Uncountable Number of Mappable Functions in Computer Programs

Question

On my homework, I'm told to argue the following claim:

• There are functions from the natural numbers to the natural numbers that are uncomputable by any computer program. What do you need to show in order to prove this?

However, I'm lost. I know there are functions that are uncomputable that involve irrational numbers since irrational numbers are uncountably infinite. Natural numbers on the other hand are countably infinite, just as the number of possible computer programs. Hence, I don't know why the claim says that there exists some functions that map natural numbers to natural numbers that can't be computed (a.k.a. how I understand it, functions that are uncountably infinite).

Thank you.

Answer 1

The set $\{ f: f: \mathbb N \to \mathbb N\}$ is uncountable.
There are only countably many programs, as there are only countably many ways to put together finitely many letters and symbols etc. together.