in the language of , and similarly for , and . In a sense, two theories that are bi-interpretable are very much “the same”, only differing in their presentation.