I don't understand that argument. But in any case, Cantor's argument is very constructive. It literally gives you the decimal expansion of the new number not in your set.
https://en.wikipedia.org/wiki/Definable_real_number#Definabi...
They start with a stronger definition of a definable number, so they find that they do exist.
I think given the argument above there must be a hole in my own argument, I'd have to go beyond ZFC.