Cantor's theorem has important consequences for the theory of propositional functions. Consider a model for a (first-order) logical language that has a domain D. The variables of the language range over members of D. Now let us add predicate variables to the language. These stand for propositional functions. How are we to interpret them in the model? The standard way of doing so—that is inherited from the extensional logic tradition—is to have predicate variables range over subsets of the domain. A model in which predicate variables range over all subsets of the domain is called a ‘standard model’ for second-order logic. Cantor's theorem tells us that the domain for predicate variables in the standard model is larger than the domain for individual variables.  If we have predicates of predicates, then the domain for third order predicates is even larger. And so on.