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

A few relevant comments:

There is a classical example of a complete theory with exacly $3$ 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.

Some less relevant comments:

I would like to know if finite axiomability has ever be asked in this context.

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

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    $\begingroup$ @James pointed out my misreading of the Question that these 3 countable nonisomorphic models are the only models, interpreting instead that among its countable models, there are exactly three isomorphism classes. $\endgroup$ – hardmath Aug 29 '14 at 15:24
  • $\begingroup$ Thanks for the comments, which make the question much more accessible to someone who hasn't thought about such things before. $\endgroup$ – Carl Mummert Aug 29 '14 at 18:01
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    $\begingroup$ Wilfred Hodges' Model Theory theorem 12.2.18 states that a totally categorical theory $T$ (a) is not finitely axiomatisable and (b) is quasi-finitely axiomatisable; where the latter property means that it is definitionally equivalent to a single sentence together with all the sentences "there are at least $n$ elements" for each $n$. And further that an $\omega$-stable and $\omega$-categorical $T$ is not finitely axiomatisable either. (So your second question--has anybody thought about these questions before?--has answer "yes". Hodges says the theorem answers a conjecture of Vaught.) $\endgroup$ – HTFB Mar 18 '15 at 13:03
  • $\begingroup$ Note though that the usual Ehrenfeucht example of a 3-model theory is unstable. $\endgroup$ – HTFB Mar 22 '15 at 17:06
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    $\begingroup$ @DanielV There are 3 models up to isomorphism. "Up to isomorphism" is usually left implicit. If you do not count "up to isomorphism" there is always a proper class of models. $\endgroup$ – Primo Petri Mar 17 '16 at 13:37

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