### Axiomatic Set Theory 4: Subsets

#### Posted by Tom Leinster

*Previously: Part 3. Next: Part 5*

This phase of the course is all about building up the basic apparatus. We’ve stated our axioms, and it might seem like they’re not very powerful. It’s our job now to show that, in fact, they’re powerful enough to do just about everything with sets that mathematicians ever want. We began that job this week, with a chapter on subsets.

For example, in the axioms, we only asked for the existence of preimages (inverse images) of *one-element* subsets of the codomain of a function. But we didn’t ask for the existence of preimages of arbitrary subsets. Nor did we ask for any kind of images at all. Nevertheless, we prove here that all preimages and images exist.

We also prove that $\mathbf{2}$ has two elements! One of the axioms (loosely stated) is that there exists a set $\mathbf{2}$ such that for all sets $X$, functions $X \to \mathbf{2}$ correspond to injections into $X$ (taken up to isomorphism over $X$). We didn’t assume that $\mathbf{2}$ has two elements. But we prove here that this does in fact follow from the axioms. In other words, it’s a consequence of the axioms that we’re in a world with exactly two truth values.

Then we build up familiar constructions like unions and intersections of subsets, and prove that obey the usual algebraic laws. As we go further down this road, the arguments look more and more like the elementary set theory that everyone’s used to. Put another way, the ETCS approach starts to converge about here with other axiomatizations of sets.

One of the more subtle points in this chapter is that there are two kinds of elementhood (if that’s a word) or membership:

The first is a type declaration, like when someone begins a conversation with “Let $z \in \mathbb{C}$”. That’s not a true/false statement; you can’t very well reply “No it’s not!”

The second kind of membership is about whether an element of a set belongs to a particular subset, and that

*is*a true/false statement, like if you’ve got your element $z \in \mathbb{C}$ and you ask whether $z \in \mathbb{R}$. (It either is or isn’t.) In this course, we’d write it as “$z \in_{\mathbb{C}} \mathbb{R}$”.

And in much the same way, there are two notions of subset (Example 4.1.13). But I won’t write more here, as I’ve already explained everything as well as I can in the notes!

## Re: Axiomatic Set Theory 4: Subsets

I’m enjoying reading these notes, and have started writing my own “Sets and Categories” lecture notes for next semester. I’m also reading a PhD thesis in forcing and this reminds me that the mathematical ideas you need to do forcing are rather different (almost disjoint?) from those developed, at least so far, in your notes. I will be developing subsets in an element-based way rather than a categorical one, and then hopefully I have time to point out the other way of doing it too.

My question: is there a reasonable way in your setup to get the right analogue of the separation axiom, that given a set $X$ and a “reasonable” formula $\phi(x)$, the subset $\{x \in X : \phi(x)\}$ exists? Maybe one can associate a characteristic function to $\phi$?