Tarski-Vaught Theorem

An elementary chain is a chain of models
 * $$ M_1 \subset M_2 \subset \cdots $$

such that $$M_i \preceq M_j $$ for i ≤ j. (The chain could have transfinite length). The Tarski-Vaught Theorem on unions of elementary chains says that the union structure
 * $$ \bigcup_i M_i $$

is an elementary extension of Mj for each j.

Applications
The Tarski-Vaught theorem plays a key role in the proofs of the following facts:
 * The uniqueness of model companions.
 * The characterization of inductive theories as ∀∃-theories.
 * The construction of $$\kappa$$-saturated models by repeatedly realizing types.
 * Robinson joint consistency.

Proof
Let N denote the limit structure. We prove by induction on the number of quantifiers in the prenex form of $$\phi(x)$$ that for every i, and tuple a from Mi
 * $$ M_i \models \phi(a) \iff M \models \phi(a).$$

The base case where $$\phi(x)$$ is quantifier-free is easy. For the inductive step, suppose $$\phi(x)$$ has n quantifiers. Replacing $$\phi(x)$$ with $$\neg \phi(x)$$ if necessary, we may assume that the outermost quantifier is existential. Then we can write $$\phi(x)$$ as $$ \exists y : \chi(x;y)$$, where $$\chi(x;y)$$ has n - 1 quantifiers. Now suppose that $$\phi(a)$$ holds for some tuple a from some Mi. Then in Mi we can find some tuple b such that
 * $$M_i \models \chi(a;b)$$

By the inductive hypothesis,
 * $$ M \models \chi(a;b)$$ and therefore $$M \models \phi(a) $$

as desired.

Conversely, suppose that a is a tuple from some Mi such that
 * $$ M \models \phi(a)$$ or equivalently $$M \models \exists y : \chi(a;y) $$

Take a tuple b from M such that
 * $$ M \models \chi(a;b) $$

Since M is the union of the Mj, there is some j > i such that b is in Mj. By induction,
 * $$ M_j \models \chi(a;b) $$,

and in particular,
 * $$ M_j \models \exists y : \chi(a;y)$$ or equivalently $$M_j \models \phi(a) $$

Because $$M_i \preceq M_j$$, we have
 * $$ M_i \models \phi(a) $$

as well.

In conclusion,
 * $$ M_i \models \phi(a) \iff M \models \phi(a)$$,

completing the inductive step, as well as the proof. QED