11.2.1 The algebraic structure of Dedekind reals

11.2.1 The algebraic structure of Dedekind reals

The construction of the

algebraic

and order-theoretic

structure

of Dedekind reals proceeds
as usual in

intuitionistic logic

. Rather than dwelling on details we point out the

differences

between the classical and intuitionistic setup. Writing

L

x

and

U

x

for
the lower and upper cut of a

real number

x

:

ℝ

𝖽

, we define

addition

as

L

x

+

y

⁢

(

q

)

:

≡

∃

(

r

,

s

:

ℚ

)

.

L

x

(

r

)

∧

L

y

(

s

)

∧

q

=

r

+

s

,

U

x

+

y

⁢

(

q

)

:

≡

∃

(

r

,

s

:

ℚ

)

.

U

x

(

r

)

∧

U

y

(

s

)

∧

q

=

r

+

s

,

and the

additive inverse

by

L

-

x

⁢

(

q

)

:

≡

∃

(

r

:

ℚ

)

.

U

x

(

r

)

∧

q

=

-

r

,

U

-

x

⁢

(

q

)

:

≡

∃

(

r

:

ℚ

)

.

L

x

(

r

)

∧

q

=

-

r

.

With these

operations

(

ℝ

𝖽

,

0

,

+

,

-

)

is an

abelian group

. Multiplication is a bit
more cumbersome:

L

x

⋅

y

⁢

(

q

)

:

≡

∃

(

a

,

b

,

c

,

d

:

ℚ

)

.

L

x

⁢

(

a

)

∧

U

x

⁢

(

b

)

∧

L

y

⁢

(

c

)

∧

U

y

⁢

(

d

)

∧

q

<

min

⁡

(

a

⋅

c

,

a

⋅

d

,

b

⋅

c

,

b

⋅

d

)

,

U

x

⋅

y

⁢

(

q

)

:

≡

∃

(

a

,

b

,

c

,

d

:

ℚ

)

.

L

x

⁢

(

a

)

∧

U

x

⁢

(

b

)

∧

L

y

⁢

(

c

)

∧

U

y

⁢

(

d

)

∧

max

⁡

(

a

⋅

c

,

a

⋅

d

,

b

⋅

c

,

b

⋅

d

)

<

q

.

These

formulas

are related to multiplication of

intervals

in interval arithmetic, where
intervals

[

a

,

b

]

and

[

c

,

d

]

with

rational

endpoints

multiply to the interval

[

a

,

b

]

⋅

[

c

,

d

]

=

[

min

⁡

(

a

⁢

c

,

a

⁢

d

,

b

⁢

c

,

b

⁢

d

)

,

max

⁡

(

a

⁢

c

,

a

⁢

d

,

b

⁢

c

,

b

⁢

d

)

]

.

For

instance

, the formula for the lower cut can be read as saying that

q

<

x

⋅

y

when there are intervals

[

a

,

b

]

and

[

c

,

d

]

containing

x

and

y

, respectively, such
that

q

is to the left of

[

a

,

b

]

⋅

[

c

,

d

]

. It is generally useful to think of an
interval

[

a

,

b

]

such that

L

x

⁢

(

a

)

and

U

x

⁢

(

b

)

as an approximation of

x

, see

\autoref

ex:RD-interval-arithmetic.

We now have a

commutative ring

with unit

(

ℝ

𝖽

,

0

,

1

,

+

,

-

,

⋅

)

. To treat

multiplicative inverses

, we must first introduce order. Define

≤

and

<

as

(

x

≤

y

)

:

≡

∀

(

q

:

ℚ

)

.

L

x

(

q

)

⇒

L

y

(

q

)

,

(

x

<

y

)

:

≡

∃

(

q

:

ℚ

)

.

U

x

(

q

)

∧

L

y

(

q

)

.

Lemma 11.2.1

.

For all

x

:

R

d

and

q

:

Q

,

L

x

(

q

)

⇔

(

q

<

x

)

and

U

x

(

q

)

⇔

(

x

<

q

)

.

Proof.

If

L

x

⁢

(

q

)

then by roundedness there merely is

r

>

q

such that

L

x

⁢

(

r

)

, and since

U

q

⁢

(

r

)

it follows that

q

<

x

.

Conversely

, if

q

<

x

then there is

r

:

ℚ

such
that

U

q

⁢

(

r

)

and

L

x

⁢

(

r

)

, hence

L

x

⁢

(

q

)

because

L

x

is a

lower set

. The other half
of the proof is

symmetric

.
∎

The

relation

≤

is a

partial order

, and

<

is

transitive

and

irreflexive

. Linearity

(

x

<

y

)

∨

(

y

≤

x

)

is valid if we assume excluded middle, but without it we get weak linearity

(

x

<

y

)

⇒

(

x

<

z

)

∨

(

z

<

y

)

.

(11.2.2)

At first sight it might not be clear what (

11.2.2

) has to do with

linear order

. But if we take

x

≡

u

-

ϵ

and

y

≡

u

+

ϵ

for

ϵ

>

0

, then we get

(

u

-

ϵ

<

z

)

∨

(

z

<

u

+

ϵ

)

.

This is linearity “up to a small numerical error”, i.e., since it is unreasonable to
expect that we can actually compute with

infinite

precision, we should not be surprised
that we can decide

<

only up to whatever finite precision we have computed.

