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.
1. Defining the reals from the rationals
Definition.
A set
of rationals is said
to be a cut or a Dedekind cut if the following hold.

is bounded above:
(
)
.

has no greatest element:
(
)
.
The idea is that the number 2 can be thought of
as making a cut
in the rationals
separating the set of rationals below it,
(
(,0))
^{22
}
,
from the set of rationals above it. Therefore we should define
the real number 2 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 2,
and indeed only one cut for any other real number, so we don't have
to equivalenceanythingout. 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 twostep version
(which is probably more elegant and less painful)
can easily be put together from details given on other web pages here.
Definition.
We let be the set of all Dedekind cuts.
Definition.
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
Definition.
For cuts ,
, we write
to mean
.
Addition is easy to define.
Definition.
For cuts ,
, we write +
for the cut
(
+)
.
Multiplication is quite a bit more tricky to define.
Definition.
(a) For cuts ,
with 0
,
,
we write
for the cut


(
)
.
(b) For cuts ,
with
0
,
we write
for the cut

(
)
.
(c) For cuts ,
with
0
,
we write
for the cut

(
)
.
(d) For cuts ,
with ,
0
,
we write
for the cut
(
)
.
Proposition.
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.
Theorem.
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.
Proof.
Another long exercise (sigh).