THE L. E. J. BROUVVER CENTENARY SYMPOSIUM Proceedings of the Conference held in Noordwijkerhout, 8- 13 June, 198 1
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THE L. E. J. BROUVVER CENTENARY SYMPOSIUM Proceedings of the Conference held in Noordwijkerhout, 8- 13 June, 198 1
Edited by
A. S. TROELSTRA Universiteit van Amsterdam Amsterdam. The Netherlands
and
D. VAN DALEN Rijksuniversiteit Utrecht Utrecht, The Netherlands
I9b2
NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK OXFORD
@
NORTH-HOLLAND PUBLISHING COMPANY, 1982
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
ISBN: 0 4 4 4 86494 6
Published by:
-
North-Holland Publishing Company- Amsterdam New York * Oxford
Sole distributors for the U.S.A.and Canada: Elsevier Science Publishing Company, Inc. 52 Vanderbilt Avenue NewYork,N.Y. 10017
PRINTED IN T H E NETHERLANDS
PREFACE
From time to time a singularly gifted man with a purpose leaves his mark on the development of mathematics. In our century one of those men was Luitzen Egbertus Jan Brouwer. He made a fundamental contribution to topology which became part of the mainstream of mathematics almost immediately. This was different in the case of his second contribution, involving a reappraisal of mathematics, logic and language, and the relationship between them. Although Kronecker and the French semi-intuitionists, E. Bore1 in particular, had already advocated various forms of constructivism, Brouwer's radical approach was a novelty. At a time that Formalism in mathematics, and Neo-Positivism in the sciences and philosophy were the leading fashion, Brouwer presented his view of mathematics as a mental activity of man -with rather unpopular consequences for the r81e of language and logic. Brouwer's programme -the rebuilding of mathematics according to intuitionistic principles- was laid down in a series of papers, already foreshadowed by his doctoral thesis. Now, a hundred years after Brouwer's birth, his achievements have found recognition and ideas from intuitionism have established themselves firmly in logic and the foundations of mathematics. During the last few decades, intuitionism and constructivism have increasingly been brought into relation with independent disciplines such as recursion theory, proof theory and category theory. This development started already in the thirties; the late Arend Heyting's best-known contribution to intuitionism, to wit the formalization of intuitionistic logic and mathematics, were soon followed by proof-theoretic researches (Gentzen, GEdel) and the introduction of topological and algebraical semantics (Stone, Tarski). The study of realizability, initiated by Kleene, and functional interpretations, such as for example GEdel's Dialectica interpretation, linked the study of intuitionistic formalisms with recursion theory. The first objects of metamathematical study were intuitionistic logic and arithmetic, but in the sixties the investigation of the intuitionistic continuum was undertaken by, among others, Kleene, Kreisel, Myhill and D.Scott. . In recent years a good deal of attention has also been given to the development of formalisms that could do for constructive and intuitionistic mathematics what axiomatic set theory did for classical mathematics: intuitionistic set theory, theories of operators and classes, and type theories. In their turn these formalisms have led to many metamathematical studies. In the model theory of intuitionistic theories we have lately witnessed the advance of categorical logic and topos theory -initially created by Lawere- which
vi has provided a common generalization of the earlier semantics of Beth and Kripke, and topological semantics. Categorical logic, and in particular topos theory is now studied and used by people with a variety of backgrounds; it provides a unifying and flexible framework for many metamathematical researches. We only mention sheaf models over Grothendieck topoi and realizability toposes as examples. Thus methods ultimately deriving from algebra and algebraic geometry are introduced in the study of intuitionism. Alongside intuitionism various other schools of constructive mathematics, at least partly deriving their initial impetus from intuitionism, albeit with a somewhat different outlook, have developed: constructivism in the style of Markov, and E . Bishop's form of constructivism. Last, but not least; it should not be forgotten that also in "ordinary" mathematics there are problems and results directly connected with the search for constructive solutions; see for example the contributions of C. Delzell, G . Kreisel and A. MacIntyre in this volume. The preceding brief and incomplete sketch i s enough to illustrate that there is a wide range of activities connected with intuitionism and constructive mathematics, and so it seemed a fitting tribute to the founding father at his centenary t o bring together the rich diversity of constructivists into a memorial symposium. The opening address of the conference was presented by his excellency the minister of education, dr. A . Pais. The organizing committee consisted of A.S. Troelstra(chairman), D. van Dalen (secretary), K. Koymans(treasurer), M. Euwe and G. Renardel de Lavalette. Max Euwe, himself a student and friend of Brouwer, has made crucial contributions towards the organization of the congress; we lament the l o s s that his death in November last year has brought u s . The conference, held under the auspices of the Royal Dutch Academy of Sciences and the Wiskundig Genootschap (Dutch Mathematical Association) was most generously supported by VOLMAC Automation at the occasion of its third lustrum; VOLMAC's contribution made the organization o f the symposium possible. We are also grateful for financial support by AMEV, and for a contribution of Shell Nederland towards the social activities. Valuable assistance has been given by North-Holland Publishing Company. The organizing committee wishes to express its gratitude to the many individuals who have contributed towards the success of the meeting, the participants, the staff of the Conference Centre De Leeuwenhorst, and in particular Doke van Dalen who ran the office. The Teyler Foundation has kindly opened its museum to u s , and the City of Amsterdam has commemorated Brouwer's Centenary at a reception at the Historical Museum. These Proceedings contain most of the invited talks, some o f the contributed talks, and in addition some contributed papers by persons who were unable to attend the meeting. The editors.
THE LEJ. BROUWER CENTENARY SYmOSIuM AS. TkoeLFlra and D. van Dalen (editors) 0 North-Holland Publishing Company, 1982
1
THE TYPE THEORETIC INTERPRETATION OF CONSTRUCTIVE SET THEORY: CHOICE PRINCIPLES
P e t e r Aczel Department of Mathematics Manchester University England
In an e a r l i e r paper I gave an i n t e r p r e t a t i o n of a system CZF of c o n s t r u c t i v e s e t theory w i t h i n an extension of Martin-Lgf's types.
i n t u i t i o n i s t i c theory of
I n t h i s paper some a d d i t i o n a l axioms, each
a consequence of the axiom of choice, a r e shown t o hold i n t h e i n t e r p r e t a t i o n .
The mathematical
deductions a r e presented i n an informal, but I hope rigorous s t y l e . INTRODUCTION
The axiom of choice does n o t have an unambiguous s t a t u s i n constructive mathematics.
the one hand i t i s s a i d t o be an immediate consequence of t h e
On
Any proof of
constructive i n t e r p r e t a t i o n of t h e q u a n t i f i e r s . Vx cA3y E B F(x,y)
must y i e l d a function
f
This i s c e r t a i n l y t h e case i n Martin-Lb;f's
E
A
+
B
such t h a t 'Vx c A F ( x , f ( x ) ) .
i n t u i t i o n i s t i c theory of types.
On
t h e o t h e r hand, from t h e very e a r l i e s t days, t h e axiom of choice has been c r i t i c i s e d as an excessively non-constructive p r i n c i p l e even f o r c l a s s i c a l s e t theory.
Moreover, i n more r e c e n t y e a r s , i t has been observed t h a t t h e f u l l axiom
of choice cannot be added t o systems of c o n s t r u c t i v e set theory without y i e l d i n g constructively unacceptable cases o f excluded middle (see e.g. Diaconescu C19751). In Myhill C19751 a system o f c o n s t r u c t i v e s e t theory was put forward as a s u i t a b l e s e t t i n g f o r t h e s t y l e of c o n s t r u c t i v e mathematics pursued by Bishop and h i s school ( s e e Bishop C19671 and Bridges C19791 and a l s o t h i s proceedings). y l h i l l argued informally t h a t t h e axiom constructively acceptable. CZF
DC
of dependent choices was
Aczel C19781 contains an i n t e r p r e t a t i o n o f a system
of c o n s t r u c t i v e s e t theory t h a t i s c l o s e l y r e l a t e d t o Myhill's system.
claimed t h e r e , without proof, t h a t
w i l l b e shown i n 15 of t h i s paper.
DC
was t r u e i n t h e i n t e r p r e t a t i o n .
I
This
I n 1 7 of Aczel C19781 I a l s o put forward an
axiom c a l l e d t h e p r e s e n t a t i o n axiom, which I thought t o be a p l a u s i b l e a d d i t i o n t o constructive s e t theory. interpretation.
But t h e r e I was unable t o v e r i f y i t s t r u t h i n t h e
A fundamental aim of t h i s paper i s t o show t h a t t h e p r e s e n t a t i o n
P.H.G. ACZEL
2
axiom i s indeed t r u e i n the i n t e r p r e t a t i o n .
In f a c t a strengthening,
IICI-PA,
of the p r e s e n t a t i o n axiom w i l l be v e r i f i e d i n 57. The i n t e r p r e t a t i o n of
given i n Aczel [1978] was c a r r i e d out w i t h i n a
CZF
n a t u r a l extension of Martin-Lb;f's Martin-Lof
C19751.
i n t u i t i o n i s t i c theory of types a s presented i n
Type theory i s intended t o be a fundamental conceptual
framework f o r t h e b a s i c notions of c o n s t r u c t i v e mathematics.
For t h i s reason I
beliege t h a t t h e i n t e r p r e t a t i o n of c o n s t r u c t i v e s e t theory i n type theory can l a y claim t o give a good c o n s t r u c t i v e meaning t o t h e s e t t h e o r e t i c a l notions.
In
view of t h i s i t i s very n a t u r a l t o explore f u l l y which s e t t h e o r e t i c a l axioms a r e true i n the interpretation.
To do t h i s I have found i t convenient t o develop a
f l e x i b l e informal s t y l e f o r p r e s e n t i n g deductions i n type theory.
Moreover, i f
type theory i s t o be a p r a c t i c a l v e h i c l e f o r t h e p r e s e n t a t i o n of mathematical deductions of c o n s t r u c t i v e mathematics i t s e l f , such an informal s t y l e w i l l be essential.
So I have t r i e d t o w r i t e t h i s paper r e l y i n g only on the informal
d e s c r i p t i o n of t h e type t h e o r e t i c notions given i n 5 1 .
There i s the danger t h a t
I may be c r i t i c i s e d f o r lack of r i g o u r , as i t may not be always transparent t o
the reader how each s t e p could b e formalised.
But my primary aim has been t o
present t h e mathematical r e s u l t s s t r i p p e d of excessive formalism.
Of course,
while w r i t i n g t h i s paper, I have had i n mind a formal language f o r type theory such a s t h a t presented i n Martin-Lgf C19791, and t h e reader may f i n d i t h e l p f u l t o r e f e r t o t h a t paper both f o r t h e l i s t of r u l e s of t h e formal language and f o r the explanations of fundamental notions.
Martin-Lof
119751 and 119791 give some
information concerning t h e pre-history of the framework of type theory used here. The reader should a l s o r e f e r t o Aczel C19781 f o r a more extensive discussion of than w i l l be given here.
CZF
intended t o be self-contained.
Apart from these t o p i c s t h e present paper i s I n view of t h i s , t h e paper s t a r t s with an informal
d e s c r i p t i o n of type theory i n 5 1 , and i n 52 and 53 goes on t o present t h e i n t e r p r e t a t i o n of
CZF
i n t h e framework described i n 5 1 .
o t h e r s e c t i o n s the discussion i n 54 c o n s t r u c t i v e s e t theory.
In c o n t r a s t t o t h e
takes place i n an informal framework f o r
That d i s c u s s i o n i s concerned with formulating t h e axioms
t h a t w i l l be v e r i f i e d i n t h e remaining s e c t i o n s .
Dependent choices i s v e r i f i e d
i n 55, and t h i s s e c t i o n a l s o introduces some e s s e n t i a l i d e a s t o be used i n t h e l a s t two s e c t i o n s .
