modified realizability: specific realizabilities in HRO, HEO as X: Kleene's recursive realizability. Clearly X is an abstract interpretation of HA in a suitable theory of partial functionals; then Kleene's realizability will result by an interpretation analogous to HRO. And, the very realizability which we need will result by an interpretation analogous to HEO. The details of this program are so similar to what is known already that we give only a sketch. First we describe a suitable theory HA of partial functionals. These are to be partial functionals of partial arguments, not only partial functionals of total arguments. We have the same type structure as for HA. We have an relation App(/, x, y) at each sensible triple of types, which is meant to stand for f(x) ^ y. So we need the axiom App(/, x, y) and App(/, x, z) —> x = z. Exactly as in Feferman's theories, we introduce application or for short, terms, which are not officially part of the language. These terms are built up from variables and constants; we include the same constants as in