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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?

BlackSwan
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  • What do you mean with the claim that natural numbers are not unique in first order logic ? – Peter Aug 12 '20 at 11:15
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    "No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axiom systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as second-order logic." https://en.wikipedia.org/wiki/First-order_logic – BlackSwan Aug 12 '20 at 11:38
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    But the numbers itself are unique, we just cannot characterize the set of all natural numbers. PA is too weak for that, we need ZF. – Peter Aug 12 '20 at 11:41
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    ZFC, as a first-order logic, cannot characterize the set of all natural numbers, either. – BlackSwan Aug 12 '20 at 11:43
  • Hm, I read something different. Joerg Resag ( a german author, but not a mathematician ) wrote that ZFC is strong enough to classify the set of natural numbers. – Peter Aug 12 '20 at 11:45
  • Can you provide the reference? What I have read is that ZFC, as a first-order logic, determines uniquely (up to isomorphism) the natural numbers, when working from a given model of ZFC. But "in general" (i.e. using arguments valid for any model), it is not true that ZFC determines uniquely the natural numbers. – BlackSwan Aug 12 '20 at 11:47
  • Well, I admit, he did not claim that there is only one possibility for the classification. However, it appears weird that there should be more than one possibility upto isomorphism. I must however admit, that I am no expert in set theory either. – Peter Aug 12 '20 at 11:49
  • That is exactly my point. I think many (most?) people give a lot of importance to be able to convince themselves that their preferred choice of axioms for the natural numbers determines uniquely those natural numbers (up to isomorphism). But my question is: why should uniqueness be so important? – BlackSwan Aug 12 '20 at 12:07
  • ... In the end, the axioms of the natural numbers are just properties of the "real natural numbers" that we "elevate" to truths, with the hope that these properties (now, axioms) gather the main characteristics of the natural numbers. But hoping that our choice of properties is so good, that these properties really define uniquely the natural numbers is possibly "too much". – BlackSwan Aug 12 '20 at 12:07
  • @Peter is wrong. If ZFC is consistent, then so is ZFC+¬Con(ZFC), which proves an obviously false sentence about the natural numbers. Yet ZFC+¬Con(ZFC) is an extension of ZFC, so obviously ZFC failed to pin down the natural numbers. Besides Asaf's answer, which I completely agree with, see also this post. – user21820 Aug 18 '20 at 07:09

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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.

Asaf Karagila
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    (I think Platonistically those are completely different situations.) – Andrés E. Caicedo Aug 12 '20 at 12:32
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    (Which situations?) (Also why are we whispering?) – Asaf Karagila Aug 12 '20 at 12:32
  • When you write "(...) prove the uniqueness of these objects, up to isomorphism, within a given universe of set theory", what do you mean by "a given universe of set theory"? – BlackSwan Aug 12 '20 at 12:34
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    (Because it is tangential to your main point. Anyway, Platonistically, universes of ZFC with nonstandard natural numbers are just false, and there is no need to argue about them. The status of CH still requires some arguing.) – Andrés E. Caicedo Aug 12 '20 at 12:37
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    (@AndrésE.Caicedo: If I don't believe in Woodin cardinals, and you do, then we fundamentally disagree on the natural numbers. And if we're both Platonists, that throws a wrench into the natural numbers.) – Asaf Karagila Aug 12 '20 at 13:24
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    @BlackSwan: ZFC is a first-order theory, it can have different models, which are de facto different universes of set theory. – Asaf Karagila Aug 12 '20 at 13:26
  • (Crazy Platonist, that one. The problem in that case is not with Platonism.) – Andrés E. Caicedo Aug 12 '20 at 13:28
  • Asaf, this is probably the part that I do not understand: since as you argue, ZFC has different universes of set theory, it must have "different natural numbers". How do you reconcile that? You pick a given universe, the standard one? Or anything else? – BlackSwan Aug 12 '20 at 13:45
  • @BlackSwan: Platonists would reconcile this by saying that there is a universe, and that's it. The numbers there are the numbers there, and ZFC does in fact prove that. Others might say that there are multiple universes, and these may share the same natural numbers (e.g. in the context of the generic multiverse where all the universes are related by a forcing extension), or they may disagree on the natural numbers because we are considering another kind of multiverse. But within each universe of set theory, which itself is a universe of mathematics, the natural and real numbers are unique. – Asaf Karagila Aug 12 '20 at 14:26
  • @AsafKaragila But how do these universes look like? Granted, it is not possible to know, since it is not possible to give a model for ZFC. But Von Neumann universe (the hierarchy of sets) would be such a model, if V were a set (it is not; it is a class). Do I understand correctly that "our natural numbers" are the ones appearing in the Von Neumann universe, in ZFC? And in other universes, how these "alternative" natural numbers would "look like"? – BlackSwan Aug 12 '20 at 15:51
  • @BlackSwan: I wouldn't say that "our natural numbers" appear in V in ZFC. Rather, ZFC has some axioms that imply that a certain minimal inductive set exists, and we choose to call that set $ω$, and from that we can build the structure $(\mathbb{N},0,1,+,·)$ and prove that that structure is the unique one satisfying PA with the second-order induction axiom under full semantics, up to isomorphism. How do we know that that structure is "our natural numbers" (in a platonic sense)? We can't. But if you believe ZFC is meaningful, then you do believe that it is. – user21820 Aug 18 '20 at 07:15
  • If ZFC turns out to be Σ1-unsound, or worse still inconsistent, then what it says about what structures exists won't be sensible anymore. If you believe there is in fact a distinguished platonic structure of natural numbers, then of course you should desire your foundational system to prove only true sentences about that structure. This is partially captured by the notion of ω-models of ZFC. Just by definition, all ω-models of ZFC have isomorphic naturals... An example of an extension of ZFC that has no ω-model is the one I gave in another comment, namely ZFC+¬Con(ZFC). – user21820 Aug 18 '20 at 07:21