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...
There are sentences $\phi$ that are true in every finite $L$-structure but such that there are infinite $L$-structures in which $\phi$ is not true.
Here is one example. Let $L$ have a single binary predicate symbol $\lt$. Let $\alpha$ be the conjunction of all the sentences that together say that $\lt$ is a total order. Let $\beta$ be the sentence that says there is a smallest element under $\lt$, and let $\phi$ be the sentence $(\alpha\to\beta)$.
Then $\phi$ is true in every finite $L$-structure. However, there are infinite totally ordered sets that do not have a least element, so there are infinite $L$-structures in which $\phi$ is false.
Remark: I am puzzled by the phrase "satisfiable in every (finite) $L$-structure. Given any $L$-structure $\mathbb{M}$, any sentence is either true of false in $\mathbb{M}$.