Second-order logic satisfies strong κ-compactness when every second-order theory, every subset of size less than κ of which has a model, has itself a model.

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 \(>\omega\), 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 (\(>\omega\)) and that every \(\lambda\)-tree (a tree of height \(\lambda\) with all level