Enumerate the countable atomic model as $M=\{a_0,a_1,\ldots\}$ and fix any $N\models T$. We recursively construct finite partial elementary maps $f_n$ from the first $n$ elements of $M$ into $N$.

Suppose $f_n(\bar a)=\bar b$. Since $M$ is atomic, choose a formula $\theta(\bar x,y)$ isolating $\operatorname{tp}(\bar a,a_n)$. Its existential consequence $\exists y\,\theta(\bar x,y)$ belongs to $\operatorname{tp}(\bar a)$, so partial elementarity gives
$$
N\models\exists y\,\theta(\bar b,y).
$$
Choose a witness $b_n$. Because $\theta$ isolates the complete joint type, extending $f_n$ by $a_n\mapsto b_n$ remains partial elementary. The union of the recursive maps is an elementary embedding $M\to N$. Thus every countable atomic model is a <prime model>, proving the <countable atomic model is prime> result.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 144 / 1 / c / Solution

A countable atomic model of a complete theory elementarily embeds into every model. Enumerate it and recursively extend a finite partial elementary map; an isolating formula for the next joint type guarantees a realization in the target.


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