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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.

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    Are you using $\phi$ to denote the empty set ($\emptyset$)? Do you implicitly have equality in your language? – 2017-02-04
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    's supposed to be an empty set, but got edited. And there's nothing else into it, so can't really tell – 2017-02-04
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    OK, thanks. What about equality? Can you use equality in your formulae? EDIT: I saw your edit. Let's assume you can use equality. How would you say, "there are *at least* two distinct elements in this structure?" – 2017-02-04
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    Yeah, I'm allowed to – 2017-02-04
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    Well, (exists x, y)(x != y), I suppose – 2017-02-04
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    Exactly. The key idea is that each sentence uses $n$ existentially quantified variables to state the existence of $n$ distinct elements. The last step is to say that there are no more than $n$ elements; that is, every element of the structure is one of those $n$ elements. – 2017-02-04

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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.