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.
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