Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 120 / 2

A complete $n$-type over $X\subseteq M$ is a maximal set $p(\bar{x})$ of $L(X)$-formulas consistent with the theory of $M$ with parameters from $X$. Equivalently, for every formula $\phi (\bar{x})$, exactly one of $\phi ,\neg \phi$ belongs to $p$.

The type space $S_n^M(X)$ has these types as points and basic open setsSince $[\phi {]}^c=[\neg \phi ]$, these sets are clopen. An isolated type is a point $\{p\}=[\phi ]$ for some formula $\phi \in p$.

Add new constants $\bar{c}$ and considerEvery finite subset is satisfiable because $p$ is consistent over $M$. By the compactness theorem it has a model $N^{\prime }$. The constants naming $M$ give an elementary embedding $M\to N^{\prime }$, and ${\bar{c}}^{N^{\prime }}$ realizes $p$. Replacing $N^{\prime }$ by an isomorphic copy containing $M$ gives the required realization of a type in an elementary extension.

If $p\ne q$, completeness gives a formula $\phi$ with $\phi \in p$ and $\neg \phi \in q$. Thenand these two disjoint open sets cover $S_n^M(X)$. This proves the total disconnectedness of a type space.

Write the Ehrenfeucht-Mostowski model as the Skolem hull of its order-indiscernible skeleton $(a_i{)}_{i\in \eta }$. Every element is $t(a_{i_1},\dots ,a_{i_r})$ for a Skolem term $t$. Choose supports for the elements of $X$ and let $J\subseteq \eta$ be their union. Then

When $\eta$ is well ordered, the type over $X$ of $t(a_{i_1},\dots ,a_{i_r})$ is determined by $t$ and the finite order pattern of the indices $i_k$ relative to $J$. There are at most $|L|$ terms and at most $|J|$ such finite patterns. Therefore the number of realized complete one-types is at most

Assume $M$ is a prime model. By the downward Lowenheim-Skolem theorem, $T$ has a countable model, and the elementary embedding of $M$ into it makes $M$ countable. If a tuple $\bar{a}\in M$ had a nonisolated type, the omitting types theorem would give a countable model of $T$ omitting that type. An elementary embedding of $M$ into this model would realize it, a contradiction. Thus $M$ is atomic.

Conversely, let $M$ be countable and atomic, enumerate it as $(a_i{)}_{i<\omega }$, and let $N\models T$. Construct an elementary embedding recursively. Suppose $a_0,\dots ,a_{n-1}$ have been mapped to $\bar{b}$. Let $\psi (\bar{x})$ isolate the type of $(a_0,\dots ,a_{n-1})$ and let $\theta (\bar{x},y)$ isolate the type of $(a_0,\dots ,a_n)$. Since the latter extends the former and is realized in $M$, completeness givesThe tuple $\bar{b}$ realizes $\psi$, so a suitable image of $a_n$ exists in $N$. The union of the finite partial elementary maps is an elementary embedding $M\to N$. Hence $M$ is prime.

Let $M$ be a prime model and let $[\phi ]\subseteq S_n(T)$ be a nonempty basic open set. Some model of $T$ realizes $\phi$, so completeness of $T$ givesTherefore $M$ realizes $\phi$ by some tuple $\bar{a}$. The prime-model characterization in part (e) says ${\mathrm{tp}}^M(\bar{a}/\varnothing )$ is isolated, and it lies in $[\phi ]$. Every nonempty basic open set thus contains an isolated point, proving density of isolated types from a prime model.