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In Martin Ziegler: A Course in Model Theory it is stated on page 53 that

Definition 4.3.8. A theory $T$ is small if $S_n(T)$ are at most countable for all $n<\omega$.

And it is followed by these two statements I don't understand:

A countable theory with at most countably many non-isomorphic at most countable models is always small. The converse is not true.

I cannot see a direct connection between isomorphism classes of models of $T$ and types. (Something like: realisation sets $\leftrightarrow$ clopen sets $\leftrightarrow$ types?)

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Every $n$-type (over a countable language) is realized in some countable model, by Lowenheim-Skolem. So the number of $n$-types extending $T$ is at most the number of $n$-tuples of elements of countable models of $T$ up to isomorphism. Each countable model has only countably many $n$-tuples of elements, so if there are only countably many countable models up to isomorphism, then $S_n(T)$ is countable.

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  • $\begingroup$ Thanks, it was very instructive. (I had a misunderstanding of types.) For the second statement I came up with following (unnatural) example: A language $L=\{R_i\;|\;i<\omega\}$ of $\aleph_0$-many unary relation symbols and the theory $\{\forall x\neg R_i(x)\vee \exists^{=1}x R_i(x)\;|\;i<\omega\}$. Then I get a distinct isomorphism class for every subset of $\omega$ while having only $\aleph_0$ many $n$-types. $\endgroup$ – Zikrunumea Sep 10 '18 at 4:02

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