Chapter 3 Notes

Chapter 3 Notes

The fact that it is possible to define sets, mere propositions, and contractible types in

type theory

, with all higher homotopies automatically taken care of as in §3.1 (

http://planetmath.org/31setsandntypes

),§3.3 (

http://planetmath.org/33merepropositions

),§3.11 (

http://planetmath.org/311contractibility

), was first observed by Voevodsky.
In fact, he defined the entire hierarchy of

n

-types by

induction

, as we will do in

http://planetmath.org/node/87580

Chapter 7.

Theorem

3.2.2 (

http://planetmath.org/32propositionsastypes#Thmprethm1

),Corollary 3.2.7 (

http://planetmath.org/32propositionsastypes#Thmprecor1

) rely in essence on a classical theorem of Hedberg,
which we will prove in §7.2 (

http://planetmath.org/72uniquenessofidentityproofsandhedbergstheorem

).
The

implication

that the propositions-as-types form of

𝖫𝖤𝖬

contradicts univalence was observed by Martín Escardó on the

Agda

mailing list.
The proof we have given of Theorem 3.2.2 (

http://planetmath.org/32propositionsastypes#Thmprethm1

) is due to Thierry Coquand.

The propositional

truncation

was introduced in the extensional type theory of

NuPRL

in 1983 by Constable

[

4

]

as an
application of “subset” and “

quotient

” types. What is here called the
“propositional truncation” was called “squashing” in the

NuPRL

type theory

[

5

]

.
Rules characterizing the propositional truncation directly, still in extensional type theory, were given in

[

1

]

.
The intensional version in

homotopy type theory

was constructed by Voevodsky using an

impredicative

quantification, and later by Lumsdaine using

higher inductive types

(see §6.9 (

http://planetmath.org/69truncations

)).

Voevodsky

[

14

]

has proposed resizing rules of the kind considered in §3.5 (

http://planetmath.org/35subsetsandpropositionalresizing

).
These are clearly related to the notorious

axiom of reducibility

proposed by Russell in his and Whitehead’s

Principia Mathematica

[

10

]

.

The adverb “purely” as used to refer to untruncated logic is a reference to the use of

monadic

modalities to model effects in programming languages; see §7.7 (

http://planetmath.org/77modalities

) and the Notes to

http://planetmath.org/node/87580

Chapter 7.

There are many different ways in which logic can be treated relative to type theory.
For

instance

, in

addition

to the plain propositions-as-types logic described in §1.11 (

http://planetmath.org/111propositionsastypes

), and the alternative which uses mere propositions only as described in §3.6 (

http://planetmath.org/36thelogicofmerepropositions

), one may introduce a separate “sort” of

propositions

, which behave somewhat like types but are not identified with them.
This is the approach taken in logic enriched type theory

[

3

]

and in some

presentations

of the internal

languages

of toposes and related categories (e.g.

[

7

]

,

[

6

]

), as well as in the proof assistant

Coq

.
Such an approach is more general, but less powerful.
For instance, the principle of unique choice (§3.9 (

http://planetmath.org/39theprincipleofuniquechoice

)) fails in the category of so-called setoids in

Coq

[

11

]

, in logic enriched type theory

[

3

]

, and in

minimal

type theory

[

8

]

.
Thus, the univalence axiom makes our type theory behave more like the internal logic of a topos; see also

http://planetmath.org/node/87584

Chapter 10.

Martin-Löf

[

9

]

provides a discussion on the history of

axioms of choice

.
Of course, constructive and intuitionistic mathematics has a long and complicated history, which we will not delve into here; see for instance

[

12

]

,

[

13

]

.

References

1

Awodey,Steven and Bauer,Andrej. Propositions as [types].

Journal of Logic and Computation

, 14:447–471 2004,Bdsk-Url-1

2

Aczel,Peter and Gambino,Nicola.

Collection

principles in dependent type theory. In emphTypes for Proofs and

Programs

,International Workshop,TYPES 2000,Durham,UK,December 8-12,2000,Selected Papers, pages 1–23Springer, .

3

Aczel,Peter and Gambino,Nicola. Collection principles in dependent type theory. In emphTypes for Proofs and Programs,International Workshop,TYPES 2000,Durham,UK,December 8-12,2000,Selected Papers, pages 1–23Springer, .

4

Constable,Robert L.. Constructive Mathematics as a Programming Logic I: Some Principles of Theory. In

Annals of Mathematics

,volume 24 pages 21–37. Elsevier Science Publishers,B.V. (North-Holland), 1985.

5

Robert L. Constable and Stuart F. Allen and H. M. Bromley and W. R. Cleaveland and J. F. Cremer and Robert W. Harper and Douglas J. Howe and T. B. Knoblock and N. P. Mendler and P. Panangaden and James T. Sasaki and Scott F. Smith,

Implementing Mathematics with the NuPRL Proof Development System

Prentice Hall,1986

6

Peter T. Johnstone,

Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2

Oxford Science Publications,2002

7

Bart Jacobs,

Categorical logic and type theory

Elsevier,1999

8

Maietti,Maria Emilia and Sambin,Giovanni. Toward a minimalist

foundation

for constructive mathematics. In

From Sets and Types to

Topology

and

Analysis

: Practicable Foundations for Constructive Mathematics

,volume 48 of Oxford Logic Guides, pages 91–114. Clarendon Press, 2005.

9

Martin-Löf,Per. 100 years of Zermelo’s axiom of choice: what was the problem with it?.

The

Computer

Journal

, 49:345–350 2006

10

Whitehead,Alfred North and Russell,Bertrand,

Principia mathematica,3 vol.s

Cambridge University Press,1910–1913; Second edition,1925–1927

11

Arnaud Spiwack.

A Journey Exploring the Power and Limits of Dependent Type Theory

. PhD thesis, École Polytechnique,Palaiseau,France, 2011.

12

Troelstra,Anne Sjerp and van Dalen,Dirk,

Constructivism in mathematics. Vol. I

North-Holland Publishing Co.,1988

13

Troelstra,Anne Sjerp and van Dalen,Dirk,

Constructivism in mathematics. Vol. II

North-Holland Publishing Co.,1988

14

Vladimir Voevodsky. A

universe

polymorphic type system.

\url

http://uf-ias-2012.wikispaces.com/file/view/Universe+polymorphic+type+sytem.pdf, 2012.

Title

Chapter 3 Notes

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