57 respectively.
The axioms
IIZI-AC
and
IICI-PA
w i l l be v e r i f i e d i n 56
and
I n t h e last s e c t i o n i t turns out necessary t o use a method not
contemplated i n Aczel C19781 o r a v a i l a b l e i n t h e formal languages of Martin-LGf C19751 and C19791, although t h e p o s s i b i l i t y of the method i s mentioned i n Martin-LSf
C19751.
This method involves making e x p l i c i t the conception of t h e
type U of small types as a type i n d u c t i v e l y s p e c i f i e d by a s p e c i f i c l i s t of r u l e s f o r forming s m a l l types. The method i s t o allow d e f i n i t i o n s of functions on
U
by a t r a n s f i n i t e recursion following t h e inductive generation of the small
Constructive set theory
types.
3
The method i s used t o c o n s t r u c t a s u i t a b l e r e p r e s e n t a t i o n of each small
type a s an element of t h e type
of i t e r a t i v e s e t s .
V
A s pointed out i n Aczel
C19781 such a r e p r e s e n t a t i o n i s what i s needed t o v e r i f y t h e p r e s e n t a t i o n axiom. A s with any a c t i v e research program t h e type t h e o r e t i c approach t o c o n s t r u c t i v e
mathematics has been under steady development over t h e years and the publications I have r e f e r r e d t o only represent s t a g e s i n t h a t development.
There have been
some s i g n i f i c a n t changes between t h e s t a g e s represented by Martin-Lgf Aczel 119781 and the more recent s t a g e s represented by Martin-LGf paper.
A minor n o t a t i o n a l change has been t h a t t h e symbols
interchanged t h e i r meaning. i n Martin-Lgf the type
and
V
have
A more s i g n i f i c a n t change has been t h e introduction
C19791 of t h e new form of type
(Wx
A)BCxI.
E
of i t e r a t i v e s e t s can be defined simply a s
V
U
119751 and
C19791 and t h i s
With t h i s new form
(Wx
E
U)x.
But i n t h i s
paper the new form w i l l not b e used, and i t s e f f e c t on t h e i n t e r p r e t a t i o n of constructive s e t theory w i l l b e l e f t t o another paper. above d e f i n i t i o n of
I have used t h e n o t a t i o n
V
f o r the i t e r a t i v e s e t s y d o l i s e d by
I
{b(x)
x
E
I n keeping with the
sup(A,b)
or
(supxcA)b(x)
A1 i n Aczel C19781.
Perhaps t h e In
most s i g n i f i c a n t change has been t h a t involving t h e treatment of e q u i l i t y .
Martin-Lgf 119751 t h e r e i s one r e l a t i o n of d e f i n i t i o n a l e q u a l i t y which i s defined t o be t h e equivalence r e l a t i o n which i s generated by the p r i n c i p l e s t h a t a definiendum i s always d e f i n i t i o n a l l y equal t o i t s d e f i n i e n s and t h a t d e f i n i t i o n a l e q u a l i t y is preserved under s u b s t i t u t i o n .
I n t h e p r e s e n t version of type theory
each notion of object must c a r r y with i t an e q u a l i t y r e l a t i o n which i s not So t h e r e i s an
n e c e s s a r i l y t o be understood a s a d e f i n i t i o n a l e q u a l i t y r e l a t i o n . e q u a l i t y r e l a t i o n f o r types, and a l s o f o r each type A.
f o r elements of type
A,
a r e l a t i o n of e q u a l i t y
Martin-L6f w r i t e s a = b c A
to express the judgement t h a t
a
and
b
example i f
f
and
g
A.
a r e equal elements of type
form of type has i t s own c r i t e r i a f o r when such a judgement holds. a r e elements of t h e type
A
-t
B
Each
So f o r
of functions'from
A
toB,
f = g E A + B means t h a t
f
and
g
a r e e x t e n s i o n a l l y equal, i . e .
I n p r a c t i c e when a judgement the context t h a t
a
and
b
a
=
b
E
A
f ( x ) = g(x)
E
B
for
x E A.
i s made i t i s usually e i t h e r c l e a r from
a r e elements of type
A
o r e l s e i t does not matter
exactly what t h e type of a and b i s . For t h i s reason I propose t o follow t h e s t a n d a r d convention of w r i t i n g simply
a = b.
Whenever two expressions a r e d e f i n i t i o n a l l y
equal, and i n a given context one of them r e f e r s t o an o b j e c t of some s o r t then the o t h e r expression w i l l r e f e r t o an equal object of t h a t s o r t .
For t h i s reason
it i s s a f e t o follow t h e standard convention, when making d e f i n i t i o n s , cf simply
4
P.H.G. ACZEL
writing equalities.
This w i l l b e done here.
The mathematical r e s u l t s i n t h i s paper were obtained while preparing a s e r i e s of t a l k s on t h e type t h e o r e t i c i n t e r p r e t a t i o n of c o n s t r u c t i v e s e t theory given i n Munich i n October 1980.
I am g r a t e f u l t o Prof. Schwichtenberg f o r h i s i n v i t a t i o n
which was t h e spur t o a f r e s h look a t t h e topic. The contents of t h i s paper do n o t e x a c t l y r e f l e c t t h e t o p i c of my t a l k a t the I f e l t unable t o w r i t e on t h a t t o p i c before having completed t h i s
conference.
paper, and I am g r a t e f u l t o t h e e d i t o r s f o r accepting t h i s s u b s t i t u t e . 5 1 . AN INFORMAL DESCRIPTION OF TYPE THEORY SOME NOTATION
If
...,
bCxl,
v a r i a b l e s then
i s an expression and
xnl
..., xn)
(xl,
bCxl,
xl,
..., xnl
..., x
i s a non-repeating l i s t of
w i l l denote t h e n-place function
f
having d e f i n i n g equation f(xl,
So whenever
bCal,
..., an]
... , xn)
... , xnl.
= bCxl,
r e f e r s t o an o b j e c t of some s o r t then
f(al,
..., an)
will r e f e r t o an equal object of t h a t s o r t and we can w r i t e f(al,
..., an] ..., x i n
..., an)
..., an].
= bCal,
i s t h e r e s u l t of simultaneously s u b s t i t u t i n g
Here bCal, for xl,
bCx,,
..., x n l ,
ais
.**’
making s u i t a b l e changes i n t h e bound
v a r i a b l e s when necessary.
If
c
i s t h e ordered p a i r
components
a
and
b
( a , b)
then
p(c)
and
q(c)
respectively.
TYPES AND THEIR ELEMENTS
1.1.
The fundamental notions of type theory a r e type
and i f
A
i s a type element of A.
If
a
i s an element of t h e type
A
then we s h a l l w r i t e a
E
A.
w i l l be t h e two
an
5
Constructive set theory
1.2.
We s t a r t our survey of the forms of type by considering the f a m i l i a r forms A-+B
A
-+
B
i s t h e type of functions
A
X
B
i s t h e type of p a i r s
A x B
(x)bCxl
( a , b)
N.
such t h a t
such t h a t
a
bCxl A
E
E
and
B
b
for
x
B.
N
E
and
A,
E
i s the
type of n a t u r a l numbers and i s i n d u c t i v e l y s p e c i f i e d using t h e r u l e s n e N s(n) E N
OeN Associated with
N
is t h e noethod of d e f i n i t i o n by recursion over
example, given a type
a
C,
and
C
E
f
N x C
E
-+
C
For
N.
we may define
h e N
by
C
-+
recursion so t h a t h(0) = a h(s(n)) We s h a l l w r i t e
a
on
and
1.3.
f
f(n, h(n))
=
R(n, a , f )
for
n
for
N.
E
i f we wish t o make t h e dependence of
h(n)
h(n)
explicit.
The notions w e have introduced so f a r already s u f f i c e f o r t h e type But h e r e w e
s t r u c t u r e of Godel's p r i m i t i v e r e c u r s i v e functions of f i n i t e types.
wish t o have a r i c h e r notion of type and i n p a r t i c u l a r we wish t o have types t h a t For t h i s reason i t i s u s e f u l t o have
a r e beyond the l e v e l of t h e f i n i t e types. types whose elements a r e themselves types.
While a type of a l l types i s
unreasonable i t i s s e n s i b l e t o have a type
U
s u i t a b l e e x t e n t t h e notion of type i t s e l f .
Reflecting on t h e forms of type we
have considered so f a r leads us t o have A, B
U
E
and a l s o t o have
N
E
A
+
U
t h a t a r e b u i l t up out of
a r e l a r g e types.
by recursion functions such as
B E U
and
A x B
E
U
whenever
Note t h a t we do not wish t o have
U.
i s n a t u r a l t o c a l l types i n U
of types t h a t r e f l e c t s t o a
small types and types such as F
E
+
N
N
Using the type
U
E
U
It
U.
U
and
U
one may form
-+
U
where
+ U
F(0) = N F ( s ( n ) ) = F(n) F
for
i s an example of a family of types.
is a function 1.4.
F
t h a t a s s i g n s a type
n
A.
If
be w r i t t e n
(Cx
E
II(A, F)
(nx
E
A)BCxl
and
i s t h e type of functions
and t h e d i s j o i n t union
C(A, F)
N.
In general a family of types over a type A F(a) t o each
We now c o n s i d e r t h e forms of type
family of types over t h e type
E
II(A, F) F
is
A)BCxl
(x)bCxl
and
(x)BCxl
a
E
A.
E(A, F)
respectively. such t h a t
i s t h e type of p a i r s
where
is a
F
then t h e s e types w i l l a l s o bCxl
The Cartesian product E
( a , b)
F(x)
for
such t h a t
x
A,
E
a
E
A
P.H.G. ACZEL
6 b
and
F(a).
E
forms when
The forms
A
B
+
i s t h e function
F
and
A x B
(x)B
a r e t h e s p e c i a l c a s e s of t h e s e new
having as c o n s t a n t value t h e type
A
+
B = (nx
E
A)B,
A
x
B = (Cx
E
A)B.
L e t u s consider more c l o s e l y t h e elements of t h e new forms of type.
f
E
II(A, F)
then
p(c)
then E
q(c)
A,
by r e c u r s i o n on considered. where a
E
N
F(0)
f
for
F(p(c))
x
and
E
c
A
f = (x)f(x).
(p(c), q(c)).
=
If
If c
(nz
E
h
N + C
E
was d e f i n e d from
a
and
C
E
h
We may u s e t h e same e q u a t i o n s t o d e f i n e F)) F ( s ( p ( z ) ) ) ,
X(N,
E
where
F
t o r e f l e c t them i n t h e type
E
TI(N,
TI(A, F)
and
L(A, F)
t h e l a s t two having s p e c i a l i n s t a n c e s
1.6.
and
No
I(A, a , b)
It has an element
c
E
If
1.7.
Z(A, a, b ) .
If
and
A
of t h e two types. f o r each
b
E
F
has t h e
A.
A -+ B
N, U, TI(A, F)
and
A x B.
B.
and
Our p r e s e n t I ( A , a , b),
V.
i s a type provided t h a t
r
provided
a = b.
a
So
and
b
c = r
a r e elements of t h e type
and
a = b
A.
whenever
The s i g n i f i c a n c e of t h i s form w i l l perhaps become c l e a r e r l a t e r .
i s s m a l l then so i s
A
from
we should remember
survey of forms of type w i l l b e completed by c o n s i d e r i n g t h e forms A + B,
F)
So we have t h a t b o t h t h e s e types a r e small
U.
So f a r we have considered t h e p r i m i t i v e forms of type
C(A, F ) ,
x C + C
i s a family of types
i s a family o f small types over t h e s m a l l type
F
N
f E
C.
