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(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 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?

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.

Indeed. It makes me wonder: there are all kinds of theories; there is a very general notion of interpretation of one theory in another; 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...?

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    $\begingroup$ Lawvere proposed using categories as an all-ecompassing basis. $\endgroup$
    – Arturo Magidin
    Commented Apr 16, 2018 at 5:35
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    $\begingroup$ The point is that a model is a collection of objects, and sets are the way we mathematically interpret the notion of a collection. $\endgroup$
    – Asaf Karagila ♦
    Commented Apr 16, 2018 at 6:53
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    $\begingroup$ In my mind, it's clear that we don't. If you have to ask, you're just proving my point. $\endgroup$
    – Asaf Karagila ♦
    Commented Apr 16, 2018 at 9:13
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    $\begingroup$ It is just an empirical fact that set theory has the strongest interpretive power, because it has been around a while and makes the strongest assumptions that add interpretive power (large cardinals). $\endgroup$
    – Monroe Eskew
    Commented Apr 16, 2018 at 9:29
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    $\begingroup$ I have voted to close, since six vague questions is too many for a clear answer. $\endgroup$
    – user44143
    Commented Apr 16, 2018 at 11:17

3 Answers 3

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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.

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    $\begingroup$ "from the ground up": the trouble is, there is rather rough terrain out there. $\endgroup$
    – Mikhail Katz
    Commented Apr 17, 2018 at 9:06
  • $\begingroup$ Let me mention that, because of what you say, it is in principle possible that sticking to widespread paradigms created by "strong" concepts like sets, categories or types, we might be completely overlooking the whole possible directions mathematics could expand in. What if, say, my models should describe some aspects of a vortex, or a rainbow, or, say, a person with some specific neurological disroder? What does that have to do with either sets or homotopy types? Or, say, consider apparent (to me) helplessness of modern mathematics in dealing with natural language analysis... $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Apr 18, 2018 at 5:44
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    $\begingroup$ @მამუკაჯიბლაძე : This is of course always possible but I don't think that set theory is particularly worrisome here. Maybe we're missing the boat because so much of our sensory input is visual rather than olfactory or sonar, or because our brains are wired wrong, or carbon atoms have only four bonds... $\endgroup$
    – Timothy Chow
    Commented Apr 18, 2018 at 15:58
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It's not clear to me whether your question is more about the role of sets versus other foundational objects, or about how set theory can be extended with large cardinal axioms to discuss models of stronger and stronger theories. Monroe's answer addresses the latter. Regarding the former, maybe I'll take this opportunity to push an analogy that I think deserves more currency. In computer science there are notions such as:

  • A Turing-complete programming language. All Turing-complete languages are expressive enough to simulate each other, and this is essentially the definition of the class. In particular, a language is Turing-complete as soon as (1) it can simulate at least one known Turing-complete language and (2) it can be simulated by at least one known Turing-complete language. Thus, to define Turing-completeness it suffices to give one example of such a language; the classical example is Turing machines.

  • An NP-complete problem is, by definition, one that is solvable in polynomial time by a nondeterministic algorithm, and any other such problem can be reduced to it in polynomial time (by a deterministic algorithm). This means that a problem is NP-complete if, and only if, (1) it can be reduced in polynomial time to some known NP-complete problem, and (2) some known NP-complete problem can be reduced to it in polynomial time. So we could equivalently define the class of NP-complete problems by giving one example.

I suggest that we should recognize an analogous notion of a mathematics-complete theory: a theory which is expressive enough that all of mathematics can be encoded into it (modulo suitable extensions such as large cardinal axioms, use of stronger or weaker logics, etc.) To define the class of mathematics-complete theories, it suffices to give one example thereof, and empirical evidence suggests that set theory is one such choice (indeed, the first to be discovered).

Mathematics-complete theories are often called foundations for mathematics. I don't propose to do away with this terminology, but it sometimes produces confusion as some people seem to sometimes mean something more by it. The phrase "mathematics-complete" emphasizes that we mean nothing more (or less) than that all of mathematics can be encoded into a theory, whether or not this encoding is "natural" or "intuitive".

So I guess my answer to your questions is that one such formal property is "set theory can be encoded into it, and it can be encoded into set theory". The apparent primacy of set theory in this definition is just an artifact of the fact that set theory was the first example of such a theory; an equivalent definition would be "type theory can be encoded into it, and it can be encoded into type theory".

