First, it would be nice if one could go from rationals to hyperreals directly without having to define the reals in between (especially for people with limit allergies, as the reals are sometimes defined as limits of Cauchy sequences). I don’t see a straightforward way to do so though, you can hardly allow people to encode their reals as sequences of rationals, otherwise the(1/n) sequence would have to be equivalent to zero instead of an infinitesimal.
Also, one could split the hyperreals into equivalence classes within which the Archimedian property holds. Using the big-O adjacent notation, the reals would be Θ(1), and the hyperreal called Ω above would be Θ(n). Stretching the big-O notation, one could call the equivalence class of ϵ something like Θ(1/n). So one has a rather large zoo of these equivalence classes. This would imply that there is no Archimedian equivalence class for the smallest infinite hyperreal. If a hyperreal Θ(f(n)) is infinite (that is, f(n) diverges), then Θ(ln(f(n)) is a smaller infinite hyperreal.
I am well used to there being no biggest infinity, but there being no smallest infinity would indicate that these things are neither equivalent to cardinals nor ordinals.
Some random thoughts.
First, it would be nice if one could go from rationals to hyperreals directly without having to define the reals in between (especially for people with limit allergies, as the reals are sometimes defined as limits of Cauchy sequences). I don’t see a straightforward way to do so though, you can hardly allow people to encode their reals as sequences of rationals, otherwise the(1/n) sequence would have to be equivalent to zero instead of an infinitesimal.
Also, one could split the hyperreals into equivalence classes within which the Archimedian property holds. Using the big-O adjacent notation, the reals would be Θ(1), and the hyperreal called Ω above would be Θ(n). Stretching the big-O notation, one could call the equivalence class of ϵ something like Θ(1/n). So one has a rather large zoo of these equivalence classes. This would imply that there is no Archimedian equivalence class for the smallest infinite hyperreal. If a hyperreal Θ(f(n)) is infinite (that is, f(n) diverges), then Θ(ln(f(n)) is a smaller infinite hyperreal.
I am well used to there being no biggest infinity, but there being no smallest infinity would indicate that these things are neither equivalent to cardinals nor ordinals.