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This is a "well-known fact," but I'm at a loss to finding a proof. I could swear I've read it somewhere, but checking the handful of places I'm used to checking doesn't help. Google gives nothing, so let's have a proof here!

Theorem: Suppose $T$ is a (countable complete) $\omega$-stable theory. Then $T$ is either $\aleph_0$-categorical or $T$ has infinitely many countable models.

Does anyone know a proof?

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I won't give a proof, but I will give some historical background and references.

Let $I(T,\kappa)$ be the number of isomorphism types of models of $T$ of cardinality $\kappa$. A countable complete theory $T$ with $1<I(T,\aleph_0)<\aleph_0$ is called an Ehrenfeucht theory.

Theorem (Baldwin-Lachlan, 1971): No Ehrenfeucht theory is uncountably categorical.

Indeed, the Baldwin-Lachlan analysis shows that models of an uncountably categorical theory are determined up to isomorphism by the dimension of a strongly minimal set, and the possible dimensions for a countable model are upwards-closed in $\{0,1,2,\dots,\aleph_0\}$.

Now uncountably categorical theories are $\omega$-stable, and $\omega$-stable theories are superstable. Just two years later, Lachlan pushed the lower bound on the complexity of Ehrenfeucht theories all the way to superstability:

Theorem (Lachlan, 1973): No Ehrenfeucht theory is superstable.

I don't know of any proof in the $\omega$-stable case which is easier than the general superstable case, but I would be very interested to see one!

You can find Lachlan's proof in his paper On the number of countable models of a countable superstable theory (if you can find the paper - I couldn't!), but it's apparently quite complicated. The proof was substantially simplified by Lascar in his important 1976 paper Ranks and definability in superstable theories, as an application of his newly-minted $U$-rank.

Since then, several other proofs have been given. If you want to read a proof in a textbook, Lachlan's theorem is the very last theorem in Pillay's little book Introduction to Stability Theory, and it appears as Theorem 5.6.2 in Buechler's Essential Stability Theory.

Lachlan's theorem was generalized by Kim:

Theorem (Kim, 1999): No Ehrenfeucht theory is supersimple.

Kim's paper On the number of countable models of a countable supersimple theory is very short and self-contained (modulo the basic properties of forking in simple theories), so it might actually be the most direct place to read the proof, even for the $\omega$-stable case!

An important question is whether we can remove the "super":

Lachlan's Problem: Is there a stable Ehrenfeucht theory?

Hrushovski addressed this problem in his thesis, where he showed that no Ehrenfeucht theory is stable and "finitely based".

In 2001, Sudoplatov announced a positive solution to Lachlan's Problem, which he subsequently laid out in his book The Lachlan Problem. I have not read the book (though I am interested in reading it at some point!).

You can find a very complete history of Ehrenfeucht theories (and information about the proposed solution to Lachlan's problem) starting on p. 9 of these slides by Sudoplatov.

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  • $\begingroup$ Thank you! Also, I believe the proof in Buechler was the one I was thinking of, so thanks for finding the reference, too. Excellent! $\endgroup$ – Richard Rast Dec 10 '15 at 1:35

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