Closing the gap between the continuous functionals and recursion in $^3E$

Abstract.

We show that the length of a hierarchy of domains with totality, based on the standard domain for the natural numbers ${\Bbb N}$ and closed under dependent products of continuously parameterised families of domains will be the first ordinal not recursive in $^3E$ and any real. As a part of the proof we show that the domains of the hierarchy share important properties with the types of continuous functionals. The main result can also be viewed as a representation theorem for recursion in $^3E$.