Is there a name for defining recursive functions as an infinite list of input/output pairs?
Well it's the "set-theoritic" perspective of a function where a function
from A onto B is really just the a subset of A x B where each a in A appears only once.

The problem is that some types are uncountable so enumerating them in this manner isn't possible; for some types a finite member may not be at a finite index.

Consider for example, how you would do this for Double.