# On the Baldwin-Lachlan theorem

Let $T$ be a complete countable theory with infinite models. The Baldwin-Lachlan theorem states that $T$ is uncountably categorical iff $T$ is $\omega$-stable and has no Vaughtian pair.

My question is somehow connected to the discussion on Morley's categoricity theorem which can be found here. I would be interested in seeing an example (either a simple exercise or a reference to a well-known result) of a situation where one can use this theorem to show that a certain theory is uncountably categorical. In other words, I'd like to see a $T$ which is not strongly minimal (because otherwise you don't need this theorem, and you can see directly the notion of dimension appear -- see Alex Kruckman's answer in the aforementioned post) and which can be proven to be uncountably categorical by proving its $\omega$-stability and its lack of Vaughtian pairs. For example, one could imagine a situation where, thanks to some quantifier elimination result, counting types over countable sets of parameters is not that difficult, and where for some trivial reason there cannot be a Vaughtian pair (for example, as in the case of real closed fields, because the cardinality of an infinite definable set is forced to be the same as that of the whole model).

In this post, two nontrivial examples of uncountably categorical theories are mentioned (the direct sum of infinitely many cyclic groups of order 4 and the example in the answer), but I was unable to prove that they are uncountably categorical, so I don't know if they would provide an answer to my question here.

Another related (less philosophical) question is the following: I've recently learned that $DCF_0$ (which is known to be $\omega$-stable) has uncountably many countable models. By a consequence of the Baldwin-Lachlan theorem, one can deduce that this theory is not uncountably categorical. Is it easy to exhibit a Vaughtian pair?

• This question is long and rambling. Please state your question concisely. – Rene Schipperus Mar 3 '18 at 17:51
• @Rene Schipperus. One thing at least is clear in this question: an example of a theory which is not strongly minimal, but is omega-stable and without Vaughtian pair, would be good to know. Any thought? – ratalan Mar 3 '18 at 19:51
• @ReneSchipperus I don't think this is rambling at all; having a bit of motivation is useful. – Noah Schweber Mar 3 '18 at 20:58
• @ratalan Two such examples are given in the linked question: the equivalence relation with 2 infinite classes and a bijection between them, and the direct sum of countably many copies of Z/4Z. – Alex Kruckman Mar 3 '18 at 23:46
• Let me add my two cents. I think the main value of Baldin-Lachlan theorem is not helping one to establish uncountable categoricity, but in characterising it. The finer version of Baldwin-Lachlan theorem says that a theory $T$ is categorical in an uncountable cardinal $\kappa$ iff it is $\omega$-stable and without Vaughtian pairs. In particular, the right hand side of this does not depend on $\kappa$. So it implies Morley's theorem: if a theory is categorical in one uncountable cardinality, then it is categorical in all of them. – Levon Haykazyan Mar 4 '18 at 19:14