I think it’s highly debatable whether the natural numbers are built at all. Arguably they’re just there (in some sense). One can construct particular “implementations” of the natural numbers, and there are many ways to do it; for instance, the usual way to do it in NF is a Frege-like construction: natural numbers are equivalence classes of finite sets under the relation “can be put in bijection with”; “finite” means “can’t be put in bijection with any finite subset”. (You can’t do that in ZF(C) because there are too many finite sets, but perhaps you can do it in a system that adds proper classes, like NBG.)
I don’t have strong feelings about how the natural numbers “should” be built, or what they “really” are. I’m happy thinking of them as “sizes of finite sets”, or as the things you get if you start at 0 and add 1 as often as you like (though there’s a certain circularity about that definition), or even as finite sequences of bits (though, again, there’s some circularity there). I don’t think it’s coincidence that these all lead to the same theorems, but I don’t feel any particular need to pick one definition and say that the others are all somehow parasitic on it.
Incidentally, when Frege came to define the natural numbers (this was in 1884, a few years before the usual Peano axioms were formulated, and I think he was the first person to do anything of the kind) he did it by (1) defining cardinal numbers as equivalence classes of sets under the same-size relation, and then (2) saying that a natural number is anything you can get to 0 from by counting downwards. Make of that what you will.
I think it’s highly debatable whether the natural numbers are built at all. Arguably they’re just there (in some sense). One can construct particular “implementations” of the natural numbers, and there are many ways to do it; for instance, the usual way to do it in NF is a Frege-like construction: natural numbers are equivalence classes of finite sets under the relation “can be put in bijection with”; “finite” means “can’t be put in bijection with any finite subset”. (You can’t do that in ZF(C) because there are too many finite sets, but perhaps you can do it in a system that adds proper classes, like NBG.)
I don’t have strong feelings about how the natural numbers “should” be built, or what they “really” are. I’m happy thinking of them as “sizes of finite sets”, or as the things you get if you start at 0 and add 1 as often as you like (though there’s a certain circularity about that definition), or even as finite sequences of bits (though, again, there’s some circularity there). I don’t think it’s coincidence that these all lead to the same theorems, but I don’t feel any particular need to pick one definition and say that the others are all somehow parasitic on it.
Incidentally, when Frege came to define the natural numbers (this was in 1884, a few years before the usual Peano axioms were formulated, and I think he was the first person to do anything of the kind) he did it by (1) defining cardinal numbers as equivalence classes of sets under the same-size relation, and then (2) saying that a natural number is anything you can get to 0 from by counting downwards. Make of that what you will.