Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 23 / 1 / d / Solution

Let $S$ be the set of all existential sentences entailed by $T$. Certainly every model of $T$ satisfies $S$. Conversely, take $C\models S$. Part (c) gives $A\models T$ embedded in an elementary extension $D$ of $C$. After identifying $A$ with its image, $D$ is a superstructure of a model of $T$. The assumed closure under superstructures implies $D\models T$. Since $C\preccurlyeq D$, it follows that $C\models T$ as well. HenceThis supplies an existential axiomatization. Conversely, an existential sentence is preserved under embeddings because its finite tuple of witnesses and the quantifier-free facts about them persist, explaining why this is the appropriate syntactic class.