The Philosophy of Cardinality: Pathologies or not?

What are numbers? For the layman numbers are those things we use for counting and measuring. The complex numbers are on the edge of being numbers, but that’s only because they are taught in high-schools and many people still consider them imaginary (despite them having some reasonably applicative uses).

But a mathematician knows that a number is basically a notion which represents a quantity. We have so many numbers that I don’t even know where to begin if I wanted to list them. Luckily most of the readers (I suppose) are mathematicians and so I don’t have to.

In a recent evening of boredom I skimmed across some old math.SE and MO comments, and I saw several times where I argued that cardinality is not “strictly defined as ordinals” and therefore “every cardinal is an ordinal” is a bad notion without the axiom of choice.

And indeed if we insist that all cardinals are ordinals then without the axiom of choice we immediately forfeit two important notions: Not every set needs to have a cardinality; cardinality exponentiation is not necessarily well-defined.

But first what are cardinal numbers and what is a cardinality? Well, as I wrote above, numbers signify some quantity that we can somehow measure. This form of measure doesn’t have to make sense, especially because if you really think about it — a lot of things in mathematics don’t make sense. At least not before you have hammered down your intuition. Cardinality is based on the notion that two sets have the same size if there is a bijection between them. So cardinal numbers should somehow represent this form of size.

Let me take a sidebar from that argument for a moment. Let us think about another excellent way of measuring size of sets, in particular sets of real numbers. Lebesgue measure. The axiom of choice tells us that not all sets are measurable. Not all sets can be fitted with a size. In a good sense we can even say that most sets cannot be fitted with size. But we don’t really care, the universe of mathematics is mainly interested in those measurable sets, so we slap on a little restriction and require our sets to be measurable.

Returning now to the notion of cardinal numbers, we want a reasonable system of numbers which we can really see as numbers, and we want them to fit the idea of measuring size of sets. So we have the initial ordinals which make an excellent choice of system for well-ordered sets. But why should we bother with sets which cannot be well-ordered? I mean, obviously we can focus on the good parts and just require that every cardinal is an $\aleph$ and get it over with. If measure theory can have its pathologies, so can “cardinal theory”. In fact, if we said that most sets are non-measurable, why should this sort of measurement be different?

Furthermore, if we take a Vitali set, and apply any measure-preserving bijection and the result is still non-measurable. If pathologies are preserved under well-behaved maps, and in the case of cardinality all bijections are well-behaved maps, should we expect any difference in the case of cardinal numbers and cardinality?

The answer is yes. We should expect difference, for two reasons. The first is that in the notion of Lebesgue measurability not all bijections are well-behaved. When we restrict the notion of size we would also like to restrict the relevant bijections (computable functions, measurable functions, etc.), but in the case of cardinality – being very raw and structureless – we really have every bijection in our arsenal.

My second reason is an excellent one. Many of the sets mathematicians actually care about can have very peculiar cardinalities. For example $\mathbb R$ might have a very strange cardinality in models where all sets have Baire property, or Lebesgue measurability, or even worse in models of $\ZF+\AD$. In fact in those models the cardinality of $[\mathbb R]^\omega$ is strictly larger than that of $\mathbb R$, although the real numbers could still be mapped onto that set. It is a frightening thought, that surjections from the real numbers cannot be reversed like that.

This also shows why cardinal exponentiation is important to have ready, otherwise how can we prove that $(2^{\aleph_0})^{\aleph_0}=2^{\aleph_0}$? How can we argue that something has size continuum? Or even define what size continuum is?

I hope this post serves as food for thought to anyone teaching a set theory course any time soon. Some year and a half ago, some visitor (a model theorist in a post-doc position somewhere in the states) in BGU told me that he was assigned to teach the course in set theory, and he defined a cardinal number as an initial ordinal. Since then I recall having this discussion at least twice more with various people. Please, if you write a book, a paper, or teach a course, beware not to confine your students only to well-ordered cardinalities.

On Leinster’s “Rethinking set theory”

There has been a lot of recent discussions regarding Tom Leinster’s paper “Rethinking set theory” (arXiv). Being an opinionated person, I only found it natural that I had an opinion on the paper. Now that I have a blog, I have a place to write this opinion as well.

The paper challenges the hegemony of $\ZFC$ as the choice set theory. It offers an alternative in the form of $\newcommand{\ETCS}{\axiom{ETCS}}\newcommand{\ETCSR}{\axiom{ETCS+R}}\ETCS$, a categories based set theory. The problem with $\ETCS$ is that it is slightly weaker than $\ZFC$. But we also know how much weaker: it lacks the expressibility of the full replacement schema. In this case we can just add a replacement schema-like list of axioms to have $\ETCSR$.

The list of axioms of $\ETCS$ is not surprising at all, much like the list of axioms of $\ZFC$ should not be very surprising. Foundational set theories should, in my opinion, be very “obvious” in formalizing what is a set. We all have this vague idea about sets, and this should be enough. From this point of view $\ETCS$ and $\ZFC$ are very similar and that’s good.

