What are some good reasonably rigorous texts on the mathematics of infinity?

Question

The Infinite Book is too light and not focused enough on the mathematics of infinity, and Everything and More: A Brief History of Infinity has too much focus on the history of infinity instead of the mathematics of it. Is it best to just read a text on set theory and learn about infinity from that?

Answer 1

I assume that a book on set theory could kill your motivation to learn it, mostly because a lot of the books do not come with historical tidbits and reasons to do such activities. On these books, they assume that you're mature enough to know why you're doing that or that you're having lectures about their contents.

For a first book, I'd really recommend you to read Stillwell's: Road to Infinity. It teaches well, contains motivation and tells you the history of set theory. The book is also full of references, in which one of them is Thomas Jech's: Set Theory, a classical in the field of set theory (perhaps you can read it after reading Stillwell's book).

Another book I really liked is Moschovakis': Notes on Set Theory, although it's a bit harder than the first one. And the last one I have to recommend is Halbeisen's: Combinatorial Set Theory, it's a very unique and interesting book. It's a bit different of a set theory book because it employs the generalization of some results in combinatorics to set theory. All these books are packed with motivation and discussion about the need of these concepts (which in my opinnion, are very important to learn the subject).

Answer 2

Kunen's Set Theory: An Introduction to Independence Proofs. If you want to seriously understand modern notions of infinite sets, this will do it. I must admit I found it pretty hard. It's not a light read.