Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 144 / 4

An atomic model is a model in which the type over the empty set of every finite tuple is isolated. Let $\bar{a}\mapsto \bar{b}$ be partial elementary and take $c\in M$. Choose a formula $\theta (\bar{x},y)$ isolating $\mathrm{tp}(\bar{a},c)$. Since $M\models \exists y\theta (\bar{a},y)$, elementarity gives $N\models \exists y\theta (\bar{b},y)$. Any witness $d$ realizes the isolated joint type, so $\bar{a},c\mapsto \bar{b},d$ is partial elementary. This is the one-point extension between atomic models.

Alternately applying this extension property in the two directions to enumerations of countable atomic models constructs a back-and-forth isomorphism.

Suppose first that $T$ is aleph-zero-categorical. If a type in some $S_n(T)$ were nonisolated, the omitting types theorem would produce a countable model omitting it, while a countable elementary submodel of a model realizing it would be another countable model. This contradicts categoricity. Thus every type is isolated. The compact Stone space $S_n(T)$ is then discrete and therefore finite.

Conversely, if every $S_n(T)$ is finite, every type is isolated. Every countable model is consequently atomic, and part i says that any two countable models are isomorphic. This proves the Ryll-Nardzewski theorem.