for all infinite sets X X and Y Y. 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 ...

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 ...

Earlier this month the Mathematics Institute at Uppsala University hosted a conference called Categorification in Algebra and Topology, clearly a theme close to our collective heart. As yet there are ...

Nov 15, 2024 The penultimate week of this axiomatic set theory course, based on Lawvere’s Elementary Theory of the Category of Sets.

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

Nov 7, 2024 00:45 Here’s a way to argue that Gerard’s solution to my puzzle is correct. Thurston showed that any star of the ...

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 ...

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 ...

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 ...

Are you interested in using category-theoretic methods to tackle problems in topics like quantum computation, machine learning, numerical analysis or graph theory? Then you might like the Adjoint ...

Thurston’s paper Shapes of polyhedra and triangulations of the sphere is really remarkable. I’m writing about it in my next column for the Notices of the American Mathematical Society. Here’s a draft ...