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Let's say we're doing ordinary mathematics, and we want ZFC to be our foundations, such that all of our mathematical objects are sets.

I have long had the idea in my head that these sets that represent our mathematical objects come to be as follows--when the axioms of ZFC are stated as a theory or used as a foundation, instantly we have given to us every single set that ZFC allows for. That is, ZFC has "generated" these sets all at once, and now they exist somewhere "along ZFC" or "on top of ZFC" or "in ZFC," and when we do our 'informal' math, everything we talk about like $\mathbb{R}$ or the number $2$ are really just sets "along/on top of/in ZFC." I suspect that none of this is really correct.

So where do these sets "live"? Anywhere? Nowhere? In a model of ZFC? In all models of ZFC? Does this question even make sense/do I have the wrong understanding of how ZFC or a first-order theory in general works?

Maybe another way to ask this is (or maybe not, and the following is actually a different idea), when in the ZFC axioms, we have something like $(\exists y(y \in x) \implies \ldots)$, where is this $\exists y$ ranging over or "looking"?

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    $\begingroup$ Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. $\endgroup$ Commented Oct 9 at 0:37
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    $\begingroup$ In Peano arithmetic, all objects are numbers. There are no sets, only numbers. Where do those numbers live? $\endgroup$ Commented Oct 9 at 0:58
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    $\begingroup$ Number theory doesn't state where numbers live. It could be encoded in set theory, but the one encoding we've chosen there is just one of infinitely many choices of encoding. We are always dealing with abstractions - we never escape that in math. Asking where set theory resides is a bit like asking what is "the chair." There is no one chair. ZFC defines the set of conditions required for us to do a certain kind of set theory. Anything that satisfies the axioms is a different instance, just as the word "chair" describes many different chairs. $\endgroup$ Commented Oct 9 at 1:00
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    $\begingroup$ This appears to be a question about ontology. I don't know if there is any answer better than the buzzwords "Platonism" and "Formalism", among others. See this potentially related post on MO. Also this is not a question that's specific to set theory. For example, where do the past and future "live"? Notably, in GR, the space-time manifold is a pre-existing and static object. See the discussion on physics.stackexchange. $\endgroup$ Commented Oct 9 at 5:26
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    $\begingroup$ They live in Mathland. $\endgroup$ Commented Oct 9 at 6:11

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I have always viewed this as a kind of strategic ambiguity in the way mathematicians communicate.

  • Some people think there is a real universe of "all sets". For them, ZFC is really about that universe, the same way that Peano Arithmetic is really about the standard natural numbers. So they read the ZFC quantifiers with this intended interpretation. This viewpoint can be referred to simplistically as a kind of Platonism.

  • Some people think that it's better to think of some unspecified model of the ZFC axioms as being fixed in the background, so the quantifiers range over that unspecified model. We don't know what model it is, so all we can say is what we can prove about it from ZFC or any other set theoretic axioms we assume. This viewpoint can be described simplistically as a different kind of Platonism.

  • Some people don't like models at all, and they view the formulas as purely syntactic. This can be viewed simplistically as a kind of formalism.

The remarkable thing about the way set theory is written is most of it can be read in any of these three viewpoints. So the people with these very different views don't have to come to an agreement, but they can still communicate meaningful ideas about ZFC that are mutually understood.

For example, set theorists understand how to convert forcing proofs that seem very model theoretic into purely syntactic relative consistency proofs, even if they don't emphasize that interpretation in most of what they write.

This ambiguity is related to the concept of "working realism", e.g. as described in the Stanford Encyclopedia article on Platonism in the Philosophy of Mathematics.

