Why can almost all ordinary mathematics be formalized by sets?

Question

there must exists a reason of why the idea 'collection' is so powerful that it can formalize nearly all mathematics.

subquestion: is there any which can not be formalized by this perspective? if so, what properties make them informalizable?

Answer 1

Almost all mathematics can be formalized without sets. For example, second-order arithmetic is good enough for most analysis and algebra. Particular theorems are usually provable in very weak formal systems, far weaker than set theory. (For instance, even though Andrew Wiles's proof of Fermat's theorem uses fancy set theory it is suspected that it can be transformed into a proof that only needs arithmetic, and we already know that only a bit of the set-theoretic universe is needed.)

When mathematics is actually formalized set theory is almost never used. Instead, people who formalize mathematics use proof assistants such as Agda, Coq, Isabelle, HOL, which are based on type theory. There are certainly similarities between sets and types, but types are more general than sets.

I have no doubts that it is possible to formalize mathematics in insanely unusual formal systems, as long as they are sufficiently expressive. So, to answer your question: mathematics can be formalized using sets because it can be formalized in any number of formal systems and there is nothing special about sets in this regard. It is just that humans invented sets and use them.

It's a bit like wondering why everyone drives cars that run on fossil fuels. Well, there are certainly many other ways to move around the planet, but this is the one we currently have, hopefully not for much longer.

We can further speculate why humans invented sets and types (which are similar to sets for the purposes of this discussion). This is not something that mathematics can answer by itself. We need to look at how people's minds work, how our language is structured, etc. I suspect we will discover that there is something very natural for us to think about "collections of things which are alike". Of course, the elements of a set need not be alike and can be quite arbitrary. So there are some surprising sets out there that we do not know how to cope with very well.