Having introduced t h e new forms o f type
whenever
C(A, F)
E
The method of d e f i n i t i o n
The e a r l i e r formulation i s now simply t h e s p e c i a l c a s e when
N.
constant value
1.5.
and
i.e.
a p p l i e s i n a s l i g h t l y m r e general c o n t e x t than t h a t p r e v i o u s l y
Previously
and
F(x)
E
E
w a s a type.
C
over
f(x)
B,
I(A, a, b).
a r e types then
A + B
I t has an element
i(a)
B
If
f
(TIx
E
i s a family of types over
A
E
A) C ( i ( x ) )
+B
i s a t y p e c a l l e d t h e d i s j o i n t union f o r each and
g
E
a
A
E
(IIy
then we have a f u n c t i o n
E
and an element
B) C ( j ( y ) ) h
E
j(b)
where
TI(A + B , C)
C
defined
by cases s o t h a t
h(c) and
h(i(a)) = f(a)
for
a E A
h ( j ( b ) ) = g(b)
for
b E B
w i l l a l s o be w r i t t e n
g
explicit.
If
A
D(c, f , g) and
B
i f w e wish t o make i t s dependence on
a r e small types then s o i s
A + B.
f
I
Constructive set theory
1.8. Ro
No
i s t h e empty type.
n(No, C).
E
For
Whenever
...
k = 1, 2 , 3 ,
is a family o f types over
C
t h e k-element
type
No
then
can be d e f i n e d as
Nk
follows N 1 = I(N, 0, 0 ) N2 = N1 + N1 Ng
=
+ N1
N,
...
etc. The type
If h
al
E
Nk
..., ak
C(lk),
n(Nk, C )
E
has elements
E
lk,
..., kk
C(kk)
where
where
i s a family of types over
C
Nk
then
can be d e f i n e d s o t h a t
h ( k ) = ak. k I f we wish t o make t h e dependence of write
\(c,
1.9.
..., %)
al,
for
h(c).
F i n a l l y we come t o t h e type
V
h(c)
on
Each
al,
\
...,
ak
e x p l i c i t then we
can be defined a s follows
of ( i t e r a t i v e ) s e t s .
This type is
inductively specified via the rule
A E U b c A + V sup(A, b) E V We s h a l l a l s o w r i t e No
E
U
and
Ro
s h a l l abbreviate
E
No
9.
(supx +
V
E
A)b[xl
for
sup(A, ( x ) b [ x l ) .
w e c e r t a i n l y have t h e s e t
More g e n e r a l l y , given s e t s
al,
Note t h a t because
sup(No, R o ) ,
... , g,
which we
we may form t h e s e t
P.H.G. ACZEL
8 (supx
k % (x, a,,
N )
E
..., a,)
which w e s h a l l abbreviate
{a,,
..., ak}.
In t h i s
way the h e r e d i t a r i l y f i n i t e s e t s can be represented a s elements of t h e type
i s t h e following
kssociated with t h e r u l e i n d u c t i v e l y s p e c i f y i n g t h e type
V
method of d e f i n i t i o n by t r a n s f i n i t e recursion on
d
d(A, b , e )
function such t h a t e
E
(nx
h
E
n(C, V)
for
A
E
A)C(b(x)),
E
If
for a l l
A
E
i s a family of types over
C
i s a three place b
U,
b
E
A
+
+ V
When we wish t o make t h e dependence of
V.
e x p l i c i t i t w i l l be w r i t t e n
T(c, d).
Note t h a t
V
and
h(c)
d
on
must be considered a l a r g e
type a s t h e r u l e used i n s p e c i f y i n g i t makes e x p l i c i t reference t o
1.10.
A
E
then we have
V,
defined so t h a t
and
U
where
C(sup(A, b ) )
E
V.
V.
Let us review what we have s a i d concerning t h e type
elements a r e themselves types, c a l l e d the smll types.
U.
It i s a type whose
U.
The following schemes
express our r u l e s f o r forming s m a l l types
N0eU
N E U
FEA-+U II(A, F) E U
A E U F E A + U Z(A, F) E U
A E U B E U A + B c U
A E U a , b E A I(A, a , b) E A
A E U
*
In t h i s paper we s h a l l need t o consider t h e s e r u l e s as giving an i n d u c t i v e s p e c i f i c a t i o n of t h e type
This means t h a t we have t h e following method of
U.
d e f i n i t i o n by t r a n s f i n i t e recursion on t h e method allows us t o form h and
a, b
E
E
II(U,
If
U.
C)
C
i s a family of types over
such t h a t f o r
A, B
E
U,
F E A
-+
U
U
A
In t h e s e equations
dN0
E
C(N ),
place functions such t h a t i f
A, B
E E
C(N)
U, F
E
and
A
+ U
d,, and
dZ, d+
a, b
E
and
dI
A
then
a r e four
Constructive set theory
for
c
E
C(A),
d
E
C(B)
and
e E (nx
E
A)C(F(x)).
PROPOSITIONS AS TYPES
LOGICAL NOTION
TYPE THEORETIC EXPLICATION
proposition
type
proof of A
element of A
A is true
A has a n element
A 3 B
A-+B
A & B
A X B
A V B
A + B
A E B
(A + B) x (B + A)
1
NO A + N0 I(A, a , b)
i A a =A b
FUNDAMENTAL THEOREM.
(vx
E
A)B[xl
(nx
E A)BCxl
(3x
E
A)BCxl
(Ex
E
A)B[xl
For every i n s t a n c e i n type t h e o r y of a n a t u r a l deduction
rule for i n t u i t i o n i s t i c p r e d i c a t e c a l c u l u s with e q u a l i t y , i f t h e premises a r e t r u e then s o i s t h e conclusion. Rather than give a d e t a i l e d formulation and proof of t h i s r e s u l t I s h a l l j u s t
9
10
P.H.G. ACZEL
examine a s e l e c t i o n of the r u l e s .
IMPLICATION INTRODUCTION
If x
E
B
i s t r u e on the assumption t h a t
A
so that
(x)bCxl
A
E
-f
B
is t r u e then t h e r e i s
A
bCxl
E
for
8
and hence t h e conclusion i s t r u e .
IMPLICATION ELIMINATION ( i .e. m d u s poneus) A 2 B B
I f the premises a r e t r u e then t h e r e a r e
A
f E A
and
+ B
a
so t h a t
A
E
f(a)
E
B
and hence t h e conclusion i s t r u e . UNIVERSAL QUANTIFICATION INTRODUCTION
Tx (Vx If
is true for
BCxl
(x)bCxl
E
(nx
E
x
A)BCxl
E
A
E
A1
BCxl E A)BCxl
then t h e r e i s
bCx1
E
B[xl
for
x
E
A
so t h a t
and hence t h e conclusion i s t r u e .
EQUALITY ELIMINATION a =A b
B[al
BCbl I f the f i r s t premise i s t r u e t h e r e i s some element of a = b
so that
Bra]
=
Cbl.
an element and hence s o does 1.13.
I ( A , a , b)
and hence
I f t h e second premise i s a l s o t r u e then
BCbl
BCal
ias
so t h a t t h e conclusion i s t r u e .
I n a d d i t i o n t o the purely l o g i c a l p r i n c i p l e s considered i n t h e fundamental
theorem t h e r e a r e a number o f o t h e r p r i n c i p l e s t h a t can be j u s t i f i e d i n type theory and w i l l be needed i n l a t e r s e c t i o n s of t h i s paper.
We l i s t them below
with an i n d i c a t i o n of why they a r e c o r r e c t . 1.14.
If
F
is a family of propositions ( i . e . types) over t h e type A i t i s
natural to c a l l
F
a species over
A.
11
Constructive set theory
N-INDUCTION.
For every s p e c i e s
F
over
N
V-INDUCTION.
For every s p e c i e s
F
over
V
(VA
U)(Vb
E
A
E
-f
V)[(Vx E A)F(b(x)) E V)F(a)
F(sup(A, b ) ) ]
3
(Va
U-INDUCTION.
For every s p e c i e s
F
F(NO)
F(N)
Oc
,
Qz
QTI
(VA where
U
over
E
@I
@+
U)F(A)
is
0,
is like
(VA
E
QII
with
U)(VB E A
C
(VA
-f
U)[F(A) & (Vx
replacing
E
U)(VB
E
TI
and
A)F(b(x))
E
Q+
U)[F(A) & F(B)
3
and
3
F(Il(A,B))l,
are
F(A+B)I
and
These p r i n c i p l e s a r e j u s t i f i e d by s u i t a b l e d e f i n i t i o n s by r e c u r s i o n . N-induction, i f t h e premises a r e t r u e then t h e r e a r e b E (TIn
E
N)(F(n)
+
F(s(n))
and t h e conclusion i s t r u e . V-induction then
1.15.
If
B
Similarly i f
d
THE AXIOM OF CHOICE (AC)
F(0)
Thus f o r
and
i s an element of t h e premise of
(a)T(a, (A, b, e ) d ( A ) ( b ) ( e ) )
F(x, y )
E
so that
i s a family of types over t h e type
f u n c t i o n such t h a t
a
is a proposition f o r
i s a n element of t h e conclusion. A
and
X E
A
F
i s a two p l a c e
and
y
E
B(x)
then
12
P.H.G. ACZEL
THE DEPENDENT CHOICES AXIOM (DC)
If
f
is an element of the premise of
h = (x)q(f(x)).
Then
g
E
II(A, B)
i s an element of the conclusion. If
AC
and So
Then
g
Now l e t
C
E
a
+
E
C,
A
where and
b
is
C
B(a).
E
X(A, B),
then l e t (Vx
E
E
g = (x)p(f(x))
A)F(x, g ( x ) )
and
so that
(g, h)
i s correct.
AC
i s an element of t h e premise of
f
h
DC
then l e t
and
k
E
(nu
C)F(p(u), p ( g ( u ) ) ) .
E
Then by recursion over
e
we may d e f i n e
N
E
N
+
so t h a t
e ( 0 ) = (a, b) e(s(n)) = g(e(n))
i s an element of
(32
E
N
+
A)G(a, z ) .
words we have t h e conclusion of 1.16. f, g
EXTENSIONALITY. E
II(A, B)
then
If
B
n
E
N.
Thus given an element of
has an element we have found an element of
B(a)
for
(32
E
A
such t h a t
N + A)G(a, z ) .
In o t h e r
DC.
i s a family of types over the type A
and
C
Constructive set theory
If c is an element of the premise then c(a)
E
13
I(B(a),
f(a),
g(a))
for a
E
A
that f(a) = g(a) for a E A and hence f = g. As f = g the type I(II(A, B), f, g) has the element r and hence the conclusion is true.
so
1.17. Let F be a species over the type C. X-EXISTENCE. then
If C is X(A, B)
(32
where B is a family of types over the type A
( 3 E A)(%
C)F(z)
E
B(x))F((x,
E
y))
+-EXISTENCE. If C is A + B where A and B are types then
%-EXISTENCE.
If C is Nk (32
where in case k =
0
for E
k =
C)F(z)
I
0,
1,
F(lk) v
the right hand side is
...
then
... v
F(kk),
1.
In each of these equivalences the implication from right to left involves a simple direct application of the existence introduction rule. For the other direction let us just consider +-existence. If a E A then
where D is the right hand side of the +-existence equivalence. Hence (IIx
E
A) (F(i(x))
has an element
+
so
D)
has an element.
Similarly (Ily
that using definition by cases
element. So (Vz E A + B)(F(z) (32 E A + B)F(z) 3 D is true.
3
D)
E
B)(F(j(y))
(IIz E A + B)(F(z)
+
D)
+
D)
also has an
is true, and using intuitionistic logic
A WARNING
Of fundamental importance in understanding type theory is an awareness of the distinction between the notions of judgement and proposition. The distinction is critical and attempts to avoid it are liable to lead to confusion. Nevertheless, from the practical point of view it seems convenient to leave the distinction implicit inour informal deductions. Before doing so it may be worthwhile to give the distinction our explicit attention.
