Reddit mentions of Infinity and the Mind: The Science and Philosophy of the Infinite (Princeton Science Library)

Rucker's non-fiction book on infinity, Infinity and the Mind: The Science and Philosophy of the Infinite, is great, too. I read that when I was about 20 and it blew my mind.

Applications? If you're a set theorist or a logician there are applications. Otherwise transfinite numbers are of intrinsic interest and have no known applications.

On the other hand if you google around you can find papers trying to relate set theory to physics. Not much out there but there's a little.

For a really cool introduction to transfinite numbers and their philosophy, I strongly recommend Infinity and the Mind by Rudy Rucker.

https://www.amazon.com/Infinity-Mind-Philosophy-Infinite-Princeton/dp/0691121273

same Rudy Rucker?

I really enjoyed this book when i found it in the free bin a couple years ago.

I'd only recommend Everything and More if you are a Wallace completist (and I am), but the math needs some work.

Rudy Rucker's Infinity and the Mind covers the same material and is more mind-blowing. I used to be comfortable with the natural numbers and he ruined that (in a good way).

Rudy Rucker's Infinity and the Mind changed my life when I read it in high school. It's a supremely approachable introduction to set theory and the mathematics of infinity, delightful in its eager concept-evangelism and sometimes loopy backgrounding but absolutely rigorous when it comes to the actual mathematical content. I come back to it every few years and still love it.

Edit: I also loved Marcus du Sautoy's Symmetry ("Finding Moonshine" in some parts of the world), which is a historical and popular survey of the title topic, intertwined with relevant travelogues from the life of a professional mathematician. James Gleick's The Information is an outstanding book about the development of information theory and his landmark book Chaos: The Making of a New Science is also terrific — and if you value lucid, gripping nonfiction writing, his prose is an utter pleasure to read. He has an almost supernatural facility to pick out the perfect analogy to explain an idea. None of these have chapter-ending exercises like the Rucker book, but they do develop some very sophisticated ideas that'll leave your head thrumming for months afterward.