11.3.1 Construction of Cauchy reals

11.3.1 Construction of Cauchy reals

The construction of the Cauchy reals

ℝ

𝖼

as a

higher inductive type

is a bit more subtle than that of the free

algebraic structures

considered in

\autoref

sec:free-algebras.
We intend to include a “take the limit” constructor whose input is a

Cauchy sequence

of reals, but the notion of “Cauchy sequence of reals” depends on having some way to

measure

the “

distance

” between

real numbers

.
In general, of course, the distance between two real numbers will be another real number, leading to a potentially problematic circularity.

However, what we actually need for the notion of Cauchy sequence of reals is not the general notion of “distance”, but a way to say that “the distance between two real numbers is less than

ϵ

” for any

ϵ

:

ℚ

+

.
This can be represented by a family of

binary relations

, which we will denote

∼

ϵ

:

ℝ

𝖼

→

ℝ

𝖼

→

𝖯𝗋𝗈𝗉

.
The intended meaning of

x

∼

ϵ

y

is

|

x

-

y

|

<

ϵ

, but since we do not have notions of

subtraction

,

absolute value

, or

inequality

available yet (we are only just defining

ℝ

𝖼

, after all), we will have to define these

relations

∼

ϵ

at the same time as we define

ℝ

𝖼

itself.
And since

∼

ϵ

is a type family

indexed by

two copies of

ℝ

𝖼

, we cannot do this with an ordinary mutual (higher) inductive definition; instead we have to use a

higher inductive-inductive definition

.

Recall from

\autoref

sec:

generalizations

that the ordinary notion of inductive-inductive definition allows us to define a type and a type family indexed by it by simultaneous

induction

.
Of course, the “higher” version of this allows both the type and the family to have path constructors as well as point constructors.
We will not attempt to formulate any general theory of higher inductive-inductive definitions, but hopefully the description we will give of

ℝ

𝖼

and

∼

ϵ

will make the idea transparent.

Remark 11.3.1

.

We might also consider a

higher inductive-recursive definition

, in which

∼

ϵ

is defined using the

recursion

principle of

R

c

, simultaneously with the

inductive

definition of

R

c

.
We choose the inductive-inductive route instead for two reasons.
Firstly, higher inductive-recursive definitions seem to be more difficult to justify in homotopical

semantics

.
Secondly, and more importantly, the inductive-inductive definition yields a more powerful induction principle, which we will need in order to develop even the basic theory of Cauchy reals.

Finally, as we did for the discussion of Cauchy completeness of the Dedekind reals in

\autoref

sec:RD-cauchy-complete, we will work with

Cauchy approximations

(

\autoref

defn:cauchy-approximation) instead of Cauchy sequences.
Of course, our Cauchy approximations will now consist of Cauchy reals, rather than Dedekind reals or

rational numbers

.

Definition 11.3.2

.

Let

R

c

and the relation

∼

:

Q

+

×

R

c

×

R

c

→

U

be the following higher inductive-inductive type family.
The type

R

c

of

Cauchy reals

is generated by the following constructors:

•

rational points:

for any

q

:

ℚ

there is a real

𝗋𝖺𝗍

⁢

(

q

)

.

•

limit points

:
for any

x

:

ℚ

+

→

ℝ

𝖼

such that

∀

(

δ

,

ϵ

:

ℚ

+

)

.

x

δ

∼

δ

+

ϵ

x

ϵ

(11.3.2)

there is a point

𝗅𝗂𝗆

⁢

(

x

)

:

ℝ

𝖼

. We call

x

a

Cauchy approximation

.

•

paths:

for

u

,

v

:

ℝ

𝖼

such that

∀

(

ϵ

:

ℚ

+

)

.

u

∼

ϵ

v

(11.3.3)

then there is a path

𝖾𝗊

ℝ

𝖼

⁢

(

u

,

v

)

:

u

=

ℝ

𝖼

v

.