Leinster’s opinion is that $\ETCS$ is much more natural to work with because it allows to put an emphasis on the universal property, being a structural property rather than material, which is a very important feature in many places in modern mathematics.

The argument presented by Leinster is that this should clear the so-called circularity from $\ZFC$ (using sets to describe sets — the circularity, I think, exists only in the natural language level), and it will help to clear out the obvious issues with having every object as a set. The question $x\in\pi$ seems ridiculous to mathematicians in most cases (except if one is writing an exercise about Dedekind cuts, perhaps). With structural set theory we don’t have this issue, it seems. Surely not because one can’t express such equation, but because the language makes it much more difficult to do so. He continues to argue, most mathematician has a very naive concept of sets, and they use it without much thinking. Sets of numbers, sets of functions, and so on. $\ETCS$ have this, and it should allow everyone to keep doing what they did so far, only without seeing the questions “If $\sqrt2$ is a set, because everything is a set, what are its elements?” or “$\langle 1,2\rangle = 3\setminus1$?”

The main problem right now, Leinster writes at the end of the paper, is that there aren’t enough books written on the topic and not enough capable teachers. And of course if someone would like to research set theory then learning $\ZFC$ would be a better choice anyway because much of the current research is done through the eyes of $\ZFC$.

I have a problem with this approach, and I am certain that I am not the only one. The first, and the most personal problem, I don’t like diagrams. When people draw what they mean I lose my ability to understand. People also tell me that diagrams are simpler to follow, but when you draw a diagram with ten points and fifteen arrows (and those are the simpler ones) it would be a lie to yourself to say this is easy to understand. It is also my impression that this gives some sort of false sense of security into the [much loathed] sentence “It is easy to verify that this diagram is commutative.“. But that is my problem with categories, and I accept that many people would find it easier to approach mathematics like that.

But it brings me to the first of my two main points. People who are used to diagrams could easily adjust to $\ETCS$ (if they haven’t done so thus far), and that’s fine. But often those people would forget that sometimes diagrams are not suitable. I recall an algebraic geometry grad student where I’ve studied for my M.Sc. who proudly showed me a diagram which his advisor proved to be commutative. What’s the big deal? Well apparently it generalizes a lot of theorems from analysis. But to many analysts this diagram is going to be helpful as explaining $2_\mathbb Q\in\pi$. To people with a hammer everything looks like a nail.

The second point, in which my strong take on material vs. structural set theory is going to kick in (at long last!), is the fact that much of the mathematics being done has nothing to do with set theory. Leinster notes that as well. For most mathematicians there are sets of functions and so on. For some of them the functions are not even sets, they are as atomic as the real numbers are. For these people $\langle\pi,0\rangle\in\sin$ makes little sense, even though nowadays we all agree that functions can be clearly presented as ordered pairs. Many of these people have so little use of more than naive set theory, that the materialistic approach is better. You have a concrete $f\in L^2$, or a particular sequence of sequences $\{x_n\}\subseteq\ell^\infty$.

True, you can write all these things with $\ETCS$ and you could easily keep a short dictionary for translating arguments back and forth from one representation to another. But for people who are interested in concrete sets. In specific kind of spaces, or even a particular vector space, there is an amount of abstraction which is going to do more harm than good. And there are people like that. There are mathematical physicists, and there are people interested in the particular space $L^2$ over the $p$-adic numbers. There are people interested in $\mathbb R^3$. Mathematics should give them the tools to deal with those problems.

It is often the case that mathematicians forget that in other fields (physics, chemistry, etc.) people see them as developing tools for others. Mathematicians are blacksmiths. We make hammers, and we make knives. These people often forget that there are blacksmiths who make hammers and anvils for other blacksmiths, and there are people who make the hammer and anvil for those blacksmiths, and so on. Eventually you reach down to the core argument and find someone trying to understand how forcing with amorphous sets into $L$ works. And yes, we’re still in the blacksmith metaphor.

So it is my opinion that challenging the hegemony of $\ZFC$ is fine, and suggesting a more structural alternative which is “in sync” with the current fashionable field of algebraic geometry is just as fine. But I have serious doubts that many analysts would start to fiddle with categories and diagrams, and with “Why is this even better than using naive and material set theory? Oh, just because it is a “junk theorem” that $1\in\pi$? Who cares, we’re trying to solve actual problems here. You boys, go play outside”.

(Note that I hardly mentioned any set theory based arguments like Francois Dorais did, on that I have a whole other post planned.)

Some relevant things to read through:

  1. “Rethinking set theory” arXiv link to the paper
  2. the n-Category cafe “Rethinking set theory” post.
  3. Francois Dorais’ reply to the n-Category post, “Back to Cantor?”.
  4. Mathoverflow discussion about the paper.
  5. Two Mathoverflow discussion relevant to this point, Set theories without “junk theorems” and Can ZFC prove “false theorems”, and still be consistent? (was “Junk Theorems” follow up).

As I said about categories before, “while a screwdriver is a very useful tool to carry, you don’t really need it if you are making a sandwich.”