11. Real numbers

11. Real numbers

Any

foundation of mathematics

worthy of its name must

eventually

address the construction of real numbers as understood by

mathematical analysis

, namely as a

complete

archimedean

ordered field

.
There are two notions of completeness. The one by Cauchy requires that the reals be

closed under

limits of

Cauchy sequences

, while the

stronger

one by Dedekind requires

closure

under

Dedekind cuts

.
These lead to two ways of constructing reals, which we study in

\autoref

sec:dedekind-reals and

\autoref

sec:cauchy-reals, respectively. In

\autoref

RD-final-field,

\autoref

RC-initial-Cauchy-complete we characterize the two constructions in terms of

universal properties

: the Dedekind reals are the final archimedean ordered field, and the Cauchy reals the initial Cauchy complete archimedean ordered field.

In traditional constructive mathematics,

real numbers

always seem to require certain compromises. For example, the Dedekind reals work better with

power sets

or some other form of impredicativity, while Cauchy reals work well in the presence of

countable choice

.
However, we give a new construction of the Cauchy reals as a higher inductive-inductive type that seems to be a third possibility, which requires neither power sets nor countable choice.

In

\autoref

sec:comp-cauchy-dedek we compare the two constructions of reals. The Cauchy reals are included in the Dedekind reals. They coincide if excluded middle or countable choice holds, but in general the

inclusion

might be proper.

In

\autoref

sec:compactness-interval we consider three notions of

compactness

of the

closed interval

[

0

,

1

]

. We first show that

[

0

,

1

]

is metrically

compact

in the sense that it is complete and

totally bounded

, and that uniformly continuous maps on metrically

compact spaces

behave as expected. In contrast, the Bolzano–Weierstraß property that every

sequence

has a

convergent

subsequence

implies the limited principle of omniscience, which is an

instance

of excluded middle. Finally, we discuss Heine-Borel compactness. A naive formulation of the finite

subcover

property does not work, but a proof relevant notion of inductive covers does.
This

section

is basically standard constructive analysis.

The

development

of real numbers and

analysis

in

homotopy type theory

can be easily made

compatible

with classical mathematics. By assuming excluded middle and the

axiom of choice

we get standard classical analysis: the Dedekind and Cauchy reals coincide, foundational questions about the

impredicative

nature of the Dedekind reals disappear, and the interval is as compact as it could be.

We close the chapter by constructing Conway’s surreals as a higher inductive-inductive type in

\autoref

sec:surreals;
the construction is more natural in

univalent

type theory

than in classical

set theory

.

In

addition

to the basic theory of

\autoref

cha:basics,

\autoref

cha:logic, as noted above we use “higher inductive-inductive types” for the Cauchy reals and the surreals: these combine the ideas of

\autoref

cha:hits with the notion of inductive-inductive type mentioned in

\autoref

sec:

generalizations

.
We will also frequently use the traditional logical notation described in

\autoref

subsec:prop-trunc, and the fact (proven in

\autoref

sec:piw-pretopos) that our “sets” behave the way we would expect.

Note that the

total space

of the

universal cover

of the circle, which
in

\autoref

subsec:pi1s1-homotopy-theory played a role

similar

to “the real numbers” in
classical algebraic topology, is

not

the type of reals we are looking for. That
type is

contractible

, and thus

equivalent

to the singleton type, so it cannot be equipped
with a non-trivial

algebraic structure

.

Title

11. Real numbers

\metatable

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