P.H.G. ACZEL
14
Examples of judgements a r e N i s a type, 0
E
N,
O = O E N ,
s(x)
E
for
N
x
E
N.
Examples of propositions a r e
Note t h a t each proposition false i f
A + No
when f a l s e .
A
i s a type, which i s t r u e i f i t has an element and Such a p r o p o s i t i o n is a meaningful o b j e c t even
has an element.
On t h e o t h e r hand judgements a r e n e c e s s a r i l y c o r r e c t a s such, and an
i n c o r r e c t l y formed judgement i s meaningless. Martin-LSf's
formal language has a system of f i n i t a r y r u l e s f o r deriving
judgements.
This i s i n c o n t r a s t t o t h e standard formal systems (e.g.
or
CZF) which involve f i n i t a r y r u l e s f o r deriving propositions.
for
HA
Nevertheless,
as we have seen, the l o g i c a l notions a r e represented i n type theory and t h e standard r u l e s f o r d e r i v i n g propositions a r e a l s o represented i n type theory. So, conceptually, t h e r e a r e two d i s t i n c t l e v e l s of d e r i v a t i o n .
There is t h e
fundamental l e v e l where judgements a r e derived, and t h e r e i s t h e secondary l e v e l concerned with t h e d e r i v a t i o n of propositions.
In our informal p r e s e n t a t i o n t h e
d i s t i n c t i o n between these l e v e l s w i l l n o t always be e x p l i c i t .
For example, we
have the notion of ( i t e r a t i v e ) s e t . and i n l a t e r s e c t i o n s we s h a l l make a l o t of use of t h e notion of an i n j e c t i v e l y presented set. a set is simply t h e judgement
a
E
V.
a is
Now the statement t h a t
On t h e o t h e r hand i f
a
E
V
then t h e
a i s i n j e c t i v e l y presented i s a p r o p o s i t i o n , which may be f a l s e .
statement t h a t
In following t h e arguments i n t h i s paper, which a r e presented i n a combination of the English language and symbolic expressions, i t i s necessary t o be aware of t h i s distinction. levels.
I n c e r t a i n cases t h e r e i s an easy i n t e r p l a y between t h e d i f f e r e n t
For example i f
the proposition
a
and
( a =A b ) ; i . e .
a c o r r e c t l y formed j u d g e w n t . proposition t h a t t h e type formed.
a = b A
i s true.
I(A, a , b)
b
a r e elements o f t h e type
t h e type
I ( A , a , b).
I f i t i s then
r
E
A
I ( A , a , b)
Conversely i f we know t h a t
then we may form
a = b
Now
a =A b
has an element, then the judgement
E
A
may not b e
s o t h a t the
is true, i.e.
a = b
E
A
can be
Constructive set theory
15
82. V AS A TYPE OF EXTENSIONAL SETS In t h i s s e c t i o n we d e f i n e t h e s p e c i e s on
of e x t e n s i o n a l e q u a l i t y , and
V x V
e x t e n s i o n a l membership, and show t h a t v a r i o u s set t h e o r e t i c a l p r o p e r t i e s hold. axiom system
CZF
An
f o r c o n s t r u c t i v e s e t theory w i l l be formulated i n t h e n e x t
s e c t i o n and t h e work i n t h i s s e c t i o n shows t h e t r u t h of t h o s e axioms.
In this
s e c t i o n and t h e n e x t we a r e r e t r a c i n g the ground of 8 5 4 - 6 of Aczel 119781, but now within t h e informal s e t t i n g of t h e previous s e c t i o n .
2.1.
THEOREM.
a
E
a
= b.
Moreover
PROOF. for
Define
a
E
a
a
= b.
E
E
2.2.
U
E
V + V
and
b
V + (Zx
a
a
E
and
U
b
A
E
a
a
E
U
A
E
A
-f
where
E
and
U
A
+
V.
b
r
E
whenever
g(a) = a
To show t h a t
a
a for
E
This
2 . 3 . THEOREM. V
V
E
then
a
+
a
= A
to
V
and
so t h a t
V
and
If
F
far all
a
E
then
V
Define
=
sup(A, b)
a
E
I(V, g(sup(A, b ) ) , sup(A, b ) )
V
a V a=A
for
+
and
g = (x)sup(x, x ) . f o r some
we need t o argue for a l l
V , T(a, (x, y , z ) r )
E
A
E
U
and
I ( V , g(a), a)
V.
i s a proposition f o r
E
g(a) = a
Hence, by t r a n s f i n i t e r e c u r s i o n on
THEOREM.
+
by t r a n s f i n i t e r e c u r s i o n on
= q ( T ( a ) ) . Then c l e a r l y
and by t h e above
g(a) =
BCxl
a, 6
E
-
a
and
V.
E
U)(x + V)
E
= sup(A, b)
W e know t h a t
A + V.
PROOF.
to
E
A
where
for
I t remains t o prove t h e f i n a l p a r t of t h e theorem. g
so t h a t If
- -
= sup(~~ a ),
= p ( ~ ( a ) ) and
as follows. b
= sup(A, b)
a
U
E
b c A + V
Also, i f
V.
Then
T
and
A E U
Now l e t
A
a
such t h a t i f
V
a
There a r e one-place f u n c t i o n s a s s i g n i n g
x
E
then d e f i n e
V
i s a s p e c i e s over
then
V
s an immediate a p p l i c a t i o n of V-induction from 1 . 2 . There i s a s p e c i e s on such t h a t
v
x
v
a s s i g n i n g a small p r o p o s i t i o n
(a,: 6 )
P.H.G.ACZEL
16
6) = CVx
(a
a3y
E
E
y)l&"dy
B(x
Define three-place functions
PROOF.
u + (V
-+
U),
E
u3y
E
v w(x)(y),
E
v3x
E
u w(x)(y),
w
E
u
(V
+
U).
-+
Gi(u, v, w) so t h a t
It follows t h a t
T(a, G)
T(sup(A, b), Now d e f i n e
(a
6)
E
-+
U
G) = G(A,
6) = T(a, G)(B)
(a
for
6 , f ) = vy
E
B3x
E
for
a(x L y),
s o t h a t we get t h e d e s i r e d r e s u l t . 2.4.
LEMMA.
For
(i)
a L a,
(ii)
a
6
a , 6, y
3
6 I a,
(iii) a L 6 8 B
PROOF.
V
E
(i) For a
L y 3
E
V
a
y.
a
E
u
for
a, 6 then
U, v
u
E
E
U, z
E
and
E
V
u
-+
V
U, b
E
and
and
V
b, (u)T(b(u), G))
and s i m i l a r l y
Gp(a,
V + U
E
f = (u)T(&(u), G)
and i f
U
V
E
v, w) & G2(u, v, w)).
is a small type f o r
G(u, z, w)
s o that
Gg
v, w) = Vx
i = 1, 2
E
and
G1, Gp
G(u, z, w) = ( v ) ( G l ( u ,
Then f o r
63x E a ( x f y)].
G 2 ( u , v, w) = Vy
G1(u,
w
E
E
V.
for
A
Then f o r
E
a,
B
E
V,
A
-+
V.
'(K
x)a
3
x p
3
KA
c
(x
X)(D
3 XA)
P.H.G. ACZEL
18
2.5. DEFINITION.
For
a, 8 E V
let
(x
8 = (3y
E
a 58 Then b o t h
a
A species
F
E
8 over
and
V
a
58
2.6.
THEOREM.
If
F
V)(F(x)
E
THEOREM.
For
a, 8, y
8).
E
E
V
a, 8
E
V.
if
3
(Vy
E
V)(y
i s an e x t e n s i o n a l s p e c i e s o v e r
PROOF.
2.7.
B ) ( a L y)
a r e small propositions for
i s extensional (Vx
E
= (Vx E a ) ( x
x
3
V
F(y))).
then f o r
a
E
V
Constructive set theory
- 5 -c a E vx
Similarly,
E
V(x
5
E
x
3
E
19
a).
Hence, by 2 . 3 and 2 . 4 ( i i )
( a ~ B & 5 ~ a )
a = B Z E
vx
E
E
V
=
V(x E a
x
5).
E
2.8. THEOREM. (i)
Unordered P a i r s .
a, 6
If
then t h e r e i s
y
E
V
such t h a t f o r a l l
n E V
n E y - ( n = a v T l = 5 ) .
(ii)
Union.
If
a E V
then t h e r e i s
I- E
( i i i ) Small-Separation. that for a l l
n
E
If
E
V
y z 3x
E
a
(n
and
F
E
V
=
E
a(F(x) &
V
LY E
E
y
3x
using t h e N - e x i s t e n c e p r i n c i p l e from 1 . 1 7 . k Let
a E V.
Then
y
E
V
where
n
E
V
y
E
E X).
+
U
then there i s
V
n
(ii)
such t h a t f o r a l l
y
n
x).
V
such
20
P.H.G. ACZEL
(iii)
y
2.9.
a
Let
that
E
E
and
V
where
V
THEOREM.
F
V
E
y = (supz
If
Then
U.
E
A)a(p(z)).
i s a s p e c i e s on
F
A
-f
V
x
where
U,
E
Now i f
let
V
A
(Cx
=
E
so
a)F(G(x)),
q E V
F'
be t h e s p e c i e s on
V x V
given by F'(x,
Then ( i )
(ii)
y) = 'du E x 3v
a, 6
If
Strong C o l l e c t i o n .
(Vx
a
If
a)(3y
E
-a
such t h a t
V
E
y F(u, v) & Vv
E
( i i i ) Subset C o l l e c t i o n .
V) F ( x , y)
E
a, 6
If
-6
y 3u
E
x F(u, v).
then
then
V
E
=
E
36
6).
V F'(a,
E
then there i s
V
E
3
y
E
n o t depending on
V,
such t h a t
F,
VX
E
Vy
E
(Y 3y 6 3x
(ii)
Let
Hence by let
a, 6
( i ) Let
PROOF.
6).
(iii)
Let
a=3
such t h a t
V
3
F(a(x),
$(y))
so that
E
a F ( x , y).
So
F ' ( a , 6).
a
E
V
such t h a t
Vx
i n 1.15 t h e r e i s
AC
6 = sup(;,
F'(a,
E
E
b).
a, 6
E V
So
6 E V,
and d e f i n e
Vx
E
a 3y
b
E
a
-
Vx
and
a 3y
E
such t h a t
and Vx
E
E
a F(&(x),
5 F ( x , y).
V F ( x , y).
E
a+V = 6
E
Then
Vx
E
a F(a(x),
8(x)).
Then
Similarly
Vx
E
(1. 3y
E
V F ( & ( x ) , y).
F(a(x), b(x)). i(x))
Then
s o t h a t by ( i )
Constructive set theory
It i s easy t o see t h a t vx E a 3y
E
B F(x, y).
there i s
f
E
6 = (supx and
vx
2.10.
a
-t
!
y
Then Then
F(a(x), 6(x))
a
+ N1)D(x,
for
a
E
V , a'
(iii) for
a
E
V
PROOF.
+
( i ) That
3y
Vx
E
(x F(G(x), s ( f ( x ) ) ) .
E
4
n
i n 1.15
AC
so that
6 E y.
Also
a
a
and f o r (I E V
E
define
V
and f o r
n
V
E
V
E
n
For
4 :a x
Hence by Let
F ' ( a , 6).
Then (i)
and f o r
E
y(f)
6
= sup(No, Ro)
(y)a).
Now assume t h a t
!F ( & ( x ) , E(y)).
E
and
V
is clear.
V
E
a,
V
E
a
so t h a t by ( i )
Recall t h a t
THEOREM.
= (supx E
(ii)
6
F.
