I'm studying the lowenheim- skolem theorem but i am a bit confused this when it comes to cardinality, in some definitions they use the cardanality of the language, in others they use the cardinality of the model.
Could someone then give me an example of how the cardinality of a model or language is calculated?
I shall give a mainstream account (there may be some variations with authors, but they do not affect the resultant cardinalities). Let us consider a structure $\mathfrak{A}$ of a familiar definition:
$\langle\mathcal{A}, \mathcal{R}, \mathcal{F}, \mathcal{C}\rangle$
where $\mathcal{A}$ is the universe (domain of discourse), $\mathcal{R}$ is the set of relations, $\mathcal{F}$ is the set of functions, $\mathcal{C}$ is the set of individual constants.
The cardinality of a structure (for that reason, of a model) $\mathfrak{A}$ is the cardinality of its universe $\mathcal{A}$. Thus,
$\lvert\mathfrak{A}\rvert =\lvert\mathcal{A}\rvert$
The cardinality of a language $\mathcal{L}$ is the cardinality of the set of well-formed formulas in $\mathcal{L}$. Thus,
$\lvert\mathcal{L}\rvert =\lvert\mathcal{R}\cup\mathcal{F}\cup\mathcal{C}\rvert\cup\omega$
(Following the preference in Chang and Keisler's Model Theory, I've formulated with ordinal numbers).
In case that infinitary languages are also included in the context, a subscript notation is appended such that, for any two cardinal numbers $\kappa$ and $\lambda$, $\mathcal{L}_{\kappa, \lambda}$ is a language of first-order logic with wffs composed of conjunctions and disjunctions of length $<\kappa$ and blocks of quantifiers of length $<\lambda$.
Accordingly, $\mathcal{L}_{\omega,\,\omega}$ is the ordinary language of first-order logic, which has wffs of finite length, while
$\mathcal{L}_{\omega_{1},\,\omega}$ is the language of first-order logic composed of countably infinite conjunctions (and so, disjunctions) and finite quantifiers;
$\mathcal{L}_{\omega,\,\omega_{1}}$ is the language of first-order logic composed of finite conjunctions (and so, disjunctions) and countably infinite quantifiers.