Suppose all the models of a first order theory are finite, have the same cardinal number, and are
isomorphic. Is the theory then necessarilly complete? Normally I would not ask such a"specialized"
question on "mathoverflow.net", a question whose answer must be well known. However I have been unable
to find an answer after quite a bit of searching through various books and articles that deal with
finite models. I did find a converse-completeness implies categoricity for finite models. Nevertheless, I am almost sure my question has a "yes" answer though I don't quite see how to prove it.



A question about first order theories having only finite models.
Yes. Any sentence in the language of the theory is either true in all models or false in all models, since all models are isomorphic. By the completeness theorem, each sentence is either provable or refutable in the theory.


