Acceptable Numerations of Function Spaces

We study when a numeration of the set of morphisms from a numeration to the other is well-behaved. We call well-behaved numerations ``acceptable numerations''''. We characterize acceptable numerations by two axioms and show that acceptable numerations are recursively isomorphic to each other. We also show that for each acceptable numeration a fixed point theorem holds. Relation between Cartesian closedness and S-m-n property is discussed in terms of acceptable numerations. As an example of acceptable numerations, we study directed indexings of effective domains.