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    $\begingroup$ The Platonist views make sense, but I can't quite see how the 'formalist' would think about these quantifiers. Maybe I am stuck with bad ideas in my head, but I really think of the quantifiers wanting to range over sets somewhere. There's a notion of somewhere in viewpoints (1) and (2), but seemingly not for (3). $\endgroup$ Commented Oct 9 at 1:33
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    $\begingroup$ @mareli Of course, this risks simplifying and stereotyping, but from some formalist perspective it is not at all necessary for the quantifiers to range over objects somewhere. The formulas are simply strings of symbols we can manipulate according to various rules. To make an analogy: is it necessary for character names in a book to refer to characters that truly exist in some non-physical sense? Can we just read a novel from the words that are in it without assuming there has to be some kind of "existence" for fictitious characters? If we can do it with a novel, why not set theory? $\endgroup$ Commented Oct 9 at 1:39
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    $\begingroup$ @mareli According to the formalist you can only make use of such a formula by applying the rules of logic to it - that is, existential elimination: going from $\exists y (y \in x \implies \phi(y))$ to naming the element $a$ and writing $\phi(a)$ (and going the other way, existential introduction). The quantifier wouldn't have a meaning other than the fact that you can instantiate it with a constant. $\endgroup$ Commented Oct 9 at 2:12
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    $\begingroup$ @CalebStanford Yes OK, so it's logical rules and manipulations that allows them to "make use" of the quantifier in the axioms. I see the formalist viewpoint on this matter now, thank you $\endgroup$ Commented Oct 9 at 2:29
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    $\begingroup$ @mareli: Actually, restricted quantifiers have in principle nothing to do with set theory. Many-sorted FOL is a very natural variant of FOL that much more accurately captures logical reasoning than the conventional one-sorted FOL, and the syntax of sorts and quantification over some sort often uses the symbol "∈", but it is not the same as the binary relation-symbol used by ZFC. In this post I give a complete deductive system for many-sorted FOL, which works without sets. Even when used with ZFC, we can quantify over non-sets like "set". $\endgroup$ Commented Oct 9 at 9:11
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Why should the sets live somewhere? :-)

In ZFC (rather unlike in usual mathematics), sets are collections of other sets. So if they do live somewhere, it's that they are abstractions over existing sets that have already been created. In other words, sets live only in relation to other sets.

You can think of sets as living in a model of ZFC as you suggest in your post, but that's imposing a metatheory (and assuming that ZFC is consistent), and it may be somewhat unsatisfying as the obvious question is where that model of ZFC itself lives (and the answer will be that it lives in another model of ZFC). So I wouldn't necessarily recommend this viewpoint, even though it's technically correct from the perspective of metamathematics (and is a view that can be adopted in model theory, when considering the model theory of ZFC).

An alternative viewpoint (which is adopted by ideas in set theory like the von Neumann universe and Gödel's constructible universe) may be a bit more intuitive: The idea is that sets are always created as collections of already existing sets. So in the beginning (or day 0), there is only one set: the empty set $\varnothing$. Then on each day, we create as a new set all collections of existing sets:

  • On day 1, we create all collections of $\varnothing$ which don't already exist. In particular we create $\{\varnothing\}$, the collection containing the empty set. And this is the only new set we create; a collection can either contain $\varnothing$, or not; and if it doesn't, it's just equivalent to the empty set, which already exists.

  • On day 2, we create all collections of $\varnothing$ and $\{\varnothing\}$ that don't already exist. In particular, that means we create two new sets, $\{\{\varnothing\}\}$ and $\{\varnothing, \{\varnothing\}\}$. (We also get $\varnothing$ and $\{\varnothing\}$, which already exist.)

And so on. To get all of ZFC though, we also need to add the set $\omega$, representing all integers - you can either add it on day 0, or you can imagine adding it after infinitely many days and continuing. Many of the other axioms, such as Union and Powerset, immediately hold in such a construction. (As pointed out in the comments, some sets — in particular those created by Replacement — require more care to be handled appropriately, for example by extending the construction to all ordinals.)

If done correctly, this construction results in all sets that you need, and only such sets; that is, it's itself a model of ZFC. But it's a very concrete one, and one that may make sense to you intuitively. It also can be visualized since the new sets at each stage are just collections of the existing ones, for example, by drawing arrows from each new set into any existing sets that it contains. It's therefore a valid mental model of some sort of "universe" in which all sets can be thought of as living.

