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In model theory, $\aleph_1$-categorical (first order) theories (in a countable language) are very important, and I am studying them at the moment. However, it seems that the only examples I can find are the typical ones: infinite sets, infinite vector spaces over, algebraically closed fields (and related theories, like fixing the characteristic of the field, or fixing some constants).

What are other examples of $\aleph_1$-categorical theories? I would like to know if there are other examples, specially ones that are relevant in other areas of maths.

Reading about stability theory is hard, because most books lack examples, so I would like to get more if possible.

I'm also working in continuous model theory, where the theory of infinite dimensional Hilbert spaces is $\aleph_1$-categorical (which is basically the same examples as with vector spaces). Are there more examples there?

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    $\begingroup$ Non-trivial divisible torsion-free abelian groups? Half joking. ;) $\endgroup$
    – blargoner
    Commented Jul 8 at 5:03

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I'll discuss classical examples. Maybe someone else can write an answer about examples in continuous logic.

The easiest examples of $\aleph_1$-categories theories are strongly minimal theories. Here the classic examples are:

  • The theory of infinite sets.
  • The theory of infinite-dimensional vector spaces over a fixed field (or division ring).
  • The theory of algebraically closed fields of a fixed characteristic.

For some time, it was believed that all strongly minimal theories might be closely related to the examples above. This was disproved by Hrushovski in his paper A New Strongly Minimal Set. Hrushovski's method was found to be quite a flexible tool for producing counterexamples in model theory, and a wide variety of new theories (both $\aleph_1$-categorical and otherwise) were built using "Hrushovski constructions". See Section 10.4 of A Course in Model Theory by Tent and Ziegler for a beautiful exposition of one of these examples.

"Closely related" is also a bit vague. The class of strongly minimal theories with trivial geometry (which are those "closely related" to the theory of infinite sets) is wider than it might first appear. For example, the theory of an equivalence relation with infinitely many classes, all of which have size $n$ for a fixed finite $n$, or the theory of $(\mathbb{N},S)$, or $(\mathbb{Z},S)$, where $S$ is the successor function, are all strongly minimal with trivial geometry.

For some more examples that are $\aleph_1$-categorical but not strongly minimal, you can see the answers to this question.

In general, however, being $\aleph_1$-categorical is a very very special property of a theory. You should not expect to find a wealth of diverse examples of mathematical structures with $\aleph_1$-categorical theories - they should be quite rare in the mathematical landscape.

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