Completion of nonstandard numbers allowed and isomorphic to non-negative, non-infinitesimal, integer hyperreals?

Question

I recently learned about nonstandard number systems satisfying Peano's axioms (when they use only first order logic). And this gives us what is often described by "$\mathbb N$ with densely many copies of $\mathbb Z$."

So we have that $\mathbb N=\{0,1,2,3,\ldots\}$ (all of the standard natural numbers) and (probably being quite loose and informal) let us define addition with $\mathbb Z$ (the standard integers) according to successor/predecessor rules, e.g. $n+1$ is the successor of number $n$ and $n-2$ is the predecessor of the predecessor of number $n$, etc. Let $\mathbb Z_1$ be a nonstandard copy of $\mathbb Z$ that is "beyond $\mathbb N$" and $\mathbf c$ be some (privileged) nonstandard number in $\mathbb Z_1$. Now let $\mathbb Z_q$ be the nonstandard copy of $\mathbb Z$ corresponding to $q\in\mathbb Q$, i.e. that $\mathbb Z_q=\{q\mathbf{c}+z \mid z\in\mathbb Z\}$. So the full collection of nonstandard numbers is $\mathbb N \cup \left(\bigcup_{q\in\mathbb Q_{>0}}\mathbb Z_q\right)$ with the appropriate ordering.

My questions: Can we "complete" this set of nonstandard numbers into something like $\mathbb N \cup \left(\bigcup_{r\in\mathbb R_{>0}}\mathbb Z_r\right)$? And if so, then is this completion simply a subset of the hyperreals or isomorphic to a subset (e.g. the non-negative, non-infinitesimal, integer hyperreals)?

We could just define equivalence classes of sets of nonstandard numbers or dedekind-like cuts of the set of sets of nonstandard numbers (with an order relation on the sets $\mathbb Z_q$ and whatever other structures are necessary). In this way we create a semi-infinite "continuum" of sets of nonstandard numbers where each point on the continuum corresponds to a nonstandard copy of $\mathbb Z$ (except the origin corresponding to the standard $\mathbb N$). I envision the structure being very much like the surreal numbers except that each point in $\mathbb R$ gets a copy of $\mathbb Z$ (a discrete set of "large numbers") instead of a copy of a continuum of infinitesimals.

The structure of such a completion seems like it would be similar to part of the set of hyperreal numbers (i.e. take only the non-negative hyperreals and no infinitesimals). Is the completion of the nonstandard natural numbers isomorphic to the non-negative, non-infinitesimal, integer hyperreals, i.e. $\{x \mid x\geq0, x\in\mathbb N \text{ or } x=r\omega+z \text{ for some } r\in\mathbb R_{>0}, z\in\mathbb Z\}$?

I'm mostly looking to confirm if this intuition is correct. I always welcome any insight whether prose or rigorous. Of course, let me know if I am making some grave error in anything above. I know that the notation and language above is not rigorous at all, but my hope is that someone sufficiently experienced here can understand what I am trying to get at.

Answer 1

First of all, there are lots of nonstandard models of $\mathsf{PA}$, including uncountable ones. Of course the ordertype of an uncountable model of $\mathsf{PA}$ can't be $\mathbb{N}+\mathbb{Z}\cdot\mathbb{Q}$ just for cardinality reasons.$^1$

The following is true however:

If $M\models\mathsf{PA}$ is countable and nonstandard, then the ordertype of $M$ is $\mathbb{N}+\mathbb{Z}\cdot \mathbb{Q}$.

So the "start" of your question is basically fine, you just need to restrict attention to the countable models.

Now on to the actual question. Surprisingly, it turns out that the class of possible ordertypes of uncountable models of $\mathsf{PA}$ is extremely complicated, and in particular we have:

There is no model of $\mathsf{PA}$ which has ordertype $\mathbb{N}+\mathbb{Z}\cdot\mathbb{R}$.

See e.g. here. So any completion process of the type you describe has to result in a structure quite different from what you start with, and indeed the argument shows that even very simple algebraic properties must be lost. In particular, you won't find an "algebraically nice" subset of the hyperreals (or more accurately, a hyperreal field - there is no such thing as "the" hyperreals) which will do the desired job.

---

$^1$Note that the ordertype alone doesn't determine the algebraic structure - we can see this e.g. by noting that the linear order $\mathbb{N}+\mathbb{Z}\cdot\mathbb{Q}$ is "computably describable" and then contrasting that with Tennenbaum's theorem. And the gap between order-theoretic and semiring-theoretic structure is exactly the obstacle to defining a "completion" process of the type you have in mind.

And in fact this points to an issue with your proposed construction of a nonstandard model of $\mathsf{PA}$. It's true that we can give a system of "names" to the elements of such a model which is relatively simple; however, that alone doesn't determine the algebraic structure (think about multiplying elements) and indeed there is no simple way to whip up an appropriate multiplication notion.