Comment by wiml

I think (I am not a mathematician) that depends on whether you accept non-constructive proofs as valid. Normally you reason that any mapping from natural numbers onto the reals is incomplete (eg Cantor's argument), and that the sets of computable or describable numbers are countable, and therefore there exist indescribable real numbers. But if you don't like that last step, you do have company:

https://en.wikipedia.org/wiki/Constructivism_%28philosophy_o...