Let a constructable number be one which can be unambiguously described in a finite string. Because we are working from a finite alphabet, we can trivially see that their is a bijection between the constructables and the integers (if we have n symbols, then each string can be read as an integer in base n, so the amount of constructables is no larger than the integers. We can also show that all integers are constructable, so the amount of constructables is no smaller than the integers). Now, take the set of all constructables, and use the diagonal arguement to construct a new number. We can see that this number is not constructable, however, it would appear that I have just unambigously described it, meaning that it must be constructable.

The only potential hole I see is that the ordering of the constructables when I apply the diagonal arguement is ambigous, but we can unambiguously order them by the lexical ordering of their 'canocial' description, and we can unambiguous define the canonical description as the smallest one when translated into a base n integer.

I suspect that doing the above will run into problems with computable numbers (as it likely involves the halting problem), however it appears to be an unambiguous description of a real number that is not constructable. Obviously there is some flaw in this reasoning.