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.
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.
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) $
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) $
- $ M_i \models \phi(a) \iff M \models \phi(a) $,
completing the inductive step, as well as the proof. QED