### Axiomatic Set Theory 1: Introduction

#### Posted by Tom Leinster

*Next: Part 2*

I’m teaching Edinburgh’s undergraduate Axiomatic Set Theory course, and the axioms we’re using are Lawvere’s Elementary Theory of the Category of Sets — with the twist that everything’s going to be done directly in terms of sets and functions, without invoking categories. That is, I’ll neither assume nor teach the general notion of category.

I thought I’d share my notes so far.

Here are the planned chapter headings for the course:

- Introduction
- The axioms, part one
- The axioms, part two
- Subsets
- Relations
- Coproducts and families
- Number systems
- Well ordered sets
- The axiom of choice
- Cardinal arithmetic.

It’s one chapter per week, and we’re one week in, which means that so far we’ve just covered the introduction.

I’m really excited to be teaching this course, because as far as I know, nothing like it has ever been done before.

Plus, lots of people — even category theorists and set theorists — don’t realize it can be done! For example, two people I know who are knowledgeable in both subjects assumed I’d have to get into toposes, and didn’t realize it was possible to do everything in a completely elementary way. I want the world to know!

I like to explain this point by analogy with number theory and rings. If you’re going to teach an introductory number theory course, you have a choice. You could say:

“the integers form a ring, number theory eventually needs rings other than the integers, and rings are important throughout mathematics anyway, so I’m going to begin my course by introducing rings and then specialize to the integers”.

Alternatively, you could say:

“basic number theory doesn’t require the general notion of ring, so let’s just talk directly about addition, subtraction and multiplication of integers without ever mentioning rings”.

Both kinds of course are valuable. The first is like teaching ETCS via the
general notion of category, more or less as in Lawvere and Rosebrugh’s book
*Sets for Mathematics*. The second is like my course: no categories, just
sets and functions done directly.

The classes are based on the students’ questions; each student brings one written question to each class, and I try to answer them all. It’s always fascinating to discover what things the students find challenging, easy, instinctive, counterintuitive, or puzzling, and what captures their imagination. They’ve asked some excellent questions so far.

But I thought it might also be interesting to share the notes here and see what others make of them. So here they are:

Tom Leinster,

Axiomatic Set Theory, undergraduate lecture notes in progress.

If all goes well, I’ll keep sharing here every week as I add new
chapters. But I’m not entirely sure: the notes are for *my students*, and
it may be that sharing them publicly distorts how I write, in which case
I’ll have to pull the plug. (But even so, I’d hope to share the notes once the course is over.) Let’s see how it goes.

## Re: Axiomatic Set Theory 1: Introduction

Thanks, this is great! I’m teaching a course that’s new to me: modern Geometry. I’m starting with Hilbert’s axioms, but that requires sets and relations, so we did some review.

That led to Foundations, and there is a lot of foundational set theory going on these days! First there is Version 10 of this paper proving consistency of Quine’s New Foundations (NF). My quick elevator pitch for NF is that it avoids Russell’s paradox by requiring stratified formulas. The stratification of formulas prevents x not in x, since for x in y, y must be a strata above x.

Then I started reading this new series of papers by Quinn, with more of the categorical approach that your notes are (barely) hiding: he defines sets as “logical domains that support quantification.”

So I am looking forward to hearing more!