Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2009 / B / 4 / d

Existential sentences are preserved under extensions: witnesses remain present and quantifier-free truth is unchanged by an embedding. Therefore any existential axiomatization implies the specified extension-closure property.

For existential axiomatization from extension closure, let $E$ be all existential consequences of $T$, and suppose $A\models E$. We claim that $T\cup {\mathrm{Th}}_{\forall }(A)$ is satisfiable. Otherwise compactness supplies finitely many universal sentences ${\theta }_1,\dots ,{\theta }_m$ true in $A$ such that $T\models \neg ({\theta }_1\wedge \cdots \wedge {\theta }_m)$. The finite conjunction is universal after renaming bound variables, so its negation is existential. It belongs to $E$ but is false in $A$, a contradiction.

The preceding universal-theory embedding result now yields $B\models T$ and $A^{\ast }\equiv A$ with $B\subseteq A^{\ast }$. By the assumed extension closure, $A^{\ast }\models T$; elementary equivalence implies $A\models T$. Thus $E$ axiomatizes exactly the models of $T$, proving existential axiomatizability is equivalent to closure under extensions. This includes the inconsistent case, whose empty model class is axiomatized by an existential contradiction.