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I've heard the expression "a translation from theory $A$ to theory $B$" thrown around a bit, but never encountered a formal definition. What is a translation between first-order theories?

Furthermore, does this make the class of all theories over a signature into a category, where the morphisms are translations?

goblin GONE
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  • Take a look at Caicedo's answer in http://math.stackexchange.com/questions/315399/how-does-zfc-infinitythere-is-no-infinite-set-compare-with-pa – boumol Apr 16 '13 at 11:51

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This is one of those cases where the devil is in the details, and there is a variety of notions hereabouts of different strengths. Can I suggest two resources, both freely available online?

Giorgi Japaridze and Dick de Jongh's contribution to the Handbook of Proof Theory deals with 'Notions of interpretability' in their §11, starting at p. 500: http://www.csc.villanova.edu/~japaridz/Text/prov.pdf

For category theoretic considerations, see Albert Visser's 'Categories of theories and interpretations' available at http://igitur-archive.library.uu.nl/lg/2008-0403-201345/preprint228.pdf

Peter Smith
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