 A particular consequence is that the first measurable cardinal cannot be second-order characterizable as a model of the empty vocabulary (a fact that was referred to above in  §7.1).  Theorem 7 resembles remarkably the Downward Löwenheim-Skolem Theorem of first order logic. The only difference is that one has to start with a big model, a model of the size of a measurable cardinal. Smaller cardinals need not work. For example, if \(\lambda\) is the least weakly compact cardinal \(&gt;\omega\), then there is a sentence \(\phi\) which has \(\lambda\) (with the empty vocabulary) as a model but no smaller models. The  sentence[10]  \(\phi\) says that \(\lambda\) is inaccessible (\(&gt;\omega\)) and that every \(\lambda\)-tree (a tree of height \(\lambda\) with all levels of size \(&lt;\lambda\)) has a branch of length \(\lambda\).