= Solution
An $\aleph_0$-saturated <first-order model>[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 <first-order model>[model].
For any <first-order model>[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 first-order theory>[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 <first-order model>[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>[elementary-chain theorem] makes $M_\omega=\bigcup_nM_n$ a <first-order model>[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 first-order theory>[consistent] with $M_\omega$ is <consistent first-order theory>[consistent] with $M_n$, since the expanded complete <first-order theory>[theories] at those parameters agree by elementarity. It is realized in $M_{n+1}$, hence in the <union>. Therefore \b[an $\aleph_0$-saturated <first-order model>[model] always exists], possibly with uncountable <cardinality>.
Back to article page
