# You're reading: Posts Tagged: Georg Cantor

### Mathematical myths, legends and inaccuracies: some examples

I’m teaching a first-year module on the history of mathematics for undergraduate mathematicians this term. In this, I’m less concerned about students learning historical facts and more that they gain a general awareness of history of maths while learning about the methods used to study history.

Last week, I decided I would discuss myths and inaccuracies. Though I am aware of a few well-known examples, I was struggling to find a nice, concise debunking of one. I asked on Twitter for examples, and here are the suggestions I received, followed by what I did.

### 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.

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 []