An elementary chain is a chain of models

such that for ij. (The chain could have transfinite length). The Tarski-Vaught Theorem on unions of elementary chains says that the union structure

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:

## Proof

Let N denote the limit structure. We prove by induction on the number of quantifiers in the prenex form of that for every i, and tuple a from Mi

The base case where is quantifier-free is easy. For the inductive step, suppose has n quantifiers. Replacing with if necessary, we may assume that the outermost quantifier is existential. Then we can write as , where has n - 1 quantifiers. Now suppose that holds for some tuple a from some Mi. Then in Mi we can find some tuple b such that

By the inductive hypothesis,

and therefore

as desired.

Conversely, suppose that a is a tuple from some Mi such that

or equivalently

Take a tuple b from M such that

Since M is the union of the Mj, there is some j > i such that b is in Mj. By induction,

,

and in particular,

or equivalently

Because , we have

as well.

In conclusion,

,

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