To see that (

11.2.2

) holds, suppose

x

<

y

. Then there merely exists

q

:

ℚ

such that

U

x

⁢

(

q

)

and

L

y

⁢

(

q

)

. By roundedness there merely exist

r

,

s

:

ℚ

such that

r

<

q

<

s

,

U

x

⁢

(

r

)

and

L

y

⁢

(

s

)

. Then, by locatedness

L

z

⁢

(

r

)

or

U

z

⁢

(

s

)

. In the first case we get

x

<

z

and in the second

z

<

y

.

Classically, multiplicative inverses exist for all numbers which are different from zero.
However, without excluded middle, a

stronger

condition is required. Say that

x

,

y

:

ℝ

𝖽

are

apart

from each other, written

x

#

y

, when

(

x

<

y

)

∨

(

y

<

x

)

:

(

x

#

y

)

:

≡

(

x

<

y

)

∨

(

y

<

x

)

.

If

x

#

y

, then

¬

⁡

(

x

=

y

)

.
The

converse

is true if we assume excluded middle, but is not provable constructively.
Indeed, if

¬

⁡

(

x

=

y

)

implies

x

#

y

, then a little bit of excluded middle follows; see

\autoref

ex:reals-apart-neq-MP.

Theorem 11.2.3

.

A real is

invertible

if, and only if, it is apart from

0

.

Remark 11.2.4

.

We observe that a real is invertible if, and only if, it is merely
invertible. Indeed, the same is true in any ring, since a ring is a set, and
multiplicative inverses are unique if they exist. See the discussion
following

\autoref

cor:UC.

Proof.

Suppose

x

⋅

y

=

1

. Then there merely exist

a

,

b

,

c

,

d

:

ℚ

such that

a

<

x

<

b

,

c

<

y

<

d

and

0

<

min

⁡

(

a

⁢

c

,

a

⁢

d

,

b

⁢

c

,

b

⁢

d

)

. From

0

<

a

⁢

c

and

0

<

b

⁢

c

it follows
that

a

,

b

, and

c

are either all

positive

or all negative.
Hence either

0

<

a

<

x

or

x

<

b

<

0

, so that

x

#

0

.

Conversely, if

x

#

0

then

L

x

-

1

⁢

(

q

)

:

≡

∃

(

r

:

ℚ

)

.

U

x

(

r

)

∧

(

(

0

<

r

∧

q

r

<

1

)

∨

(

r

<

0

∧

1

<

q

r

)

)

U

x

-

1

⁢

(

q

)

:

≡

∃

(

r

:

ℚ

)

.

L

x

(

r

)

∧

(

(

0

<

r

∧

q

r

>

1

)

∨

(

r

<

0

∧

1

>

q

r

)

)

defines the desired

inverse

. Indeed,

L

x

-

1

and

U

x

-

1

are inhabited because

x

#

0

.
∎

The

archimedean principle

can be stated in several ways. We find it most illuminating in the
form which says that

ℚ

is

dense in

ℝ

𝖽

.

Theorem 11.2.5

(Archimedean principle for

R

d

).

For all

x

,

y

:

R

d

if

x

<

y

then there merely exists

q

:

Q

such that

x

<

q

<

y

.

Proof.

By definition of

<

.
∎

Before tackling completeness of Dedekind reals, let us state precisely what

algebraic
structure

they possess. In the following definition we are not aiming at a

minimal

axiomatization, but rather at a useful amount of structure and properties.

Definition 11.2.6

.

An

ordered field

is a set

F

together with
constants

0

,

1

, operations

+

,

-

,

⋅

,

min

,

max

, and mere relations

≤

,

<

,

#

such that:

1.

(

F

,

0

,

1

,

+

,

-

,

⋅

)

is a commutative ring with unit;

2.

x

:

F

is invertible if, and only if,

x

#

0

;

3.

(

F

,

≤

,

min

,

max

)

is a

lattice

;

4.

the

strict order

<

is transitive, irreflexive,

and weakly linear (

x

<

y

⇒

x

<

z

∨

z

<

y

);

5.

apartness

#

is irreflexive, symmetric and cotransitive (

x

#

y

⇒

x

#

z

∨

y

#

z

);

6.

for all

x

,

y

,

z

:

F

:

x

≤

y

⇔

¬

⁡

(

y

<

x

)

,

x

<

y

≤

z

⇒

x

<

z

,

x

#

y

⇔

(

x

<

y

)

∨

(

y

<

x

)

,

x

≤

y

<

z

⇒

x

<

z

,

x

≤

y

⇔

x

+

z

≤

y

+

z

,

x

≤

y

∧

0

≤

z

⇒

x

⁢

z

≤

y

⁢

z

,

x

<

y

⇔

x

+

z

<

y

+

z

,

0

<

z

⇒

(

x

<

y

⇔

x

z

<

y

z

)

,

0

<

x

+

y

⇒

0

<

x

∨

0

<

y

,

0

<

1

.

Every such field has a canonical

embedding

Q

→

F

. An ordered field is

archimedean

when for all

x

,

y

:

F

, if

x

<

y

then there merely exists

q

:

Q

such that

x

<

q

<

y

.

Theorem 11.2.7

.

The Dedekind reals form an ordered archimedean field.

Proof.

We omit the proof in the hope that what we have demonstrated so far makes the

theorem

plausible.
∎

Title

11.2.1 The algebraic structure of Dedekind reals

\metatable

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