Ryll-Nardzewski theorem

Remarks

If some $S_n(T)$ is infinite, then by compactness of $S_n(T)$ it has a non-isolated point. The omitting types theorem gives a countable model omitting it, while another countable model realizes it, so $T$ is not $\aleph_0$-categorical. Conversely, if all types are isolated, every countable model is atomic and also realizes every type, and a back-and-forth argument between two countable models gives an isomorphism.

Example: the theory of dense linear orders without endpoints is $\aleph_0$-categorical. In $(\Q, <)$, two $n$-tuples lie in the same orbit of the automorphism group exactly when they have the same order pattern, and there are finitely many patterns.

Non-example: the theory of $(\Z, <)$ is not $\aleph_0$-categorical. The formulas "there are at least $k$ elements strictly between $x_1$ and $x_2$", for $k \in \N$, are pairwise inequivalent, so clause 3 fails for $n = 2$.

The theorem is named after Czesław Ryll-Nardzewski; it was also found independently by Erwin Engeler and Lars Svenonius.

Going deeper

For a student

The theorem turns a question about models into a question about counting types or formulas. Clause 5 links model theory with permutation groups: the countable models of $\aleph_0$-categorical theories are exactly the countable structures with oligomorphic automorphism groups. Fraïssé limits of classes with finitely many structures of each finite size in a finite relational language, such as the Rado graph, are $\aleph_0$-categorical.