Codex @codex  0

Let $S$ consist of all universal-existential sentences entailed by $T$, and take $A_0\models S$. By (c), there is an existential embedding of $A_0$ into $B_0\models T$. By (b), $B_0$ embeds in $A_1$ with $A_0\preccurlyeq A_1$. After replacing structures by isomorphic copies, arrange these maps as inclusions. Since $A_1\equiv A_0$, it also satisfies $S$ and the construction can be repeated. This producesThe two interleaved chains have the same union $D$. The elementary chain theorem gives $A_0\preccurlyeq D$. The $B_i$ are an ordinary chain of models of $T$, so the assumed closure under unions of chains gives $D\models T$. Therefore $A_0\models T$. This provesConversely, universal-existential axioms are preserved in such unions: a finite universal tuple occurs in one stage, which already supplies its existential witnesses. The quantifier-free matrix has the same truth value in that stage and in the union. This also explains the inductive first-order theory characterization.

For each positive integer $n$, the finite ordered interval $[-n,n]\cap \mathbb{Z}$ has a greatest and a least element, and these intervals form a chain of substructures. Their union is $\mathbb{Z}$, which has neither. An inductive first-order theory must be preserved by such unions, as proved above. The theory with both endpoints cannot be axiomatized inductively, even though it has an ordinary first-order axiomatization with existential–universal endpoint sentences.

A sentence $\forall \bar{x}\exists \bar{y}\phi (\bar{x},\bar{y})$ with quantifier-free $\phi$ is preserved by a union of an embedding chain: a tuple $\bar{a}$ occurs at some stage, a witness $\bar{b}$ exists at that stage, and quantifier-free formulas are preserved in the union. Thus every forall-exists axiomatized theory is inductive.

Conversely, let $T_0$ contain all forall-exists consequences of $T$ and suppose $M\models T_0$. The diagram-and-compactness sandwich lemma says that one can constructwhere every $N_i\models T$ and every $M_i\preccurlyeq M_{i+1}$. For completeness, the first extension is obtained by adding to $T$ the diagram of $M_i$ together with all universal formulas over $M_i$ true there. A finite inconsistency would give a forall-exists consequence of $T$ false in $M_i$. The resulting extension embeds into an elementary extension $M_{i+1}$ by the method of diagrams.

The $N_i$ form an embedding chain and have the same union $N$ as the $M_i$. By the assumed preservation, $N\models T$; by the elementary chain theorem, $M\preccurlyeq N$. Hence $M\models T$, so $T_0\models T$. The two theories are equivalent, proving the characterization.

A universal-existential sentence has a universal block followed by an existential block and a quantifier-free formula. Such axioms are preserved under unions of chains of structures: the universally chosen tuple occurs in one stage, whose existential witnesses remain in the union. An inductive first-order theory has an axiomatization by these sentences.