$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\U}{\mathfrak{U}}$ $\newcommand{\hR}{^\ast\R}$ $\newcommand{\A}{\mathbb{A}}$ Below I will define everything for the sake of rigor, completeness, and lack of ambiguity, but first:
Questions:
Is there standard terminology for the notion called "Archimedean Growth Class" below? If so, would it be possible to provide one or more references defining the notion and discussing some of its basic properties?
Given two positive real number sequences $(a_n)$, $(b_n)$ with subsequences $(a_k)$, $(b_k)$ such that $a_k = \omega(b_k)$ as $k \to \infty$, or equivalently $b_k = o(a_k)$ as $k \to \infty$, is it true that there exists a choice of free ultrafilter on $\mathbb{N}$ such that, for the resulting construction of the hyperreal numbers $\hR$, we have that the Archimedean Growth Class $\A(\U(a_n))$ of $\U(a_n)$ is greater than the Archimedean Growth Class of $\A(\U(b_n))$, i.e. $\A(\U(a_n)) > \A(\U(b_n))$, i.e. $\U(a_n) / \U(b_n)$ is infinite, i.e. $\U(b_n) / \U(a_n)$ is infinitesimal? If so, are stronger versions of this result also true? (E.g. logical equivalences, weaker conditions, etc.)
The answer to 2. might be false based on this answer to a related question, although to be honest I don't think I understand the definition of free ultrafilter well enough to be sure, nor do I understand that question's emphasis on specifically Cauchy sequences (given that those are used for constructing the reals from the rationals using Cantor's construction).
Here and below I would like to restrict consideration (at least temporarily) to only positive real and hyperreal numbers for the sake of simplicity.
Definitions (Archimedean Growth Classes):
Assume that the hyperreal numbers $\hR$ and the (standard) real numbers $\R$ have been defined. Then a positive hyperreal $x > 0$ is "infinitesimal" if and only if $x < r$ for every (standard) real number $r \in \R$. Similarly, a positive hyperreal $x > 0$ is "infinite" if and only if $x > r$ for every (standard) real number $r \in \R$.
Call a positive hyperreal $x > 0$ "finite" if and only if $x = r + \epsilon$, $x = r$, or $x = r - \epsilon$, for some (standard) real $r \in \R$ and some infinitesimal hyperreal $\epsilon$. Notice that the finite (positive) hyperreals are closed under both pairwise addition and pairwise multiplication, and hence also under (standardly) finitely many iterations of both operations.
Given two positive hyperreal numbers $x_1, x_2 > 0$, call $x_1$ and $x_2$ "Archimedean equivalent", $x_1 \sim x_2$, if and only $\frac{x_1}{x_2}$ is a finite hyperreal number (or equivalently $\frac{x_2}{x_1}$ is a finite hyperreal number). This is an equivalence relation, so for any positive hyperreal $x > 0$ define its equivalence class under this relation its "Archimedean Growth Class", denoted $\A(x)$. In other words, the set of all positive hyperreal numbers such that $\frac{y}{x}$ is finite.
Observations (Archimedean Growth Classes):
Notice that the finite hyperreals (trivially) form an Archimedean Growth Class, and that they are the only Archimedean Growth Class closed under pairwise multiplication (again, trivially). However, all Archimedean Growth Classes are closed under pairwise addition (which is the source of the name, i.e. referring to the Archimedean property).
It is also possible to define a total order on Archimedean Growth Classes as follows: say that $\A_1 > \A_2$ if and only if for all $x_1 \in \A_1$ and all $x_2 \in \A_2$ one has that $\frac{x_1}{x_2}$ is infinite, or equivalently $\frac{x_2}{x_1}$ is infinitesimal.
The infinitesimal (positive) hyperreals correspond to all hyperreals belonging to Archimedean Growth Classes strictly smaller than the finite Archimedean Growth Class, while the infinite (positive) hyperreals correspond to all hyperreals belonging to Archimedean Growth Classes strictly larger than the finite Archimedean Growth Class. As far as I can tell, the total order is dense both when restricted to infinitesimal (positive) hyperreals and when restricted to infinite (positive) hyperreals, consider e.g. all $x^{\alpha}$ for all $0 < \alpha < 1$, for $x$ infinitesimal or infinite respectively.
In "standard" / "trivial" applications of the hyperreals, e.g. defining limits, derivatives, or continuity, the use of infinitesimal hyperreals, i.e. those belong to Archimedean Growth Classes strictly smaller than the finite Archimedean Growth Class, can "always" be replaced by using $o(1)$ sequences in the "delta-epsilon" definitions of those concepts using real numbers only. Analogously for infinite hyperreal numbers and $\omega(1)$ sequences.
This suggests that the notion of "Archimedean equivalence" in the hyperreal numbers is heuristically similar / analogous to asymptotic equivalence (or even the "big Theta" relationship) between sequences of (positive) real numbers. But then it occurred to me that this correspondence might actually be "semi-rigorous", or even could possibly made rigorous, by thinking about hyperreal numbers as equivalence classes of sequences of real numbers via the ultrapower construction.
Definitions (Ultrapower Quotient Map $\U$):
So assuming a choice of free ultrafilter has already been made, we get a quotient map $\U$ from the space of all real sequences to the hyperreal numbers. So given a real sequence $(a_n)$, $\U(a_n)$ denotes the hyperreal number corresponding to the equivalence class containing $(a_n)$ for the given (implicit) choice of free ultrafilter. (Really the notation for $\U$ should be indexed by the choice of free ultrafilter but I'm lazy and didn't want to complicate the notation further.)
Observations (Ultrapower Quotient Map $\U$):
Anyway the observations above seem to be that, for any sequence $(a_n)$ of positive real numbers for which $a_n = \omega(1)$ as $n \to \infty$, one should have $\U(a_n)$ belongs to an Archimedean Growth Class strictly greater than the finite Archimedean Growth Class (even regardless of the choice of free ultrafilter). And likewise for sequences $(b_n)$ with $b_n = o(1)$ as $n \to \infty$ corresponding to hyperreals belonging to Archimedean Growth Classes strictly smaller than the finite Archimedean Growth Classes follow.
And so then all of the standard questions follow, e.g. to what extent can this be generalized, to what extent do generalizations depend on the choice of free ultrafilter (e.g. does it determine which subsequences we can ignore vs. which are "authoritative"), what about "nasty" sequences without well-defined asymptotic growth rates (e.g. $c_n = \log(n)$ for $n$ odd and $c_n = n$ for $n$ even), to what extent do logical converse results also hold, etc.
In general: to what extent are the similarities between "Archimedean equivalence" of (positive) hyperreals and "Big Theta equivalence" of sequences of (positive) reals more than just analogies?
If there are references where a notion corresponding to "Archimedean Growth Classes" is defined and study somewhere, it seems likely that such a reference would answer most of these questions, which is why I am interested in looking for one.
