11.3 Cauchy reals

11.3 Cauchy reals

The Cauchy reals are, by intent, the

completion

of

ℚ

under limits of

Cauchy sequences

.
In the classical construction of the Cauchy reals, we consider the set

𝒞

of all Cauchy sequences in

ℚ

and then form a suitable

quotient

𝒞

/

≈

.
Then, to show that

𝒞

/

≈

is Cauchy

complete

, we consider a Cauchy sequence

x

:

ℕ

→

𝒞

/

≈

, lift it to a

sequence

of sequences

x

¯

:

ℕ

→

𝒞

, and

construct

the limit of

x

using

x

¯

. However, the lifting of

x

to

x

¯

uses
the

axiom of countable choice

(the

instance

of (

LABEL:eq:ac

) where

X

=

ℕ

) or the

law of excluded middle

, which we may wish to avoid.
Every construction of reals whose last step is a quotient suffers from this deficiency.
There are three common ways out of the conundrum in constructive mathematics:

1.

Pretend that the reals are a setoid

(

𝒞

,

≈

)

, i.e., the type of
Cauchy sequences

𝒞

with a coincidence

relation

attached to it by
administrative decree. A sequence of reals then simply

is

a sequence of Cauchy
sequences representing them.

2.

Give in to temptation and accept the axiom of countable choice. After all, the axiom
is valid in most models of constructive mathematics based on a computational viewpoint,
such as realizability models.

3.

Declare the Cauchy reals unworthy and construct the Dedekind reals instead.
Such a verdict is perfectly valid in certain

contexts

, such as in sheaf-theoretic models of constructive mathematics.
However, as we saw in

\autoref

sec:dedekind-reals, the constructive Dedekind reals have their own problems.

Using

higher inductive types

, however, there is a fourth

solution

, which we believe to be preferable to any of the above, and interesting even to a classical mathematician.
The idea is that the Cauchy real numbers should be the

free complete metric space

generated by

ℚ

.
In general, the construction of a free gadget of any sort requires applying the gadget

operations

repeatedly many times to the

generators

.
For instance, the elements of the

free group

on a set

X

are not just binary

products

and

inverses

of elements of

X

, but words obtained by iterating the product and inverse constructions.
Thus, we might naturally expect the same to be true for Cauchy completion, with the relevant “operation” being “take the limit of a Cauchy sequence”.
(In this case, the iteration would have to take place transfinitely, since even after infinitely many steps there will be new Cauchy sequences to take the limit of.)

The

argument

referred to above shows that if excluded middle or countable choice hold, then Cauchy completion is very special: when building the completion of a space, it suffices to stop applying the operation after

one step

.
This may be regarded as analogous to the fact that

free monoids

and free groups can be given explicit descriptions in terms of (

reduced

) words.
However, we saw in

\autoref

sec:free-algebras that higher inductive types allow us to construct free gadgets

directly

, whether or not there is also an explicit description available.
In this

section

we show that the same is true for the Cauchy reals (a

similar

technique would construct the Cauchy completion of any

metric space

; see

\autoref

ex:metric-completion).
Specifically, higher inductive types allow us to

simultaneously

add limits of Cauchy sequences and quotient by the coincidence relation, so that we can avoid the problem of lifting a sequence of reals to a sequence of representatives.

Title

11.3 Cauchy reals

\metatable

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