I think it’s much simpler than that. If you remember your maths well enough to remember the proof that the reals are uncountable and the rationals are countable, consider that the list of programs that produce a number is countable. You don’t need to check if the program halts or not, even if we “count” the “numbers” from the programs that don’t halt _it’s still countable_. But we’re still left with an uncountable set of numbers for which we can’t even write a program.

Easiest way to construct one is to use the diagonal argument that’s used to prove, amongst other things, the halting problem.