Construction of the Hyperreal numbers

Question

Several times I have seen questions/answers here about using the correct definition of derivatives. There are also questions about whether or not $1/0$ is defined. Sometimes there is a discussion about the concept of infinitesimal and infinite numbers which I guess are related to Hyperreal numbers ${}^*\mathbb{R}$ (correct me if I am wrong).

My question is

How do we construct/define the set of hyperreal numbers?

I am specifically interested in understanding what the hyperreals are as a set and how the hyperreals become an ordered field (This would necessarily include specifying what the inverses of the "extra" elements in ${}^*\mathbb{R}$ are). How, for example, does $\mathbb{R}$ sit inside ${}^*\mathbb{R}\,?$

Edit: I see that the question "Create the hyperreals using ..." already asks about this, but I am wondering if the construction of the set might be explained clearly.

Answer 1

There are a couple of Monthly articles that give reasonably accessible introductions. See

Wm. Hatcher. Calculus is Algebra, AMM 1982.

D.H. Van Osdol. Truth with Respect to an Ultrafilter or How to make Intuition Rigorous. AMM, 1972..

For a much more comprehensive introduction to ultraproducts see

Paul Eklof. Ultraproducts for Algebraists, 1977.

Answer 2

To motivate the construction of the hyperreals, one could compare it with the construction of the real numbers as equivalence classes of Cauchy sequences of rational numbers. Namely, hyperreals can be similarly constructed as equivalence classes of Cauchy sequences of real numbers. To motivate the construction, note that there is a lot of "collapsing" going on when one passes from a Cauchy sequence to its equivalence class. Namely, one loses all information about anything related to the rate of convergence of the sequence.

In fact, the construction of the hyperreals can be achieved by refining the Cauchy sequence construction. The refined equivalence relation will declare two sequences $(u_n)$ and $(v_n)$ to be equivalent if they agree on a "dominant" set of indices; i.e. the subset of $\mathbb{N}$ given by $\{n\in\mathbb{N} : u_n=v_n\}$ is "dominant". I will comment on the nature of "dominant" sets of indices in a moment. Now taking the set of equivalence classes will give you only finite (more precisely, limited) hyperreals. This includes infinitesimals, i.e. hyperreals represented by null sequences $(u_n)$ (i.e., sequences that tend to zero).

To make this into a field, one also needs the elements represented by sequences $(\frac{1}{u_n})$ where $(u_n)$ represents an infinitesimal. Note that such sequences $(\frac{1}{u_n})$ are no longer Cauchy. The punchline is that if one assumes CH then the result is the full hyperreal field (given an appropriate choice of ultrafilter, namely a P-point filter). Moreover a hyperreal field constructed via the ultrapower is unique up to isomorphism assuming CH (in particular, independent of the ultrafilter used).

To comment briefly on the notion of a "dominant" set of indices: here a finite set is never "dominant", and a cofinite set is always dominant. To get the job done one needs to choose, out of every pair of complementary infinite subsets of $\mathbb{N}$, precisely one that will be declared "dominant" in a coherent fashion. For the details you would need to look up the notion of an ultrafilter.

Edit 1. To respond to your question concerning the way $\mathbb{R}$ sits inside ${}^{\ast}\mathbb{R}$, note that the constant sequences $(u_n)$ where $u_n=r$ give an imbedding, namely a real number $r$ goes to the constant sequence $(u_n)$ as above.