E
Vx
such t h a t
a)s(f(x)):
E
E
and i s independent of
V
E
21
E
E
V
No)(n
I Ro(x))
- 1
by No-existence i n 1.17. If
(ii)
a
and
a'
E
V.
E
a V
If
V
E
TI
then
y
for E
V
E
N1
a
E
U
so t h a t
so t h a t
D(x,
then, using 1.17
a a,
+ N1
E
(y)a)
U. E
Also V
for
&(x) x E
a
E
V
for
+ N1,
x
E
Hence
=
6
P.H.G. ACZEL
22 (iv)
a, B
For
V
E
6'
a'
J)
a E B' & B
J)
(a
8
E
V
E
by ( i i )
a',
8)
a
&
(8
E
a
V
8 &a)
2 a L B .
The l a s t s t e p
uses
THEOREM.
For
n E N
for
n
2.11.
Then A h )
V
E
(a
E
5 & B
E
a)
3
which can be proved using 2 . 2 .
I
let
so t h a t
c N,
w E V
where
w = ( s u p x ~N)A(x),
and
E
w
,
( i i ) (Wa
E
w)(a'
0
(i)
E
w),
( i i i ) for every s p e c i e s
PROOF.
As
and
$ E V
a'
F
V
for
for
n
E
V
on
a E V
i t follows t h a t
A(n)
E
V
for
n
E
N.
(Ah))'
E
w
for
n
E
N
As N E U , W E V .
(Ah))'
( i i ) As and hence (iii) (Wn
E
A(s(n))
(Va
Assume
E
N)(F(A(n))
3
(Vx
and
F(4)
(Vn E N) F ( A ( n ) ) ,
E
N,
i t follows t h a t
w) (a' E w ) .
E
(Vx
E
3
F(x')).
Then
F(A(0))
and
so t h a t by N-induction from 1.14 w e g e t
F(A(s(n))))
i.e.
w)(F(x)
w)
F(x).
13. THE CZF AXIOM SYSTEM
3.1.
In t h i s s e c t i o n I review t h e language and axioms o f t h e system CZF of
c o n s t r u c t i v e set theory. s e c t i o n show t h a t t h e type model of
It w i l l then be clear t h a t t h e r e s u l t s of the previous V , with extensional e q u a l i t y and membership, i s a
CZF.
The language o f
CZF
i s e s s e n t i a l l y a standard one for s e t theory.
As t h e
23
Constructive set theory
underlying l o g i c i s t o be i n t u i t i o n i s t i c , a l l t h e l o g i c a l o p e r a t i o n s (vx
E
V)
(3x
(3x
E
V)
r a t h e r than t h e more customary
$
$J
5
and
E
V)
w i l l be t r e a t e d as p r i m i t i v e .
w i l l abbreviate x
$
3
(4
and
I
3
(Vx)
and
& ($I
$J)
Note t h a t I use (3x). $)
3
take as primitive t h e r e s t r i c t e d q u a n t i f i e r s
is r e s t r i c t e d
(Vx
E
y)
As usual
respectively.
w i l l be t r e a t e d a s p r i m i t i v e atomic formulae.
y
I & v
(Vx
and
16
3
V)
E
and
and
x
Both
E
(3x
E
y).
i f i t has been b u i l t up without using t h e q u a n t i f i e r s
A formula
(Vx
E
V)
V).
and
(3x
CZF
i s axiomatised u s i n g a s t a n d a r d a x i o m a t i s a t i o n of i n t u i t i o n i s t i c p r e d i c a t e
logic.
3.2.
E
The remaining axioms a r e p r e s e n t e d below.
STRUCTURAL AXIOMS
R e s t r i c t e d Q u a n t i f i e r axioms
f o r e v e r y formula $ [ X I ,
vx
E
VD
[XI
3
where
v z E V(x
is
O
k
z
3
4 Cz1)l.
E x t e n s i o n a l i t y axioms
S e t Induction
Vx
E
V(Vy
f o r every formula
E
x $ Cyl
$[XI.
3 . 3 . SET EXISTENCE AXIOMS
3
$ [XI)
3
vx
E
V$[xl,
y
In addition I s h a l l
24
P.H.G. ACZEL
Union Va
E
V 3y
V Vn
E
(n
E
V (n
E
y I
3
E
a
E
V
E
y
3x
E
a ($[XI & rl
x)).
E
R e s t r i c t e d Separation Va
E
V 3y
V Vn
E
(rl
x))
f o r every r e s t r i c t e d formula $ Cxl. Strong Collect ion Va
E
v
(VX €
f o r every formula
Vx
E
a 3y
a 3y
E
$Cx, y l ,
v $ cx,
yl
where
$‘[a, 21
E
z $Cx, y l & Vy
E
v
E
3
32
z 3x
E
E
v
$“a, 21)
is a $[x, y l .
Subset Collect ion Va
E
v
VB
f o r every formula
3y
E
v
Ocx, y l
Vu
E
v
(VX
E
a 3y
E
B $Cx, y l
’ 32
€
y
$“a,, 21)
( t h a t may contain f r e e occurrences of t h e v a r i a b l e
u).
Infinity
3y where
v
E
(3x
succ(x, y)
E
y Vy
If
Nx,,
x 1
&‘VX E y
3y
E
y succ(x, y)
is
X € Y & V U E X
3.4.
E
(uoy) & V u E y ( u E X V U ~ y )
..., xnl
i s a formula, a l l of whose f r e e v a r i a b l e s have been
displayed then by i n t e r p r e t i n g t h e formula i n type theory i t y i e l d s an n-place function a s s i g n i n g a proposition of elements of t h e type proposition OCx,,
$[al,
..., xnl
is
..., a
V.
OCa,,
.,., an]
Note t h a t when
t o each n-tuple
..., xnl
QCx,,
1 w i l l always be small.
valid i f
the proposition
(Vx,
al,
..., a
i s r e s t r i c t e d the
I say t h a t t h e formula E
V)
... (Vxn
E
V) OCx,,
..., xnl
i s true.
The language and a x i o m t h a t w e have given f o r
CZF have been chosen s o t h a t t h e
following r e s u l t has as d i r e c t a proof as possible.
25
Constructive set theory
THEOREM.
PROOF.
Every theorem of
i s valid.
CZF
The c o r r e c t n e s s of i n t u i t i o n i s t i c p r e d i c a t e l o g i c h a s a l r e a d y been
discussed i n 8 1 .
The v a l i d i t y of t h e s t r u c t u r a l axioms follows from theorems
2.6, 2 . 7 and 2.2.
2.11.
For t h e s e t e x i s t e n c e axioms use theorems 2.8, 2.9,
2.10 and
The d e t a i l s a r e l e f t t o t h e reader.
3.5. REMARKS.
I n t h e r e s t r i c t e d q u a n t i f i e r axioms t h e formula
the s p e c i e s over
d e f i n e d by t h e formula
V
induction on t h e way t h a t t h e formula
$[XI
$[XI
expresses t h a t
0
is e x t e n s i o n a l .
A routine
i s b u i l t up s u f f i c e s t o prove 0 It follows t h a t
using only t h e r e s t r i c t e d q u a n t i f i e r and e x t e n s i o n a l i t y axioms.
can be dropped from t h e r e s t r i c t e d q u a n t i f i e r axioms without
the assumption
0
altering
s o t h a t t h e r e s t r i c t e d q u a n t i f i e r s can be t r e a t e d i n t h e s t a n d a r d
way.
CZF,
Because o f t h i s t h e r e s t r i c t e d s e p a r a t i o n axiom can be rephrased i n t h e more
f a m i l i a r form u s i n g
(0 E
a & $Cql)
3x
i n s t e a d of
E
a ($[XI &
x).
0
14. CHOICE PRINCIPLES FOR CONSTRUCTIVE SET THEORY
4.1.
I n t h i s s e c t i o n we s h a l l work e x c l u s i v e l y i n an informal framework f o r
CZF.
The s t a n d a r d conventions and n o t a t i o n s of c l a s s i c a l s e t theory w i l l be used. ordered p a i r s a r e d e f i n e d as usual.
A, B
A
The C a r t e s i a n product
x
B
So
of t h e s e t s
can b e shown t o e x i s t a s t h e s e t
using replacement (which i s a consequence of s t r o n g c o l l e c t i o n ) and t h e union axiom.
More g e n e r a l l y we can d e f i n e t h e d i s j o i n t union
of s e t s indexed by t h e s e t
A + B
The d i s j o i n t union
A s usual a r e l a t i o n such t h a t
A function
< x , y> f
<x, y > , <x, z> domain
A
E
R
A
A, B
of s e t s
i s a s e t of ordered p a i r s whose domain i s t h e s e t of y
such t h a t
i s a s i n g l e valued r e l a t i o n , i . e . one where E
f.
f
of a family
i s a f u n c t i o n from A
and range a s u b s e t of
of s e t s indexed by t h e s e t A.
B
i s defined t o he t h e s e t
and whose range is t h e s e t of
R
Z(A, B)
as t h e s e t
B.
to
B
if
y f
z
<x, y,
L
x R.
whenever
is a f u n c t i o n with
Above we have used t h e n o t i o n of a fami1.y
B i s simply a f u n c t i o n w i t h domain
A.
B
P.H.G. ACZEL
26 4.2.
I N D U C T I V E DEFINITIONS
We s h a l l use t h e informal n c t i o n of c l a s s a s i n c l a s s i c a l s e t theory. class
@
< a , A>
E
THEOREM.
the class
is
X
@-closed i f
A
implies
X
a
X
E
For any
f o r every p a i r
The following r e s u l t i s u s e f u l .
0.
For any c l a s s
t h e r e i s a s m a l l e s t @-closed c l a s s
@
I(@),
called the
c l a s s of @-generated s e t s .
Call a relation
PROOF SKETCH.
such t h a t
E
g
i f whenever
<x, y>
E
g
there i s a s e t
C a l l a s e t @-generated i f i t i s i n t h e range of some good r e l a t i o n .
I(@)
the class s e t s , where
of @-generated s e t s i s @-closed l e t
< a , A>
A 3g
E
i s good & 3x
(g
By s t r o n g c o l l e c t i o n t h e r e is a s e t
and
L UG u {
-
E
g
so t h a t
a g
G
To s e e t h a t
be a s e t of @-generated
V (<x, y>
E
E
8)).
o f good s e t s such t h a t
x L {x
a > ] where
i s a @-closed c l a s s and
A
Then
E 0.
Vy
Now i f
A
and
Q
I
3y
E
i s @-generated.
V<x, y >
E
UG}
i s good
then
Thus I(@) is @-closed.
Now i f
i s good then an easy proof by s e t induction on
X
will
x
show t h a t <x, y>
I(@)
so that
E
g
=
y
E
x,
5 X.
4 . 3 . THE SET OF NATURAL NUMBERS
This may be c h a r a c t e r i s e d a s t h e unique s e t
where
4
i s t h e empty s e t a n d
The e x i s t e n c e of such a s e t
w
y'
w
such t h a t f o r every s e t
is t h e set of
z
such t h a t
z
E
y
x
V
z
y.
follows from t h e axiom of i n f i n i t y u s i n g r e s t r i c t e d
Constructive set theory
separation.
21
Its uniqueness can be proved by s e t induction, a s can the scheme of
mathematical induction f o r
Our f i r s t choice p r i n c i p l e i s the following.
w.
4 . 4 . DEPENDENT CHOICES (DC)
f o r a l l formulae
e[x]
and
$Ex, y l .
OCa, y l
Here
expresses t h e conjunction
of t h e following statements. (i)
y is a function with domain
(ii)
E y .
