My texts give a proof Gödel's Completeness Theorem for the predicate calculus for countable domains. The theorem briefly says, if a predicate letter formula is valid in the domain of the natural numbers, then it is provable in the predicate calculus.
How do we extend this theorem to arbitrary (i.e. potentially uncountable) domains? The Löwenheim-Skolem theorem only says that if a sentence is satisfiable in some non-empty domain, then it is satisfiable in the natural numbers. Is there a theorem that says if a sentence is valid in any non-empty domain, it is valid in the natural numbers?