0
$\begingroup$

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.

$\endgroup$

1 Answer 1

4
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ Many thanks for finally clearing this up. $\endgroup$ Oct 17, 2012 at 14:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.