I'm having trouble with this specific task:
Let's assume, that signature of $L$ language is: ($\emptyset, \emptyset, \emptyset)$. Find an infinite collection of phrases $(\varphi_n \: : \: n \in N)$ in $L$ and an infinite collection of models $\lbrace M_n : n \in N \rbrace$ which satisfy: $$ M_k \models \varphi_n \Longleftrightarrow k = n$$
Can't really find a way to get on this, any tips? Thanks in advance.
Without anything but equality in your language, a model is characterized by how many elements it has - that is, the only interesting thing you can say is "there are at least this many elements" or "there are only this many elements". With that in mind, it would make sense to try to make it so that each $\varphi_n$ is specifying the size of the structure.
So think about how you would say "there is exactly one object", "there are exactly two objects", and so on. If you can see how to do that, it'll be an excellent starting point for finding the $\varphi_n$ and the $M_k$ that you want.