Construction of the reals via Dedekind cuts

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 x < y 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 A of rationals is said to be a cut or a Dedekind cut if the following hold.

  1. A is bounded above: B   a A   ( a b ) .
  2. A has no greatest element: a A   b A   ( a < b ) .

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 ) ) { q : q 2 < 2 } , 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 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.

Definition.

We let be the set of all Dedekind cuts.

Definition.

For q we associate with q the cut ( - , q ) . Clearly distinct q give rise to distinct cuts by this method. We identify q with this cut ( - , q ) , 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 A , B , we write A B to mean A B .

Addition is easy to define.

Definition.

For cuts A , B , we write A + B for the cut { q : a A   b B   ( q < a + b ) } .

Multiplication is quite a bit more tricky to define.

Definition.

(a) For cuts A , B with 0 A , B , we write A · B for the cut { q : a - A   b - B   ( q < a b ) } .

(b) For cuts A , B with A 0 B , we write A · B for the cut { q : a A   b - B   ( q < a b ) } .

(c) For cuts A , B with B 0 A , we write A · B for the cut { q : a - A   b B   ( q < a b ) } .

(d) For cuts A , B with A , B 0 , we write A · B for the cut { q : a A   b B   ( q < a b ) } .

Proposition.

For all cuts A , B the sets A + B and A · B 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).