I'm not a mathematician, but constructivists aim to define mathematics without uncomputable numbers, see

https://en.wikipedia.org/wiki/Computable_analysis

and

https://en.wikipedia.org/wiki/Computable_number#Use_in_place...

As far as I can understand, the set of all computable numbers (including all algebraic numbers and many transcendental numbers, such as Pi), even has the same cardinality as the rationals, and thus the natural numbers.

The reason we consider uncomputable numbers "numbers" include some definitions about infinite series and analysis that would need to have stricter requirements for convergence when looking only at the computable numbers, not the real numbers.

And defining a concrete bijection between the natural numbers and the computable numbers would also solve the halting problem and is impossible, we only know that such a bijection exists: defining it would mean to have an algorithm that can prove for a specific Turing machine that it is the minimal one computing it's output, among a given set of universal Turing machines / UTM encoding.

(please take this with a grain of salt as I'm stepping outside the bounds of my knowledge here)