Simultaneously, the type family

∼

:

R

c

→

R

c

→

Q

+

→

U

is generated by the following constructors.
Here

q

and

r

denote rational numbers;

δ

,

ϵ

, and

η

denote

positive

rationals;

u

and

v

denote Cauchy reals; and

x

and

y

denote Cauchy approximations:

•

for any

q

,

r

,

ϵ

, if

-

ϵ

<

q

-

r

<

ϵ

, then

𝗋𝖺𝗍

⁢

(

q

)

∼

ϵ

𝗋𝖺𝗍

⁢

(

r

)

,

•

for any

q

,

y

,

ϵ

,

δ

, if

𝗋𝖺𝗍

⁢

(

q

)

∼

ϵ

-

δ

y

δ

, then

𝗋𝖺𝗍

⁢

(

q

)

∼

ϵ

𝗅𝗂𝗆

⁢

(

y

)

,

•

for any

x

,

r

,

ϵ

,

δ

, if

x

δ

∼

ϵ

-

δ

𝗋𝖺𝗍

⁢

(

r

)

, then

𝗅𝗂𝗆

⁢

(

x

)

∼

ϵ

𝗋𝖺𝗍

⁢

(

r

)

,

•

for any

x

,

y

,

ϵ

,

δ

,

η

, if

x

δ

∼

ϵ

-

δ

-

η

y

η

, then

𝗅𝗂𝗆

⁢

(

x

)

∼

ϵ

𝗅𝗂𝗆

⁢

(

y

)

,

•

for any

u

,

v

,

ϵ

, if

ξ

,

ζ

:

u

∼

ϵ

v

, then

ξ

=

ζ

(propositional truncation).

The first constructor of

ℝ

𝖼

says that any rational number can be regarded as a real number.
The second says that from any Cauchy approximation to a real number, we can obtain a new real number called its “limit”.
And the third expresses the idea that if two Cauchy approximations coincide, then their limits are equal.

The first four constructors of

∼

specify when two rational numbers are close, when a rational is close to a limit, and when two limits are close.
In the case of two rational numbers, this is just the usual notion of

ϵ

-closeness for rational numbers, whereas the other cases can be derived by noting that each approximant

x

δ

is supposed to be within

δ

of the limit

𝗅𝗂𝗆

⁢

(

x

)

.

We remind ourselves of proof-relevance: a real number obtained from

𝗅𝗂𝗆

is represented not
just by a Cauchy approximation

x

, but also a proof

p

of (

11.3.2

), so we
should technically have written

𝗅𝗂𝗆

⁢

(

x

,

p

)

instead of just

𝗅𝗂𝗆

⁢

(

x

)

.
A

similar

observation also applies to

𝖾𝗊

ℝ

𝖼

and (

11.3.3

), but we shall write just

𝖾𝗊

ℝ

𝖼

:

u

=

v

instead of

𝖾𝗊

ℝ

𝖼

⁢

(

u

,

v

,

p

)

:

u

=

v

. These abuses of notation are
mitigated by the fact that we are omitting mere propositions and information that is
readily guessed.
Likewise, the last constructor of

∼

ϵ

justifies our leaving the other four nameless.

We are immediately able to populate

ℝ

𝖼

with many real numbers. For suppose

x

:

ℕ

→

ℚ

is a traditional Cauchy sequence of rational numbers, and let

M

:

ℚ

+

→

ℕ

be its
modulus of convergence. Then

𝗋𝖺𝗍

∘

x

∘

M

:

ℚ

+

→

ℝ

𝖼

is a Cauchy
approximation, using the first constructor of

∼

to produce the necessary

witness

.
Thus,

𝗅𝗂𝗆

⁢

(

𝗋𝖺𝗍

∘

x

∘

m

)

is a real number. Various famous
real numbers

2

,

π

,

e

, … are all limits of such Cauchy sequences of
rationals.

Title

11.3.1 Construction of Cauchy reals

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