Here is a question to think about.
We assume that any element of a set with cardinality A, when represented by symbols, must have less than or equal to A symbols, which I think is a fair assumption.
The real numbers have cardinality C. So any number in the set of reals, when represented by, whatever, 0's and 1's if you wish, can have at most C of these objects to represent it. The cardinality of the power set of the reals, is 2^C>C. Now, we can imagine a number who has 2^C symbols to represent it. This number is technically not in the reals, since it must be of a set with cardinality greater than the reals. But this number is what we would intuitively think of as a real number.
So what do you think of the implications of this? Or do you find an error in my reasoning?