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?
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.
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.