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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?


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The cardinality of the underlying set. – plm Apr 30 '12 at 8:03
@plm I edited the question.. can you help me more? Thanks. – user30272 Apr 30 '12 at 8:28
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. – Michael Hardy Apr 30 '12 at 15:38
up vote 2 down vote accepted

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

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So, what are the individuals? Isn't an infinite number a sole individual? – user30272 Apr 30 '12 at 8:29
I don't know if this is the source of your confusion, but the word "it" towards the end of your Wikipedia quote refers to the countable first-order theory, not to the infinite cardinal number. "It has a model of size $\kappa$" means that the theory has a model of size $\kappa$, i.e., a model whose underlying set is a set of cardinality $\kappa$. – Ted Apr 30 '12 at 8:46
@Ted Thanks - that's where I got confusion... stupid me... – user30272 Apr 30 '12 at 9:25

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