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  "body": "\n## Statement\n\nLet $T$ be a [[complete theory|655K8Y7VS8]] in a countable signature $L$, and suppose that $T$ has infinite models.\nThe following are equivalent:\n\n1. $T$ is [[$\\aleph_0$-categorical|1JJCGNX240]].\n2. For every $n \\ge 1$, the set $S_n(T)$ of complete $n$-[[types|9V71Q3SD1G]] of $T$ is finite.\n3. For every $n \\ge 1$, there are only finitely many $L$-formulas in the variables $x_1, \\dots, x_n$ up to equivalence modulo $T$.\n4. For every $n \\ge 1$, every complete $n$-type of $T$ is isolated.\n5. $T$ has a countable model $M$ whose [[automorphism group|0K1SCPBNKJ]] has only finitely many [[orbits|0YQPZB5Q1G]] on $M^n$ for every $n \\ge 1$.\n\n## Remarks\n\nIf some $S_n(T)$ is infinite, then by compactness of $S_n(T)$ it has a non-isolated point.\nThe omitting types theorem gives a countable model omitting it, while another countable model realizes it, so $T$ is not $\\aleph_0$-categorical.\nConversely, 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.\n\nExample: the theory of [[dense linear orders without endpoints|2W4S2H4VYA]] is $\\aleph_0$-categorical.\nIn $(\\Q, \u003c)$, 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.\n\nNon-example: the theory of $(\\Z, \u003c)$ is not $\\aleph_0$-categorical.\nThe 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 $n = 2$.\n\nThe theorem is named after Czesław Ryll-Nardzewski; it was also found independently by Erwin Engeler and Lars Svenonius.\n\n## Explanations\n\n### L1 human\n\nA theory is $\\aleph_0$-categorical when it has only one countable model up to isomorphism.\nThe theorem says that this happens exactly when the language can only say finitely many different things about any fixed number of elements.\nFor the rationals with their order, all one can say about two numbers is whether the first is smaller, equal or larger than the second.\n\n### L2 human\n\nThe theorem turns a question about models into a question about counting types or formulas.\nClause 5 links model theory with permutation groups: the countable models of $\\aleph_0$-categorical theories are exactly the countable structures with oligomorphic automorphism groups.\n[[Fraïssé limits|HJVRBDQSGG]] of classes with finitely many structures of each finite size in a finite relational language, such as the Rado graph, are $\\aleph_0$-categorical.\n",
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