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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 3:27 PM UTC | Permalink | Followups (5)

November 15, 2024

Axiomatic Set Theory 9: The Axiom of Choice

Posted by Tom Leinster

Previously: Part 8. Next: Part 10.

It’s the penultimate week of the course, and up until now we’ve abstained from using the axiom of choice. But this week we gorged on it.

We proved that all the usual things are equivalent to the axiom of choice: Zorn’s lemma, the well ordering principle, cardinal comparability (given two sets, one must inject into the other), and the souped-up version of cardinal comparability that compares not just two sets but an arbitrary collection of them: for any nonempty family of sets (X i) i∈I(X_i)_{i \in I}, there is some X iX_i that injects into all the others.

The section I most enjoyed writing and teaching was the last one, on unnecessary uses of the axiom of choice. I’m grateful to Todd Trimble for explaining to me years ago how to systematically remove dependence on choice from arguments in basic general topology. (For some reason, it’s very tempting in that subject to use choice unnecessarily.) I talk about this at the very end of the chapter.

Section of a surjection

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November 8, 2024

Axiomatic Set Theory 8: Well Ordered Sets

Posted by Tom Leinster

Previously: Part 7. Next: Part 9.

By this point in the course, we’ve finished the delicate work of assembling all the customary set-theoretic apparatus from the axioms, and we’ve started proving major theorems. This week, we met well ordered sets and developed all the theory we’ll need. The main results were:

  • every family of well ordered sets has a least member — informally, “the well ordered sets are well ordered”;

  • the Hartogs theorem: for every set XX, there’s some well ordered set that doesn’t admit an injection into XX;

  • a very close relative of Zorn’s lemma that, nevertheless, doesn’t require the axiom of choice: for every ordered set XX and function φ\varphi assigning an upper bound to each chain in XX, there’s some chain CC such that φ(C)∈C\varphi(C) \in C.

I also included an optional chatty section on the use of transfinite recursion to strip the isolated points from any subset of ℝ\mathbb{R}. Am I right in understanding that this is what got Cantor started on set theory in the first place?

Diagram of an ordered set, showing a chain. I know, it's not *well* ordered

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November 5, 2024

The Icosahedron as a Thurston Polyhedron

Posted by John Baez

Thurston gave a concrete procedure to construct triangulations of the 2-sphere where 5 or 6 triangles meet at each vertex. How can you get the icosahedron using this procedure?

Gerard Westendorp has a real knack for geometry, and here is his answer.

Continue reading “The Icosahedron as a Thurston Polyhedron” …
Posted at 11:18 PM UTC | Permalink | Followups (5)

November 2, 2024

Summer Research at the Topos Institute

Posted by John Baez

You can now apply for the 2025 Summer Research Associate program at the Topos Institute! This is a really good opportunity.

Details and instructions on how to apply are in the official announcement.

A few important points:

  • The application deadline is January 17, 2025.
  • The position is paid and in-person in Berkeley, California.
  • The Topos Institute cannot sponsor visas at this time.

For a bit more, read on!

Continue reading “Summer Research at the Topos Institute ” …
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November 1, 2024

Axiomatic Set Theory 7: Number Systems

Posted by Tom Leinster

Previously: Part 6. Next: Part 8.

As the course continues, the axioms fade into the background. They rarely get mentioned these days. Much more often, the facts we’re leaning on are theorems that were deduced from theorems that were deduced — at several removes — from the axioms. And the course feels like it’s mostly converging with any other set theory course, just with the special feature that everything remains resolutely isomorphism-invariant.

This week we constructed ℕ\mathbb{N}, ℤ\mathbb{Z}, ℚ\mathbb{Q} and ℝ\mathbb{R}. This was the first time in the course that we used the natural numbers axiom, and that axiom did get cited explicitly (in the first few pages, anyway). We had to use the universal property of ℕ\mathbb{N} to define sums, products and powers in ℕ\mathbb{N}, and to prove the principle of induction.

I think my highlight of the week was a decategorification argument used to prove the classic laws of natural number arithmetic. Read on…

Continue reading “Axiomatic Set Theory 7: Number Systems” …
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