Dedekind cuts

Dedekind cuts

The purpose of

Dedekind cuts

is to provide a sound

logical foundation

for the

real number

system.
Dedekind’s motivation behind this

project

is to notice that a real number

α

, intuitively,
is completely determined by the

rationals

strictly smaller than

α

and those strictly larger than

α

. Concerning the completeness or

continuity

of the

real line

, Dedekind notes in

[

2

]

that

If all points of the

straight line

fall into two classes such that every point of the first class lies to the
left of every point of the second class, then there exists one and only one point which produces this

division

of all points into two classes, this severing of the straight line into two portions.

Dedekind defines a point to produce the division of the real line if this point is either the
least or

greatest element

of either one of the classes mentioned above. He further notes that the completeness
property, as he just phrased it, is deficient in the rationals, which motivates the definition of reals as cuts
of rationals. Because all rationals

greater

than

α

are really
just excess baggage, we prefer to sway
somewhat from Dedekind’s original definition. Instead, we adopt the following definition.

Definition

.

A

Dedekind cut

is a subset

α

of the rational numbers

ℚ

that

satisfies

these
properties:

1.

α

is not empty.

2.

ℚ

∖

α

is not empty.

3.

α

contains no greatest element

4.

For

x

,

y

∈

ℚ

, if

x

∈

α

and

y

<

x

, then

y

∈

α

as well.

Dedekind cuts are particularly appealing for two reasons. First, they make it very easy to
prove the completeness, or continuity of the real line. Also, they make it quite plain to
distinguish the rationals from the

irrationals

on the real line, and put the latter on a firm
logical foundation.
In the construction of the real numbers from Dedekind cuts, we make the following definition:

Definition

.

A

real number

is a Dedekind cut. We denote the set of all real numbers by

ℝ

and we
order them by set-theoretic

inclusion

, that is to say, for any

α

,

β

∈

ℝ

,

α

<

β

⁢

if and only if

⁢

α

⊂

β

where the inclusion is strict. We further define

α

=

β

as real numbers
if

α

and

β

are equal as sets. As usual, we write

α

≤

β

if

α

<

β

or

α

=

β

. Moreover, a real number

α

is said to be

irrational

if

ℚ

∖

α

contains no least

element

.

The Dedekind completeness property of real numbers, expressed as the

supremum

property, now becomes straightforward
to prove. In what follows, we will reserve Greek variables for real numbers, and Roman variables for rationals.

Theorem 1

.

Every nonempty subset of real numbers that is bounded above has a least

upper bound

.

Proof.

Let

A

be a

nonempty set

of real numbers, such that for every

α

∈

A

we have that

α

≤

γ

for some real number

γ

. Now define the set

sup

⁡

A

=

⋃

α

∈

A

α

.

We must show that this set is a real number. This amounts to checking the four conditions
of a Dedekind cut.

1.

sup

⁡

A

is clearly not empty, for it is the nonempty union of nonempty sets.

2.

Because

γ

is a real number, there is some rational

x

that is not in

γ

. Since
every

α

∈

A

is a subset of

γ

,

x

is not in any

α

, so

x

∉

sup

⁡

A

either. Thus,

ℚ

∖

sup

⁡

A

is nonempty.

3.

If

sup

⁡

A

had a greatest element

g

, then

g

∈

α

for some

α

∈

A

. Then

g

would be a greatest element of

α

, but

α

is a real number, so
by

contrapositive

,

sup

⁡

A

has no greatest element.

4.

Lastly, if

x

∈

sup

⁡

A

, then

x

∈

α

for some

α

, so given any

y

<

x

because

α

is a real number

y

∈

α

, whence

y

∈

sup

⁡

A

.

