Category Archives: Mathematics

Some thoughts about teaching introductory courses in set theory

Dianna Crown, the physics woman on YouTube, has posted a video where she is interviewed by her editor about why and how she found herself majoring in physics in MIT.

Here is the video:

One of the question is how she found herself studying physics. The answer was that she had very enthusiastic teachers when it came to physics, and that caught on to her.

For the past few years, I’ve been telling people that undergraduate students are very impressionable. They follow the teachers that seem to be most excited about the material they teach. You need a bit of charisma, of course. But if you have that, and you teach with excitement, and with awe at all the awesome things that you teach, students are going to follow you. Sure, only about 10 people finished the axiomatic set theory course I gave last year. But the first lecture had about 30-35. And almost every year I’ve had someone student (or more) ask me about axiom of choice related problem they can write a paper for some seminar.

Because I enjoy teaching set theory. I think it’s awesome. I love it. And I want to pass this enthusiasm forward. And for this, I am willing to match my students in effort. If someone needs extra help, I am always willing to give them the extra help. Crash course in proof writing before an exam? Sure. Meeting outside of office hours to talk about the material in class? My pleasure. As long as I feel that the student is making an effort, I will put the same effort.

Now. I am not writing this to glorify myself as a teacher. I am writing this to make a point. Some years ago, if my memory serves me right, there was some panel about what to do regarding set theoretic education. One of the thing is to revive some of that energy. In a university where nobody is teaching set theory properly, no student is going to want to study set theory (well, there might be some exceptions, but these are rare). In a university where the set theory teachers are terrible teachers, there won’t be many people who would want to pick up the mantle and do research.

Dianna, just as well, mentioned that one of the things she realized is that learning physics is very different from research in physics. The same can be said about math (and probably about anything, because you usually learn something as a student after the research is done. Research takes tenacity, patience, more tenacity. Research is draining the belief out of you, and reaching into the bottom of your emotions for failure and resignation. And if you’re doing it right, research is about bouncing back from these feelings, and finding the right path to actually get things done.

And set theory is no different. Only that compared to other mathematical branches, set theory is the “odd kid out” when it comes to undergraduate material. Even as students. Basic calculus has examples, basic analysis has examples, basic linear algebra has examples, basic combinatorics has many examples. And all of these examples come from real life. They are easy to explain. Set theory, however, doesn’t have examples. It has motivations, it has “prettifications of concepts”; but set theory does not have concrete examples you can explain. We cannot comprehend infinitude, so we cannot understand intuitively why there are more real than rational numbers. This requires relinquishing intuition in favor of definitions, so new intuition can be built.

So getting students interested in basic set theory is even more complicated than it is to get them to stick with math. But it is not impossible. We’re all here, and there are grad students in set theory.

What is missing, I guess, is better, more enthusiastic, and more giving, and more willing to walk the extra mile for their students. If we cannot make that happen, then set theory students are going to dwindle down, and then the demand for set theorists in departments will dwindle down, which will cause set theory education as a whole to dwindle even more… and eventually, we will go extinct.

Let’s not go extinct. Let’s find our inner motivation, and let’s get some students interested in set theory!

Dangerous knowledge in the Information Age

Back in the days of yore, if one wanted to know mathematics, one would have to go to the university and take a course; or hire a tutor; or go to the library and open a book and learn on their own.

And that was fine. All three options are roughly equivalent, in the sense that they present you the material in a very structured way (or they at least intend to). You don’t reach the definition of $\aleph_0$ because you defined what is equipotency and cardinality. You don’t reach the definition of a derivative before you have some semblance of notion of continuity. Knowledge was built in a very structural way. Sometimes you use crutches (e.g. some naive understanding of the natural numbers before you formally introduce them later on as finite ordinals), but for the most part there is a method to the madness.

Fast forward to the information age. Everything is one Wikipedia, every entry tries to be self-contained with respect to at least a short introduction. You can now learn about Hilbert’s Grand Hotel (and his shrewd business acumen), without learning what it means for two sets to have the same cardinality. And that is an essential gap. Yes, the point of the Grand Hotel is to demonstrate that infinite sets can have different properties than finite sets when it comes to cardinality. And yes, depending on the teacher, this can be a segue into the definition of cardinality (although in my opinion not as good as the usual “do I have the same amount of fingers on each hand without counting them?” approach). But nevertheless, in an unstructured learning environment there is a high risk—which is actual reality, as witnessed by the many confused questions on the internet regarding infinity and the Grand Hotel—that the reader is not going to follow through with the definition of cardinality, since this example will already be confusing enough, or distracting enough from being just an example.

