11.4 Comparison of Cauchy and Dedekind reals

11.4 Comparison of Cauchy and Dedekind reals

Let us also say something about the relationship between the Cauchy and Dedekind reals. By

\autoref

RC-archimedean-ordered-field,

ℝ

𝖼

is an

archimedean

ordered field

. It is also
admissible for

Ω

, as can be easily checked. (In case

Ω

is the initial

σ

-frame
it takes a simple

induction

, while in other cases it is immediate.)
Therefore, by

\autoref

RD-final-field there is an

embedding

of ordered fields

ℝ

𝖼

→

ℝ

𝖽

which fixes the

rational numbers

.
(We could also obtain this from

\autoref

RC-initial-Cauchy-complete,

\autoref

RD-cauchy-complete.)
In general we do not expect

ℝ

𝖼

and

ℝ

𝖽

to coincide
without further

assumptions

.

Lemma 11.4.1

.

If for every

x

:

R

d

there merely exists

c

:

∏

q

,

r

:

ℚ

(

q

<

r

)

→

(

q

<

x

)

+

(

x

<

r

)

(11.4.1)

then the Cauchy and Dedekind reals coincide.

Proof.

Note that the type in (

11.4.1

) is an untruncated variant
of (

LABEL:eq:RD-linear-order

), which states that

<

is a weak

linear order

.
We already know that

ℝ

𝖼

embeds into

ℝ

𝖽

, so it suffices to show that every Dedekind
real merely is the limit of a

Cauchy sequence

of rational numbers.

Consider any

x

:

ℝ

𝖽

. By assumption there merely exists

c

as in the statement of the
lemma, and by inhabitation of cuts there merely exist

a

,

b

:

ℚ

such that

a

<

x

<

b

.
We construct a

sequence

f

:

ℕ

→

\setof

(

q

,

r

)

∈

ℚ

×

ℚ

|

q

<

r

by

recursion

:

1.

Set

f

(

0

)

:

≡

(

a

,

b

)

.

2.

Suppose

f

⁢

(

n

)

is already defined as

(

q

n

,

r

n

)

such that

q

n

<

r

n

.
Define

s

:

≡

(

2

q

n

+

r

n

)

/

3

and

t

:

≡

(

q

n

+

2

r

n

)

/

3

. Then

c

⁢

(

s

,

t

)

decides between

s

<

x

and

x

<

t

. If it decides

s

<

x

then we set

f

(

n

+

1

)

:

≡

(

s

,

r

n

)

, otherwise

f

(

n

+

1

)

:

≡

(

q

n

,

t

)

.

Let us write

(

q

n

,

r

n

)

for the

n

-th term of the sequence

f

. Then it is easy
to see that

q

n

<

x

<

r

n

and

|

q

n

-

r

n

|

≤

(

2

/

3

)

n

⋅

|

q

0

-

r

0

|

for all

n

:

ℕ

. Therefore

q

0

,

q

1

,

…

and

r

0

,

r

1

,

…

are both Cauchy sequences
converging to the

Dedekind cut

x

. We have shown that for every

x

:

ℝ

𝖽

there merely
exists a Cauchy sequence converging to

x

.
∎

The lemma implies that either

countable choice

or excluded middle suffice for coincidence
of

ℝ

𝖼

and

ℝ

𝖽

.

Corollary 11.4.3

.

If excluded middle or countable choice holds then

R

c

and

R

d

are

equivalent

.

Proof.

If excluded middle holds then

(

x

<

y

)

→

(

x

<

z

)

+

(

z

<

y

)

can be proved: either

x

<

z

or

¬

⁡

(

x

<

z

)

. In the former case we are done, while in the latter we get

z

<

y

because

z

≤

x

<

y

. Therefore, we get (

11.4.1

) so that we
can apply

\autoref

lem:untruncated-linearity-reals-coincide.

Suppose countable choice holds. The set

S

=

\setof

(

q

,

r

)

∈

ℚ

×

ℚ

|

q

<

r

is equivalent to

ℕ

, so we may apply countable choice to the statement that

x

is located,

∀

(

(

q

,

r

)

:

S

)

.

(

q

<

x

)

∨

(

x

<

r

)

.

Note that

(

q

<

x

)

∨

(

x

<

r

)

is expressible as an existential statement

∃

(

b

:

𝟐

)

.

(

b

=

0

𝟐

→

q

<

x

)

∧

(

b

=

1

𝟐

→

x

<

r

)

. The (curried form) of
the

choice function

is then precisely (

11.4.1

) so that

\autoref

lem:untruncated-linearity-reals-coincide is applicable again.
∎

Title

11.4 Comparison of Cauchy and Dedekind reals

\metatable

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