Ordering the hyperreals and infinitesimals

Question

I'm just getting into the hyperreals and infinitesimals and I would like to understand how one can determine when e is <, = or > g (where e and g are elements of the the infinitesimals). How does one order this field?

Answer 1

The ordering of a$^1$ hyperreal field is part of its algebraic structure: we say $a<b$ iff $b-a$ has a square root (and this works for all $a,b$ in the field). So as soon as you know the algebraic structure, you know the ordering.

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More interestingly, your question could also be asked in terms of representations, analogously to decimal representations of real numbers: is there an analogue of decimal representation for hyperreals such that comparing representations is "reasonably simple"?

Unfortunately, the answer to this question is no: in a precise sense, hyperreal fields are "less explicitly describable" than $\mathbb{R}$ itself. For example, in the usual construction of a hyperreal field $H$ via an ultrapower over $\mathbb{N}$ our hyperreals are equivalence classes of sequences of real numbers, and each sequence in a given class can be thought of as a "name" for that class. Now so far this is reminiscent of the definition of reals as equivalence classes of Cauchy sequences; however, it turns out that however we build this ultrapower there will be no good way to either pick canonical representations of hyperreals or to compare two representations in general.

In fact, it's consistent with ZF (= set theory without the axiom of choice) that no hyperreal fields exist at all. This means that we can never have too concrete a representation system for a given hyperreal field, since otherwise we could in ZF alone reconstruct that field from that representation system.

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$^1$Per Mark S.'s comment above, there isn't a single hyperreal field; roughly speaking, a hyperreal field is any field which contains $\mathbb{R}$, is sufficiently rich, and has an appropriate transfer principle.