The real numbers have to be constructed. Typically, a number is represented by a Cauchy sequence or a Dedekind cut.
To determine if a real number is representable symbolically, we simply need a finite sequence of symbols which stands for this Cauchy sequence, lets say.
Theroem: The real numbers and definable numbers are the same set.
Assume a real number exists but is not definable. This means at the very least we have a mathematical statement saying there exists a number such that some logical predicate is valid (we may not even have a construction in ZFC), which can also be constructed using a Cauchy sequence. This mathematical statement is embedded in ZFC, and since we are humans it must be finite. In fact, you could come up with a binary representation for such a statement using methods from Godel, by mapping each symbol to some binary representation. Therefore, this number can be represented as a sequence of zeros or ones, a contradiction.
QED