# Some infinities (and egos) are bigger than others

Here’s a tale of a rational (or irrational?) legal battle from the 1990s re: Cantor’s diagonal argument.

Cantor’s diagonal argument from 1891 was truly revolutionary: an ingenious way to demonstrate that no matter what proposed list of all real numbers (or, say, just those between $0$ and $1$) is put forth, it’s easy to find a number which is definitely missing from the list.1

In a nutshell, Cantor was the first to show that some infinities are bigger than others.

Cantor’s diagonalisation argument for the reals is watertight, and has proved to be a model of elegance and simplicity in the century plus that has passed since it first appeared.

That didn’t stop engineer William Dilworth publishing A correction in set theory, in which he refutes Cantor’s argument, in the Transactions of the Wisconsin Academy of Sciences in 1974.

Nor, when mathematician Underwood Dudley included Dilworth in his book Mathematical Cranks, did it stop him suing Dudley and his publishers in Milwaukee, Wisconsin, in 1995. You can read about that on Justia.com.

It says, “The complaint alleges that because Dilworth is not a professional mathematician he finds it very difficult to get his articles on mathematics published and being labeled a “crank” will create an additional obstacle.”

Luckily for sanity, “The district judge granted the motion to dismiss on the ground that the word “crank” is incapable of being defamatory; it is mere “rhetorical hyperbole.” ”

After Dilworth saw himself in Mathematical Cranks he tried to get in touch with me but I, in keeping with best cranks policy, replied once and once only. Not one to be put off, one day he showed up in my town looking for me. In an amazing coincidence, the wife of my department chair overheard his asking someone about me and, being a nice person, took him to her husband who was able to get rid of him.

If I hadn’t been so standoffish, he might not have decided to sue me, the MAA, and my school for damages. He started in federal court in Wisconsin, where his case was thrown out. My respect for lawyers went up, first because he represented himself — I assume because he could find no lawyer to take the case — and second because the MAA lawyers (the MAA has legal insurance for this sort of thing) were able to find all sorts of precedents where people called other people awful things in print, up to and including “lazy, stupid, crap-shooting, chicken-stealing idiot”, without being held liable for damages.

He then appealed, as was his right, to the Seventh Circuit Court of Appeals where Judge Posner, a smart man, threw him out again. Rather than go to the Supreme Court, he sued again, this time in Wisconsin state court. He lost again, and was made to pay the defendants’ legal expenses. That put a stop to the suing.

Nevertheless, he continued to send me letters. Why he thought I’d answer them is a mystery and I ended by throwing them away unread. A few years ago he died, I think up in his 80s, probably thinking to the end that he was right.

Dilworth wasn’t the first person who felt that his entanglements with, and pronouncements on, infinite sets led to a loss of status. According to Wikipedia, Georg Cantor himself, back in 1904,

was outraged and agitated by a paper presented by Julius König at the Third International Congress of Mathematicians. The paper attempted to prove that the basic tenets of transfinite set theory were false. (König is now remembered as having only pointed out that some sets cannot be well-ordered, in disagreement with Cantor.) Since the paper had been read in front of his daughters and colleagues, Cantor perceived himself as having been publicly humiliated. Although Ernst Zermelo demonstrated less than a day later that König’s proof had failed, Cantor remained shaken, even momentarily questioning God.

(Georg Cantor: his mathematics and philosophy of the infinite, by Joseph W. Dauben , Boston: Harvard University Press, 1979. page. 248)

1. One has to pay close attention to realise that the same proof doesn’t also establish that the rationals are uncountable, bearing in mind that the Cantor pairing function shows that the rationals most certainly are countable. See http://en.wikipedia.org/wiki/Countable_set []

### 3 Responses to “Some infinities (and egos) are bigger than others”

1. Thony C

It is a little known fact that Cantor was not the first to use the diagonal argument in a proof. That honour goes to the German mathematician Paul DuBois-Reymond