Another terrible example is the old Numberphile video about $1+2+3+\ldots=-\frac1{12}$. Yes, this can be found in many books and so on. But in all these books, I am sure, it will be mentioned explicitly that this manipulation is not the standard definition of summation, but rather obtained through other mathematically valid methods that have been subjected to abuse of notation. Stripping the context from all this, and just presenting this summation as a magic trick, is a surefire way to confuse everyone who is not already familiar enough with these topics. And of course that it has, I even had students of mine asking me about that back when the video first hit tsunami sized waves across the web.

What’s the problem, you might ask? Let those people go online and ask experts! Well, it turns out that there is a reason you don’t talk about Ramanujan summation or zeta regularization in the first semester of undergrad. And people come with an honest question, and they expect an easy answer to quickly dispel the dissonance they have between this weird summation and what they know (or think they know). And there are no quick answers which are clear, simple, and not entirely condescending. There is a reason why one has to work through several years of set theory before gaining the actual and intuitive understanding why you need the axiom of choice to prove there is an injection from $\omega_1$ into the real numbers. These things are complicated.

Dangerous knowledge usually refers to knowledge that is considered dangerous for other people to have. Like how at some point terrorist organizations realized that if you just teach everyone to make homemade bombs, it’s going to be a lot harder to actually stop the bomb production and hinder the organization (and even caused people who just self-identified with the cause of the organization to pick up arms and commit terrible acts).

But in the context of education, I think that a dangerous knowledge is knowledge which you obtain without a structured set up. You are not ready for that sort of knowledge, and you do not have the means of placing it in the bigger picture. I had this problem, through all my life, I have gone to read about things, and I skipped and jumped ahead, and I tried to learn further and better. And every time I jumped and made an unstructured “discovery” I eventually had to go back and correct the err of my ways.

The question, from an educational point of view, is how can you fight this? How can you make sure that dangerous knowledge is kept to a minimum?

One way is to instil into students from a very young age the sense of curiosity and wonderment. I remember reading somewhere about someone who as a kid opened up a book, and read about some problem, then started to work backwards to obtain all the knowledge necessary for understanding it better. It could have been Feynman or Wiles, I am not sure, and it doesn’t matter. The point is that when coming across dangerous knowledge, the protagonist of that story “defused” the danger by starting to go backwards and learning the necessary framework.

In today’s modern era, where everything needs to be a click-bait-bite-size-immediately-satisfying thing, the above is difficult. It is hard to make sure that people actually sit down to read. People want the information they feel is missing, and not a long list of information they are actually missing. And not to mention that re-educating the whole planet seems like a fairly Herculean task.

But I do think that at least in academia this is possible. It should be possible to try and educate students about this. I think it is important, especially in natural sciences, where there are good chances that the students will go on to research later (either in academia, or elsewhere) to remember this. Because having dangerous knowledge can affect the way you perceive your actual knowledge. It can re-frame your knowledge incorrectly, or shift the importance of something you are currently learning (or about to) from one side of the picture to another, and not necessarily in a good way.

Another option is to educate people about the existence and dangers of dangerous knowledge. Once you are aware that learning something in an unstructured way can be problematic, you can put this knowledge in check automatically, reminding to yourself that you need to know more in order to fully appreciate some anecdotal piece of information that you read online, and heard about. This can also motivate you to go and actually study more about something, which is always a good outcome.

The transitive multiverse

There are many discussions on the multiverse of set theory generated by a model. The generic multiverse is given by taking all the generic extensions and grounds of some countable transitive model.

Hamkins’ multiverse is essentially taking a very ill-founded model and closing it to forcing extensions, thus obtaining a multiverse which is more of a philosophical justification, for example every model is a countable model in another one, and every model is ill-founded by the view of another model. The problem with this multiverse is that if we remove the requirement for genericity, then everything else can be satisfied by the same model. Namely, $\{(M,E)\}$ would be an entire multiverse. That’s quite silly. Moreover, we sort of give up on a concrete notion of natural numbers that way, and this seems a bit… off putting.

