I’ve been given the chance to teach the course in axiomatic set theory in Jerusalem this semester. Today I gave my first lecture as a teacher. It went fine, I even covered more than I expected to, which is good, I guess. I am also preparing lecture notes, which I will probably post here when the semester ends. These predicated on some rudimentary understanding in logic and basic set theory, so there might be holes there to people unfamiliar with the basic course (at least the one that I gave with Azriel Levy for the past three years).

Yesterday, however, I spent most of my day thinking about how we—as a collective of set theorists—teach axiomatic set theory. About that usual course: axioms, ordinals, induction, well-founded sets, reflection, $V=L$ and the consistency of $\GCH$ and $\AC$, some basic combinatorics (clubs, Fodor’s lemma, maybe Solovay or even Silver’s theorem). Up to some rudimentary permutation.

Is this the right way to approach axiomatic set theory? This path is not easy to justify. Sure, you can justify things like well-founded sets by arguing that this is how we justify the Axiom of Foundation. And you could argue that this is a rudimentary foray into inner model theory, and that this is important. And you are absolutely right. But on the other hand, I feel that engaging the students should involve more set theory which is “interactive”. Where you obtain actual results, rather than just consistency of axioms, especially axioms which you have very little motivation towards.

I mean, look at how we teach (or learn) about algebraic structures. We don’t spend all semester just with the axioms of groups, or rings, proving things. We also see a lot of examples, and a lot of ways where these structures interact with mathematics. Set theory doesn’t have this luxury, we don’t have natural models to work with and their interactions with mathematics is meta-theoretical, rather than direct as it is the case with groups and rings.

So set theory, in essence, should be taught in a mixture of motivating examples and consistency proofs. I am taking this from my advisor, who is a wonderful teacher, as anyone who ever sat in his lectures could witness. A couple of years ago, Menachem gave a course about stationary tower forcing. In most texts about stationary tower forcing, you spend the first several dozen pages in technical concepts like Completely Jonsson cardinals, and so on. But Menachem started with the motivation: universally Baire sets, and their properties. Once you understand those, stationary tower forcing becomes much easier to digest, because it is with purpose. Last year, and next semester, Menachem is talking about inner models, and again a lot of motivation is given into fine structural considerations, mainly square’ish ones for the basic fine structure of $L$, but also through mice we get a good intuition as to what $K^{DJ}$ is supposed to be.

Right. So the basic axiomatic set theory course. What can we do about that? Well, my initial approach is to take $\ZF$ for starters. Motivate Foundation by talking about induction, and then prove that Foundation adds no new contradictions. After that, we’ll see exactly, but the next step is again motivation for either choice or Reflection principles. In either case, I feel that having motivation interspersed with consistency proofs is key here.

So now, let me ask you, my fellow set theorists, who have taught courses in axiomatic set theory. What is your experience on the matter? What is your take on my approach? This is my first time doing this, and I will definitely be reporting again during the semester and afterwards. But I also want to hear what you have to say on the matter. I will leave the comments open, but also feel free to contact me over email.

Asaf, I think your suggestion that we’re all teaching set theory basically the same way, “up to some rudimentary permutation,” is likely far from true. My expectation is that set theorists and mathematicians generally each develop their own preferred way to present their subject, and that we are all enriched by this. So I encourage you to break out of any curriculum that you find to be confining or unmotivated, and develop your own voice.

In my own case, for example, at the graduate level I tend to dwell on the Cantor-Bendixson analysis, which both motivates the ordinal concept (it is after all the source of the ordinal concept), while also remaining connected with the reals, and it leads directly to the continuum hypothesis. Then, I aim as quickly as possible for large cardinals, getting to inaccessibles, hyperinaccessibles and Mahlos. Apart from having an inherently attractive combinatorics, the large cardinals also showcase the awesome size of the objects studied in set theory, which is inspiring and motivating on its own. In addition, the fact that we can’t prove they exist leads to profound philosophical and foundational issues and questions, and this motivates the consistency strength arguments and relative consistency proofs generally. I would expect that you would want to spend some significant time on ZF and the cases where AC fails, but I rarely spend much time on that at all.

