# Complete, Finitely Axiomatizable, Theory with 3 Countable Models

Does there exist a complete, finitely axiomatizable, first-order theory $$TT$$ with exactly 3 countable non-isomorphic models?

There is a classical example of a complete theory with exacly $$33$$ models. This theory is not finitely axiomatizable (For the trivial reason that the language is infinite).

In this post, Javier Moreno explains how to rephrase this example in a finite language. Still, the theory is not finitely axiomatizable.

There have been research in connection with stability. Lachlan has proved that a superstable theory with finitely many countable models is $$ω\omega$$-categorical. And it is still open if this can be extended to all stable theories.