Fraïssé's theorem says that for a finite relational structure $\sigma$, two $\sigma$-structures $\mathfrak{M}$ and $\mathfrak{N}$ are finitely isomorphic iff they are elementary equivalent.
I found an example of two structures that are elementary equivalent, but not finitely isomorphic if the relational signature is infinite.
Is there an example of two structures that are finitely isomorphic, but not elementary equivalent? Or does finite isomorphism imply elementary equivalence even if relational signature is infinite?
If two $L$-structures $M$ and $N$ are finitely isomorphic, then for any finite language $L'\subseteq L$, the reducts $M|_{L'}$ and $N|_{L'}$ are finitely isomorphic. By Fraïssé's Theorem, $M|_{L'}$ and $N|_{L'}$ are elementarily equivalent. But then $M$ and $N$ are elementarily equivalent (since every $L$-sentence uses only finitely many symbols and hence is an $L'$-sentence for some finite $L'$).