Everything is just so small
Often in mathematics, we have trouble talking about the sizes of things and understanding whether a thing is small enough to handle or too big to examine fully. In the most basic and natural form, a collection of objects is pretty small if it's finite because there's a computable way to list all of the objects of the collection, exhaust them, and witness that there are objects you can no longer fit into the collection. It is remarkable that, with reasonable though controversial assumptions, 'everything' is really tiny, and you can always count past all of its elements, and I mean count.
In the following, I present my informal description of the process of transfinite recursion and the astonishing implications of the axiom of choice on the notion of size.
Ordinals
The story begins with ordinals, really really gigantic 'numbers'. Ordinals start at 0, count all of the natural numbers, and then count beyond all of them and beyond that, and beyond that still, and then add 1, and again, and again, and then go beyond all of that and still counting. The ordinals are so big that they cannot even tell you how big they can get.
Let's see this more formally. Ordinals are, in fact, sets and are constructed from the ground up using two simple operations: the successor which from a set \(X\) produces the set \(X \cup \{X\}\) and the limit which from sets \(\{X_i : i \in I\}\) produces the set \(\bigcup_{i \in I} X_i\) and starting with a base ordinal 0 corresponding to the empty set.
The ordinals are defined as sets which are transitive and well-ordered. What
that means is that any subset of an ordinal \(\alpha\) has a smallest
element, and any element of \(\alpha\) (which is a set since everything is a
set!) is also itself a subset of \(\alpha\). It turns out, the collection of
ordinals is not a set! It's too big to be a set! In fact, Ord the collection
or ordinals is transitive and well-ordered, so if it was a set, it would be an
element of itself, but we can't allow that, or at least, doing that creates way
too much confusion and risks paradoxes that pretty much no one wants to deal
with (except a few people looking into non-well-founded set theories).
In the simple word, there would be no infinite set and all the ordinals would be finite and we would be stuck at the union step of all the finite ordinals as we would not be able to form a set containing all finite ordinals as there is infinitely many of them. But in this world, things are boring, and I wouldn't be writing an post on how everything is small. The surprise is that once we admit the existence of the set of natural numbers, we can get sets waaaaay bigger than that using ordinals. Here's one set that's bigger than \(\omega \), the set of natural numbers: \(\omega + 1\). It is a set containing all finite ordinals as elements and the infinite ordinal \(\omega \). You can tell that these two sets are distinct because \(\omega \) is an element of \(\omega + 1\) but not of \(\omega \). In general, any ordinal \(\alpha \) has a strictly larger ordinal which is \(\alpha + 1\), but it also has ordinals much larger than that. Here's one: \(\omega + \omega \). This ordinal is obtained as the limit of \(\omega + n\) over all finite ordinals n.
The possibility to take limits whenever you feel that you're no longer progressing with successors instills the notion of transfinite induction. Transfinite induction is a way to construct functions out of an ordinal by defining the function, inductively, on the different kinds of ordinals. First you define \(f(0)\) as some (uniquely determined) element, then you can define, for all \(\alpha \), \(f(\alpha + 1)\) uniquely determined up to the knowledge of \(f(\alpha )\), and similarly, you can define \(f(\sup_{\beta < \alpha} \beta)\) also uniquely determined up to the values of \(f(\beta)\) for all \(\beta < \alpha\).
This is an extension of induction on the natural numbers and with the axiom of choice, it allows us to do so much!
Axiom of choice
The axiom of choice, a bit controversial, states that there is a way to simultaneously pick exactly one element from each non-empty set in any family (set) of sets. The axiom of choice is not an axiom but is a trivial tautology for finite families of sets. You can always pick some element of a set, and repeating that finitely many times gives you a choice of elements from each set. Once we talk about infinite collections of sets, it's no longer obvious that you can do that infinitely many times, simultaneously. This is where the axiom of choice comes in to tell us that, yes, we can.