Is mathematics just a bunch of nested empty sets?

When in high school I used to see mathematical objects as ideal objects whose existence is independent of us. But when I learned set theory, I discovered that all mathematical objects I was studying were sets, for example:

$ 0 = \emptyset $

What does it mean to say that the number $ 1 $ is the singleton set of the empty set?

Thank you all for the answers, it is helping me a lot.

Obs.

This question received far more attention than I expected it to do. After reading all the answers and reflecting on it for a while I've come to the following conclusions (and I would appreciate if you could add something to it or correct me): Let's take a familiar example, the ordered pair. We have an intuitive, naive notion of this 'concept' and of its fundamental properties like, for example, it has two components and

$ (x, y) = (a, b) $ iff $ x = a $ and $ y = b $

But mathematicians find it more convenient (and I agree) to define this concept in terms of set theory by saying that $ (x, y) $ is a shortcut for the set { {x}, {x, y} } and then proving the properties of the ordered pair. I.e. showing that this set-theoretical ordered pair has all the properties one expect the 'ideal' ordered pair to have. Secondly, mathematicians don't really care much about these issues.

Warning: personal opinion ahead!

It depends upon what the meaning of the word "is" is.

One way to think about it - not necessarily historically correct - is the following: that the axioms of set theory are intricate enough - or, if you prefer, describe structures (their models) which are intricate enough - to "implement" all of mathematics. This is analogous to the relation between algorithms (as clear-but-informal descriptions of processes) and programs (their actual implementation), with the observation that the same algorithm can be implemented in different ways.

It's worth noting that this "in different ways" has multiple senses: there will be many ways to write a program which carries out a specific algorithm in a given programming language, and there are also lots of programming languages. Correspondingly, we have:

• There are lots of ways to implement (say) basic arithmetic inside ZFC; the von Neumann approach is just the standard one.

• There are also different theories which similarly are intricate enough to implement all of mathematics.

Asking what $1$ "is" is an ontological question, but set theory doesn't need to be thought of ontologically - the pragmatic approach ("how can we formally and precisely implement mathematics?") is sufficient.

I'm not claiming that this is the universal view; for example, one can also argue (although I don't) that the cumulative hierarchy (= the sets "built from" the emptyset) consists of all the mathematical objects which are guaranteed to exist, in a Platonic sense. But I think the view above probably more faithfully reflects the general attitude of the mathematical community, and is certainly how I approach the question.

Firstly, there is a big difference between abstract concepts and their concrete representations. It is true that we can encode and reason about natural numbers in ZFC, in the specific sense that we can reason about a model of PA (Peano Arithmetic) within ZFC. It does not mean that somehow the abstract concept of $0$ is the empty-set!

Similarly, it is true that we can express notions like "pair" and "function" and so on in terms of set-theoretic definitions in ZFC, but these notions have been around for far far longer than ZFC, and furthermore there are infinitely many viable definitions to 'make concrete' these notions in ZFC. No mathematician actually thinks of the abstract pair $\langle x , y \rangle$ as $\{\{x\},\{x,y\}\}$ or some other concrete representation.

Furthermore, even our standard mathematical notation shows that we conceive of functions in a more fundamental way than is suggested by the standard encoding as a set of input/output pairs.

For more details, see this post about abstract mathematical objects that are not sets, which also mentions urelements (non-sets such as you and me).

Secondly, it is true that from the viewpoint of ZFC, everything is a set (in the sense that given any two objects $x,y$ we can ask whether $x \in y$ or not). But even then, it is not necessarily the case that everything is "just a bunch of nested empty sets"!

The axiom of infinity is the only axiom of ZFC that asserts the (absolute) existence of some set. Every other axiom can be applied only if you already have some set. Now, informally the axiom of infinity says that there exists an inductive set, where a set $S$ is called inductive iff ( $S$ includes the empty set as a member, and is closed under the successor operation ), where the successor of $x$ is $S(x) := x \cup \{x\}$. Note that the axiom does not stipulate that there is a 'minimal' such set, nor what are the members of such a minimal set!

Well, we can use the other axioms of ZFC to construct $N$ to be the intersection of all inductive sets. But there is no way to prove that $N$ only includes as members the empty set and sets obtained by iteratively applying the successor operation. If that seems strange to you, that is unfortunately the way it is.

You see, ZFC does not have the natural numbers as primitive notions, and the axiom of infinity was concocted precisely to enable ZFC to construct a model of PA; we can define addition and multiplication on $N$ and prove that $N$ satisfies PA. But that means that $N$ is what we take to be natural numbers when working in ZFC! Nothing precludes 'our' set-theoretic universe (if it at all exists) from having an $N$ that has more members than those that you can manually write down, namely $0, S(0), S(S(0)), \cdots$, where $0 := \varnothing$.

And the curious thing about this is that ZFC itself knows that the above may happen! ZFC proves that if ZFC is consistent then there is a model of ZFC that has extra (called non-standard) members of $N$.

Finally, most mathematics is in fact independent of set theory, and can be recovered in very weak theories of arithmetic such as ACA, as briefly described here. For real-world applications, it is even better, because there is no evidence of infinitary objects in reality, and there is even a humorous grand conjecture by Harvey Friedman:

Every theorem published in the Annals of Mathematics whose statement involves only finitary mathematical objects (i.e., what logicians call an arithmetical statement) can be proved in EFA. (EFA is a fragment of PA.)