11.2 Dedekind reals

11.2 Dedekind reals

Let us first recall the basic idea of Dedekind’s construction. We use two-sided

Dedekind
cuts

, as opposed to an often used one-sided version, because the

symmetry

makes
constructions more elegant, and it works constructively as well as classically.
A

Dedekind cut

consists of a pair

(

L

,

U

)

of subsets

L

,

U

⊆

ℚ

, called the

lower

and

upper cut

respectively, which are:

1.

inhabited:

there are

q

∈

L

and

r

∈

U

,

2.

rounded:

q

∈

L

⇔

∃

(

r

∈

ℚ

)

.

q

<

r

∧

r

∈

L

and

r

∈

U

⇔

∃

(

q

∈

ℚ

)

.

q

∈

U

∧

q

<

r

,

3.

disjoint

:

¬

⁡

(

q

∈

L

∧

q

∈

U

)

, and

4.

located:

q

<

r

⇒

q

∈

L

∨

r

∈

U

.

Reading the roundedness condition from left to right tells us that cuts are

open

,
and from right to left that they are

lower

, respectively

upper

, sets. The
locatedness condition states that there is no large gap between

L

and

U

. Because cuts
are always open, they never include the “point in between”, even when it is

rational

. A
typical Dedekind cut looks like this:

{tikzpicture}

[x=]

\draw

[¡-),line width=0.75pt] (0,0) – (0.297,0) node[anchor=south east]

L

;

\draw

[(-¿,line width=0.75pt] (0.300, 0) node[anchor=south west]

U

– (0.9, 0) ;

We might naively translate the informal definition into

type theory

by saying that a cut
is a pair of maps

L

,

U

:

ℚ

→

𝖯𝗋𝗈𝗉

. But we saw in

\autoref

subsec:prop-subsets that

𝖯𝗋𝗈𝗉

is an

ambiguous

notation for

𝖯𝗋𝗈𝗉

𝒰

i

where

𝒰

i

is a

universe

. Once we
use a particular

𝒰

i

to define cuts, the type of reals will reside in the next
universe

𝒰

i

+

1

, a property of reals two levels higher in

𝒰

i

+

2

, a property of
subsets of reals in

𝒰

i

+

3

, etc. In principle we should be able to keep track of the
universe levels, especially with the help of a proof assistant, but doing so here would
just burden us with bureaucracy that we prefer to avoid. We shall therefore make a
simplifying

assumption

that a single type of propositions

Ω

is

sufficient

for all
our purposes.

In fact, the construction of the Dedekind reals is quite resilient to logical
manipulations. There are several ways in which we can make sense of using a single type

Ω

:

1.

We could identify

Ω

with the ambiguous

𝖯𝗋𝗈𝗉

and track all the universes
that appear in definitions and constructions.

2.

We could assume the propositional resizing axiom,
as in

\autoref

subsec:prop-subsets, which essentially collapses the

𝖯𝗋𝗈𝗉

𝒰

i

’s to the
lowest level, which we call

Ω

.

3.

A classical mathematician who is not interested in the intricacies of type-theoretic
universes or

computation

may simply assume the

law of excluded middle

(

LABEL:eq:lem

) for
mere propositions so that

Ω

≡

𝟐

.

This not only eradicates questions about
levels of

𝖯𝗋𝗈𝗉

, but also turns everything we do into the standard classical
construction of real numbers.

4.

On the other end of the spectrum one might ask for a

minimal

requirement that makes
the constructions work. The condition that a mere

predicate

be a Dedekind cut is
expressible using only

conjunctions

,

disjunctions

, and

existential quantifiers

over

ℚ

, which
is a

countable set

. Thus we could take

Ω

to be the initial

σ

-frame

,
i.e., a

lattice

with

countable

joins in which binary meets

distribute over

countable
joins. (The initial

σ

-frame cannot be the two-point lattice

𝟐

because

𝟐

is not

closed under

countable joins, unless we assume excluded middle.) This
would lead to a construction of

Ω

as a higher inductive-inductive type, but one
experiment of this kind in

\autoref

sec:cauchy-reals is enough.

In all of the above cases

Ω

is a set.
Without further ado, we translate the informal definition into type theory.
Throughout this chapter, we use the
logical notation from

\autoref

defn:logical-notation.

Definition 11.2.1

.

A

Dedekind cut

is a pair

(

L

,

U

)

of mere predicates

L

:

Q

→

Ω

and

U

:

Q

→

Ω

which is:

1.

inhabited:

∃

(

q

:

ℚ

)

.

L

(

q

)

and

∃

(

r

:

Q

)

.

U

(

r

)

,

2.

rounded:

for all

q

,

r

:

ℚ

,

L

⁢

(

q

)

⇔

∃

(

r

:

ℚ

)

.

(

q

<

r

)

∧

L

(

r

)

𝑎𝑛𝑑

U

⁢

(

r

)

⇔

∃

(

q

:

ℚ

)

.

(

q

<

r

)

∧

U

(

q

)

,

3.

disjoint:

¬

⁡

(

L

⁢

(

q

)

∧

U

⁢

(

q

)

)

for all

q

:

ℚ

,

4.

located:

(

q

<

r

)

⇒

L

(

q

)

∨

U

(

r

)

for all

q

,

r

:

ℚ

.

We let

isCut

⁢

(

L

,

U

)

denote the conjunction of these conditions. The type of

Dedekind reals

is

ℝ

𝖽

:

≡

\setof

(

L

,

U

)

:

(

ℚ

→

Ω

)

×

(

ℚ

→

Ω

)

|

𝗂𝗌𝖢𝗎𝗍

(

L

,

U

)

.

It is apparent that

𝗂𝗌𝖢𝗎𝗍

⁢

(

L

,

U

)

is a mere proposition, and since

ℚ

→

Ω

is a
set the Dedekind reals form a set too. See

\autoref

ex:RD-extended-reals,ex:RD-lower-cuts,ex:RD-interval-arithmetic for variants of
Dedekind cuts which lead to extended reals, lower and upper reals, and the interval
domain.

There is an

embedding

ℚ

→

ℝ

𝖽

which associates with each rational

q

:

ℚ

the cut

(

L

q

,

U

q

)

where

L

q

(

r

)

:

≡

(

r

<

q

)

and

U

q

(

r

)

:

≡

(

q

<

r

)

.

We shall simply write

q

for the cut

(

L

q

,

U

q

)

associated with a rational number.

Title

11.2 Dedekind reals

\metatable

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