At the undergraduate level, I tend to spend a big chunk of time on the back-and-forth construction for DLO, and then universality, such as building universal partial orders, and I always look quite a bit at embedding into the lattice P(N). For example, can you order-embed the real line into P(N) with subset?

Joel, there are two subtle points, I guess that we both are missing here.

The first is the assumed knowledge of the students. You can’t talk about Cantor–Bendixson analysis with people whose grasp of topology is weak to non-existent. Sure, you can define all the relevant terms, but that is time you’re not getting back from other topics. You can’t talk about motivating inaccessible cardinals and measurable cardinals with people whose grasp in model theory is weak, or with people that never saw a proof of the incompleteness theorem. Yes, you can cite these, but it doesn’t quite cut to the same depth. My class is very heterogeneous in this aspect, there are sophomore undergrads, there are seniors, there are grad students; there are some who had taken a course about the incompleteness theorem, and some of them are only taking their first logic course in parallel. If you want to retain a large class, a large heterogeneous class, it means making everyone feel that they actually follow you. And this means that it might be a good idea to stick to some basics.

For what it’s worth, the basic set theory course—for the past three years, under Azriel’s helm with me on the exercises—covered quite more than order embedding the reals into P(N). We covered choice related examples, we covered ordinal arithmetic, we covered the notion of cofinality and we covered the basics too. So now we need to talk about clubs, stationary sets, getting to know cardinal arithmetic better, and talking about why some axioms are consistent: specifically Foundation which is not introduced in the basic course, and Choice which is just saying let’s build L that I find to be important for later in set theory also.

The second issue is “follow your heart”. My first course in logic and set theory was with Uri Abraham, who is a wonderful teacher, but it was a very basic course. We covered very little in terms of set theory, and the very very basics of logic (what is a structure, what is logical entailment; without even getting into compactness, or even the axiom of choice). The rest of the courses (logic, axiomatic set theory, descriptive set theory) were taught by Matti Rubin when I took them. Now Matti is a wonderful person to talk to, and he’s very fun as a teacher. But I got out of these courses knowing almost nothing relevant. The axiomatic set theory course spent most of its time proving the equiconsistency of Foundation and its negation, in a proof that is essentially in PRA (without really talking about coding proofs into PRA). At the time I thought it was great, but when the next semester I had taken a reading course to learn forcing (there weren’t enough students to get the course to be frontal), I realized that I don’t understand almost anything about how you actually prove something in set theory, or how you do any research in set theory (we did other things, sure, but not many things, and only for a couple of weeks). For a course that is supposed to give you a taste of set theory, this was terrible. I’ve since covered the rest of the ground, of course, either with Menachem and his many courses, or by myself (by reading and by interacting with people like you online). But I much rather give my students a better taste as to what set theory is like.

And yes, you are absolutely right, that there are topics that you can use to demonstrate this to a better degree. You can show them a myriad of combinatorial constructions, or large cardinals which do not require a complicated statement (so Woodin cardinals are off the table here). You could even show them topological consequences of forcing axioms. But at the end of the day, you need to give them a breadth of ground to work with. The next semester there will be a course about forcing and independence. I need to prepare them to that course, and I need to prepare them to the future if they choose to be set theorists (many won’t be, but some will definitely be); and most importantly, those who will drop out over the next few weeks, and those that will not continue in logic-related topics, should leave this course feeling that set theory is an important and interesting topic, so when they are sitting in some committee a couple of decades from now, a set theorist might have a better chance in getting whatever the committee is voting on because they won’t be biased against the topic.

Okay, maybe the line between those two points is a bit blurred, and they make more of a point and a half. The thing I’m trying to say is that (1) I have a very heterogeneous class, and (2) doing “the things close to you, the way you want to do them” might not be in the best interest of the students for a first class in set theory.