Are all Hyperreal Infinitesimals representable by Monotonically Decreasing Sequences to 0?

Question

I know there are many possible theoretical ways to built *R, including axiomatic and set-theoretic approaches. I am limiting my attention specifically to the Superstructure approach, perhaps best outlined by Robert Goldblatt's "Lecture on the Hyperreals": involving infinite sequences of real numbers, using a non-principal ultrafilter U to combine them into representative classes. Thus any hyperreal h can be represented by a sequence $\langle h_0, h_1, h_2, \ldots \rangle$. (And, of course, this representation is not unique, as there are an infinite number of such representations, for each h.)

Since all limited hyperreals can be uniquely described as r + e (r a real number, e an infinitesimal), my interest is on infinitesimals. Specifically, my question is:

• Is it true that every positive infinitesimal can be represented by a monotonically decreasing sequence converging to 0? Or more formally: Given any positive infinitesimal e, does there exist a representation $\langle e_0, e_1, e_2, \ldots \rangle$ of e, such that: e_i > e_i+1 for all i in N, e_n's all positive, and lim e_n = 0?

Certainly some of them can (such as: $\langle 1, 1/2, 1/3, \ldots \rangle$). But are we guaranteed that this is true in all cases? And I realize that the choice of Ultrafilter U may play a part. It seems to me the answer could be one of the following:

1. Yes, this is always true.
2. No, this is always false.
3. No, it is false, but could be made true if we dropped the requirement of monotonicity.
4. It entirely depends on the choice of U.
5. It is unknown if this is true or false (or if the choice of U plays a role).

(In the event of #4 as the answer, is there anything we can say about this Ultrafilter choice?)

I am not making any assumptions about the Continuum Hypothesis, Martin's Axiom, or any other axiom at this time. (Although I would assume that CH would have to be false in the case of #4, since CH $\to$ *R models are isomorphic.) This is more for an understanding on my part as to how generalized infinitesimals can be, relative to real convergent sequences to 0.

Thanks.

Answer 1

Great question! The answer is that it can depend on the ultrafilter. (Note that even if $\mathbb{R}^\mathcal{U}\cong\mathbb{R}^\mathcal{V}$ the ultrafilters $\mathcal{U}$ and $\mathcal{V}$ may yield very different representation systems, so in fact $\mathsf{CH}$ does not immediately trivialize things.)

Suppose $\alpha=(a_i)_{i\in\mathbb{N}}$ is a sequence of positive reals with the following properties:

• $\lim_{i\rightarrow\infty}a_i=0$.

• There is no finite $S_1,...,S_n\subseteq\mathbb{N}$ such that $\mathbb{N}=S_1\cup ...\cup S_n$ and on each $S_i$ the sequence $\alpha$ is monotonic.

Constructing such an $\alpha$ (very explicitly! no complicated set theory needed) is a fun exercise:

Consider something like $$1,\quad{1\over 3}, {1\over 2}, \quad{1\over 4},\quad {1\over 7},{1\over 6},{1\over 5},\quad {1\over 8},\quad ...,$$ which goes to zero but has arbitrarily large finite increasing subsequences.

The first bulletpoint ensures that such an $\alpha$ names an infinitesimal in every ultrapower of $\mathbb{R}$, but by the second bulletpoint we can whip up an ultrafilter $\mathcal{U}$ such that for each $S\subseteq\mathbb{N}$ if $\alpha$ is monotonic on $S$ then $S\not\in\mathcal{U}$, and this $\mathcal{U}$ will have the property that $\alpha\not\sim_\mathcal{U}\beta$ for any monotonic $\beta$.

On the other hand, every Ramsey ultrafilter will have the property that every infinitesimal has a "monotonic name." Note that the existence of Ramsey ultrafilters follows from $\mathsf{CH}$, so - elaborating on my parenthetical remark above - $\mathsf{ZFC+CH}$ proves both that all ultrapowers of $\mathbb{R}$ over $\mathbb{N}$ are isomorphic and that some but not all ultrafilters on $\mathbb{N}$ have the "all-infinitesimals-are-monotonically-named" property!

Finally, it turns out that the property "all-infinitesimals-have-monotonic-names" is equivalent to P-point-ness, and $\mathsf{ZFC}$ can't prove P-points exist. See Cutland/Kessler/Kopp/Ross 1988, which I learned about from Mikhail Katz's answer to the above-mentioned near-duplicate. (EDIT: Corazza's 2015 paper $P$-points in the construction of the real line seems to be a rediscovery of this CKKR result. I wouldn't be surprised if it's been rediscovered several times since both the problem and its solution are quite natural.)