You can go farther and say that you can't even construct real numbers without strong enough axioms. Theories of first order arithmetic, like Peano arithmetic, can talk about computable reals but not reals in general.

> The idea is we can't actually prove a non-computable real number exists without purposefully having axioms that allow for deriving non-computable things.

Sorry, what do you mean?

The real numbers are uncountable. (If you're talking about constructivism, I guess it's more complicated. There's some discussion at https://mathoverflow.net/questions/30643/are-real-numbers-co... . But that is very niche.)

The set of things we can compute is, for any reasonable definition of computability, countable.