# 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.

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.