Should we be worried at all there is no uniqueness for the natural numbers in first-order ZFC?

Question

There are quite a few posts devoted to the uniqueness of the naturals numbers. Natural numbers are unique (up to isomorphism) in second-order logic. But in first-order logic (where we can prove things), the natural numbers are not categorical (i.e. unique up to isomorphism).

Explicit constructions of the real numbers (e.g. Dedekind cuts, etc) are based on a specific model of ZFC. But the same explicit construction of the real numbers, for a different model, could result (and it results) in different real numbers.

It seems many people want to have the uniqueness property of the naturals / reals, so they resort to second-order logic.

But sticking to first-order logic is important, for provability issues. So, the question is: do we lose much by not having uniqueness of the naturals / reals?

First, I would say it would be really surprising that we could find a few properties of the "real natural numbers" (in a Platonic sense), and then find that these few properties uniquely characterize the "real natural numbers". True, it would be great if this happened. But it seems we would be "too lucky".

As a consequence, I think it is natural that once we agree on the "few properties" of the "real natural numbers", and we set them as axioms, we should be too bothered about having other objects, different from the "real natural numbers", also satisfying those axioms. In fact, this should not bother us at all.

What we should bother us is, I think:

1. These axioms are properties of the "real natural numbers". This is really basic, and if it does not happen, we should throw these axioms to the bin.
2. These axioms do result in a property which is known not to be true for the "real natural numbers". If this happens, we should throw these axioms to the bin.
3. These axioms should result in finding new true results, which when applied to the "real natural numbers", make us learn something new (and true) about the "real natural numbers", that we did not know before.

If 1, 2 and, especially, 3, are correct, then we (temporarily, at least) accept these axioms as "the axioms of the real natural numbers". But of course, we could change these axioms if either 1, 2, or 3 do not apply, or even when 1, 2 and 3 being true, we find another set of axioms that satisfy 1, 2 and 3 "better".

But we should not worry at all about the fact that there are other objects (which cannot be understood as "real natural numbers", since they do not satisfy 1, 2 or 3) that also satisfy these axioms.

Does this make sense?

Answer 1

No. There's nothing to worry about. And in fact, the way you present the results is somehow skewed. Let me try and clarify.

Working in ZFC, the second-order theory of the natural numbers becomes a first-order objects internal to the universe of ZFC. And we can, in fact, prove that the uniqueness of the natural numbers as the only model of the second-order Peano axioms (up to isomorphism, of course).

You are talking about the uniqueness of the real numbers as the only Dedekind-complete ordered field, which is again something that ZFC proves. But the completeness axiom is not a first-order axiom. Indeed, any real-closed field is elementary equivalent to the real numbers, so $\overline{\Bbb Q}\cap\Bbb R$ (here $\overline{\Bbb Q}$ is the algebraic closure of the rationals) is an elementary submodel of $\Bbb R$, which is quite a lot more than just saying that we cannot characterise the real numbers with first-order axioms.

But this is exactly why we work in a stronger foundation which allows us to internalise second-order logic, and prove the uniqueness of these objects, up to isomorphism, within a given universe of set theory.

If you want to worry about something Platonistic, you may worry about how different universes of ZFC may have different theories of the natural numbers (e.g. is Con(ZFC) true there or not), but this is no different than worrying whether or not the Continuum Hypothesis is true or false, from a Platonistic point of view.