### Axiomatic Set Theory 3: The Axioms, Part Two

#### Posted by Tom Leinster

*Previously: Part 2. Next: Part 4*

This week, we finished formulating the axioms of the Elementary Theory of the Category of Sets.

This week’s axioms were the last four:

`7`

. For every function $f: X \to Y$ and element $y \in Y$, we can form the fibre $f^{-1}(y)$.Category theorists will recognize this as a special case of the existence of pullbacks. But in the presence of the other axioms, this special case alone is enough to imply the existence of all pullbacks. We’ll also be able to deduce the existence of arbitrary preimages (not just preimages of singletons), and of arbitrary images too.

`8`

. There exists a subset classifier. That is, there’s a set $\mathbf{2}$ with the property that for all sets $X$, the injections into $X$ correspond (up to iso) with the functions $X \to \mathbf{2}$.This axiom is very powerful. For instance, it’s proved in this section of the notes that every bijection is invertible and that all empty sets are isomorphic. Neither of those statements seems to have anything to do with the existence of a subset classifier. Nevertheless, we need it in the proofs.

The axiom does

*not*demand that $\mathbf{2}$ is a set with two elements. “$\mathbf{2}$” is just notation for now. But we’ll prove next week that the axioms do in fact imply that $\mathbf{2}$ is a two-element set.`9`

. There exists a natural number system.We’re not going to need this axiom for several weeks.

`10`

. Every surjection has a section.This is the axiom of choice, and we’re not going to need it until nearly the end of the course — Chapter 9 out of 10.

You can find the notes here.

## Re: Axiomatic Set Theory 3: The Axioms, Part Two

In case anyone’s interested in the pedagogical side of this, here’s some information.

There are 51 students enrolled in the class, nearly all of them fourth or fifth year undergraduates. (Our BSc is a four-year programme, and those doing an integrated masters degree — an MMath — do a further year.) Most of them are doing a maths degree, with a few doing joint maths + something degrees, e.g. maths + physics or maths + informatics.

In yesterday’s class, we’d just finished the axioms, so I asked the students to do the following little exercise. They got into pairs, each student chose the topic they’d found hardest so far, and then the other member of the pair tried to explain that topic to them for five minutes. Then they swapped roles.

I went round the room asking everyone what they’d chosen as their topic, and there was a clear winner: function sets (a.k.a. exponentials). In retrospect, I can see that that section of the notes is a bit lacking explanation and examples, but I suspect ot would have been the winner anyway.

From what the students said, I think there were several related difficulties with function sets. One was that they’re not that used to thinking about sets of functions. (Or perhaps they are, but they don’t realize they are?) Another was the correspondence between functions $A \times X \to Y$ and functions $A \to Y^X$. A third was the role of the evaluation function $Y^X \times X \to Y$. And finally, the formal definition of function set, via the universal property, takes some getting used to.

I think all these things are an important part of learning mathematics anyway. For instance, if you don’t understand in your bones how you can collect together functions to form a set, then you’re never going to understand permutation groups. But there’s a more optimistic converse: if you

dounderstand permutation groups, then youdounderstand that functions can form a set! That’s what I meant when I wrote above that some students might be more familiar with function sets than they realize.If any of my students are reading this and want to share their thoughts on what they found challenging, I’d very much welcome that.

As I said in class, I wouldn’t be surprised if the previous week or two turns out to be the hardest part of the course. That’s quite unusual: normally the hardest part of a course is at the end. Perhaps it’s like the shape of an introductory analysis course that starts with the definition of continuity — something that many learners need time to get used to. In fact, there are some shared difficulties, in that both the definition of continuity and universal properties both have somewhat complicated quantifiers.