Are all non-computable functions from $\mathbb{N}$ to $\mathbb{N}$ denumerable?

Question

In the Cutland book called Computability, there is a very interesting exercise on page 81.

Show that the set of all non-computable total functions from $\mathbb{N}$ to $\mathbb{N}$ is not denumerable.

I have already shown that the set of all computable functions from $\mathbb{N}$ to $\mathbb{N}$ is denumerable. But i have not yet shown that the set of all total functions from $\mathbb{N}$ to $\mathbb{N}$ is uncontable.

Thanks!

I want to ask another question related to this. I have developed the next conjecture.

CONJECTURE: The set of all partial functions $f:(0,n)\rightarrow (0,n)$, being n a finite natural number, are computable, and the cardinality of this set is $n^n$ (finite).

I think that there are no NON-computable functions $f:(0,n)\rightarrow (0,n)$ because the set is finite. Could you please tell me if this is correct?

Thanks.

Answer 1

Well, the set ${\Bbb N}^{\Bbb N}$ is not denumerable (use Cantor's second diagonal argument to show this), while the set of all computable functions ${\Bbb N}\rightarrow{\Bbb N}$ is denumerable (the set of all Turing/Goto/Registermachine programs or the set of all C-programs over a finite alphabet is denumerable).