> The problem is that while a superset of an uncountable set is uncountable, a superset of a computably uncountable set may instead be computably countable. The partial functions over the integers show that this is indeed the case.
> The computable countability of the partial maps from N to N.
Can anybody give an example?
does this have any to do with rationals?
or is it more related to limits and calculus?
> does this have any to do with rationals? or is it more related to limits and calculus?
No.
I'm going to use Haskell, which I'm going to assume you know. I'm using it because it seems closer to the math. The type for naturals is:
data Nat = Zero | Succ Nat
and then all Haskell functions of type `Nat -> Nat` represent partial functions. They're not total functions because they might enter an infinite loop for some inputs. You can clearly enumerate all Haskell functions which are syntactically correct and which have type Nat->Nat, so the partial functions of that type are computably enumerable.
But consider the total functions of type Nat->Nat (i.e. those that never enter an infinite loop). Assume you can have a function `en :: Nat -> (Nat -> Nat)` which can output every total function of type Nat->Nat.
Then the function `counterexample n = Succ (en n n)` is a function that cannot be outputted by `en`, and therefore `en` fails to enumerate all total functions.
I've got other things to do, unfortunately, so I can't say more than this.
> The computable countability of the partial maps from N to N.
Can anybody give an example?
does this have any to do with rationals? or is it more related to limits and calculus?