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 *M _{j}* for each

*j*.

## ApplicationsEdit

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.

## ProofEdit

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 *M _{i}*

- $ 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 *M _{i}*. Then in

*M*we can find some tuple

_{i}*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 *M _{i}* 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 *M _{j}*, there is some

*j*>

*i*such that

*b*is in

*M*. By induction,

_{j}- $ 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