Recently the RJLipton blog had two interesting and contentious posts about people who dispute Cantor's diagonal argument (that real numbers have different cardinality than natural numbers), which I'm pretty sure generated more comments than anything else to date on the blog. Apparently this is one of the more popular topics for math-cranks to extensively argue that they've proven the other way -- read for yourself here and here.
I wish that I had the opportunity to address issues like this in the classes I teach, but unfortunately at the moment I don't have any such opportunity. It would be nice to have a venue to refine the argument with a fresh audience every so often, and to work to ferret out the criticisms that arise. If we do so, with a disputatious subject like this (namely: the first few times a student deals with infinite sets and their counterintuitive by-products), then I think it's extra-important that we carefully lay out initial definitions at the start, break down the argument into very atomic numbered steps (so that we can refine discussion and disputes as they come up later), and also give explicit justifications for each step.
Here's another issue which I feel has the same flavor to it: the fact that 0.999... = 1 (or more generally, that any terminating decimal has two different, equivalent representations: the normal one, and a second one that ends with an endless sequence of "9"'s). Here's a suggestion on the careful way that I'd want to do it (again -- not having had this battle-plan encounter the enemy yet):
Definition of 0.999...
(a) The number has infinitely repeating digits.
(b) After every "9" digit, there is another "9".
(c) There is no end to the "9"'s.
Proof that 0.999... = 1 (by algebra)
(1) Let x = 0.999...
(2) Then 10x = 9.999... (multiply each side by 10)
(3) So 9x = 9 (subtract step 1 from step 2; note decimals cancel)
(4) Which means x = 1 (divide each side by 9)
(5) Therefore 0.999... = 1 (substitute from step 1)
And then when the arguments arise you can at least ask your interlocutor to focus on one single step or definition in which they think there's a logical gap.