 Combined, the proofs of the downward and upward Löwenheim-Skolem theorems show that for any satisfiable set \(\Gamma\) of sentences, if there is no finite bound on the models of \(\Gamma\), then for any infinite cardinal \(\kappa\), there is a model of \(\Gamma\) whose domain has size exactly \(\kappa\). Moreover, if \(M\) is any interpretation whose domain is infinite, then for any infinite cardinal \(\kappa\), there is an interpretation \(M'\) whose domain has size exactly \(\kappa\) such that \(M\) and \(M'\) are equivalent.