An elementary chain is a chain of models

{\displaystyle M_{1}\subset M_{2}\subset \cdots }

such that {\displaystyle 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

{\displaystyle \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 {\displaystyle \phi (x)} that for every i, and tuple a from Mi

{\displaystyle M_{i}\models \phi (a)\iff M\models \phi (a).}

The base case where {\displaystyle \phi (x)} is quantifier-free is easy. For the inductive step, suppose {\displaystyle \phi (x)} has n quantifiers. Replacing {\displaystyle \phi (x)} with {\displaystyle \neg \phi (x)} if necessary, we may assume that the outermost quantifier is existential. Then we can write {\displaystyle \phi (x)} as {\displaystyle \exists y:\chi (x;y)}, where {\displaystyle \chi (x;y)} has n - 1 quantifiers. Now suppose that {\displaystyle \phi (a)} holds for some tuple a from some Mi. Then in Mi we can find some tuple b such that

{\displaystyle M_{i}\models \chi (a;b)}

By the inductive hypothesis,

{\displaystyle M\models \chi (a;b)} and therefore {\displaystyle M\models \phi (a)}

as desired.

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

{\displaystyle M\models \phi (a)} or equivalently {\displaystyle M\models \exists y:\chi (a;y)}

Take a tuple b from M such that

{\displaystyle 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,

{\displaystyle M_{j}\models \chi (a;b)},

and in particular,

{\displaystyle M_{j}\models \exists y:\chi (a;y)} or equivalently {\displaystyle M_{j}\models \phi (a)}

Because {\displaystyle M_{i}\preceq M_{j}}, we have

{\displaystyle M_{i}\models \phi (a)}

as well.

In conclusion,

{\displaystyle M_{i}\models \phi (a)\iff M\models \phi (a)},

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