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.