I'm not sure about this problem. I should determine if this if true or false:
If L is languege $L-$sentence $\phi$ is satisfiable in every finite $L-$structure, is it satisfible also in every infinite $L-$structure?
I would say that's true. In my opinion, from the Compactness theorem it holds that if $\phi$ is satisfiable in every finite $L-$structure, it has to be satisfiable in any infinite $L-$structure. However, I'm not sure why it has to be satisfiable in every.
I've tried to prove it like that: Let ve have $\neg \phi$ and let suppose it has some infinite model (that means there exist some $L-$structure in which $\neg \phi$ is satisfiable). Then, according to the Compactness theorem, there has to exist some finite subset of our $L-$structure in which $\neg \phi$ is satisfiable, which is a contradiction.
Is it possible to do it like that or am I wrong in some point? Because I'm not really sure about it, so I'm afraid I'm doing some mistake without knowing it...