It's always worth recalling why we bother with set theory. Philosophical objections like Leśneivski's are extremely valuable and good insights which we cannot discard trivially. Maybe sets are not good things to study. The main reason that we study sets today is because they are a place where we could study ordinals and the rest of number theory. We know about two bananas, two apples, two trees, etc. but what is two itself? Set theory provides a capable if unsatisfying answer: Two is anything which is uniquely isomorphic to the second ordinal number, which happens to be a particular finite set, and since sets formally contain nothing but other sets, we can manipulate two as a set without having to know about bananas, apples, trees, etc.
The modern way to talk about this stuff is via categorification; [0] is a good high-level introduction.
Set theory does not answer the question, "what is two itself?" This is a common misconception. It just provides a way to translate statements about the natural numbers into statements about sets, which then you can use for proofs and formal constructions. It's an encoding, not a definition. In particular, the definition you give of two is not correct and I'd be interested to learn who told you it.
Further, the "modern" way to discuss this is not categorification, despite what Baez says; people who work and publish on the foundations of mathematics almost universally do so in the language of set theory.
Since sets are 0-categories, we can't escape set theory when talking about structures like the natural numbers. A natural numbers object is a feature of a topos, preserved by topos functors (geometric morphisms). Nobody told me this definition; it's something I had to absorb for myself when learning topos theory.
Formally, let N be the natural numbers object (in some topos), let z : 1 -> N be the zero arrow, and s : N -> N be the successor arrow. Then z;s;s : 1 -> N is the arrow which chooses 2 as an element of the NNO. Since z and s are unique up to unique isomorphism, so is 2. Moreover, since geometric morphisms between topoi preserve finite limits, the NNO should also be preserved, and that includes 2.
When the topos we choose/define is (equivalent to) Set, then we get the standard ordinal-number definition of 2.
To use a pun, this lets us upgrade from Dedekind-categoricity to a more modern and natural sort of categorical categoricity.
Of course we can escape set theory (and categories) when talking about the natural numbers. The concept of "two" predates the concept of a set by at least a thousand years. People were happily manipulating and investigating the natural numbers before set theory ever came along.
As I said previously, it is true that set theory provides a way to encode the natural numbers as sets (or features of a topos, etc.), so that questions about natural numbers can be stated as questions about sets. It is further true that this endeavor can be incredibly fruitful, for instance for studying the foundations of mathematics. But it does not mean that natural numbers "are" sets (or objects in a topos), any more than Quicksort "is" a piece of C++ code.
> It's always worth recalling why we bother with set theory.
Because we limit ourselves to First Order logic, which can quantify over only individuals.
In order to quantify over multitudes (sets, groups, whatever) you either need Second Order logic or else a First Order axiomatization of how these multitudes behave.
We chose the latter rather than the former because Second Order logic has no Sound, Complete, and Decidable proof theory. First Order logic does.
That is cheating, though, since you cannot even formulate well-ordering conditions categorically in first-order logic, neither can first-order logic distinguish between finite and infinite domains.
Obviously sets are useful things, so there are those who study them in depth. For the rest of us, sets are just freaking convenient if you want to talk about something without too much hand waving.
The original philosophical foundation of mathematics would be Pythagoreanism [1].
For instance, that there is a special, transcendent meaning to "oneness" or "twoness" — or more generally, that there is a basic harmony within mathematics that manifests in the harmonies of the cosmos.
That's fine. It's just that when you get to the very bottom of axioms, it can get a little weird. Like, is the underlying basis the one, the nothing or the all? Is it being or not being? That is metaphysical -- and has implications for the foundation of any axiom, no?
I'm not sure what it even means for the underlying basis to be one or nothing or everything (basis of what specificly?). That doesn't sound like an axiom thing to me.
The modern way to talk about this stuff is via categorification; [0] is a good high-level introduction.
[0] https://math.ucr.edu/home/baez/quantization_and_categorifica...