(iii) sets
(
E
y &
E 6.
by 5 . 3 ( i ) , and
F
A c(n)
E V
IA h ) ,
n
and f o r
a
so that
6
g(a)
such t h a t x'
E
is a
5, y
E
V
DC
N
6 = S(w,
so t h a t i f
E
6
and a l l
w
E
Then by t h e type t h e o r e t i c a l v e r s i o n of
with
x E w y
6
i s i n j e c t i v e l y presented,
w
Then f o r some
6
there is
and f o r every
B(a).
rl E V
Finally, let
6
F b e an
V (B(y) & F(x, y ) ) ) .
E
B(a) E
and l e t
V
e x t e n s i o n a l i n each argument) such t h a t
such t h a t
V
and by 5.3 ( i i ) , as
V
domain
B
c
3y
3
G$, a>
such t h a t
V
(i.e.
such t h a t
V
E
V (B(x)
E
f u n c t i o n with domain
E
( i v ) a r e needed.
be an e x t e n s i o n a l s p e c i e s over
B
Vx
6
and w i t h i n
N
is a base, and we have proved CC, t h e countable choice
w
e x t e n s i o n a l s p e c i e s over
Let
E
But we have t h e following s t r o n g e r r e s u l t .
5 . 6 . THEOREM.
PROOF.
n
f i r s t on
A t t h e key s t e p s 2.10 ( i i i ) and 2.10
N.
E
A(s(n) = ( A h ) ) '
and
n =N m.
3
This can be shown by a r o u t i n e double N-induction, t h a t on
Q
=
N
q)
then
is a f u n c t i o n w i t h i t follows t h a t <x, B> e 6
A(s(n))
and
and hence,
Hence, by (*) and t h e ' a s s u m p t i o n t h a t
B(5) & F(8, y ) .
Constructive set theory
33
I f w e apply t h i s r e s u l t t o the extensional p r e d i c a t e s defined by formulae i n t h e language of s e t theory we get the main r e s u l t of t h i s s e c t i o n .
5.7. THEOREM.
The s e t t h e o r e t i c a l
DC
i s valid.
16. THE IIXI-AXIOM OF CHOICE The aim of t h i s s e c t i o n i s t o show t h a t every IIXI-generated s e t i s a base.
We do
t h i s using t h e following notion.
6.1. DEFINITION.
B
presented
E
a
E
V
i s a s t r o n g base
if
6
a
f o r some i n j e c t i v e l y
V.
Note t h a t t h i s i s an extensional v e r s i o n of the notion of an i n j e c t i v e l y presented It will be convenient t o extend our use of t h e terminology associated with
set.
c l a s s e s t o such e x t e n s i o n a l notions, even though they may n o t be definable i n the purely s e t - t h e o r e t i c a l language.
By t h e r e s u l t s i n 12 the axiom schemes of
CZF
As the n o t i o n of base i s extensional i t
s t i l l apply t o t h i s wider context.
follows from 5 . 4 t h a t every s t r o n g base i s a base.
Hence t h e r e s u l t t h a t we a r e
aiming a t w i l l be a consequence of t h e r e s u l t t h a t every IIXI-generated s e t i s a strong base, and f o r t h i s i t s u f f i c e s t o show t h a t t h e c l a s s of s t r o n g bases is IIXI-closed.
We s h a l l need t o foruiulate e x p l i c i t i n t e n s i o n a l versions of the set
theoretical IIZI
operations.
X(a, 6)
In t h e following d e f i n i t i o n s we s h a l l d e f i n e a, 6
E
-
-
a = 6.
such t h a t
V
E
II(a, 6) E V
and
V
for
i s i n j e c t i v e l y presented
a
We then show t h a t when
then t h e s e a r e t h e d i s j o i n t union and Cartesian product, r e s p e c t i v e l y , of the family of sets
Let
a, 6
E
B) indexed by t h e s e t a.
S(a,
i = is.
with
V
6.2. DEFINITION OF Z(a, C
E
U, where
i(p(z))
E
V
C = (Zx E
,
q(z)
E
B).
As
a
E
U and
a) m. For
8(p(z))
so t h a t
z
E
f(z)
B(x)
E
E
V
for
U
C, p(z) E
u
x
a
E
so t h a t
where
i t follows t h a t a(p(z)) E V
f(z) = (e(p(z)))"(q(z)).
Now d e f i n e
6 . 3 . DEFINITION OF n(a, 6). z
E
D.
For
x
E
a,
Note t h a t
& ( x ) E V, B(x)
E
V
D
E
and
U,
where
z(x)
E
and
D = (IIx
B(x)
E
so that
a)
i(x).
Let
P.H.G. ACZEL
34
6.4.
-
-
Let
THEOREM.
a
E
be i n j e c t i v e l y presented and l e t
V
E
such t h a t
V
E
E = a. (i)
E(a, E)
is i n j e c t i v e l y presented f o r a l l
(ii) If
B(x)
a r e i n j e c t i v e l y presented.
By 5.3 ( i i ) S(a, 6)
indexed by t h e s e t (i)
Let
rl
i s t h e Cartesian product of t h e
S(a, 6).
n(a, 8)
PROOF.
n(a, 6 )
i s t h e d i s j o i n t union and
family of s e t s
E
a
then both
i s a function with domain
Z(a,
8)
and
a, i . e . a family of s e t s
a. Then
V.
E
x
S(a, 8) & 3y
E
a 36
E
V (<x, 6 >
:3x
E
a
36
E
V (
5
3x
E
cr
36
E
v
E
3x
E
a
3y
E
! 32 E
c (n
:n
Z(a,
E
S(a, E)
i s i n t h e d i s j o i n t union of
q
:3x
(6
h
E
E
S(a,
$ ( x ) & 3y
E
E
(n
6
E) & 3y
E
6
L
<x, y > ) )
(n
) )
6(ll 2 < a x ) , y>))
i ( x ) ( n A .&x) , Y>) )
8).
The t h i r d of t h e above :uses 5.3 ( i ) and t h e f i f t h uses t h e Z-existence p r i n c i p l e of 1 . 1 7 .
For the Cartesian product f i r s t n o t e t h a t by 5.3 ( i i )
with domain
a i f and only i f 3y
Hence
rl
v (7
E
=
E
&
n
S(a, 6 ) ) .
i s i n t h e Cartesian product of 3y
where, i f
is a f u n c t i o n
rl
y
E
V
with
F(a, B, y) = ‘dx
E
(7 7 = E, V
E
a 36
Using 5.3 ( i ) twice, t h e
E
AC
=
&
n
V (<x, 6 >
E
S(a,
8)
S(a,
Y) 8. F ( a , 8, Y)),
S(a,
6) & 3y
i f and only i f
E
6 ( < x , y>
from 1.15 and 5.3 ( i i ) we get t h a t
E
S(a,
y))).
35
Constructive set theory
I t follows t h a t E
D (n
(ii)
Let
32
y = C(a, 8 )
show t h a t
7(zl)
i s i n the Cartesian product of
II
S(a, g(z))),
z1
y(z2)
=c z 2 .
and hence i f and only i f and l e t
Let
zl,
z2
xi = p(zi)
i f and only i f
n
8).
E n(a,
q(zl)
such t h a t
C
E
S ( a , 8)
and
yi = q ( z i )
for
;(z2).
i = 1, 2.
We must
As
i t follows t h a t
< a x 1 ) , (8(X1))"(Yl)>
so t h a t a(xl) L G(x2) (8(xp) 1 -(YZ).
(B(Xl))-(yl) As
i s i n j e c t i v e l y presented
a
x1 =_ x p a
(i(xl))-(yl) g(xl)
As z1
let
y = n(a, 0)
that
and l e t Note t h a t
z1 = D_'2'
5 y(z,),
(*)
VXl
a 3x2 a
then
then
x1
=_
;(zl)
Hence
;(z2).
Now
We must show
= ( s u p x ~a ) < k ( x ) , ( g ( z i ) ) - ( x ) > .
s o that i f also
x2,
..a
(B(xl))-(z1(xl))
z l ( x l ) =B(xl) z 2 ( x l )
* ;(XI).
i s i n j e c t i v e l y presented.
such t h a t
D
E
B
( < a x l ) , (g(zl))-(xl)> L
V
E
6.
a.
As
S(a,
5) 5 6
5
and
II(a, 8)
E
V
x
E
and
i.
and d i s j o i n t union of
y E V
B,,
=
x
x
-
E
t h e r e i s an i n j e c t i v e l y
a
By the type t h e o r e t i c
6.
a
Without loss we may assume
a.
E
i,
5 = sup(;,
S ( a , 8)
By 5.3 ( i i )
Z(u, 5) S(a,
AC of 1.15
i s i n j e c t i v e l y presented and
f(x)
where
i s a f u n c t i o n with domain
6
and
product and d i s j o i n t union of a s t r o n g base and
E
such t h a t , f o r each Then
i n j e c t i v e l y presented f o r
By 6 . 4 both
For each
f),
and
a
8 ) , i.e. of
Hence t h e Cartesian product
6.
a r e s t r o n g bases. It remains t o show t h a t i f E
rl E
such t h a t V
S(a, 5)
w e m u s t have
a r e i n j e c t i v e l y presented and a r e t h e Cartesian
6,
such t h a t f o r a l l
is
i(x)
is a f u n c t i o n with domain
6
V
V
E
B,,
5,
E
a
a
E
V
is
then t h e r e i s ' a s t r o n g base
6.
31
Constructive set theory
Without loss w e may assume t h a t where
a.
-
E
a
6. = a ( a i )
is i n j e c t i v e l y p r e s e n t e d and t h a t
a
By t h e lemma i t s u f f i c e s t o l e t
..-
y = I(a, al, a2).
The ILXI-axiom of choice i s v a l i d .
6.8. THEOREM.
PROOF.
i = 1, 2.
for
By t h e previous theorem every IIXI-generated s e t i s a s t r o n g base and hence
i s a base. 57. THE IICI-PRESENTATION AXIOM The main a i m of t h i s f i n a l s e c t i o n i s t o show t h a t every s e t has a IIZI-presentation. This i s done by coding each s m a l l t y p e set
T(A)
set
S(T(;),
= A.
such t h a t
a s a n i n j e c t i v e l y presented HCI-generated
A
It then e a s i l y follows t h a t f o r each s e t
i s a f u n c t i o n with range
a)
a
a, t h e
and domain t h e ILCI-generated
set
T(;).
The s e c t i o n ends w i t h a r e s u l t g i v i n g s e v e r a l c h a r a c t e r i s a t i o n s of t h e n o t i o n of a base.
7.1.
THEOREM.
and
a, b
E
There is a f u n c t i o n
T E
U
-t
V
such t h a t f o r
A, B
E
U,
F
E
A
+
A T ( N ~ )=
0
r(N) = w T(II(A, F ) ) = ~'I(T(A),( s u p x ~A)T(F(x))) T(C(A, F)) = C(T(A), ( s u p x ~A)T(F(x))) T(A + B) = T(A) + T ( B ) T ( I ( A , a , b ) ) = I(A, a , b ) . Moreover
-
T(A) = A
for a l l
Note t h a t t h e symbols
II, C
A
E
and
U. +
equations r e f e r t o t h e o p e r a t i o n s on
PROOF.
Define
T(A) = sup(A, u(A))
defined by t r a n s f i n i t e r e c u r s i o n over
t h a t occur on t h e r i g h t hand s i d e of t h e s e introduced i n 56.
V
for U
A
E
U where
so t h a t
u
E
(I[A E U)(A
+
V)
is
U
38
P.H.G. ACZEL o(I(A, a , b ) ) = ( z ) $
for
A, B
E
U,
F
A
E
+
and
U
a, b
E
4 , w , II,
follow using t h e d e f i n t i o n s of
be i n c o r r e c t t o use t h e equations f o r recursion over Z(a,
B)
7.2.
THEOREM.
