>>16712714 (OP)
There are several ways to think about this, so this is largely arbitrary. Take your pick.
The axiom of infinity in ZF gives the precise definition of the natural numbers as a set.
[eqn]\exists\mathbb{N}:\varnothing\in\mathbb{N}\land\forall{n}\left(n\in\mathbb{N}\implies n\cup\{n\}\in\mathbb{N}\right)[/eqn]
As you can see from the recursive definition, you need the empty set there as the first element in the recursion. You could take any other set, but then not all natural numbers correspond to finite cardinals.
If you want algebraic flavor to it, then the natural numbers can be defined as either a semigroup or a monoid under addition. The way you would canonically construct these is as free objects generated by the singleton set. This is more of a category-theoretic construction. If you choose it to be a semigroup, then zero isn't included. But then you're missing out on neat category-theoretic properties since composition in categories acts like a monoid.
