An elementary chain is a chain of models

*M*_{1} ⊂ *M*_{2} ⊂ ⋯

such that *M*_{i} ≼ *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

$\bigcup\limits_{i}M_{i}$

is an elementary extension of *M*_{j} 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 *ϕ*(*x*) that for every *i*, and tuple *a* from *M*_{i}

*M*_{i} ⊨ *ϕ*(*a*) ⇔ *M* ⊨ *ϕ*(*a*).

The base case where *ϕ*(*x*) is quantifier-free is easy. For the inductive step, suppose *ϕ*(*x*) has *n* quantifiers. Replacing *ϕ*(*x*) with ¬*ϕ*(*x*) if necessary, we may assume that the outermost quantifier is existential. Then we can write *ϕ*(*x*) as ∃*y* : *χ*(*x*; *y*), where *χ*(*x*; *y*) has *n* - 1 quantifiers. Now suppose that *ϕ*(*a*) holds for some tuple *a* from some *M*_{i}. Then in *M*_{i} we can find some tuple *b* such that

*M*_{i} ⊨ *χ*(*a*; *b*)

By the inductive hypothesis,

*M* ⊨ *χ*(*a*; *b*) and therefore *M* ⊨ *ϕ*(*a*)

as desired.

Conversely, suppose that *a* is a tuple from some *M*_{i} such that

*M* ⊨ *ϕ*(*a*) or equivalently *M* ⊨ ∃*y* : *χ*(*a*; *y*)

Take a tuple *b* from *M* such that

*M* ⊨ *χ*(*a*; *b*)

Since *M* is the union of the *M*_{j}, there is some *j* > *i* such that *b* is in *M*_{j}. By induction,

*M*_{j} ⊨ *χ*(*a*; *b*),

and in particular,

*M*_{j} ⊨ ∃*y* : *χ*(*a*; *y*) or equivalently *M*_{j} ⊨ *ϕ*(*a*)

Because *M*_{i} ≼ *M*_{j}, we have

*M*_{i} ⊨ *ϕ*(*a*)

as well.

In conclusion,

*M*_{i} ⊨ *ϕ*(*a*) ⇔ *M* ⊨ *ϕ*(*a*),

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