In class I saw this version of the compactness theorem:
If every finite subset of a theory $\Gamma$ in the language $(\mathbb P,\mathbb F)$ has a model, then $\Gamma$ has a model with cardinality less than or equal to $|\mathbb P|+|\mathbb F|$.
There must be something that I don't understand about the statement: in fact, in the language with$\ \mathbb P=\mathbb F=\emptyset,\ $if I take as $\Gamma$ the theory of the sets with cardinality a fixed $n>0$, every model of $\Gamma$ has cardinality $n$ (they are all isomorphic). Maybe I have not too clear the meaning of the sum of two cardinalities, but I'd say that $n>|\emptyset|+|\emptyset|$. Thanks in advance for any clarify, I didn't find anything online and we have no suggested texts for the course (we just said that the proof of this form of the compactness basically follows from the proof of the completeness).
In the absence of further context, this is indeed incorrect - and is probably a typo. What is true is that an $\mathcal{L}$-theory (I've not seen the notation you use for languages before) which is finitely satisfiable has a model of size at most $\color{red}{\aleph_0+}\vert\mathcal{L}\vert$, or perhaps more intuitively $\color{red}{\max\{\aleph_0,\color{black}{\vert\mathcal{L}\vert}\}}$.
That "$\aleph_0$" is crucial. For example, consider the $\{<\}$-theory "$<$ is a linear order with no greatest element." This is a finitely satisfiable theory (more to the point, it's finite and satisfiable) but all of its models are infinite.
(EDIT: Alternatively, as Primo Petri comments below we can use "$\mathcal{L}$" to refer to the set of $\mathcal{L}$-formulas as well as the signature itself. In this case of course the "$\aleph_0$" is unnecessary. However, I personally still prefer having it for two reasons. First, it's a bit clearer. Second, if one is considering multiple different logics at once, then it is crucial to keep the signatures and the sets of formulas separate. Since I do this a lot, I like to avoid this overload.)
We can also get an example in the empty language, although now we need to use an infinite theory: consider the theory containing the sentence "There are at least $n$ distinct elements" for each $n$.