Does there 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 ω-categorical. And it is still open if this can be extended to all stable theories.

**Answer**

**Attribution***Source : Link , Question Author : Primo Petri , Answer Author : Community*