Poor Georg Cantor.

During his life, he suffered from dreadful depression. He was mocked by

his mathematical colleagues, who didn't understand his work. And after his

death, he's become the number one target of mathematical crackpots.

As I've mentioned before, I get a *lot* of messages either from or

about Cantor cranks. I could easily fill this blog with nothing but

Cantor-crankery. (In fact, I just created a new category for Cantor-crankery.) I generally try to ignore it, except for that rare once-in-a-while that there's something novel.

A few days ago, via Twitter, a reader sent me a link to a new monstrosity

that was posted to arxiv, called Cantor vs Cantor. It's novel and amusing. Still wrong,

of course, but wrong in an amusingly silly way. This one, at least, doesn't *quite*

fall into the usual trap of ignoring Cantor while supposedly refuting him.

You see, 99 times out of 100, Cantor cranks claim to have

some construction that generates a perfect one-to-one mapping between the

natural numbers and the reals, and that therefore, Cantor must have been wrong.

But they never address Cantors proof. Cantors proof shows how, given *any*

purported mapping from the natural numbers to the real, you can construct at example

of a real number which isn't in the map. By ignoring that, the cranks' arguments

fail: Cantor's method still generates a counterexample to their mappings. You

can't defeat Cantor's proof without actually addressing it.

Of course, note that I said that he didn't *quite* fall for the

usual trap. Once you decompose his argument, it does end up with the same problem. But he at least *tries* to address it.