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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.

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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.

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    $\begingroup$ it could perhaps be of use to say "$k$-ary atomic formula" instead of "relation symbol" : this doesn't change the argument and helps to deal with the case where $L$ has some function symbols (without having to argue that you can change the language to contain no function symbols, which of course is possible) $\endgroup$ – Max Jul 10 at 13:40

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