Of all of the work in the history of mathematics, nothing seems to attract so much controversy, or even outright *hatred* as Cantor’s diagonalization. The idea of comparing the sizes of different infinities – and worse, of actually concluding that there are different infinities, where some infinities are larger than others – drives some people absolutely crazy. As a result, countless people bothered by this have tried to come up with all sorts of arguments about why Cantor was wrong, and there’s only one infinity.

Today’s post is another example of that. This one is sort of special. Unless I’m very much mistaken, the author of this sent me his argument by email last year, and I actually exchanged several messages with him, before he concluded, roughly “We’ll just have to agree to disagree.” (I didn’t keep the email, so I’m not certain, but it’s exactly the same argument, and the authors name is vaguely familiar. If I’m wrong, I apologize.)

Anyway, this author actually went ahead and wrote the argument up as a full technical paper, and submitted it to arXiv, where you can download it in all it’s glory. I’ll be honest, and admit that I’m a little bit impressed by this. The proof is still completely wrong, and the arguments that surround it range from wrong to, well, not even wrong. But at least the author has the Chutzpah to treat his work seriously, and submit it to a place where it can actually be reviewed, instead of ranting about conspiracies.

For those who aren’t familiar with the work of Cantor, you can read my article on it here. A short summary is that Cantor invented set theory, and then used it to study the construction of finite and infinite sets, and their relationships with numbers. One of the very surprising conclusions was that you can compare the size of infinite sets: two sets have the same size if there’s a way to create a one-to-one mapping between their members. An infinite set A is larger than another infinite set B if every possible mapping from members of B to members of A will exclude *at least* one member of A. Using that idea, Cantor showed that if you try to create a mapping from the integers to the real numbers, for any possible mapping, you can generate a real number that isn’t included in that mapping – and therefore, the set of reals is larger than the set of integers, even though both are infinite.

This really bothers people, including our intrepid author. In his introduction, he gives his motivation:

Cantor’s theory mentioned in fact that there were several dimensions for infinity. This, however, is questionable. Infinity can be thought as an absolute concept and there should not exist several dimensions for the infinite.

Philosophically, the idea of multiple infinities is uncomfortable. Our intuitive notion of infinity is of an absolute, transcendent concept, and the idea of being able to differentiate – or worse, to be able to compare the sizes of different infinities *seems* wrong.

Of course, what *seems* wrong isn’t necessarily wrong. It *seems* wrong that the mass of something can change depending on how fast it’s moving. It seems even more wrong that looked at from different viewpoints, the same object can have *different* masses. But that doesn’t change the fact that it’s true. Reality – and even worse, abstract mathematics – isn’t constrained by what makes us comfortable.

Back to the paper. In the very next sentence, he goes completely off the rails.

Continue reading The Continuum Hypothesis Solved: All Infinities are the Same? Nope.