Do the axioms of set theory actually define the notion of a set?

In Henning Makholm's answer to the question, When does the set enter set theory?, he states:

In axiomatic set theory, the axioms themselves are the definition of the notion of a set: A set is whatever behaves like the axioms say sets behave.

This assertion clashes with my (admittedly limited) understanding of how first-order logic, model theory, and axiomatic set theories work. From what I understand, the axioms of a set theory are properties we would like the objects we call "sets" to have, and then each possible model of the theory is a different definition of the notion of a set. But the axioms themselves do not constitute a definition of set, unless we can show that any model of the axioms is isomorphic (in some meaningful way) to a given model.

Am I misunderstanding something? Is the definition of a set specified by the axioms, or by a model of the axioms? I would appreciate any clarification/direction on this.

Update: In addition to all the answers below, I have written up my own answer (marked as community wiki) gathering the excerpts from other answers (to this question as well as some others) which I feel are most pertinent to the question I originally posed. Since it's currently buried at the bottom (and accepting it won't change its position), I'm linking to it here. Cheers!

Solution 1:

This is the commonplace clash between the semi-Platonic view of the laymathematician and the foundational approach for mathematics through set theory.

It is often convenient, when working in "concrete" mathematics, to assume that there is a single, fixed universe of mathematics. And everyone who took a course or two in logic and set theory should be able to tell you that we can assume this universe is in fact a universe of $\sf ZFC$.

Then we do everything there, and we take the notion of "set" as somewhat primitive. Sets are not defined, they are just the objects of the universe.

But the word "set" is just a word in English. We use it to name this fickle, abstract, primitive object. But how can you ensure that my intuitive understanding of "set" is the same as yours?

This is where "axioms as definitions" come into play. Axioms define the basic ground rules for what it means to be a set. For example, if you don't have a power set, you're not a set: because every set has a power set. The axioms of set theory define what are the basic properties of sets. And once we agree on a list of axioms, we really agree on a list of definitions for what it means to be a set. And even if we grasp sets differently, we can still agree on some common properties and do some math.

You can see this in set theorists who disagree about philosophical outlooks, and whether or not some conjecture should be "true" or "false", or if the question is meaningless due to independence. Is the HOD Conjecture "true", "false" or is it simply "provable" or "independent"? That's a very different take on what are sets, and different set theorists will take different sides of the issue. But all of these set theorists have agreed, at least, that $\sf ZFC$ is a good way to define the basic properties of sets.

As we've thrown Plato's name into the mix, let's talk a bit about "essence". How do you define a chair? or a desk? You can't really define a chair, because either you've defined it along the lines of "something you can sit on", in which case there will be things you can sit on which are certainly not chairs (the ground, for example, or a tree); or you've run into circularity "a chair is a chair is a chair"; or you're just being meaninglessly annoying "dude... like... everything is a chair... woah..."

But all three options are not good ways to define a chair. And yet, you're not baffled by this on a daily basis. This is because there is a difference between the "essence" of being a chair, and physical chairs.

Mathematical objects shouldn't be so lucky. Mathematical objects are abstract to begin with. We cannot perceive them in a tangible way, like we perceive chairs. So we are only left with some ideal definition. And this definition should capture the basic properties of sets. And that is exactly what the axioms of $\sf ZFC$, and any other set theory, are trying to do.

Solution 2:

In axiomatic set theory, the axioms themselves are the definition of the notion of a set: A set is whatever behaves like the axioms say sets behave.

I half-agree with this. But recall that the axioms of group theory don't axiomatize the concept "element of a group." Rather, they axiomatize the concept "group." In a similar way, the axioms of ZFC don't axiomatize the concept "set." They axiomatize the concept "universe of sets" (or "von Neumann universe" or "cumulative hiearchy", if you prefer).

Solution 3:

The axioms of ZFC (or any other sufficiently strong first-order formal system) cannot define the notion of "set", in the sense that you're looking for, namely that ZFC cannot pin down a unique structure that satisfies ZFC. Why so? Because ZFC cannot prove its own consistency, by Godel's incompleteness theorem, and hence ZFC cannot prove that there is a model of ZFC. Furthermore, ZFC can prove that if ZFC is consistent then it has infinitely many models, not at all a single one.

Similarly, no recursive extension of first-order PA can (completely) define the natural numbers, because we can prove (in our meta-system that is usually ZFC) that PA has non-standard models. However, second-order PA is categorical (has a unique model up to isomorphism) and arguably captures completely the natural numbers. The catch is that you need to be in a meta-system that already has the standard model of PA before you can prove this fact about second-order PA, so in a way there is a priori no way to define the natural numbers.

You might want to read this post that I wrote about what every usable formal system (as of today) ultimately depends on, which cannot be further broken down into simpler notions.

There's a partial way to get around the circularity, that appears to be what many mathematicians do in practice. We can use natural language and define "set" to be a type of object such that the ZFC axioms hold, and insist that we can only call something a set when we have proven its existence in ZFC. Note that there is no need to have a model of ZFC here, because we're saying that if you can't prove it then I don't accept that it exists (but neither am I insisting that it doesn't), so it becomes a purely syntactic notion.

In other words, if we define "set" syntactically using the axioms of ZFC, then we've escaped most of the circularity (except what we already need to know about string manipulation). What we still can't escape is that we can't define "model" in the usual sense without collections of some sort, and so we can't even articulate that ZFC has successfully defined any structure. (Unless of course our natural language is so powerful, but then we're in trouble.)

Solution 4:

There are thorough answers by Asaf Karagila and user21820. I want to point out a different issue: there is more than one concept or notion of "set". Paradoxes like Russell's paradox showed that the original "naive" concept of "set" is inconsistent, and so some of the properties of these sets needed to be discarded. But there are many ways to do that.

ZFC set theory is intended to capture the idea of the iterative cumulative hierarchy. This means that the kind of "set" that ZFC is intended to study -- the "ZFC-set" -- has several limitations:

• The only elements of ZFC-sets are other ZFC-sets. For example, I am not a member of any ZFC-set.
• The nature of ZFC-sets is that no ZFC-set can contain every ZFC-set (there is no ZFC-set that is universal for ZFC-sets)
• ZFC sets are well-founded: there is not an infinite sequence $x_1, x_2, \ldots$ of ZFC-sets with $x_1 \ni x_2 \ni \cdots$.

These limitations are inherent in the specific choices made in the formulation of ZFC, which are not part of other conceptions of "set":

• In the most naive conception of "set", sets can contain not only other sets, but also other objects like me, or like my car. This concept of "set" is likely to be more recognizable to non-mathematicians. Even among mathematicians, there are some who dislike the idea that all mathematical objects might be viewed as sets.

• In the set theory known as New Foundations, NF, there is a NF-set that contains all NF-sets. NF attempts to avoid the paradoxes by restricting the set-existence axiom scheme instead.

• There are set theories that are not well founded. One example uses the Aczel's anti-foundation axiom. These are of particular interest for some applications in computer science.

So the key point of the ZFC axioms is to define, in a particular sense, the notion of a "ZFC-set" (actually, the notion of a "universe of ZFC-sets" or a "ZFC-universe of sets"). It is easy to forget that there are other notions of "sets", if everything that you see uses ZFC, and these all say "set" instead of "ZFC-set". Indeed, some set theorists completely internalize the notion of "ZFC-set", so that to them a "set" is a "ZFC-set", and the other notions of "set" are not actually "sets".