'I believe, and hope, that a future generation will laugh at this hocus pocus.' - Wittgenstein.
It seems like the mainstream view today is that Cantor showed something wonderful and surprising, namely that there are 'higher infinities'. Anyone who is suspicious or negative about this is liable to be written off as a crank.
There is a paper by Wilfrid Hodges on crank attempts to disprove Cantor's theorem called 'An Editor Recalls Some Hopeless Papers' (there's currently a copy on scribd). He actually dedicates the paper to them, but I wonder about his sincerity. More importantly, he doesn't in my opinion shed any real light on the question of what motivates the cranks.
Of course, to try to refute Cantor's theorem is folly - the theorem certainly holds. But I think what the cranks who do try this are dimly feeling is something I feel too, only I have the good sense to realize that it lies not in the mathematics itself being false, but our interpretation of it being idiotic.
David Foster Wallace wrote a book which includes stuff on Cantor's work called Everything and More. I seem to recall a reference to it as Beyond Infinity. Maybe that was the working title. In any case, I think 'Beyond Infinity' is a horrible title. It almost offends me, which I find interesting.
It is very hard to put your finger on what the difference is between people who like the paradoxical-sounding talk surrounding Cantor's theorem and those who hate it. I wonder what else they are likely to divide on.
Part of what ticks me off about it is that people act almost as if you can, before explaining Cantor's special definition of equinumerousness, report his result in ordinary language, e.g. 'Some infinite sets have more members than others' or whatever. Even after the definition is on board, the paradoxical flavour - what Wittgnestein called the 'slight reeling' which the thought produces - still comes from our pre-Cantorian grasp of the concept being smuggled in (or rather, not being thoroughly replaced by the Cantorian development). But then this is treated as though it is some sort of perception of the wonders of mathematics.
I want to put my hand up as someone who takes issue with this way of regarding and talking about Cantor's results. It seems cheap, sensationalistic and stupid to me. I don't know of any contemporaries who have the same view about this (apart from mathematical cranks).
I can't recommend enough Wittgenstein's material on this matter, particularly in Philosophical Grammar and the Remarks on the Foundations of Mathematics. That isn't to say it's all on the right track - I suspect some of it is short-sighted and that that has made it all seem bad to some people. There are some excerpts here. Regarding the quote I opened with: I wonder if that will come to pass. It certainly hasn't yet. It can be bitterly thrilling to catch glimpses of ways in which ours is a dark age.