  Return, now, to the original version of the Löwenheim-Skolem theorem—the one which simply claimed that any theory which has an infinite model also has a model which is countably infinite. Skolem's Paradox arises when we notice that the standard axioms of set theory can themselves be formulated as (countable) collection of first-order sentences. If these axioms have a model at all, therefore, then the Löwenheim-Skolem theorem ensures that they have a model with a countable  domain.[8]  But this seems quite puzzling. How can the very axioms which prove Cantor's theorem on the existence of uncountable sets be satisfied by a model which is itself only countable? How can a countable model satisfy the first-order sentence which “says that” there are uncountably many things?