For reasons that I never understood, the fact that there are several sizes of infinity and, in particular, the fact that the reals are uncountable is unacceptable for certain persons. My question is: why is it that those who try to refute that assertion always (or so it seems to me) point their guns at Cantor's diagonal proof? After all:
- Cantor himself proved (before creating the diagonal proof) that the reals are uncountable by another method (based upon the fact that, in $\mathbb R$, every bounded monotonic sequence converges).
- There are other proofs that the reals are uncountable.
A few years ago, Wilfrid Hodges, a logician, wrote an interesting article about nearly the same question, called An Editor Recalls Some Hopeless Papers, but his article was about the validity (or lack thereof) of certain “refutations” of Cantor's diagonal argument. But my question is: why don't they try to refute the other arguments? Is it just because the diagonal argument is so well-known? Or is there some other reason?