Chapter 11 Notes

Chapter 11 Notes

Defining algebraic operations on Dedekind reals, especially

multiplication

, is both somewhat tricky and tedious.
There are several ways to get

arithmetic

going: each has its own advantages, but they all seem to require some technical work.
For

instance

, Richman

[

Richman:reals

]

defines multiplication on the Dedekind reals first on the

positive

cuts and then extends it algebraically to all

Dedekind cuts

, while Conway

[

conway:onag

]

has observed that the

definition

of multiplication for

surreal numbers

works well for Dedekind reals.

Our treatment of the Dedekind reals borrows many ideas from

[

BauerTaylor09

]

where the Dedekind reals are constructed in the

context

of Abstract Stone Duality.
This is a (restricted) form of simply typed

λ

-

calculus

with a distinguished object

Σ

which classifies

open sets

, and by

duality

also the closed ones. In

[

BauerTaylor09

]

you can also find detailed proofs of the basic

properties

of arithmetical

operations

.

The fact that

ℝ

𝖼

is the least Cauchy

complete

archimedean

ordered field

, as was proved in

\autoref

RC-initial-Cauchy-complete, indicates that our Cauchy reals probably coincide with the Escardó-Simpson reals

[

EscardoSimpson:01

]

.
It would be interesting to check whether this is really the case. The notion of Escardó-Simpson reals, or more precisely the corresponding

closed interval

, is interesting because it can be stated in any category with finite

products

.

In constructive set theory augmented by the “regular extension axiom”, one may also try to define Cauchy completion by closing under limits of

Cauchy sequences

with a transfinite

iteration

.
It would also be interesting to check whether this construction agrees with ours.

It is constructive folklore that coincidence of Cauchy and Dedekind reals requires dependent choice but it is less well known that

countable choice

suffices. Recall that

dependent choice

states that for a

total relation

R

on

A

, by which we mean

∀

(

x

:

A

)

.

∃

(

y

:

A

)

.

R

(

x

,

y

)

, and for any

a

:

A

there merely exists

f

:

ℕ

→

A

such that

f

⁢

(

0

)

=

a

and

R

⁢

(

f

⁢

(

n

)

,

f

⁢

(

n

+

1

)

)

for all

n

:

ℕ

. Our

\autoref

when-reals-coincide uses the typical trick for converting an application of dependent choice to one using countable choice. Namely, we use countable choice once to make in advance all the choices that could come up, and then use the

choice function

to avoid the dependent choices.

The intricate relationship between various notions of

compactness

in a constructive
setting is discussed in

[

bridges2002compactness

]

. Palmgren

[

Palmgren:FT

]

has a
good comparison between

pointwise

analysis

and
pointfree topology.

The surreal numbers were defined by

[

conway:onag

]

, using a sort of inductive definition but without justifying it explicitly in terms of any foundational system.
For this reason, some later authors have tended to use sign-expansions or other more explicit presentations which can be coded more obviously into

set theory

.
The idea of representing them in

type theory

was first considered by Hancock, while
Setzer and Forsberg

[

forsbergfinite

]

noted that the surreals and their inequality relations

<

and

≤

naturally form an inductive-inductive definition.
The

higher

inductive-inductive version presented here, which builds in the correct notion of equality for surreals, is new.

Title

Chapter 11 Notes

\metatable

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