Constructing the reals from the rationals

Question

Dr. H. Jerome Keisler, in his book Elementary Calculus: An Infinitesimal Approach, states on page 24:

Just as the real numbers can be constructed from the rational numbers, the hyperreal numbers can be constructed from the real numbers.

In what sense is this true? Since $\mathbb{R}$ is uncountably infinite, and $\mathbb{Q}$ is countably infinite, I would think that it is not possible to construct $\mathbb{R}$ from $\mathbb{Q}$, at least in the sense that $\mathbb{Q}$ can be constructed from $\mathbb{N}$.

Answer 1

There are two classical (and equivalent) possibilities:

1. Dedekind cuts. Represent every real by the set of rational numbers that are smaller than it -- such sets can be characterized without already knowing $\mathbb R$: they are the downwards closed subsets of $\mathbb Q$ that are neither empty nor $\mathbb Q$ itself and don't have a largest element.

2. Cauchy sequences. Let the reals be equivalence classes of sequences of rational numbers that "ought to" have a limit according to the Cauchy criterion. Two such sequences are equivalent (and so represent the same real number) if their term-by-term difference converges to $0$.

Answer 2

For variety, I will also give another construction.

From the rational numbers, you apply the ideas of non-standard analysis to construct the hyperrational numbers. You can then construct the real numbers as the standard parts of limited hyperrational numbers.

More precisely,

• Any limited (i.e. not infinite) hyperrational number is the name of a real number
• Every real number can be named in this way
• Two limited hyperrational numbers are both names for the same real number if and only if they differ by an infinitesimal.

EDIT: to answer the question of the post, from one point of view, central idea behind the constructions is that the real numbers fill in the "holes" between rationals -- i.e. the real numbers are a continuum whereas the rationals are not. The method I describe here and the two classic ones Henning mention are all schemes for locate the holes. If you know where the holes are, then you know what the irrational numbers need to be, and so you can define the reals in that fashion.

The difference as compared to constructing the integers and rationals from the natural numbers is, in some sense, how much data is needed to identify a real number.

Identifying integers as things that ought to be the difference between natural numbers only requires two natural numbers to identify the integer. Similarly for the rational numbers as quotients of integers.

The classic constructions using Dedekind cuts or Cauchy sequences use an infinite amount of data to pick out a real number. Since the data can be chosen in a sufficiently arbitrary way, the number of ways to select data is a greater cardinal number than the number of rationals.

As for the version I describe, only two hyperrationals are needed to locate a real, but the hyperrationals are already (externally) uncountably infinite.