There is also Väänänen’s multiverse, which is more abstractly defined, and I cannot for the life of me recall its definition and its details.

Some time ago Ur Ya’ar gave a seminar talk about Hamkins’ multiverse in the logic seminar in Jerusalem. It was interesting, and afterwards Yair Hayut and myself talked with Ur about these multiverses. One idea that came up, and I don’t think that I ever ran into it, is sort of a combination between the generic multiverse and Hamkins’ multiverse. Consider the following axiom “Every real is an element of a transitive model”. Now look at $\cal M$, the set of all the countable transitive models, we get the following axioms are satisfied by $\cal M$:

  1. If $M\in\cal M$, then every generic extension and every ground of $M$ is also in $\cal M$.
  2. If $M\in\cal M$, then every inner model of $M$ is also in $\cal M$.
  3. If $M\in\cal M$, then there is some $N\in\cal M$ such that $M\in N$ and $N\models M\text{ is countable}$.
  4. For all $M,N\in\cal M$, $L^M$ and $L^N$ are comparable.

So what do we have here? We have a multiverse of sets, it is closed under generic extensions and grounds, and it is even closed under definable inner models. It also has the property that we can always find bigger models that think a given model is countable.

Now, I have no idea what useful things can come out of this multiverse. And I would imagine that one should first refine this notion a bit more before it becomes actually useful for something. But nonetheless, it seems like an interesting interpretation of the whole notion of multiverse.

Strong coloring

I am sitting in the 6th European Set Theory Conference in Budapest, and watching all these wonderful talks, and many of them use colors for emphasis of some things. But yesterday one of the talks was using “too many colors”, enough to make me make a comment at the end of the talk after all the questions were answered. Since I received some positive feedback from other people here, I decided to write about it on my blog, if only to raise some awareness of the topic.

There is a nontrivial percentage of the population which have some sort of color vision deficiency. Myself included. Statistically, I believe, if you have 20 male participants, then one of them is likely to have some sort of color vision issues. Add this to the fairly imperfect color fidelity of most projectors, and you get something that can be problematic.

Now, I’m not saying “don’t use any colors”. Not at all. Just keep in mind that some people might have problems with your choice of colors. Using too many colors can be distracting, and one of the slides in the said talk had black text almost on par with the rest of the colored text. This is far from ideal. But since color deficiency can vary from one to another, let me only give an account of my own personal experience. I cannot do anything more, after all.

I have a mild red-green issue. But this means also that yellow and bright green, or light orange, all mix together sometimes; and darker greens can be red or brown (which themselves are often mixed); and blues can mix with purple, and sometimes with pink as well. One other effect of color deficiency is that you are more sensitive to brightness and darkness (the eye compensates the damaged cones by having better rods, so your night vision gets somewhat better, for example).

So when you have a slide with some pink/purple and green/yellow/orange and some blue and some red and some black, my brain will not read the text. My brain will try to make sense of the colors. Not to mention the terrible eye strain coming from the brighter colors (here the quality of the viewing media is important, I’m sure that I’d be fine watching the same slides on a proper computer monitor). There were slides that I had to turn my eyes away from the talk. Yes, it was pretty bad.

What can you do about it? Don’t use colors when you don’t have to. Use boldface or italics for emphasis when possible, or different font family entirely. If you want to use colors, using them sparingly, and try to avoid relatively close colors together and certainly try to avoid brighter colors like light green or yellow. If you know a color blinded person, you can maybe ask them to give some critique on your choice of colors.

Some people commented to me after my remark that they prefer the colors, and they are helpful. I understand that. Again, the point is not to get people to use colors. Just… to use them intelligently. Colors are like spices. I’m not trying to get you to cook without spices, but you’re not going to serve a dish entirely made of cinnamon and cumin.

In the name of all color vision deficient people, thanks in advance for your consideration!

What a long strange trip it’s been…

As some of you may have noticed, I don’t use this blog to write about my papers in the “traditional way” math bloggers summarize and explain their recent work. I think my papers are prosaic enough to do that on their own. I do use this blog as an outlet when I have to complain about the arduous toil of being a mathematician (which has an immensely bright light side, of course, so in the big picture I’m quite happy with it).

