An elementary chain is a chain of models

$ M_1 \subset M_2 \subset \cdots $

such that $ M_i \preceq M_j $ for ij. (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:


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