Let $M$, $N$ be two countable structures in an uncountable language $L$. Suppose that for any countable sub-language $L'$ of $L$, $M$ and $N$ are isomorphic $L'$-structures. Then, can $M$ and $N$ be non-isomorphic $L$-structures?
I found this problem when I thought about Does any uncountable complete theory have exactly two countable models?. Understanding restrictions to countable languages might help to understand uncountable languages.
Any help would be appreciated. Thank you.
If $M$ and $N$ are isomorphic in every countable sublanguage of $L$, then they are isomorphic in $L$.
For each natural number $k$, and for each choice of $\langle a_1,\dots,a_k\rangle\in M^k$ and $\langle b_1,\dots,b_k\rangle\in N^k$, choose a $k$-ary relation symbol $P$ (if any exist) such that $P(a_1,\dots,a_k)$ in $M$ and $P(b_1,\dots,b_k)$ in $N$ have opposite truth values. Let $L'$ be the countable sublanguage consisting of the relation symbols so chosen. If $f:M\to N$ is an $L'$-isomorphism, then it is also an $L$-isomorphism.