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November 1, 2024

Axiomatic Set Theory 7: Number Systems

Posted by Tom Leinster

Previously: Part 6.

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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