Thus,

sup

⁡

A

is a real number. Trivially,

sup

⁡

A

is an upper bound of

A

, for every

α

⊆

sup

⁡

A

.
It now suffices to prove that

sup

⁡

A

≤

γ

, because

γ

was an arbitrary
upper bound. But this is easy, because
every

x

∈

sup

⁡

A

is an element of

α

for some

α

∈

A

, so because

α

⊆

γ

,

x

∈

γ

. Thus,

sup

⁡

A

is the least upper bound of

A

. We call this real number the

supremum

of A.
∎

To finish the construction of the real numbers, we must endow them with algebraic operations, define the

additive

and

multiplicative identity

elements, prove that these definitions give a field, and prove further results about the order of the reals (such as the totality of this
order) – in short, build a

complete

ordered field

. This task is somewhat laborious, but we include here the
appropriate definitions. Verifying their correctness can be an instructive, albeit tiresome, exercise. We use the
same symbols for the

operations

on the reals as for the rational numbers; this should cause no confusion in

context

.

Definition

.

Given two real numbers

α

and

β

, we define

•

The

additive identity

, denoted

0

, is

0

:=

{

x

∈

ℚ

:

x

<

0

}

•

The

multiplicative identity

, denoted

1

, is

1

:=

{

x

∈

ℚ

:

x

<

1

}

•

Addition

of

α

and

β

denoted

α

+

β

is

α

+

β

:=

{

x

+

y

:

x

∈

α

,

y

∈

β

}

•

The

opposite

of

α

, denoted

-

α

, is

-

α

:=

{

x

∈

ℚ

:

-

x

∉

α

,

but

-

x

⁢

is not the least element of

⁢

ℚ

∖

α

}

•

The

absolute value

of

α

, denoted

|

α

|

, is

|

α

|

:=

{

α

,

if

⁢

α

≥

0

-

α

,

if

⁢

α

≤

0

•

If

α

,

β

>

0

, then

multiplication

of

α

and

β

, denoted

α

⋅

β

, is

α

⋅

β

:=

{

z

∈

ℚ

:

z

≤

0

⁢

or

⁢

z

=

x

⁢

y

⁢

for some

⁢

x

∈

α

,

y

∈

β

⁢

with

⁢

x

,

y

>

0

}

In general,

α

⋅

β

:=

{

0

,

if

⁢

α

=

0

⁢

or

⁢

β

=

0

|

α

|

⋅

|

β

|

if

⁢

α

>

0

,

β

>

0

⁢

or

⁢

α

<

0

,

β

<

0

-

(

|

α

|

⋅

|

β

|

)

if

⁢

α

>

0

,

β

⁢

<

0

⁢

or

⁢

α

>

⁢

0

,

β

<

0

•

The

inverse

of

α

>

0

, denoted

α

-

1

, is

α

-

1

:=

{

x

∈

ℚ

:

x

≤

0

⁢

or

⁢

x

>

0

⁢

and

⁢

(

1

/

x

)

∉

α

,

but

⁢

1

/

x

⁢

is not the least element of

⁢

ℚ

∖

α

}

If

α

<

0

,

α

-

1

:=

-

(

|

α

|

)

-

1

All that remains (!) is to check that the above definitions do indeed define a complete ordered field, and that
all the sets implied to be real numbers are indeed so.
The properties of

ℝ

as an ordered field follow from these definitions and the properties of

ℚ

as an
ordered field.
It is important to point out that in two steps, in showing that inverses and

opposites

are properly defined,
we require an extra property of

ℚ

, not merely in its capacity as an ordered field.
This requirement is the

Archimedean property

.

Moreover, because

ℝ

is a field of

characteristic

0

, it contains an

isomorphic

copy of

ℚ

. The rationals
correspond to the Dedekind cuts

α

for which

ℚ

∖

α

contains a least member.

References

1

Courant, Richard and Robbins, Herbert.

What is Mathematics?

pp. 68-72 Oxford University Press, Oxford, 1969

2

Dedekind, Richard.

Essays on the Theory of Numbers

Dover Publications Inc, New York 1963

3

Rudin, Walter

Principles of

Mathematical Analysis

pp. 17-21 McGraw-Hill Inc, New York, 1976

4

Spivak, Michael.

Calculus

pp. 569-596 Publish or Perish, Inc. Houston, 1994

Title

Dedekind cuts

Canonical name

DedekindCuts

Date of creation

2013-03-22 12:38:34

Last modified on

2013-03-22 12:38:34

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

26

Author

rmilson (146)

Entry type

Definition

Classification

msc 26A03

Synonym

Schnitt

Related topic

RealNumber

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