Asaf Karagila
I don't have much choice...

In a recent Math.SE question about the foundations of category theory without set theory, someone made a claim that $$\ZF$$ makes it hard to learn mathematics, because in $$\ZF$$ the questions "is $$\RR\subseteq\pi$$?" and "is $$\RR\in\pi$$?" can be phrased. They continued to argue that there are questions like whether or not hom-sets are disjoint or not, which are hard to explain to people who are "drunk on ZF's kool-aid".

So I raised a question in the comment, and got replies from two other people who kept repeating the age old silly arguments of what are the elements of $$\RR\times\RR$$ or what are these or that elements. And supposedly the correct pedagogical answer is "It does not matter what are the elements of $$\RR\times\RR$$." With that I strongly agree, and when I taught my students about ordered pairs on the very first class of the semester, I made it very clear that there are other ways to define ordered pairs and that we only do that because we want to show that there is at least one way in which ordered pairs can be realized as sets; but ultimately we couldn't care less about what way they encode ordered pairs into sets, as long it is a "legal" way.

And it seems to me that a lot of the flak set theory gets comes from the inability, or rather the unwillingness of people outside of set theory to study just the little bit more than they absolutely have to. Perhaps set theory wasn't properly introduced to them in their undergrad, maybe too early and maybe too late (after they had heard bad things about it from their algebra teachers complaining about proper classes and the axiom of choice).

I don't know why people don't do that, and I am very happy to have had the opportunity to work with Azriel Levy in the past three years and see how so many of our students love the topic. And this is not because Israel is a friendly place towards set theory, this becomes increasingly less true. It's because they have a teacher and a teaching assistant (who is teaching another complementary half-course in the exercises) which come from set theory and do not shy away from questions like "Can we define ordered pairs in a different way? What happens then?" and it's because they have teachers which are truly enthusiastic about set theory.

Students are impressionable. If you have a good teacher and they tell you something, this will stick with you. And if that good teacher tells you that set theory is the root of all evil, then you will continue to think so until, if you're lucky, you'll learn otherwise. Or, if you have a good teacher that tells you in your sophomore year that the axiom of choice is a dormant research topic, but it has a lot of beautiful mathematics and a lot of interesting open questions left... well, you end up like me.

So what's the whole issue here? Is $$\pi\subseteq\RR$$? That depends, you haven't given me a definition for the sets $$\pi$$ and $$\RR$$. But once you will give me a definition, my answer is that it doesn't matter, because we are interested in structures with a certain property, not with specific sets. And that given any other way to interpret the real numbers as sets can yield different answers to the question.

And I truly don't understand why this bothers people. There are three paths from here to the nearest groceries store, and all take about the same time and effort to cross. It means that if you ask me to go to the groceries store you can ask me if I passed this or that place, and the answer will depend on which path I took. And thank goodness I am not afraid of questions like "is $$\pi\subseteq\RR$$?", because then the choice of path to the store would have terrified me, and I would be forced to move to the building right next to it, which I am told is really shitty.

In short, if there's one thing to take from this post is that the people who are terrified of the ability to make sense of the questions like "is $$\pi\subseteq\RR$$?" are the people who are too terrified to understand that even set theorists don't care about these questions. And if you worry about them, and find that to be a good argument against $$\ZF$$, then don't. There are reasonable arguments against $$\ZF$$, but this is not one of them.

### There are 5 comments on this post.

By
(Nov 10 2015, 00:11)

I placed a comment on that post about the Kool-aid comment, which I find to be unnecessarily pejorative.

By Peter
(Nov 10 2015, 10:23)

Thanks for a lovely read. Now I want to know what's so bad about that building next to the grocery store... Gotta come visit you one day.

By Tim Campion
(Nov 19 2015, 05:55)

There do seem to be at least two camps here of people who don't understand each other well enough. We need to talk about these things to get past that! I really like Carl Mummert's answer to the same Math.SE question comparing the two kinds of foundations to the difference between typed and untyped programming languages. I have two points I'd like to make:

1.

I agree that it would be nice if people were interested in learning more set theory. Personally, I've never taken an introductory set theory course, but I'm always interested in learning more about it. Maybe it's a chicken-and-egg thing -- people might be more interested in learning set theory if they didn't think it was all about arbitrary encodings like when it comes to ordered pairs!

