What is the use of Tarski-Vaught test?

Question

As title says, what is the use of Tarski-Vaught test? I do understand that it is necessary and sufficient criteria for $N$ to be an elementary substructure of $M$, but beside that I don't see how this can be further used. Can anyone give an example?

Answer 1

Since elementary submodels are quite important, this is an important use. The whole point is to give a criteria for when a submodel is elementary submodel, so there is no further use.

Perhaps a nice example. If $M\subseteq\Bbb N$ is an elementary submodel of $(\Bbb N,\leq)$ (the ordered set with the usual order) then $M=\Bbb N$.

Proof. Suppose that $M$ is an elementary submodel, and assume towards contradiction that $M\neq\Bbb N$. Let $m\in M$ be the least such that $\{0,\ldots,m\}\nsubseteq M$. That is, $m-1\notin M$, and let $m'=\max M\cap\{0,\ldots,m-1\}$.

Then $\Bbb N\models\exists x(m'<x\land x<m)$, but $M$ does not satisfy the same existential sentence. Therefore $M$ is not elementary, in contradiction to the assumption. $\square$