Convincing proofs, proofs by contradiction and countability Disclaimer: I have a (modest) background in mathematical physics, not logic, so I know very little of the latter.
Although when I understood Cantor's argument for the first time (from one of Martin Gardner's books) I was immediately taken by its beauty, recently I have grown slightly suspicious of it. I realize that it sounds like a beginning of a very cranky opinion, but I'll state it anyway: Cantor's argument, an argument by contradiction, seems to be very unlike the direct arguments for countability of $\mathbb{Q}$ and $\mathbb{Z}$. It is surprising, especially taken in its historical context, and, from a naive point of view that I do not necessarily espouse, almost unconvincing: why can't this ingenious construction be toppled with some other, even more ingenious?
Given these, admittedly dubious, considerations, I'll try to pose the question: is there a possibility of direct argument that would prove the uncountability of reals? And if there is not, what would be the reasons for the lack of its existence?
P.S.: I have just discovered this question and think that another way to reformulate my question would be: is the uncountability of reals provable in minimal logic?

There is nothing dubious about a proof by contradiction if you are trying to prove the non-existence of some object, for example "There is no bijection from $\mathbb N$ to $\mathbb R$". Assume there exists such a bijection; derive a contradiction; and you have established the result, even for the strictest of constructivists.
It is only if you are trying to prove the existence of some object that a proof by contradiction is less satisfactory than a constructive proof.
