Structure of Hyperreal Numbers

By now we know that the system of natural numbers can be extended to include infinities while preserving algebraic properties of the former. A similar statement holds for the real numbers that may be extended to include the infinitely large but also the infinitely small. We used the notation PA1 for Peano Arithmetic of first-order and PA1_∞ for its hyperinteger extension. The set of the hyperreals did not actually have analogous notations. Here we correct this asymmetry and introduce more common notations.

From now on, we'll use N and N to denote the set of integers and the theory of the first-order arithmetic (formerly PA1.) N^* and N^* will denote the set of all hyperintegers and the extension of N that was formerly denoted as AP1_∞. For the real numbers, whose set will be denoted R, R, R^*, R^* stand respectively for the (first-order) theory of real numbers, the set of the hyperreal numbers, and the theory of the hyperreals that extends R.

R^* contains infinitely large, infinitely small but also the "regular" reals R. When talking of R^*, the terminology is as follows:

the elements of R are called standard,
the elements of R^* - R are non-standard,
the elements s∈R^*, for which there is a real bound r∈R (|s| < r), are said to be finite; the rest are infinite.
s∈R^* is infinitesimal if, for any positive r∈R, |s| < r. In particular, by that definition, 0 is infinitesimal. The reciprocals of infinite numbers is infinitesimal.

The sum of standard numbers is standard; the sum of non-standard numbers may be either standard or non-standard. The same holds for the difference, product, and the ratio.

Two (hyperreal) numbers whose difference is infinitesimal are said to approximate each other. The relation of approximation is an equivalence. Indeed,

Reflexivity holds because 0 is infinitesimal, so that, for any hyperreal s, s is an approximation of itself.
Symmetry holds because of the presence of the absolute value in the definition of the infinitesimal: if |s - t| < r, for any positive real r, then also |t - s| < r.
To prove transitivity, assume three hyperreal numbers s, t, and u satisfy |s - t| < r and |t - u| < r, for any real r > 0. Then, by the Triangle inequality, |s - u| < |s - t| + |t - u| < 2r, for any real r > 0. However, 2r is very much as arbitrary as r itself, thus proving the transitivity.

It follows that all hyperreals are split into the sets (classes of equivalence) of mutual approximations: any two numbers in one of the classes approximate each other, whereas the numbers from different classes do not approximate each other: their difference is anything but infinitesimal. We write u ≈ v to indicate that u and v belong to the same equivalence class. In other words u ≈ v if and only if u - v is infinitesimal.

An important observation is that no two distinct real numbers may approximate each other (as their difference is a non zero real and, hence, is not infinitesimal.) In other words, a class of mutual approximations may contain one and only one real number. Also, if such a class contains a finite number, then (as an immediate consequence of the definitions and the Triangle inequality) all of its elements are also finite. Let's call such a class finite. We have the following

Theorem

Any finite class of mutual approximations contains a real number.

Proof

Given a finite class C, let s be any element of C. For definiteness sake, assume s is positive. Since s is finite, |s| < r, (or s < r) for a real positive r. Let R be the set of all real numbers r that satisfy this condition. All numbers in R are positive, so that R is bounded from below. By the completeness property of real numbers, R has the least low bound, say, ρ. The claim is that ρ∈C and, since it is real, it is exactly the one whose existence is claimed by the theorem. Indeed, ρ ≥ s, and the difference ρ - s is infinitesimal; for, if it is not, it would exceed a positive real number, which when subtracted from ρ would still give an upper bound for s and be less than ρ - in contradiction with the definition of ρ.

As a practical application of the theorem, any interval (r - ε, r + ε), where r is real while ε > 0 infinitesimal, contains no other real different from r.

We see that any finite class of mutual approximations contains a unique real number. This real number ρ is called the standard part of s (any hyperreal from the equivalence class C of s): ρ = st(s). s ≈ t is equivalent to st(s) = st(t). The standard part has several important properties. For example,

For a (standard) real r, st(r) = r.
For finite, s, t,
- st(s + t) = st(s) + st(t).
- st(s × t) = st(s) × st(t).
- If s ≤ t then st(s) ≤ st(t).

For a real number ρ, the set {s∈R^*: s ≈ ρ} is called the monad of ρ; harking back to Leibniz's terminology. The terms cloud and shadow are also being used. In the following, Mon(ρ) = {s∈R^*: s ≈ ρ} will denote the monad of a real number ρ.

The infinite hyperintegers have a pretty complex structure: if κ is one such then so are the ones below (and yet some...):

... κ - 2, κ - 1, κ, κ + 1, κ + 2, ..., 2κ, 3κ, ..., κ² - κ, ..., κ² - 1, κ², ...

The structure of the infinite hyperreals is even more complex. Now, as we know, the reciprocal of an infinite hyperreal is infinitesimal. The infinitesimals inherit the complexity of the aforementioned structure. It may be hard to imagine, but around every standard real number there is a cloud of its hyperreal approximations, all on the same number line, if you will.

Reference

A. H. Lightstone, Infinitesimals, The American Mathematical Monthly, Vol. 79, No. 3 (Mar., 1972), pp. 242-251

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