Can the hyperreal numbers have a property akin to completeness by considering hypernatural sequences?

Question

I’ve seen that $\mathbb{R}^*$ isn’t complete as many Cauchy sequences won’t converge, and that includes power series. In other stack exchange posts, I’ve seen that even the exponential function won’t converge if the sum is only taken over the natural numbers, but will converge when taken over all hypernatural numbers. This got me thinking if that’s the answer to the completeness problem on the hyperreals. Would using these longer sequences ensure completeness?

Answer 1

Working in nonstandard analysis, it is customary to extend many notions, such as that of a sequence or of a Cauchy sequence, by taking the *-transform of the corresponding notion in the standard world (a disclaimer: for the sake of simplicity, in this answer I use "standard" as a synonym for "belonging to the superstructure based on the usual real numbers". This is, unfortunately, not the only meaning of this term).

The standard notion of a sequence is that of a function $f: \mathbb{N}\to \mathbb{R}$, so the corresponding notion of *sequence in nonstandard analysis is $f:\ \!^*\mathbb{N}\to\ \!^*\mathbb{R}$ that satisfy an additional technical hypothesis that I will discuss below. Once this has been done enough times, the * is usually omitted: in nonstandard analysis, the word "sequence" is shorthand for "*sequence". One extends the notion of Cauchy sequence in the same way. With these caveats, the transfer principle entails that all Cauchy sequences converge (actually, there is no need to use the transfer principle since the *-transform of the usual proof is correct as well).

I mentioned the transfer principle because it ties in with the answer to the second part of your question: it is not the use of "long sequences" that ensures completeness of $\ \! ^*\mathbb{R}$, but the use of internal sequences (this is the technical hypothesis that was needed above). Basically, the transfer principle states that rules, laws or procedures valid for the real numbers, still apply (i.e., are ‘transferred’) to the hyperreals, provided that one works only with internal objects.

Functions $f: \mathbb{N}\to\ \! ^*\mathbb{R}$ are not internal, so the transfer principle does not apply to them. Notice also that there are also some functions $f:\ \!^*\mathbb{N}\to\ \!^*\mathbb{R}$ that are not internal, such as $f(n)=0$ if $n \in \mathbb{N}$ and $f(n)=n$ if $n \in \ \!^*\mathbb{N} \setminus \mathbb{N}$. The transfer principle does not apply to those, as well.

The definition of an internal object is deceivably simple: $T$ is internal if $T \in \ \! ^*A$ for some standard $A$. However, most of the time this is not sufficient to get a feeling of what objects are internal and what objects are not. For this topic, it's best to refer to some introductory books to nonstandard analysis.