Suppose that we wanted to construct a mathematical universe where all objects were computable in some sense. How would we do it? Well, we could certainly allow the set \mathbb{N} into our universe: natural numbers are the most basic computational objects there are. (Notation: I’ll use N to refer to \mathbb{N} when we’re considering it as part of the universe we’ll building, and just \mathbb{N} when we’re talking about the set of natural numbers in the “real” world.) What should we take as our set of functions N^N from N to N? Since we want to admit only computable things, we should let N^N be the set of computable functions from \mathbb{N} to \mathbb{N}, which we can represent non-uniquely by their indices (i.e., by the programs which compute them).