Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2009 / B / 6 / c

An ${\aleph }_0$-saturated model realizes every complete type in finitely many variables over every finite parameter subset of itself. It suffices equivalently to require one-variable types and then realize finite tuples successively. The definition puts no countability restriction on the size of the model.

For any model $M$, collect all complete types over its finite parameter sets. Add to its elementary diagram, for each such type, a new tuple of constants satisfying all that type's formulas. Any finite selection of these requirements is realizable in $M$: each finite type fragment has its existential closure true there, since it is consistent with the complete theory of the corresponding named finite tuple. Distinct new tuples have no additional mutual restrictions. Compactness gives an elementary extension realizing all these types simultaneously.

Start from any model $M_0\models T$ and apply this construction repeatedly to obtain $M_0\preccurlyeq M_1\preccurlyeq \cdots$, with each successor realizing all types over finite subsets of its predecessor. The elementary-chain theorem makes $M_{\omega }={\bigcup }_n{M}_n$ a model of $T$ with every stage elementary in it. A finite parameter set $A\subseteq M_{\omega }$ lies in some $M_n$. Every type over $A$ consistent with $M_{\omega }$ is consistent with $M_n$, since the expanded complete theories at those parameters agree by elementarity. It is realized in $M_{n+1}$, hence in the union. Therefore an ${\aleph }_0$-saturated model always exists, possibly with uncountable cardinality.