The Fraenkel-Mostowski construction produces a Henkin (set-theoretic) model in which the Axiom of Choice fails and the relevant partition property fails.

For more on the proof theory of second-order logic, see Buss (1998). There is a translation of many sorted logic further to single sorted first order logic due essentially to Herbrand (1930), see also Wang (1952) and Schmidt (1951). This can be used to obtain many of the basic properties of first order logic first for many sorted logic and then further for second-order logic with general models. The most important application of general models is the Completeness Theorem: Theorem 14 (Henkin 1