With newfound freetime during the pandemic, I have been studying non-standard analysis. I wasn't too fond of ultrafilters, so I've gravitated toward Nelson's internal set theory and Hrbacek set theory. Although I prefer the latter, I have more experience with Nelson's work, so I will phrase things in terms of IST.
I have basic knowledge of ordinal numbers in set theory, of which $\omega$ is the first. I want to know where the set fits into IST. Is it simply a standard hyperfinite number? Intuitively, the fact that $\omega > n$ for every natural number $n$, caused me to assume that $\omega$ could be a member of ${}^*\mathbb{N}$, as this is the defining property of these natural numbers. I found a paper (Taras Kudryk et al., 2004) mentioning standard hyperfinite integers that proved in its Proposition 2.1 that:
There exists a $\mathbf{standard}$ R-infinite [i.e. in ${}^*\mathbb{N}\setminus\mathbb{N}$] hypernatural number.
As I understand it, every set uniquely defined in ZFC without reference to the standard predicate is standard. Hence, the first transfinite ordinal $\omega$ is a standard set. With this, I have been hoping to prove that $\omega\in{}^*\mathbb{N}\setminus\mathbb{N}$. However, at the same time, I recall that there is no least hyperfinite natural number. This seems to contradict the fact that $\omega$ is the least ordinal number.
At this point, my lack of experience with set theory is probably showing. Looking at questions discussing the differences between $\omega$ and $\mathbb{N}$ makes me realize that I might be over my head here. Could I have some clarification from those with more experience with set theory and its non-standard extensions? Where does $\omega$ (and really the ordinal numbers in general) fit into IST?
Best Answer
The smallest transfinite von Neumann ordinal $\omega$ and the elements of ${}^*\mathbb{N}\setminus\mathbb{N}$ are different sorts of objects altogether. Asking "does $\omega$ belong to the set ${}^*\mathbb{N}\setminus\mathbb{N}$?" does not make much sense, the same way asking "does the group $S_3$ contain the set $\mathbb{R}$ as an element?" does not make much sense.
I can arrange a situation where the answer to the latter question is technically yes. E.g. by defining the group $S_3$ as the group with underlying set $S_3 = \{A,B,C,D,E,\mathbb{R}\}$ and with multiplication table
we have not only that $\mathbb{R} \in S_3$, but also that $\mathbb{R}$ is the identity element of $S_3$. This is of course a meaningless technicality, and should not be mistaken for a mathematical relationship between the group $S_3$ and the real numbers $\mathbb{R}$.
Depending on your construction of the extension ${}^*\mathbb{N}$, you can similarly arrange to make $\omega \in {}^*\mathbb{N} \setminus \mathbb{N}$ hold, but that does not teach you anything about the ordinals: you could arrange e.g. $\mathbb{R} \in {}^*\mathbb{N}$ in exactly the same way.
With that out of the way, is there a natural mathematical way in which the ordinal $\omega$ corresponds to some fixed nonstandard natural number? The answer to that question is no, and it remains no even if we replace "fixed nonstandard natural number" with "fixed standard element of ${}^*\mathbb{N}\setminus\mathbb{N}$ where ${}^*\mathbb{N}$ denotes some standard hyperextension of $\mathbb{N}$" (in fact, I'd suggest avoiding these mixed IST and Robinsonian NSA notions until you become much more comfortable with both formalisms).
The same goes for your implicit question about obtaining "concrete" nonstandard numbers: you won't be able to pin down any concrete nonstandard number using the IST axioms. The only way to construct nonstandard numbers is via Idealization (if you omit the Idealization axiom from IST, it's consistent with the resulting system that all objects are standard), and one can construct models of IST where every specification by Idealization (essentially every non-isolated 1-type) is realized by at least two different elements of the model.