This morning I woke up to see that my paper about the Bristol model was announced on arXiv. But unbeknownst to the common arXiv follower, this also marks the end of my thesis. The Hebrew University is kind enough to allow you to just stitch a bunch of your papers (along with an added introduction) and call it a thesis. And by “stitch” I mean literally. If they were published, you’re even allowed to use the published .pdf (on the condition that no copyright infringement occurs).

My dissertation is composed of three papers, all of which are on arXiv (links in the “Papers” page of this site):

1. Iterating symmetric extensions;
2. Fodor’s lemma can fail everywhere; and
3. The Bristol model: an abyss called a Cohen real.

Of course, the ideal situation is that all three papers have been accepted for publication, but all three of them are still under review. So it puts me at this odd situation where I will have essentially four sets of referees (one for each paper, and then two additional referees for my thesis), and so the output can end up oddly different between the resulting dissertation and the published papers. But that’s fine.

In any case. Those of you who are interested in reading my thesis can find it in those three papers. I am probably going to post the final thesis online when it will be approved, but the only thing you’re currently missing out is an introduction with some minor historical background and a summary of the three papers. So if you read all three, you don’t really need that introduction anyway.

Good. So what next? I have a few things lined up. More news will follow as reality unfolds itself like a reverse origami.

Stationary preserving permutations are the identity on a club

This is not something particularly interesting, I think. But it’s a nice exercise in Fodor’s lemma.

Theorem. Suppose that $\kappa$ is regular and uncountable, and $\pi\colon\kappa\to\kappa$ is a bijection mapping stationary sets to stationary sets. Then there is a club $C\subseteq\kappa$ such that $\pi\restriction C=\operatorname{id}$.

Proof. Note that the set $\{\alpha\mid\pi(\alpha)<\alpha\}$ is non-stationary, since otherwise by Fodor's lemma there will be a stationary subset on which $\pi$ is constant and not a bijection. This means that $\{\alpha\mid\alpha\leq\pi(\alpha)\}$ contains a club. The same arguments shows that $\pi^{-1}$ is non-decreasing on a club. But then the intersection of the two clubs is a club on which $\pi$ is the identity. $\square$

This is just something I was thinking about intermittently for the past few years, but now I finally spent enough energy to figure it out. And it’s cute. (Soon I will post more substantial posts, on far more exciting topics! Don’t worry!)

Mathematical philosophy on YouTube!

If you follow my blog, you probably know that I am a big fan of Michael Stevens from the VSauce channel, who in the recent year or so released several very good videos about mathematics, and about infinity in particular. Not being a trained mathematician, Michael is doing an incredible task.

Non-mathematicians often tend to be Platonists “by default”, so they will assume that every question has an answer and sometimes it’s just that we don’t know that answer. But it’s out there. It’s a fine approach, but it can somewhat fly in the face of independence if you are not trained to think about the difference between true and provable.

This morning, as I was watching the new video of Physics Girl, there was an announcement about a new math channel. So of course I went to look into that channel. It’s fairly new, and there are only a handful of videos, but they already tackled some nice topics. The videos are written and hosted by a Cornell grad student, Kelsey Houston-Edwards. I watched the one about a hierarchy of infinities, and while I was a bit skeptic after the first minute, I was quite happy at the end, when the discussion went from just the fact that the reals are uncountable (although without a proof, and that’s fine, I guess, there are plenty of those on the internet), to a discussion about the continuum hypothesis.

In another video, Kelsey tackles mathematical Platonism and its somewhat-opposite, Formalism. And it’s done well. Kelsey doesn’t lean into one side or another, because at the end of the day, mathematicians—as opposed to mathematical philosophers—do mathematics, and that is their main concern. The philosophy is mostly a spice to add some taste and meaning to your work.

In any case, I enjoyed watching the few videos that I have, and I hope that you will as well. I’m sure that this is not the last that you’ll see me talk about Kelsey and her channel.

Some thoughts about teaching advanced set theory

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.