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    $\begingroup$ @მამუკაჯიბლაძე It seems to me that it would be impossible in principle to know that any formal definition definitely encompasses every new kind of mathematics that might be invented some day in the future. That said, it seems unlikely to me that a totally new kind of mathematics would be able to gain any foothold without at least a translation into existing mathematics. $\endgroup$
    – Mike Shulman
    Commented Apr 16, 2018 at 11:53
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    $\begingroup$ Conway says something similar in On Numbers and Games. "It seems to us, however, that mathematics has now reached the stage where formalisation within some particular axiomatic set theory is irrelevant, even for foundational studies. It should be possible to specify conditions on a mathematical theory which would suffice for embeddability within ZF...but which do not otherwise restrict the possible constructions in that theory...This appendix is in fact a cry for a Mathematicians' Liberation Movement!" $\endgroup$
    – Timothy Chow
    Commented Apr 16, 2018 at 15:02
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    $\begingroup$ @Mike Shulman. Is ZF biinterpretable in ZFC? $\endgroup$
    – André Henriques
    Commented Apr 18, 2018 at 20:17
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    $\begingroup$ @AndréHenriques Yes. Any model of ZFC is also a model of ZF, and inside any model of ZF you can build Godel's constructible universe L which satisfies ZFC. $\endgroup$
    – Mike Shulman
    Commented Apr 18, 2018 at 21:03
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    $\begingroup$ @AndréHenriques ZF and ZFC are mutually interpretable, not bi-interpretable. $\endgroup$
    – user76284
    Commented Oct 26, 2021 at 15:34
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It is my understanding that set theory's interpretive power is a quasi-empirical fact, and that at present there is no grand theoretical explanation of the phenomenon. By "set theory," I mean the mathematical research area, not a specific formal theory.

Set theory has strong interpretive power in the sense that it provides a "measuring stick" for gauging the relative consistency (aka interpretability) of mathematical theories in general--the large cardinal hierarchy. We do not have an explanation for why this system should appear as a measuring stick rather than a partial order, but it appears to be a well-ordered hierarchy. Large cardinals have had enough success in proving relative consistency results that we have come to accept them as the arbiters of consistency (of strong enough theories).

The fact that large cardinal concepts are formalized in set theory rather than some other foundational theory is a historical fact. Things could have conceivably turned out differently. The study of interpretability of weaker theories is typically done in the context of subsystems of second-order arithmetic, where the notion of set plays a much smaller role.

One could raise the following objection to the large cardinal hierarchy as an ultimate arbiter. There are many unsolved relative consistency questions. Proving these statements consistent/independent relative to large cardinals would be considered a solution. But perhaps there are statements that will not be measured by large cardinals because their consistency strength is not comparable to large cardinals. Our belief in the supremacy of large cardinals (or social conformity to the practice) just prevents us from designating other statements as important nodes in the sea of consistency strengths. An incomparable statement would remain invisible to us as such.

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    $\begingroup$ I don't think this idea explains large cardinals up to a certain level, like measurable cardinals. These have implications for the "width" of the universe, not merely "height," i.e. how far up induction takes you. $\endgroup$
    – Monroe Eskew
    Commented Apr 16, 2018 at 13:36
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    $\begingroup$ It's not at all clear to me that the existence of a measurable cardinal is an "induction principle." Measurable cardinals are not measurable in all inner models (classes satisfying ZF with the same ordinals and thus the same notion of well-foundedness), so this seems like a counterexample to the claim that they are some kind of "induction principle." $\endgroup$
    – Monroe Eskew
    Commented Apr 16, 2018 at 16:59
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    $\begingroup$ Ok well this is quite well known. If M is a submodel of N with the same ordinals, both satisfying ZF, and R is a relation in M, then M and N agree on whether R is well-founded. (To be fair, this is an important and nontrivial absoluteness theorem.) $\endgroup$
    – Monroe Eskew
    Commented Apr 16, 2018 at 18:16
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    $\begingroup$ I’m talking about a situation in which the ordinals between the two models are identitical (although isomorphic will do just fine). This means that they do inducitive constructions of exactly the same lengths. Furthermore, all the objects they have in common which are capable of some kind of inductive analysis in one model are this way in the other model. (This is the absoluteness theorem I mentioned.) Now, what do you mean by “inductive strength”? $\endgroup$
    – Monroe Eskew
    Commented Apr 17, 2018 at 4:49
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    $\begingroup$ I agree with Monroe Eskew. Suppose I even grant Mike Shulman that the fact the two models have the same ordinals is not a "counterexample to the claim that a measurable cardinal is an induction principle." Fine. But on what grounds are you claiming that the existence of a measurable cardinal is an induction principle? It certainly doesn't look like an induction principle upon superficial examination. What makes something an induction principle? Can you give an example of an axiom that is definitely not an induction principle, and if so, why is it not an induction principle? $\endgroup$
    – Timothy Chow
    Commented Apr 17, 2018 at 17:28

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