The required equations f o r
f
+,
C,
6
a,
For a l l
A
the s e t
U
E
T(A)
by t r a n s f i n i t e
T
II(a, B )
-
without the assumption t h a t
V
E
E
5.
a =
now
T
Note t h a t i t would
given i n 16.
t o d i r e c t l y define
T
That would only work i f we had t h a t
U.
for
V
E
A.
and
V
i s i n j e c t i v e l y presented and
IIZI-
generated. PROOF.
A
Call
No
and
good and t h a t
F
A
E
a, b
7.3. THEOREM.
Let
E
E
V
B and range
a.
x
E
A.
A, B
That
are
U
II(A, F)
i s good and by 6.6
A + B
E
and
I(A, a , b)
A.
For each
a
i s good f o r
F(x) By 6.5
a
E
B
and l e t
the s e t
V
B
A s , by 7.2,
i s a IIXI-presentation of
S ( T ( ~ ) ,a )
B
E
V,
S ( 6 , a)
E
V
Then
= T(;).
So by 5.1 ( i ) and 5.3 ( i i )
presented.
F i r s t note t h a t
As induction hypothesis assume t h a t
such t h a t
U
-+
i s i n j e c t i v e l y presented and IICI-generated.
T(A)
i s good by U-induction (see 1 . 1 4 ) .
U
E
a r e good follows from 6.4.
i s good f o r
PROOF.
A
a r e good by 5.5.
N
C(A, F)
if
U
E
We show t h a t every
=
a
a.
i s injectively
and
i s a f u n c t i o n with domain
i s IICI-generated t h e r e s u l t follows.
As a consequence of t h i s theorem and 6.8 w e have t h e main r e s u l t . 7.4. THEOREM.
The IICI-presentation axiom i s v a l i d .
We next give a type t h e o r e t i c c h a r a c t e r i s a t i o n of t h e notion of a IIXI-generated set. 7.5.
THEOREM.
t h e set
T(A)
A s e t i s IIXI-generated i f and only i f i t i s e x t e n s i o n a l l y equal t o f o r some
A
E
U.
The implications from r i g h t t o l e f t follows from 7.2.
PROOF.
implication, c a l l a s e t U-generated some
A
Clearly
U.
E
w
For t h e converse for
i f i t i s e x t e n s i o n a l l y equal t o T(A)
It s u f f i c e s t o show t h a t t h e c l a s s of U-generated s e t s is IICI-closed.
i s U-generated because
w = T(N).
Now suppose t h a t
of U-generated s e t s indexed by a U-generated s e t . function with domain
Vx
Then f o r some
So
T(A). E
T(A) 3y
E
U (<x, r ( y ) > e 6),
6
V
E
A
E
U
i s a family 6
is a
39
Constructive set theory
and hence .vx
E
A 3~
E
u
( < ( T ( A ) ) - ( x ) , T ( Y ) >E 6 ) .
from 1 . 1 5 t h e r e i s
AC
By t h e type t h e o r e t i c
VX
E
A (
F
E
A
+
such t h a t
U
6).
E
It follows t h a t S(T(A), ( s u p x ~A)T(F(x)))
6 , and hence i s e x t e n s i o n a l l y e q u a l t o
is a s u b s e t of
T(A).
with domain
6
union of
a r e t h e U-generated s e t s
6
Finally let
~ ( l l ( A ,F ) )
a, 6
be U-generated and l e t
V
E
i s U-generated i t i s e x t e n s i o n a l l y equal t o a, b
A
E
Then
y
E
a
T ( A ) ( ~ ) and
V,
and f o r
~l E
z
(ll L
Il
E
6,
as both a r e f u n c t i o n s
From t h i s i t follows t h a t t h e C a r t e s i a n product and d i s j o i n t
y
6
L
T(A)(b).
and E V
T(A)
T(C(A, F))
f o r some
A
respectively.
a, 6
such t h a t E
E
As
6.
6
So f o r some
U.
y = T ( I ( A , a , b ) ) = f ( A , a, b ) .
Let
V
$ & a
s o t h a t the U-generated s e t
5)
L
I ( a , B) of 4 . 5 .
is t h e s e t
y
We end with t h e promised c h a r a c t e r i s a t i o n s of t h e n o t i o n of a base.
7 . 6 . THEOREM.
For
t h e following a r e e q u i v a l e n t
a E V
(i)
a i s i n one-one correspondence with some IIXI-generated s e t .
(ii)
a i s i n one-one correspondence with some s t r o n g base.
( i i i ) a i s a s t r o n g base. (iv)
a i s a base.
PROOF.
We show t h a t ( i )
6.5.
For
base
6.
(ii)
3
a.
and range
$
(ii)
let
a
3
(iii)
y
E
V
6
L
y l , y2
a.
(i).
3
(i)
3
(ii)
follows from
such t h a t
6
i s i n j e c t i v e l y presented.
Hence, as
y
By our
i s a one-one f u n c t i o n w i t h domain
y
y 2 S(5, 6)
i s one-one and
B
f o r some
6
E
V
5 such
i s injectively
E
8(y1)
8(y2)
3
IB(y,)
’ Y1
6
(iv)
By 5.1 and 5 . 3 i t follows t h a t and
=
presented, i f
so t h a t
3
be i n one-one correspondence with t h e s t r o n g
Without loss we may assume t h a t
assumption t h e r e i s
that
3
(iii)
B(y2)
=- Y 2 9
B i s i n j e c t i v e l y p r e s e n t e d and hence
is a consequence of 5.4.
a
i s a s t r o n g base.
By 7 . 4 we may u s e 4.6 t o g e t
(iv)
3
(i).
(iii)
3
(iv)
40
P.H.G. ACZEL
REFERENCES Aczel, P. C19781 The type t h e o r e t i c i n t e r p r e t a t i o n of c o n s t r u c t i v e s e t theory, i n Logic Colloquium ' 7 7 , eds. A . Macintyre, L. Pacholski and J. P a r i s , North Holland. Bishop, E . C19671 Foundations of Constructive Analysis, McGraw H i l l . Bridges, D.S. C19791 Constructive Functional Analysis, Research Notes i n Mathematics, Vo1.28, Pitman. Diaconescu, R. C19751 Axiom of choice and complementation. Proc. h e r . Math. SOC., Vol .5 1, pp .176-178. Martin-Lgf, P. C19751 An i n t u i t i o n i s t i c theory of types: P r e d i c a t i v e p a r t , i n Logic Colloquium '73, eds. H.E. Rose and J . C . Sheperdson, North Holland, pp. 73-118. Martin-LGf, P. C19791 Constructive Mathematics and Computer Programing. To appear i n t h e Proceedings of the 6 t h I n t e r n a t i o n a l Congress f o r Logic, Methodology and Philosophy of Science at Hannover i n 1979. Myhill, J. C19751 Constructive S e t Theory, J.S.L.,
V01.40, pp.347-382.
THE L.E.J. BROUWER CENTENARY SYMPOSIOM A.S. Troelstra and D. van Dalen (editors) 0North-HoNandPublishingCompany, 1982
41
RECENT PROGRESS IN CONSTRUCTIVE APPROXIMATION THEORY Douglas S . Bridges University College at Buckingham, Buckingham MK18 lEG, England.
Approximation theory can be regarded as a good testing-ground for constructive mathematics. Recent work on a general aspect of approximation theory, and on the particular case of best Chebyshev approximation over a compact interval, shows that constructive techniques can handle this material successfully, in such a way that new classical results and estimates are obtained as by-products of the constructive development.
1 If constructive mathematics is to gain widespread acceptance, it will have to satisfy the mathematical community on several issues quite apart from those of philosophy and foundations. One of these issues is its ability to produce significant results in functional analysis and measure theory. Although there is much to be done in these fields - for example, almost nothing constructive exists in the very important area of operator algebra theory - much has already been accomplished, particularly by Bishop cl,Zl. Certainly, enough constructive analysis has been developed to invalidate the argument of those mathematicians who are prepared to dismiss constructive mathematics on the ground that its methods cannot lead to deep theorems.
I would like to suggest that there is another test which constructive mathematics must pass before the day arrives when it "will be the accepted norm" [l, p. XI. In order to justify the constructivists' claim that their mathematics is characterised by numerical meaning and computational method 11, pp. 2-31, they must produce a fruitful constructive development of numerical analysis and related subjects. In this article, I shall outline the first steps of such a development of approximation theory. All the work below will be in the constructive spirit of Errett Bishop C1,31. Thus all the proofs will be acceptable to the intuitionist, but there may be (in fact there are) places where the use of intuitionistic principles would simplify matters considerably.
2
The fundamental theorem of classical approximation theory states that: (*)
If X is a finite-dimensional linear subspace of a real normed space a E E, then there exists b E X such that
E, and if
Ila
-
bll = dist(a,X) = infflla - xII : x
E
X}
It is a remarkable fact that, although this theorem lies at the heart of a large and growing body of computational mathematics, all its proofs are nonconstructive. The constructive content of these proofs seems to be the locatedness of the finite dimensional subspace X; the further conclusion that b exists invariably depends on an application of the nonconstructive proposition that a continuous, real-valued function on a compact space attains its infimum.
D.S. BRIDGES
42
I n f a c t , theorem (*) i t s e l f appears t o be n o n c o n s t r u c t i v e , a s i s shown by t h e following Brouwerian counterexample, which was suggested t o me by Fred Richman. Let
a
be an a r b i t r a r y r e a l number, and l e t X be t h e one-dimensional subspace
R ( c o s a , s i n a ) of Et2, where each r e a l number t , d e f i n e
lR2 i s given t h e norm
$ ( t ) : 11(0,1)
=
-
(x,y)
+
max( 1x1, l y l ) .
For
t ( c o s a , sina)ll
m a x ( J t c o s a 1 , ) l- t s i n a l )
Suppose t h a t theorem ( * ) o b t a i n s ; so t h a t we can compute a r e a l number T with $(T) = i n f { $ ( t ) : t E El}. W e s h a l l show t h a t a t 0 or a 2 0 . C l e a r l y , we may assume t h a t l a ( < Tr/4, E i t h e r 0 < T or T < l / ( c o s a + \ s i n 4 ) . I n t h e former c a s e , i f a < 0 t h e n , a s a l i t t l e computation shows, T = - l / ( c o s a s i n a ) < 0 , a c o n t r a d i c t i o n ; hence a L 0 . I n t h e o t h e r c a s e , a > 0 e n t a i l s t h e c o n t r a d i c t i o n T = l / ( c o s a + s i n a ) , and so a 5 0. Is t h e r e a reasonable c o n s t r u c t i v e s u b s t i t u t e f o r t h e fundamental theorem of c l a s s i c a l approximation theory? I n o r d e r t o answer t h i s a f f i r m a t i v e l y , w e need some d e f i n i t i o n s . Let E be a m e t r i c s p a c e , X a l o c a t e d s u b s e t of E , and a an element of E. An element b of X is a best approximant of a i n X i f d ( a , b ) = d i s t ( a , X ) . I f each element of E has a b e s t approximant i n X , w e s a y t h a t X i s proximinal i n E . On t h e o t h e r hand, a has a t most one b e s t approximant i n X i f max(d(a,x),d(a,x'))
>
dist(a,X)
whenever x , x ' belong t o X and d ( x , x ' ) > 0. The subspace X is quasiproximinal i n E i f , t o each element of E t h a t has a t most one b e s t approximant i n X , t h e r e corresponds a (unique) b e s t approximant i n X . I t i s t r i v i a l t o prove c l a s s i c a l l y t h a t q u a s i p r o x i m i n a l i t y and p r o x i m i n a l i t y a r e e q u i v a l e n t p r o p e r t i e s . That they a r e u n l i k e l y t o be e q u i v a l e n t i n a c o n s t r u c t i v e framework i s shown by t h e above Brouwerian counterexample and t h e following theorem, which i s c l a s s i c a l l y e q u i v a l e n t t o ( * ) . THEOREM 1
A f i n i t e - d i m e n s i o n a l l i n e a r subspace X of a r e a l normed space E
quasiproximinal
& I E
C71.
