(**Later edit** - tried to clarify a couple of vague places concerning interpretations of theories that became evident in comments (thanks to Andrej Bauer, Mauro ALLEGRANZA and Emil Jeřábek). (To closers and downvoters: may I humbly direct your attention to the [soft-question](/questions/tagged/soft-question "show questions tagged 'soft-question'") tag down there...))


This is an ideal question: I don't even know what I am asking


It has been inspired by reading [What "metatheory" did early set theory/logic researchers use to prove semantic results?](https://mathoverflow.net/q/297942/41291)


Specifically by



> 
> The modern approach seems to be, usually, to interpret a "model" specifically as a set in some other (typically first-order) "set metatheory." So when we talk about a model of PA, for instance, we typically mean that we are formalizing it as a subtheory of something like first-order ZFC. Models of ZFC can be formalized in stronger set theories, such as those obtained by adding large cardinals, and etc.
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Indeed. It makes me wonder: there are all kinds of theories; there is a very general notion of [interpretation of one theory in another](https://en.wikipedia.org/wiki/Interpretability); nothing about sets comes into play so far. Then all of a sudden, when one says "model", the first thing comes to mind is "a set with blablabla...". Why?


I mean, is there a rigorous way to distinguish some theories among all others by some formal property which (a) guarantees that any sensible notion of model is subsumed by interpretability in a theory with this property and (b) every theory with this property is equivalent (in some rigorous sense) to "a theory of sets"? What does this latter even mean?


Are there any other such "all-encompassing" classes of theories? This last question might seem senseless since any two "all-encompassing" theories must be mutually interpretable and thus be equivalent to each other (I think, at least in some sense?). But still maybe there is some notion of "meta-encompassing" that makes some "all-encompassings" more "all-encompassing" than others...? 



What is so special about set theory anyway?
I agree with other respondents that it is unlikely that one will be able to come up with some kind of *formal* argument that distinguishes set theory from other "mathematics-complete" systems (to use Mike Shulman's term, which I like!), because mathematicians are so good at rephrasing one language in terms of another. There is surely also some degree of what one might call "historical accident" involved; we instinctively think of a model as a set because that's how we were taught, and how our teachers were taught, etc.


That said, I think a case can be made that set theory has some *psychological* advantages when it comes to addressing certain questions, e.g., is mathematics consistent? and exactly which assumptions are used to derive which theorems? Today, most mathematicians take a rather breezy attitude towards the consistency question, assuming that it's all been sorted out by logicians and that it's no concern of theirs. If you take consistency seriously, however, then it is vitally important to try to build everything from the ground up, one small step at a time, in as simple and clear a manner as possible. As long as you limit yourself to finitary mathematics, several alternatives to set theory are available (arithmetic, syntax, type theory, ...), but for infinitary reasoning, set theory seems to be the psychologically best choice for most people.


Even for finitary reasoning, set theory has the advantage of being a very flexible and fine-grained tool. In reverse mathematics, the standard approach is to consider subsystems of second-order arithmetic. Arithmetic is certainly a natural foundation for finitary mathematics, but it quickly becomes an irresistible temptation to introduce set-theoretic reasoning because it makes it so easy to introduce any new concept or axiom that you might come up with. That is why "second-order" arithmetic becomes the de facto foundation.


There are other goals you might have for a "foundation for mathematics" for which set theory is arguably not the optimal approach. For example, maybe you have developed some conception of the mathematical universe, and you want some formal theory that directly captures the structures and concepts you have in mind. Then category theory or homotopy type theory might be more attractive. Still, I think set theory is hard to beat as a way to "get off the ground" so to speak. To fully appreciate the merits of category theory or homotopy type theory requires a lot of prior mathematical knowledge and experience, and I think that most people would have trouble acquiring all that background knowledge without ever appealing to any set-theoretic concepts along the way.


