22 1. The Real Numbers

0 1

2

1 2

)

Figure 1.4.1. A Dedekind Cut in the Rationals.

Each rational number r determines a cut – the set Lr of (1.4.1). In this case,

r is called the cut number for the Dedekind cut. Are there Dedekind cuts that are

not determined in this way? Cuts that have no rational cut number?

Example 1.4.2. Describe a Dedekind cut that is not of the form Lr for a rational

number r.

Solution: We are guided by the idea that there ought to be a number whose

square is 2, but there is no such rational number. If there were a number

√

2 with

square 2, then the set of rational numbers less than

√

2 could be described as

L = {r ∈ Q : r ≥ 0 and

r2

2} ∪ {r ∈ Q : r 0}.

We claim this is a Dedekind cut not of the form Lr for any r ∈ Q.

Certainly L is a non-empty, proper subset of Q. It has no largest element

because if

n

m

is any positive element of L, then we can always choose a larger

rational number which still has square less than 2 as follows:

kn+1

km

n

m

for every

k ∈ N and

kn + 1

km

2

=

n

m

2

+

1

km

2

n

m

+

1

km

.

By choosing k large enough, we can make the second term on the right less than

2 − (

n

m

)2

and this will imply that (

kn+1

km

)2

2. Thus, L has no largest element.

If x ∈ L and y x, then either y is negative, in which case it is in L, or

0 ≤ y x. In the latter case, y2 x2 2, and so y ∈ L in this case as well. Thus

L is a Dedekind cut.

We next show that there is no rational number r such that L = Lr. If there

is such a number r, then r is a positive rational number not in L and so r2 ≥ 2.

However, there are numbers in L arbitrarily close to r and each of them has square

less than 2. It follows that r2 ≤ 2. This means r2 = 2, which is impossible for a

rational number r.

Thus, although it might seem that every Dedekind cut ought to correspond to

a cut number, the above example shows that this is not the case. In fact, there are

a lot more cuts than there are rational cut numbers. However, we can fix this by

enlarging the number system so that there is a cut number for every Dedekind cut.

The way this is usually done is to define the new number system to actually be the

set of all Dedekind cuts of the rationals. Below, we attempt to describe this idea

in a way that is somewhat visually intuitive.

We will think of a Dedekind cut L as specifying a certain location (the location

between L and its complement R) along the rational number line. We will think

of the real number system R as being the set of all such locations. Then each real

number x corresponds to a Dedekind cut Lx, which is to be thought of as the set of