How do we know (almost) all of math can be interpreted in set theory?

Question

By "all of math" I mean geometry, groups, algebra, functions, etc; that which we don't do in set theory proper even though in principle we could. How do we know that (almost) all of math which is not set theory proper (the formalization) can be done in the formalization? Here I am essentially just pointing to the fact that we don't typically do set-theory the formalization when we do math.

I can see some broad strokes about how we expect the interpreting: As long as we restrict our predicates someway (e.g. no set of all sets), we could talk about many many sets, how functions are really sets, how the number systems are sets, and that covers much of math.

But I want to know see we know/expect that all of math is so encompassed? Searching I see demonstrations of how vast areas of math (functions, groups, all the number systems, etc) can be interpreted/formalized as sets, and that we do claim almost of all math can be done, but I haven't seen anything fully satisfying about how. I.e. should I just expect that because functions, numbers, etc can be formalized the rest of math can be to?

Answer 1

There is a good discussion of this, modulo category theory, on the MathSE:

That we can represent nearly every mathematical strcuture [sic] in terms of sets stems from the fact that many structures are in fact defined as sets with extra structure (this is especially true for algebraic structures such as groups, rings, fields, etc.). But we can also view these as objects of their respective categories, where the categories carry the information making them special (for example, the category of groups does have a zero object, while the category of sets does not). Currently I cannot think of a mathematical concepts that is not describable via sets/categories but I might be missing something out.

A possible, or at least borderline, case of a proper mathematical non-set would be a proper class, but I say (and emphasize!) "borderline" in that description since the language of class theory is effectively the same as the language of set theory. One might also consider entities like Brouwer's free-choice sequences or Freely Creating Subject, or the infinitesimals of smooth infinitesimal analysis.

But I suspect that the conceptual versatility of the elementhood relation and its symbolism can be exercised to translate most anything into a sort of set theory. The concept of a set is in itself very "thin" and so in a multiverse of such concepts, you should be able to manipulate the axioms and inference rules to get the desired result easily.

Answer 2

We know that all of mainstream mathematics can be interpreted as set theory based on model theory. That is: by developing set theoretical models of the domain of interest. We map (redefine, interpret) all primitive terms of the domain to sets (perhaps with some special properties) and all axioms to set theoretical theorems.

Model theory is one way in which mathematics is self-reflective and tries to comprehend its own meaning -- as a kind of conceptual ouroboros. Strictly speaking it's a form of meta-mathematics, but it is practiced as a branch of mathematical logic.