The cardinal trichotomy: finite, countable, and uncoutnable.

There is a special trichotomy for cardinality of sets. Sets are either finite, or countably infinite, or uncountable. It's an interesting distinction, and it has a very deep root -- at least in my perspective -- in the role of first-order logic.

Finite objects can be characterized in full using first-order logic. The fact that you can write down how many elements a set have, is a huge thing. For example, every finite structure of a first-order logic language has a categorical axiomatization. If the language is finite, then the axiomatization is finite as well.

Countable objects are the limit of finite objects. They are the infinity that we can approximate using first-order logic. Now, it is true you can approximate uncountable sets using finite sets (e.g. \([\omega_1]^{<\omega}\) approximates \(\omega_1\) using finite sets), but the fact that there is a linear, nay well-ordered, approximation of the countable set is much stronger. That tells us a lot. It means that we can really get a lot of properties simply by taking a limit of finite objects to the countable object.

By this virtue alone, I think, mathematics deals often with the countable, and the countably generated. This includes separable spaces. For example.

Then we have the uncountable. This doesn't have to be any particular cardinality. Just uncountable. The continuum is uncountable. These objects, the uncountable objects, are beyond our control in first-order logic and the axioms of \(\ZFC\). We find independence results whenever we are around uncountably generated objects. This, in my view, is again the direct result of being uncountable. Because we cannot approximate these objects in a very nice way using first-order logic, we end up with them being out of control.

Some examples:

1. The independence of the continuum hypothesis, and much more than that: the continuum can be any cardinal whose cofinality is uncountable. This is really the stronger expression of this idea in set theory.
2. The Whitehead problem. Every Whitehead group which is finitely generated is easily provable free. The countably generated case is slightly more complex. The uncountable case is unprovable, as Shelah shows. At least from the axioms of \(\ZFC\).
3. Morley's theorem. Either a theory is categorical for every infinite cardinal; or just for \(\aleph_0\); or if it is categorical for some uncountable cardinal then it is categorical for all uncountable cardinals.
4. Diamond on the successor of \(\aleph_0\) is strictly stronger than \(\sf CH\). However for uncountable cardinals it is true that \(\lozenge_{\kappa^+}\) if and only if \(2^\kappa=\kappa^+\).
5. If \(A\) is a finite set, then we cannot change its cardinality, nor if it is countable. If it is uncountable then we can always collapse it some way, or all the way. This is less of a consequence of first-order "describability", and more the fact that we can only add witnesses, not remove them. But still, it's an interesting peculiarity nonetheless.

For the readers

I want to ask you, the readers, to come up with others examples of this trichotomy. There are some examples which I have left out intentionally (e.g. \(C^*\)-algebra related ones). But it seems to me that these sort of things hide everywhere in mathematics.

One foreseeable consequence, for me, is that as we grow tired of countably generated objects we begin to explore the uncountably generated objects. The most striking example I know is the current applications of descriptive set theory to \(C^*\)-algebras, and operator theory -- true, it does a lot more than independence results, but it really brings those results out. In those cases, seeing how first-order logic is insufficient to characterize everything (even as an approximation), we begin to find more and more dependence on the axioms of set theory. This will either cause people to assume some particular theory (e.g. \(V=L\)), or consider other forms of foundations (at least for the time being). Unfortunately, I doubt that a rise in interest in set theory will occur, although I'd love to be proven wrong.

So please, share the link to this post, and post your comments, ideas and thought about this sort of trichotomy.