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I’m working through Alain M. Robert’s “Nonstandard Analysis”.

I’m intrigued by his suggestion that his axiomatic approach, rather than adding elements to ℕ to create a nonstandard model *ℕ, “discerns more structure” in ℕ itself via the idealization, transfer, standardization axioms.

But soon enough, he ends up talking about “nonstandard” elements of ℕ, such as “illimited” integers bigger than every ordinary successor of 0. This seems contradictory to me. Standard ℕ doesn’t contain any infinitely large integers. So why is the student being told “We’re not adding any new elements to ℕ“, if the approach sneaks in new elements one way or another?

Rivers McForge
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    "This seems contradictory to me." That might be so, but it's not actually contradictory. The ordinary axioms of set theory are not strong enough to rule out that $\mathbb{N}$ contains such nonstandard numbers (see e.g. any proof of the compactness theorem), which is the whole point. It allows Robert to go "all in" on them: if you can't prove they don't exist anyway, might as well add some new axioms which allow you to show that they do exist, and do something useful with them. – Z. A. K. Sep 18 '23 at 01:12
  • I've not read Robinson's work, but I think that this idea works best in a fully relativized system a la Hrbacek. See for instance Analysis with Ultrasmall Numbers by Hrbacek, Lessmann, and O'Donovan or Relative Set Theory: Internal View by Hrbacek. In these theories we have a relative observability predicate between objects. So in the natural numbers you would have some numbers that are "infinite" relative to other numbers, but there are no "globally infinite" numbers because of course a number is never infinite relative to itself. – Nicholas Todoroff Sep 18 '23 at 01:18
  • But we do have numbers which are observable in all contexts; these are essentially any of the numbers you can express explicitly in some way. – Nicholas Todoroff Sep 18 '23 at 01:18
  • @NicholasTodoroff Right, I understand the new system will be *-Archimedean, for instance, but at the same time, saying it has new natural numbers that aren’t present in the original model, contradicts saying you’re just working with the original model under some new axioms. Either it is the original, or it has new nonstandard stuff in it, but not both. – Rivers McForge Sep 18 '23 at 01:24
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    @RiversMcForge: Again, which new natural numbers are those?! Every model $(M,st,\in_M)$ of IST has both standard and nonstandard elements in its $\mathbb{N}$. Of course, using the ZFC axioms in the language $(\in)$ which lacks the $st$ predicate, you can't really say anything about them, but that doesn't mean that there's a model of IST which doesn't contain the nonstandard elements. You don't need to add any new elements to $M$, you just need a larger theory with axioms about the predicate $st$ so you can refer to them. Which is precisely what "not adding new elements" means here. – Z. A. K. Sep 18 '23 at 04:52
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    @RiversMcForge, would you agree with the following definition of $\mathbb N$: "it is the smallest inductive set" ? – Mikhail Katz Sep 19 '23 at 10:35

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The OP wrote: "Standard $\mathbb N$ doesn't contain any infinitely large integers." This is true, but $\mathbb N$ can contain unlimited integers. To clarify this, it is helpful to compare the model-theoretic and the axiomatic approaches to NSA. In the model-theoretic approach, there are three types of entities: (m1) natural extensions of standard sets, (m2) internal sets, and (m3) external sets. In axiomatic approaches such as Edward Nelson's IST (which is the basis of Robert's book) , there are only two types of entities: (a1) standard and (a2) nonstandard. Standard entities (a1) correspond to natural extensions of standard sets (m1), and nonstandard entities (a2) correspond to internal sets (m2). There are no entities in IST corresponding to external sets (m3). For example, the $\mathbb N^\ast$ of (m1) corresponds to $\mathbb N$ of (a1). Just as the $\mathbb N^\ast$ of (m1) contains a nonstandard integer $\mu$, so does the $\mathbb N$ of (a1) contain a nonstandard integer $\mu$. In the context of model-theoretic approaches, such a $\mu$ would sometimes be called infinite or unlimited. In the context of axiomatic theories, $\mu$ can be called unlimited, but it is preferable not to refer to it as infinite to avoid confusion (after all, being a member of $\mathbb N$, the number $\mu$ is finite by definition).

How does one provide a satisfying philosophical picture for such a situation? I would like to emphasize that several approaches are possible. Since we are discussing Nelson's IST, I will outline Nelson's own picture. Nelson would say that the unlimited integers have been there in $\mathbb N$ all along, it's just that we couldn't detect them because we lacked the tools to do so. The necessary tool is the standardness predicate st that in particular distinguishes between standard and nonstandard integers. So Nelson would say that it is the original $\mathbb N$ and it has nonstandard stuff in it (though the latter only seems "new"). To put it another way, the richer st-$\in$-language enables us to discern more entities than the $\in$-language of set theory such as ZF (here $\in$ is of course the membership relation).

I should emphasize that this is only one possible philosophical picture among a few; see this article for more details. Meanwhile, the precise mathematical situation is as outlined in the first paragraph above. See also this related post.

Mikhail Katz
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