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.
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asked Oct 16, 2012 at 20:31
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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.
answered Oct 16, 2012 at 20:43
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$\begingroup$ Many thanks for finally clearing this up. $\endgroup$ Commented Oct 17, 2012 at 14:57