11.2.3 Dedekind reals are Dedekind complete

11.2.3 Dedekind reals are Dedekind complete

We obtained

ℝ

𝖽

as the type of

Dedekind cuts

on

ℚ

. But we could have instead started
with any

archimedean

ordered field

F

and constructed Dedekind cuts on

F

. These would
again form an archimedean ordered field

F

¯

, the

Dedekind completion of

F

,
with

F

contained

as a

subfield

. What happens if we apply this construction to

ℝ

𝖽

, do we get even more

real numbers

? The answer is

negative

. In fact, we shall prove a
stronger result:

ℝ

𝖽

is final.

Say that an ordered field

F

is

admissible

for

Ω

when the

strict order

<

on

F

is a map

<

:

F

→

F

→

Ω

.

Theorem 11.2.1

.

Every archimedean ordered field which is admissible for

Ω

is a subfield of

R

d

.

Proof.

Let

F

be an archimedean ordered field. For every

x

:

F

define

L

,

U

:

ℚ

→

Ω

by

L

x

(

q

)

:

≡

(

q

<

x

)

and

U

x

(

q

)

:

≡

(

x

<

q

)

.

(We have just used the

assumption

that

F

is admissible for

Ω

.)
Then

(

L

x

,

U

x

)

is a Dedekind cut. Indeed, the cuts are inhabited and rounded because

F

is archimedean and

<

is

transitive

,

disjoint

because

<

is

irreflexive

, and
located because

<

is a weak

linear order

. Let

e

:

F

→

ℝ

𝖽

be the map

e

(

x

)

:

≡

(

L

x

,

U

x

)

.

We claim that

e

is a field

embedding

which preserves and

reflects

the order. First of
all, notice that

e

⁢

(

q

)

=

q

for a

rational number

q

. Next we have the equivalences,
for all

x

,

y

:

F

,

x

<

y

⇔

(

∃

(

q

:

ℚ

)

.

x

<

q

<

y

)

⇔

(

∃

(

q

:

ℚ

)

.

U

x

(

q

)

∧

L

y

(

q

)

)

⇔

e

(

x

)

<

e

(

y

)

,

so

e

indeed preserves and reflects the order. That

e

⁢

(

x

+

y

)

=

e

⁢

(

x

)

+

e

⁢

(

y

)

holds
because, for all

q

:

ℚ

,

q

<

x

+

y

⇔

∃

(

r

,

s

:

ℚ

)

.

r

<

x

∧

s

<

y

∧

q

=

r

+

s

.

The

implication

from right to left is obvious. For the other

direction

, if

q

<

x

+

y

then there merely exists

r

:

ℚ

such that

q

-

y

<

r

<

x

, and by taking

s

:

≡

q

-

r

we get the desired

r

and

s

. We leave preservation of

multiplication

by

e

as
an exercise.
∎

To establish that the Dedekind cuts on

ℝ

𝖽

do not give us anything new, we need just one
more lemma.

Lemma 11.2.2

.

If

F

is admissible for

Ω

then so is its Dedekind completion.

Proof.

Let

F

¯

be the Dedekind completion of

F

. The strict order on

F

¯

is
defined by

(

(

L

,

U

)

<

(

L

′

,

U

′

)

)

:

≡

∃

(

q

:

ℚ

)

.

U

(

q

)

∧

L

′

(

q

)

.

Since

U

⁢

(

q

)

and

L

′

⁢

(

q

)

are elements of

Ω

, the lemma holds as long as

Ω

is

closed under

conjunctions

and

countable

existentials, which we assumed from the outset.
∎

Corollary 11.2.3

.

The Dedekind reals are

Dedekind complete

: for every real-valued Dedekind cut

(

L

,

U

)

there is a unique

x

:

R

d

such that

L

(

y

)

=

(

y

<

x

)

and

U

(

y

)

=

(

x

<

y

)

for all

y

:

R

d

.

Proof.

By

\autoref

lem:cuts-preserve-admissibility the Dedekind completion

ℝ

¯

𝖽

of

ℝ

𝖽

is admissible for

Ω

, so by

\autoref

RD-final-field we have an embedding

ℝ

¯

𝖽

→

ℝ

𝖽

, as well as an embedding

ℝ

𝖽

→

ℝ

¯

𝖽

. But these embeddings must be

isomorphisms

, because their

compositions

are

order-preserving

field homomorphisms

which
fix the dense subfield

ℚ

, which means that they are the

identity

. The corollary now
follows immediately from the fact that

ℝ

¯

𝖽

→

ℝ

𝖽

is an isomorphism.
∎

Title

11.2.3 Dedekind reals are Dedekind complete

\metatable

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