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:

• The uniqueness of model companions.
• The characterization of inductive theories as ∀∃-theories.
• The construction of -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 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 ::: ::: :::