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November 22, 2024

Axiomatic Set Theory 10: Cardinal Arithmetic

Posted by Tom Leinster

Previously: Part 9.

The course is over! The grand finale was the theorem that

X×Y≅X+Y≅max(X,Y) X \times Y \cong X + Y \cong max(X, Y)

for all infinite sets XX and YY. Proving this required most of the concepts and results from the second half of the course: well ordered sets, the Cantor–Bernstein theorem, the Hartogs theorem, Zorn’s lemma, and so on.

I gave the merest hints of the world of cardinal arithmetic that lies beyond. If I’d had more time, I would have got into large sets (a.k.a. large cardinals), but the course was plenty long enough already.

Thanks very much to everyone who’s commented here so far, but thank you most of all to my students, who really taught me an enormous amount.

Part of the proof that an infinite set is isomorphic to its own square

Posted at November 22, 2024 3:27 PM UTC

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7 Comments & 0 Trackbacks

Re: Axiomatic Set Theory 10: Cardinal Arithmetic

Hurrah, and have a nice break!

Posted by: John Baez on November 22, 2024 5:58 PM | Permalink | Reply to this

Re: Axiomatic Set Theory 10: Cardinal Arithmetic

Thanks! It was intense.

The custom here is that although the teaching semester is 11 weeks long, you don’t teach anything new in the last week. At least, you don’t if the exam happens very soon after the course ends, as it does for us. We’ve just finished the 10th week, which means there’s one more week — but it’s just for revision and review.

As I keep saying, there’s so much I’d do differently next time. I don’t have the energy right now to go into this. And actually, I don’t know if there’ll be a next time! This course alternates years with Category Theory, so even if I’m teaching Axiomatic Set Theory again, it won’t be until nearly two years from now.

Maybe when you’re next in Edinburgh, you’ll cross paths with some of the students who are now ninjas in ETCS.

Posted by: Tom Leinster on November 22, 2024 6:07 PM | Permalink | Reply to this

Re: Axiomatic Set Theory 10: Cardinal Arithmetic

If I see someone dressed in black wielding a small collection of surprisingly intuitive axioms, I’ll run.

Posted by: John Baez on November 27, 2024 1:32 AM | Permalink | Reply to this

Re: Axiomatic Set Theory 10: Cardinal Arithmetic

A small typo: you say just before Proposition 10.1.5 “ways of defining finiteness”, then go on to characterise infinite sets.

Posted by: David Roberts on November 23, 2024 1:28 AM | Permalink | Reply to this

Re: Axiomatic Set Theory 10: Cardinal Arithmetic

Thanks! Fixed.

Posted by: Tom Leinster on November 23, 2024 10:49 AM | Permalink | Reply to this

Re: Axiomatic Set Theory 10: Cardinal Arithmetic

Ex- AST student here! I just finished my 4th year exams; wanted to let you know that your Axiomatic Set Theory course was by far my favorite course at uni, and I really appreciate the time and attention you poured into us, sharing your love of this gorgeous subject! Such a formative course. I’m graduating, so won’t get to take Category Theory next year :( but am excited to follow along remotely from the US!

Posted by: Graham Jackson on May 16, 2025 1:46 PM | Permalink | Reply to this

Re: Axiomatic Set Theory 10: Cardinal Arithmetic

Thanks so much for your enthusiastic feedback! It means a lot to me.

It was an amazing opportunity to be able to put together this course and then teach it — lots of work, of course, but I was propelled along by the energy and dedication of the class.

Posted by: Tom Leinster on May 16, 2025 2:13 PM | Permalink | Reply to this

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