## Cantorian Sets: "Beyond Infinity - An expedition to the outer limits of the mathematical universe" by Eugenia Cheng

*“If this be not that you look for, I have no more to say, but bid Bianca farewell for ever and a day.”*

*In*“The Taming of the Shrew” by William Shakespeare (quoted by Cheng in the book)

Eugenia Cheng starts by saying right at the beginning of the book, "Infinity is not a number," and I think it really helps to get that misconception out of the way at the start. As soon as we one gets past that hurdle the rest is just a piece of cake.

As pointed out numerous times by Cheng, Cantor is the accepted authority on this, but are there alternatives?

### Cantor Infinities

Key Idea: You can put the even numbers in one-to-one correspondence with the whole numbers and say that this demonstrates they have the same cardinality.

1->2

2->4

3->6

...

This shows that the set of whole numbers is the same size as the set of even numbers.

This seems counter-intuitive - and it's usually a real challenge to anyone encountering it for the first time, but if you do accept this then all sorts of deep and interesting mathematics follow. The way I think about this is, it's not a natural property, it's not a statement about the world*; it's Cantor's definition of infinity, let's go along with it and see what happens.

If you're into Computer Science and Math in particular, read on.