Equality between Functionals Revisited

In this note we shall try to place Friedman's remarkable little paper ([3]) in the context of what we know today about the model theory of the typed λ-calculus. In order to do this, it is appropriate to survey recent work in this area, in so far as it touches on issues raised by Friedman. We don't intend a general survey; much will be left out. In particular, we shall omit discussion of unification, fragments of L.C.F., and polymorphic types, which are areas of interest to the author, and the specialist will surely find other omissions. However, we will try to show the reader how Friedman's paper lays the foundation for the general model theory of the typed λ-calculus. The plan of this note is the following. We shall begin by a brief, informal introduction to the subject. The reader is then advised to read Friedman's paper. It is, after all, quite accessible and easy to read. We shall then proceed to consider the three major issues touched on in “Equality between functionals” as we view them today. These issues are (1) completeness theorems, (2) the solvability of higher type functional equations, and (3) logical relations.