Surely not? If there are more numbers that can be represented in any amount of bytes (as shown by the diagonal argument), then you cannot represent a constructive proof for each of them in any amount of bytes.
The standard diagonalization argument is a constructive proof, identifying a specific number. Normally a constructive proof is preferable. But with a bit of hand waving you can make it into a non-constructive proof that there are "unknowable" numbers that cannot be described in any finite amount of bytes.
The point is that such a sequence needs to be constructive in the usual sense, so there are still only a countable number of them.
The implication is that human minds wouldn't be able to represent it either, at least not with language.