= Solution
Let $U$ be all <universal sentences> true in $C$. The theory $T\cup U$ is consistent. Indeed, if a finite conjunction $u$ of sentences from $U$ were inconsistent with $T$, then $T\models\neg u$. A finite conjunction of <universal sentences> is universal, so its negation is equivalent to an <existential sentence>. By the hypothesis $C$ would satisfy $\neg u$, whereas it satisfies $u$.
Choose $A\models T\cup U$. Now expand the language by names for every element of $C$ and consider its <elementary diagram> $\operatorname{ElDiag}(C)$. Its universal consequences in the original language are exactly $U$: any such consequence is true in the named copy of $C$, while any original-language sentence true in $C$ already belongs to its elementary diagram. Part (b), applied to $A$ and this expanded theory, gives a <structure embedding> of $A$ into a model $D$ of $\operatorname{ElDiag}(C)$. The named copy of $C$ is an <elementary substructure> of $D$. Identifying that copy with $C$, we obtain
$$
\boxed{A\models T,\qquad A\hookrightarrow D,\qquad C\preccurlyeq D.}
$$
Thus the required target is an <elementary extension> of $C$, although $C$ itself need not be a model of $T$.
Back to article page