Consider a student taking an introductory course on group theory, who needs to know what a cartesian product is. They shouldn't have to consider its encodability into material set theory -- the concept is basic enough to be understood on its own. Ironically, if the student's only exposure to set theory is an arbitrary encoding, they may be turned off or frustrated by the subject.

2.

I think it's unfortunate if anyone gives the impression that set theorists are concerned with questions like “is π⊆ℝ?”. But I think we should all find it interesting that in alternative foundations, such questions cannot even be raised, even if those aren't our preferred foundations.

To provide an alternative analogy to the "ways to the grocery store" one, I think the reasons to prefer a more type-theoretical foundation are similar to the reasons to prefer doing geometry in a coordinate-free way, or linear algebra without choosing a basis. Surely, all things being equal, it's preferable to do things in a way that introduces a minimum of extraneous encodings that are conceptually irrelevant? Or to go back to the metaphor, if a zipline leading straight to the grocery store is available, why not use that instead?

By Asaf Karagila
(Nov 19 2015, 06:29 In reply to Tim Campion)

Nobody says that you should always keep in mind the encoding method. That's generally unhelpful and I don't think anyone is disputing that (in the general case, I am sure that there are some cases where carefully choosing your encoding improves how things might work).

If you look at set theoretic research it has to do with the structure of infinite sets (I hate it when people give it terrible titles like "the study of well-pointed accessible well-founded trees" or some other list of buzzwords, as this one is probably incorrect anyway). And that has little to do with foundationalism. We know how to interpret logic and we know how to define the notion of "model", which is just about enough for us to know how to interpret more or less anything in mathematics assuming ZF and additional axioms if need be.

Then set theory studies the structure of infinite sets, and it turns out that this structure has consequences "downstairs" in "working mathematics". For example all sort of determinacy results, measure theoretic results (both of which involve large cardinals), Whitehead problem, or automorphisms of operator algebras. And many others.

But as long as people outside of set theory keep insisting that set theorists are mainly quibbling about what is the best way to interpret one thing or the other as sets, it strikes me as a particularly stupid prejudice against set theory and the people who study it. Which is why I am bothered by these things.

And sure, it might be a reasonable approach to say "We do mathematics in a typed setting, therefore our foundation should be typed as well". But it's like saying that because you write code in an object-oriented language, it is better off being executed on a processor which has native OOP support (namely, it has dedicated circuits for each type and so on and so forth).

So this is exactly the doublethink exercised here. On the one hand, some people claim it's an atrocity that you should to interpret everything back into sets as it allows you to raise all sort of "wrong questions"; and on the other hand, no one disagrees that it's a real miracle of technology how we can use just a few logical gates to turn electronic signals into computers, smartphones and LOLcats. Despite the fact that these electronic signals are "arbitrary" and couldn't care less whether they carry something representing an integer, a long integer, a signed long integer, or a bignum type given by an external library.

By Tim Campion
(Nov 19 2015, 10:24 In reply to Asaf Karagila)

Again, I agree that set theory is a beautiful subject that gets an undeservedly bad rap from people who don't necessarily know a lot about it. I guess I'd argue that this is largely because set theory is being asked to do two things at once: provide a theory of sets and provide a foundation for all of mathematics. And I agree it's adequate for the latter, but, well, you can't please all the people all the time... I would surmise that most negative attitudes voiced about set theory have their source in some kind of venting against the OS one is working with and the hoops it forces one to jump through, not really against set theorists per se. I will defer to your judgment that this frustration can get transferred to set theorists themselves. I'm sorry about that.

I think your point that we use non-typed low-level languages in our computer technology is a pretty good argument that a more type-theoretic sort of foundations is not a priori better than, say, ZFC. But I think it's a bit of a stretch to call this "doublethink". After all, mathematics is not computer science. There are no hardware constraints compelling us to build our foundations in a minimalist sort of language, or in an "untyped" language, no machines which ultimately have to "run" our language -- we have the freedom to choose the language we'd prefer to read. (Although ironically, when it comes to computer formalization of mathematics, type-theoretic foundations are apparently the way to go!)

Would I be correct in saying that we've found a bit of common ground: both sorts of foundational approaches at least might be reasonable, and have something to say for them?

By the way, one thing that fascinates me about set theory is how naturally it expresses nonconstructive existence principles (e.g. the axiom of choice or the ultrafilter lemma, but also large cardinals) which seem to part of the bread and butter of mathematics but which also seem to sort of be at odds with the philosophy one sees behind, say, homotopy type theory. I'd like to understand this better.

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