What are proper classes good for in mathematics (Applications)

Pardon me if the following question is somewhat naive or stupid. I am not an expert in formal logic!

I have a question about the notion of "proper classes" coming from the realm of formal logic, see here: https://en.wikipedia.org/wiki/Class_(set_theory)

It is said that each set is also a class, but not any class need to be a set. Such classes are then called "proper classes". I am wondering if there are any applications of this notion in other areas of mathematics. For example, is there an interesting mathematical object (e.g. in Analysis, Algebra,...) which is a "proper class"? Or is it enough for other areas of mathematics to use the notion of sets given by the Zermelo–Fraenkel set theory? If so, why is it important at all to introduce the notion of "proper classes" in logic?

Best wishes

• Why introduce the notion? Because proper classes are mathematical objects, and one goal of mathematical logic is to better understand mathematical objects. – Carl Mummert Dec 5 '17 at 15:49

Russell's paradox—and many other paradoxes of naive set theory—show us that there are collections of mathematical objects that we can define, but they do not form sets.

This is a bit of a thorn in the side of set theory, since the whole point of sets is to have a mathematical object which is itself a collection of mathematical objects (or rather, at least model this idea). So in an ideal world, every collection of mathematical objects that we can define would be a set. But alas, we do not live in a perfect world.

Proper classes solve this problem by allowing us to have a notion of collection which is formal, but does not correspond to a set. The collection of all sets, the collection of all partial orders, etc. are all collections which we can describe, but do not form sets.

In modern set theory this is used in full power when talking about the ordinals, inner models, ultrapowers of the universe, and many many other collections of sets which themselves make a proper class.

So, what happens outside of set theory? Well. The basic theorems of model theory show us that if there is one infinite group, then there is one of every cardinality, and therefore there is a proper class of groups none of which is isomorphic to any other; and there is a proper class of topological spaces, no two of them are homoemorphic; and so on and so forth.

The collection of most things satisfying a certain property which is not bound to a specific set will usually end up being a proper class. And while sometimes we can sort of remedy that (e.g. the collection of all singletons is a proper class, but usually it is enough to talk about $\{\varnothing\}$), in many cases we really do need to talk about a proper class of objects.

Many people would argue that this is a hindrance rather than an important notion. This makes category theory cumbersome. This makes things like "define the algebraic closure of a field" cumbersome. This causes a lot of problems when you want to be more formal, correct, and maintain clarity without getting into set theoretic issues (which is arguably irrelevant in an algebra course).

But, I argue that the fact that proper classes are there, stuck like a bone in the throat of everyone, is a good thing. If utilized properly, they promote thinking with care, they promote attention to details, and once you understand how proper classes actually work, and what does it mean for you as a mathematician, you can see that this is mostly a large shadow cast by a small mouse.

So where can you find proper classes to be prominent?

Category theory. That's an easy one. Most "obvious categories" are in fact proper classes, since the collection of objects is a proper class, and sometimes the morphisms between objects make a proper class as well. So the categories $\bf Set, Grp, Ab, Ring, Mod_{\it R}, Vec, Top$ and many others are in fact a pair of proper classes: a class of objects, and a class of morphisms between these objects.