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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?

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A realist will tell you that that's just the way it is, particularly set theory realists (like Gödel). – Git Gud Feb 5 '14 at 8:02
There is no such thing as ordinary mathematics. – Your Ad Here Feb 5 '14 at 10:22
@TooOldForMath The OP appears to be implicitly defining ordinary mathematics as non-set theory mathematics. – Cruncher Feb 5 '14 at 13:45
Of course there is such a thing as ordinary mathematics. It includes the core subjects such as analysis, algebra, geometry and topology. The exact boundaries are fuzzy, but that does not make the concept useless. – Andrej Bauer Feb 5 '14 at 16:31
up vote 13 down vote accepted

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.

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I think Gödel would disagree about there not being anything special about sets. "But, despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true", in What is Cantor's Continuum Problem. – Git Gud Feb 5 '14 at 8:57
I think without the idea 'collection' I can not even generate the concept 'two apples', or any further abstraction like number system. – user126349 Feb 5 '14 at 9:14
@user126349, there is a long between between an intuitive notion of a "collection" and a ZFC notion of "set". This is what Andrej may have been trying to point out. – Mikhail Katz Feb 5 '14 at 13:31
There are several ways to mathematize the intuitive idea of collection, such as: set, type, space, (the extension of) a predicate, object of a category, etc. all of these may be taken as primitive. Naturally, they all resemble each other, but there really is nothing special about sets. I respect Gödel's opinion, and his position is understandable in historic context, but I disagree st sets are somehow inevitable or more primitive than other similar concepts. – Andrej Bauer Feb 5 '14 at 16:29
It is a fact that actual formalizations of mathematics tend to use types rather than sets. This is extremely strong evidence in favor of the position that sets are not more primitive. Anyone arguing otherwise must explain this phenomenon. – Andrej Bauer Feb 5 '14 at 16:33

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