For any system we could use to define a real number, (and not just because we are mortal), the system could only define or uniquely specify countably many real numbers. Any countable set of real numbers has measure 0, and therefore, almost all real numbers cannot be defined/specified using that system.

This holds for whatever system you use which defines a (possibly partial) function from [ the set of finite (but unboundedly long) sequences of characters over a finite alphabet], to real numbers

Or equivalently, from the natural numbers to the real numbers.