1.3 Universes and Families

1.3 Universes and Families

So far, we have been using the

expression

“

A

is a type” informally. We
are going to make this more precise by introducing

universes

.
A universe is a type whose

elements

are types. As in

naive set theory

,
we might wish for a universe of all types

𝒰

∞

including itself
(that is, with

𝒰

∞

:

𝒰

∞

).
However, as in

set
theory

, this is unsound, i.e. we can deduce from it that every type,
including the empty type representing the

proposition

False (see §1.7 (

http://planetmath.org/17coproducttypes

)), is inhabited.
For

instance

, using a
representation of sets as trees, we can directly encode Russell’s

paradox

[

1

]

.

To avoid the paradox we introduce a hierarchy of universes

𝒰

0

:

𝒰

1

:

𝒰

2

:

⋯

where every universe

𝒰

i

is an

element of

the next universe

𝒰

i

+

1

. Moreover, we assume that our universes are

cumulative

,
that is that all the elements of the

i

th

universe are also elements of the

(

i

+

1

)

st

universe, i.e. if

A

:

𝒰

i

then also

A

:

𝒰

i

+

1

.
This is convenient, but has the slightly unpleasant

consequence

that elements no longer have unique types, and is a bit tricky in other ways that need not concern us here; see the Notes.

When we say that

A

is a type, we mean that it inhabits some universe

𝒰

i

. We usually want to avoid mentioning the level

i

explicitly,
and just assume that levels can be assigned in a

consistent

way; thus we
may write

A

:

𝒰

omitting the level. This way we can even write

𝒰

:

𝒰

, which can be read as

𝒰

i

:

𝒰

i

+

1

, having left the
indices implicit. Writing universes in this style is referred to as

typical ambiguity

.
It is convenient but a bit dangerous, since it allows us to write valid-looking proofs that reproduce the paradoxes of self-reference.
If there is any doubt about whether an

argument

is correct, the way to check it is to try to assign levels consistently to all universes appearing in it.
When some universe

𝒰

is assumed, we may refer to types belonging to

𝒰

as

small types

.

To model a

collection

of types varying over a given type

A

, we use

functions

B

:

A

→

𝒰

whose

codomain

is a universe. These functions are called

families of types

(or sometimes

dependent types

);
they correspond to families of sets as used in
set theory.

An example of a type family is the family of

finite sets

𝖥𝗂𝗇

:

ℕ

→

𝒰

, where

𝖥𝗂𝗇

⁢

(

n

)

is a type with exactly

n

elements.
(We cannot

define

the family

𝖥𝗂𝗇

yet — indeed, we have not even introduced its domain

ℕ

yet — but we will be able to soon; see

\autoref

ex:fin.)
We may denote the elements of

𝖥𝗂𝗇

⁢

(

n

)

by

0

n

,

1

n

,

…

,

(

n

-

1

)

n

, with

subscripts

to emphasize that the elements of

𝖥𝗂𝗇

⁢

(

n

)

are different from those of

𝖥𝗂𝗇

⁢

(

m

)

if

n

is different from

m

, and all are different from the ordinary

natural numbers

(which we will introduce in §1.9 (

http://planetmath.org/19thenaturalnumbers

)).

A more

trivial

(but very important) example of a type family is the

constant

type family
at a type

B

:

𝒰

, which is of course the

constant function

(

λ

(

x

:

A

)

.

B

)

:

A

→

𝒰

.

As a

non

-example, in our version of

type theory

there is no type family “

λ

(

i

:

ℕ

)

.

𝒰

i

”.
Indeed, there is no universe large enough to be its codomain.
Moreover, we do not even identify the indices

i

of the universes

𝒰

i

with the natural numbers

ℕ

of type theory (the latter to be introduced in §1.9 (

http://planetmath.org/19thenaturalnumbers

)).

References

1

Coquand,Thierry. The paradox of trees in type theory.

BIT Numerical Mathematics

, 32:10–14 1992

Title

1.3 Universes and Families

Canonical

name

13UniversesAndFamilies

Date of creation

2013-11-12 21:01:43

Last modified on

2013-11-12 21:01:43

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PMBookProject (1000683)

Last modified by

rspuzio (6075)

Numerical id

11

Author

PMBookProject (6075)

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