  Let T be a countable collection of first-order sentences and let A be an infinite set. The Upward Löwenheim-Skolem Theorem says that if T has any infinite model, then T has a model whose domain has the same size as A (indeed, we can assume without loss of generality that the domain of this second model just is A).[5] The Downward Löwenheim-Skolem Theorem says that if N is a model of (infinite) cardinality κ and if λ is an infinite cardinal smaller than than κ, then N has a submodel of cardinality λ which satisfies exactly the same sentences as N itself  does.[6]  Finally, the Transitive Submodel Theorem strengthens the downward Löwenheim-Skolem theorem by saying that if our initial N happens to be a so-called transitive model for the language of set theory, then the submodel generated by the downward theorem can also be chosen to be  transitive.[7]