It's not a proof of anything, so it can't be adapted to the reals. Specialness has no rigorous mathematical definition, and certainly the second non-special number is not special, even if the first is. But more mathematically speaking, given a standard real number, there is no "next" real number. Induction is one of the defining properties of the natural numbers and it is induction that allows one to construct proofs that show that some property holds for all natural numbers by showing that if it holds for n, then it holds for the successor of n. On the other hand, if you assume the axiom of choice, the real numbers are well-ordered. So if there is at least one non-special real number, then there is a least non-special real number with respect to this ordering, which makes it special (assuming there is something special about this particular well-ordering). Unfortunately, no one can exhibit a well-ordering of the reals, so it's a little hard to judge if any of them are "special". On the other, other hand, it is humans that seem to decide if numbers are special, and there are finitely many humans writing finitely many papers and other communications, which are thus countable. This implies that for there to be any chance of all real numbers being special, there must be uncountable families of special numbers. Once one starts thinking along these lines, one might recognise that all real numbers are special, because they are real (and not more generally complex). I can't personally think of any uncountable sets of numbers that are more special than the reals, except perhaps the transcendental reals. On the other hand, these are basically just all the real numbers that aren't special enough to be algebraic. So they aren't very special, and certainly not more special than the natural numbers.

No. ”Assume that not every number is mathematically interesting and let X be the first such number” doesn’t work as is because a set of reals need not contain a smallest number. For example, the set

   {x | ∃ n ∈ ℕ : 10^n = 1}

   =

   {1, 0.1, 0.01, 0.001, 0.0001, 0.00001, …}

doesn’t have a smallest number.

It is impossible to adapt this proof because the set of reals is uncountable. It is doable for the rationals, though, as they _are_ countable.

Outside the set of integers, the only reals left are either only fractionally interesting or crazy </end stupid pun>