We need a new [chat], I am reading about the axiom of reducibility, hence our new [chat] topic:
Thus a predicative function of an individual is a first-order function; and for higher types of arguments, predicative functions take the place that first-order functions take in respect of individuals. We assume then, that every function is equivalent, for all its values, to some predicative function of the same argument. This assumption seems to be the essence of the usual assumption of classes [modern sets] . . . we will call this assumption the axiom of classes, or the axiom of reducibility.
It's just that simple.