Question, if two sentences A & B, are such that for all countable structures M: M⊨A iff M⊨B, and A & B be thus logically eguivalent. But how?!
I understand that I have to use Löwenheim-Skolem theory, and I have seen the proof of that there is a phi that M'⊨phi and M⊨phi then M and M' are isomorphic.But from that I can not decuce an answer. How should I think?
Let $\phi$ be the sentence $(A\longrightarrow B)\land (B\longrightarrow A)$. By hypothesis, $\phi$ is true in all countable $L$-structures, for an appropriate language $L$.
Then $\phi$ is true in all $L$-structures. For by Lowenheim-Skolem, if the sentence $\lnot\phi$ had a model, it would have a countable model.
Since $\phi$ is true in all $L$-structures, the sentences $A$ and $B$ are logically equivalent.