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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?

Thanks.

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    The way this question is phrased shows a lot of confusion. I think what is needed is not just straightening out some particular error, but rather learning the definitions from the beginning.2012-04-30

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Each model has a set of individuals. The size of the model is the cardinality of this set.

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    @Ted Thanks - that's where I got confusion... stupid me...2012-04-30