Zornian Functional Analysis or: How I Learned to Stop Worrying and Love the Axiom of Choice

Back in the fall semester of 2015-2016 I had taken a course in functional analysis. One of the reasons I wanted to take that course (other than needing the credits to finish my Ph.D.) is that I was always curious about the functional analytic results related to the axiom of choice, and my functional analysis wasn’t strong enough to sift through these papers.

I was very happy when the professor, Matania Ben-Artzi, allowed me to write a final paper about the usage of the axiom of choice in the course, instead of taking an exam.

I have decided to finally post this paper online. It covers some possible disasters in functional analysis without the axiom of choice, or with “seemingly nice” assumptions (such as automatic continuity). You can find it in the Papers section.

My goal was to make something readable for analysts, rather than to provide a retread of older set theoretic proofs (and some model theoretic proofs). So some things are left with only a reference, and other set theoretic statements are formulated in a rather unusual way. If you are interested, I’d be happy to hear any remarks on this paper, or suggestions for improvements in the comments below or over email. If you know analysts that might be interested to read this, please let them know of the existence of the paper.

For the set theorists the paper can be seen as a nice historical overview of these results, and perhaps it can be of use in other ways.

In praise of some history

Teaching pure mathematics is not a trivial thing. You have to overcome the several barriers that were constructed by the K12 education that mathematics is a bunch of “fit this problem into that mold”.

I recently had a chat with James Cummings about teaching. He said something that I knew long before, that being a good teacher requires a bit of theatricality. My best teacher from undergrad, Uri Onn, had told me when I started teaching, that being a good teacher is the same as being a good storyteller: you need to be able and mesmerize your audience and keep them on the edge of their seats, wanting more.

My answer to James was something that I had in mind for a while, but never put into words until then. You should know a bit of history of the topic you’re teaching.

If you look at [pure] mathematical education—at least undergrad—it is quite flat. You just have a list of theorems, each extending the previous, building this wonderful structure. But it’s a flat building, it’s a drawing. The theorems come one after the other, after the other… Historically, however, there were many decades (if not centuries) between one theorem to the next. Rolle’s theorem came about the late 17th century, but Lagrange’s theorem came only in the mid-19th century. So in one lecture, we covered some 150 years of mathematical progress. That is amazing, if you can point this out properly to your students. Not to mention the oppositions that people had to infinitesimal calculus in Rolle’s days, which makes it interesting, and contributes to the definitions given by Cauchy as solid foundations for analysis.

Similarly, the history of the Cantor-Bernstein theorem is incredible. As in the history behind’s König’s theorem (about cardinal arithmetic). Those things are amazing, what sort of motivations and mistakes people had made back then, when these fields were fresh.

The more I thought about it, the more I realized that there are two important reasons that one should always spice up their teaching with some historical facts.

  1. The first is that mathematical education is flat, as I remarked above. We learn the theorem, one after another. In one lecture, you can cover decades or even centuries of mathematical progress. And it feels dry, it feels like “why are you teaching me all this?” sort of thing. I still remember that as a student, I do.

    But with a bit of history, suddenly everything becomes three-dimensional, it becomes something that had actual progress. It shines a light on “there is a notion of mathematical progress”. Something that engineering students, for example, often baffled by.

  2. The second reason is that often theorems and motivations were coming from attempts to disprove something. König, if we mentioned him already, proved his lemma as an attempt to prove that the real numbers cannot be well-ordered. Baire, Borel and Lebesgue rejected the axiom of choice because they felt it is preposterous that there are non-measurable sets.

    When you explain this to students, you show them that their natural confusion about a topic, especially abstract and confusing topic, is natural. You show them that a lot of smart people made the same mistake before. And while today we know better, their instinctive recoil makes sense. This reinforces the idea that they didn’t misunderstand something, that they are not stupid, and that mathematics is often surprising (at least when dealing with the infinite).

So we have these two reasons. And I think these are excellent reasons for adding some historical references when talking about mathematics. Of course, you shouldn’t put more than a pinch of cumin in your stew, because cumin is not the main part of your dish, it’s just what makes a good meal into a great meal (well, at least good cumin). You shouldn’t talk only about history in a mathematical course. This should be the slight addition that gives taste, flavor and volume to your material.

Historical anecdotes are what turns a flat material into a fleshed form of progress, from one theorem to the other. Use them sparsely, use them well. But use them.