This section discusses the construction of the real numbers from the rationals via the idea of a Dedekind cuts. This approach is the most natural one from the point of view of Dedekind's version of the completeness axiom. It works well and generalises to many other contexts where an order relation is present, but is useless without such an order.
This web page is still under construction. The final version will contain some (but not all) of the proofs omitted here at present.
A set of rationals is said to be a cut or a Dedekind cut if the following hold.
The idea is that the number can be thought of
as making a
cut in the rationals
separating the set of rationals below it,
from the set of rationals above it. Therefore we should define
the real number to be this cut. This is a nice idea.
The best thing is that we don't have to worry about
equivalence relations: there can only be one cut for ,
and indeed only one cut for any other real number, so we don't have
to equivalence-anything-out. The bad news is that the technical details
are somewhat messy when it comes to negative numbers.
For negative numbers, one possible approach would be to use Dedekind cuts to define the nonnegative reals, and then in a second step use an algebraic construction to get all the reals from these. We shall sketch the complete construction in one step using Dedekind cuts. The two-step version (which is probably more elegant and less painful) can easily be put together from details given on other web pages here.
We let be the set of all Dedekind cuts.
For we associate with the cut . Clearly distinct give rise to distinct cuts by this method. We identify with this cut , thus viewing as a subset of .
Informally, one cut is to the right of another if it includes it as a set. So we define
For cuts , we write to mean .
Addition is easy to define.
For cuts , we write for the cut .
Multiplication is quite a bit more tricky to define.
(a) For cuts with , we write for the cut .
(b) For cuts with , we write for the cut .
(c) For cuts with , we write for the cut .
(d) For cuts with , we write for the cut .
For all cuts the sets and just defined are in fact cuts hence in .
This completes the definition part, as we have now defined with its arithmetic operations, and order. All that remains it to check the axioms.
The set with is an ordered field. It also satisfies the Archimedean property and is a complete ordered field in the sense that every nonempty set bounded above has a least upper bound.
Another long exercise (sigh).