Extending models of second order predicate logic to models of second order dependent type theory

We describe a method for constructing a model of second order dependent type theory out of a model of classical second order predicate logic. Apart from the construction being of interest by itself, this also suggests a way of proving the completeness of the formulas-as-types embedding from second order predicate logic to second order dependent type theory. Under this embedding, formulas are interpreted as types, and derivability (of a formaula) in the logic should correspond to inhabitation (i.e. the associated type being nonempty) in the type system. This correspondence works in one way (called soundness): if a formula is derivable, then the associated type is inhabited (there is a term of that type). It's an open problem whether the correspondence works in the other direction (called completeness): if the type associated with formula ' is inhabited, then ' is derivable. The completeness is proved if any model M of second order logic can faithfully be extended to a model S(M) of second order dependent type theory. That is, for all formulas ', M j = ' if and only if ' is inhabited in S(M). In this paper we show that such a faithfull extension is possible if M is a full model of classical second order predicate logic. This implies that a second order formula that is derivable in classical P 2 is true in all full models (but it may not be derivable in classical second order logic, due to the existence of non-full models of classical second order logic). We give one small application of the method to an axiomatization of nite structures.