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    $\begingroup$ This motivation of the cumulative hierarchy is often viewed as very compelling. Shoenfield presents it in the Handbook of Mathematical Logic. The only complication is the set of days needs to be identified with a set of ordinals that is quite long (so that the ultimate model satisfies the replacement axiom). The naive viewpoint would stop after $\omega$ or $\omega + \omega$ days, but we need the set of days to be Ord which, from below, will look a lot like an inaccessible cardinal. $\endgroup$ Commented Oct 9 at 1:24
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    $\begingroup$ A nitpick on this good answer: I wouldn't say "this is the viewpoint adopted in model theory". For one thing, most of the time model theorists are not working with foundational theories like ZFC at all. So it might be more accurate to say "this is the viewpoint adopted in the model theory of ZFC". For another, it's possible to have a wide range of views on the philosophy of mathematics and still use models of set theory as a mathematical tool to prove things about ZFC, like independence results. $\endgroup$ Commented Oct 9 at 2:16
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    $\begingroup$ Another nitpick: it's not right that "You may need to create some sets explicitly, such as those created with Choice." In fact every set appears in the cumulative heriarchy, i.e. by iterating the power set operation transfinitely, so there's no need to explicitly throw in things like witnesses to choice. Axioms like choice and separation can be viewed as expressing that the power set operation we use to build the cumulative hierarchy is really as rich as we think it should be. $\endgroup$ Commented Oct 9 at 2:24
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    $\begingroup$ As Carl Mummert write, the role of Replacement is to force us to continue the transfinite iteration of power set sufficiently long. If $F$ is a class function and $X$ is a set, the image $F[X]$ may contain elements that appear at later and later "days" of the construction. In order for Replacement to hold, we have to continue the construction past all of those days, so that $F[X]$ becomes a set at the first day after the supremum of the days at which its elements appear. $\endgroup$ Commented Oct 9 at 2:33
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    $\begingroup$ @Caleb Stanford One key place where Replacement comes in is at limit stages of the construction. Say we start with $\omega$ at day $0$, and we add some new set $s_i$ on each day $i$ in a way that $F[i] = s_i$ is a definable function. Then, after all those sets are added, we need to add $S = \{s_i : i \in \omega\}$ at day $\omega$, to satisfy Replacement. In particular, there has to be a day $\omega$. If we wanted to add $S$ on an earlier day, it's not clear which day that would be. The instance $F$ of Replacement that gives us $S$ is not an instance until all the $s_i$ are formed. $\endgroup$ Commented Oct 9 at 10:00
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It seems like this is a philosophical question that goes well beyond set theory. For example where do natural numbers (i.e. non-negative integers) "live"?

That is, we know that we can't (currently) say for sure whether the universe is infinite or finite, or whether space and time are truly continuous or discrete at some very small scale. So there's no sense in which a really big number like $10^{10^{1000}}$ can be said, even in principle, to definitively "exist" on the basis of counting things in physical reality (we don't know if there can ever exist that many physical things to count). So does the number $10^{10^{1000}}$ "exist"? If the answer is to be "yes", it cannot be justified on the basis of counting physical things. So in what sense does it exist? Where does it "live"?

There are many schools of philosophy of mathematics, giving different answers to such questions.

  • This Numberphile video is not a bad starting point (though I wouldn't say it's totally comprehensive). Note that the "nominalist" philsophy described there runs into the problem I just alluded to about the existence of really big numbers. Also, its description of fictionalism seems a bit thin; see here for more details.
  • A purist version of formalism would say mathematics isn't actually describing things that "exist" in any meaningful sense. Rather, mathematics is just a collection of rules for manipulating symbols on paper (literally just some rules for turning certain sequences of symbols into other sequences of symbols). I'm pretty sure this is a minority view (a quite small minority).
  • A category of philosophies which I find compelling are those which argue that mathematical entities exist (or fail to exist) on the basis of whether they are logically coherent concepts (for example here). In this sense, rainbow-colored unicorns and the number $10^{10^{1000}}$ "exist" (albeit perhaps not in the physical world) but four-sided triangles do not (since "four-sided triangle" is a logical contradiction).

All of the preceding philosophical considerations apply equally well to set theory like ZFC. Note that there are many possible models of ZFC (assuming ZFC is consistent and that one believes philosophically that models of mathematical theories exist at all). For example, some models in which the Continuum Hypothesis is true and others in which it is false. What distinguishes the Platonists is their view that only one of these models is the "true" or "best" set-theoretic universe.

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