A bunch of people have been asking me to take a look at yet another piece of Cantor crankery recently posted to Arxiv. In general, I’m sick and tired of Cantor crankery – it’s been occupying much too much space on this blog lately. But this one is a real prize. It’s an approach that I’ve never seen before: instead of the usual weaseling around, this one goes straight for Cantor’s proof.

But it does much, much more than that. In terms of ambition, this thing really takes the cake. According to the author, one J. A. Perez, he doesn’t just refute Cantor. No, that would be trivial! Every run-of-the-mill crackpot claims to refute cantor! Perez claims to refute Cantor, Gödel, Church, and Turing. Among others. He claims to reform the axiom of infinity in set theory to remove the problems that it supposedly causes. He claims to be able to use his reformed axiom of infinity together with his refutation of Cantor to get rid of the continuum hypothesis, and to eliminate any strange results proved by the axiom of choice.

Yes, Mr. (Dr? Professor? J. Random Schmuck?) Perez is nothing if not a true mastermind, a mathematical genius of utterly epic proportions! The man who single-handedly refutes pretty much all of 20th century mathematics! The man who has determined that now we must throw away Cantor and Gödel, and reinstate Hilbert’s program. The perfect mathematics is at hand, if we will only listen to his utter brilliance!

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So, remember back in December, I wrote a post about a Cantor crank who had a Knol page supposedly refuting Cantor’s diagonalization?

This week, I foolishly let myself get drawn into an extended conversation with him in comments. Since it’s a comment thread on an old post that had been inactive for close to two months before this started, I assume most people haven’t followed it. In an attempt to salvage something from the time I wasted with him, I’m going to share the discussion with you in this new post. It’s entertaining, in a pathetic sort of way; and it’s enlightening, in that it’s one of the most perfect demonstrations of the behavior of a crank that I’ve yet encountered. Enjoy!

I’m going to edit for formatting purposes, and I’ll interject a few comments, but the text of the messages is absolutely untouched – which you can verify, if you want, by checking the comment thread on the original post. The actual discussion starts with this comment, although there’s a bit of content-free back and forth in the dozen or so comments before that.

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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.

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