Today I listened to more of A Brief History of Mathematics. Most of what I heard was about Cantor. For non-enthusiasts, Cantor is famous for discovering that there are multiple sizes of infinity. The set of natural numbers (i.e. 1, 2, 3, 4, and so on) is infinite, and so is the set of real numbers (i.e. 1, $-\frac{1}{3}$, 2.5837, $\pi$, and all the other numbers on a number line). Cantor discovered, however, that the cardinality of the set of natural numbers—how many things are in it, essentially—is less than the cardinality of the set of real numbers. The infinity of natural numbers is a smaller infinity than the infinity of real numbers.
Not only that, but Cantor discovered that there are infinitely many sizes of infinity, called infinite cardinals. Listening to that idea being discussed, I was struck by a question I’d never thought of before, which was “what size of infinity describes the infinitude of sizes of infinity?” or, more intelligibly, “what is the cardinality of the set of all infinite cardinals?”
I don’t have an exciting answer to share with you, unfortunately. Basically, the answer is that the question is not intelligible. I was able to find a thread on Mathematics Stack Exchange about the topic, and the answers given there explained that there is no set of all infinite cardinals, because the collection of infinite cardinals is in fact a class. Unlike sets, classes do not have cardinality, so there is no answer even to an adjusted form of the question. I was and still am a bit confused about what a class is, but another thread on Mathematics Stack Exchange suggests that it’s any collection of things that can be described but can’t be a set for logical reasons founded in the set theory axioms.
(I also spent a bit of time today reading about categories, which are another mathematical collection of things. They were something I’d heard of but never actually encountered, and my other researches brought them to mind.)
