Theories with countably many countable models Having another question in mind (which is not yet fully worked out, but will come soon) I'd like to gather some examples of (interesting)

theories with countably many countable models

("Countably many models" means "countably many isomorphism classes of models".)
 A: I assume you're going for a big list?  Here are several:


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*Many strongly minimal theories (e.g. $\mathbb Q$-vector spaces, algebraically closed fields of a fixed characteristic)

*More generally, any $\aleph_1$-categorical theory which is not also $\aleph_0$-categorical.

*Shelah has an $\omega$-stable example with $\textrm{ENI}$-depth two (so is not $\aleph_1$-categorical) which has exactly $\aleph_0$-many countable models, which is complicated to describe (see Section 4 of Chapter 18 of Baldwin's stability theory book)


Some interesting nonexamples:


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*O-minimal theories have either finitely many or continuum-many countable models (theorem of L. Mayer) and both cases are possible

*Linear order with countably many unary predicates attached have the same property (theorem of M. Rubin) and both cases are possible


It is somewhat interesting that the uncountable models of such a theory are uninteresting in the following sense: every model of such a theory (of any cardinality) is back-and-forth equivalent to a countable model.
This means there is no definable behavior, using any $L_{\infty\omega}$-formula you like, which can occur on any model which does not occur on a countable model.
