Constructing the reals from the rationals 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 $\mathbb{N}$.

There are two classical (and equivalent) possibilities:


*

*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.

*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 $0$.
