This Wikipedia entry on the Löwenheim–Skolem theorem says:
In mathematical logic, the Löwenheim–Skolem theorem, named for Leopold Löwenheim and Thoralf Skolem, states that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ.
What does the "size" of a model referring to (or mean)? Edit: If it is referring to the cardinality of a model (set), how do you get the cardinality of one model (-> It's synonymous with interpretation, right?)? What is inside the model, then? I mean, it seems sensical to define a model of a language, as a language has some constant numbers and objects, but defining a model of a single object - a number - seems nonsensical to me. What is inside the model of an infinite number?