&
0
The proof of t h i s theorem proceeds by i n d u c t i o n on t h e dimension of X , t h e one c o n s i d e r a b l e d i f f i c u l t y (which occurs i n t h e i n d u c t i o n s t e p ) being r e s o l v e d by extending t h e n e s t e d i n t e r v a l s argument used i n t h e c a s e dim X = 1. I t i s worth n o t i n g t h a t t h i s d i f f i c u l t y can be overcome much more e a s i l y by t h e i n t u i t i o n i s t , by an appeal t o h i s theorem t h a t a continuous mapping o f a compact m e t r i c space i n t o t h e p o s i t i v e r e a l l i n e has p o s i t i v e infimum. Theorem 1 s u g g e s t s t h a t t h e next s t e p i n c o n s t r u c t i v e approximation theory i s t o c o n s i d e r c l a s s i c a l s i t u a t i o n s where t h e b e s t approximant i s unique, and t o conv e r t t h e c l a s s i c a l proof of unique e x i s t e n c e i n t o a c o n s t r u c t i v e proof t h a t t h e r e is a t most one b e s t approximant of t h e element i n q u e s t i o n . I n t h e c a s e of Chebyshev approximation over C O , l l , where E = C [ O , l ] and X i s a Chebyshev subspace of C[O,ll, t h i s procedure was c a r r i e d o u t i n C5l (cf.C41). In t h a t c a s e , we can c o n s t r u c t a mapping P which a s s i g n s t o each a i n CC0,ll i t s unique b e s t approximant i n X. Now, c o n s i d e r a t i o n s f a m i l i a r to anyone working i n c o n s t r u c t i v e mathematics suggest t h a t t h e c o n s t r u c t i o n of P should embody a proof of i t s uniform c o n t i n u i t y on compact s u b s e t s of CC0,ll. As I s h a l l i n d i c a t e below, t h i s proof of c o n t i n u i t y of P can be obtained by c a r e f u l l y a n a l y s i n g t h e p r o o f s , and thereby s t r e n g t h e n i n g the results, in [5l.
43
Constructive approximation theory
As it happens, these stronger results lead to a proof of existence of P which does not require an.application of Theorem 1. I suspect that this phenomenon may be quite general, and that every time we examine carefully a classical situation where every element of a normed space has a unique best approximant in a finitedimensional subspace, we will be able to prove constructively both existence and continuity of the best approximation process without appeal to Theorem 1. (See also the remark following [4, 3.11.) However, Theorem 1 remains of interest as both a guide for action in particular cases, and a constructive substitute for the fundamental theorem of classical approximation theory.
3 Let us now look more closely at Chebyshev approximation over [0,11. Let $1, ..., $ be elements of Cc0,ll (taken with the supremum norm). Define N
and
where l l . l l z is the Euclidean norm on lRN. positive real line as follows:
~ ( a )=
infIl$l(x)I
=
Define mappings B,y of (0,1/N1 into the
for each a :
x
E
: 0 5 xl,
J
l 0 for each a
E
(O,l/Nl.
Condition ( g ) is classically equivalent to finite linear combination of $1, ..., @ a simple Brouwerian counterexample to thys N = 2, $,(x) :1, $,(x) E a x , and it is
the well known Haar condition: every has at most N-1 zeroes in C0,ll. For equivalence, consider the case where not known if a = 0 or c1 f 0. Intui-
tionistically, (9is equivalent to the simpler condition: whenever xl, ..., xN are distinct points of C0,ll.
]detC$j(xi)l]
> 0
The best known example of a Chebyshev system is the set of polynomial functions 2
{l,x,x , sin 4nx,
.. . ,x?. ...,
Another example is the set cos h n x , sin 2nnx).
From now on, we shall assume that
I$,,
. . ., $N}
{l, cos 2nx, sin ZTx, COS 4nx,
i s a Chebyshev system, and we
shall denote by H the N-dimensional real linear subspace of Cr0,llspanned by . . . , I $ ~ } . Given a E C[O,l], we shall discuss the characterisation, existence and uniqueness of a best Chebyshev approximant of a in H: that is, an
D.S. BRIDGES
44 element b of H such that \\a- bn
=
=
dist(a,H)
-
inf{lta
c~=lci$ill : cl,
...,
CN E
m}
The classical characterisation of best Chebyshev approximants depends on the notion of "alternant". If a E C[O,l] and p E H, then an alternant of a and p is an where j E {0,11, 0 5 x1 x2 < . < XN+l 5 1 ordered pair (j,(xl,. . . ,x~+~)), and
..
Classically, a necessary and sufficient condition for b E H to be the best Chebyshev approximant of a is that there exists an alternant of a and b [12, Theorem 3-11. To see the nonconstructive nature of this characterisation, consider best Chebyshev approximation of a by constant polynomials. In this case, if m and M are respectively the infimum and supremum of a on [ O , l l , then dist(a,H) = &(M - m) and the best approximant of a in H is &(M + m). Were the above characterisation constructively valid, we would be able to find 5 in C0,ll with a(c) whence
a
- &(a + m)
=
&(M
-
would attain its supremum M at
m),
5.
To obtain a constructive characterisation of best Chebyshev approximants, we weaken the notion of "alternant" in the obvious manner. Let a E C[O,ll, P E H and E > 0. An E-alternant of a and p is an ordered pair (j,(xl,...,xN+l)), where j E {O,l}, 0 5 X1 < X2 < .. < xN + 1 5 1 and
> If also 0 < E < \la - pll and m p is an ordered pair (j,(xl, < x2m+4 = 1 ,
- pll
-
(1 5 k 5 N + 1 ) .
E
...,N-l},
0,
E
...,
Ila
x2m+4)),
an (m,E)-prealternant of a and where j E {O,l}, 0 = x1 < X2 < .
..
__ and
sup{((a
-
p)(x)J
:
x~~ s x
5
x2k+l1
0 and la - pll 5 dist(a,H) + LO(€). Ila - pll > ~ / 4 . In the former case, as
IIa - pll if
O S X
< x2
0 )
min(ct/4, ty(min(l/N,6(2t))/16NII$II))
E
CO,ll),
E
an E-alternant of
a and p. On the other hand, if f a - pll > ~ / 4then, by Lemmas 1 and 2 , there exists an (N-1,E)-prealternant (j,(tl, t2N+2)) of a and p. Choosing xk in [t2k-l,t2kl so that
...,
(-1lk-’(a we obtain an E-alternant
-
p)(xk) (j,(xl,
>
ila - PII-
..., xN+l))
(1
E/4 of
a
and P.
S
k
S
N+1),
0
D.S. BRIDGES
46
Let
THEOREM 2
a
and
C[O,ll
E
an
E-alternant
of
of
and b.
a
A
b E H.
b t o be a b e s t Chebyshev approximant
necessary
s u f f i c i e n t condition
is t h a t , f o r each
a
E
for
> 0, there exists
0
For t h e proof of Theorem 2 , s e e c4, 4.41. From P r o p o s i t i o n 1 w e can o b t a i n a v e r y u s e f u l lemma. A be5 t o t a l l y bounded s u b s e t o f
LEMMA 3 H.
and
Q: IR+
Then t h e r e i s a mapping max(lla
-
pll,Ila
-
911)
-f
IR+
dist(a,H) +
5
e
C[O,l],
B a bounded s u b s e t o f
such t h a t i f
a
E
A, p E B, q
a(€), then
Ilp
-
qll S E .
E
B,
E
>
0
Proof: As i n t h e proof of P r o p o s i t i o n 1, w e can f i n d a common modulus of c o n t i n u i t y 6 f o r t h e f u n c t i o n s i n A-B, w i t h 6 S 1 / N . With w: R+ + IR+ a s i n Propos i t i o n 1, d e f i n e mappings c , Q o f IR+ i n t o R+ a s f o l l o w s : f o r each E > 0 , ( II $11
NY (6( ~ 1 2))- ~ I I $11
c ( ~ )E
-
/y(6 ( € 1 2 ) )N+i-l
1)
and
: m i n ( ~ / 4 c ( 6 ( ~ / 2 ) )+, w ( E / ~ c ( ~ ( E / ~ ) ) ) ) .
Q(E)
Let
a E A, p
E
B, q C B
Ila
-
and
pll + Ila
-
-
IIp
911
2
911
>
€
> 0.
Ilp
-
911
>
As E,
we may assume t h a t IIa - qll > E/2. E i t h e r Ila - 911 > d i s t ( a , H ) or, by P r o p o s i t i o n 1, t h e r e i s an € / 4 ~ ( 6 ( ~ / 2 ) ) - a l t e r n a n n t ( j , ( x l , a and q . I n t h e l a t t e r c a s e , f o r 1 % k 5 N , /(a
Hence
x k+l
-
-
q)(xk+l)
-
(a
-
-
qll
-
E/~c(~(E/z)))
t
z(lla
- qll
-
~/4)
>
E/2.
1 S k 5 N.
-
min l_
dist(a,H)
+
R(E).
47
Constructive approximation theory
0
The result follows immediately.
The existence and uniqueness of best Chebyshev approximants follows simply from Lemma 3 and Theorem 1. However, Lemma 3 also enables us to prove existence without reference to Theorem 1. To each
THEROEM 3
H.
whenever
Proof:
p
E
n: R+
Moreover, with
n(E)
a
-t
and
H
E
there corresponds a
C[O,ll
as in Lemma 3, we have
R+
Ilp
unique best approximant b &
-
bll >
E
Ila - pII
t
dist(a,H) +
> 0. lim [la - p 11 = dist(a,H). By n* being finite-dimensional, is complete,
Let (p ) be a sequence in H such that
Lemma 3, (p
is a Cauchy sequence.
)
there exists b in H with
As H,
b = lim n-m pn.
Clearly,
Ila - bll
rest of the theorem follows immediately from Lemma 3.
=
dist(a,H).
The
0
In view of Theorem 3, there is a well defined mapping P - the Chebyshev projection of C[O,ll onto H which carries each element of C[O,l] to its best Chebyshev approximant in H.
The Chebyshev
THEOREM 4
projection *uniformly
continuous on each totally
bounded subset of C[O,ll. ___-and choose R > 0 s o that Proof: Let A be a totally bounded subset of C[O,l], IIall I R for each a in A . With B E {p E H : llpll I ZR), let the mapping n be as in Lemma 3. Given E > 0, let a,a' be points of A with IIP(a) - P(a')ll > E. Then IIP(a)ll
and so P(a)
B.
E
-
I
Ila
P(a)II
5
Ilall + Ilall
+ IIall
dist(a,H)
+ IIall
ZR,
I
Similarly, P(a')
IIa - P(a')ll
=
-
B.
E
=
max(lla
2
dist(a,H) +
By Lemma 3,
p(a)ll,Ila
-
p(a')ll)
a(€),
and so IIa
-
a'II t t
IIa - P(a')ll
- Ila' - P(a')ll
+
dist(a,H)
Q(E)
- dist(a',H).
Likewise, Ila'
-
all t
dist(a' ,H) +
n(E)
- dist(a,H).
Thus IIa
-
all1 2
It follows that
IlP(x)
Idist(a,H)
-
P(x')ll
-
I E
dist(a' ,H) whenever
I
+
Q(E),
x,x'
belong to A and
IIx
- x'I
0 For points in the metric complement of H , the uniqueness property in Theorem 3 can be strengthened considerably.
II a
-
bllX
whenever
H.
A
tion that b be the best Chebyshev approximant
of
THEOREM 7
~ e at
E
CLO,ll, and let
---- j (0,l) such
b
E
__-
there exists
(1 5 k i N+1)
a(xk)
E
(-l)k-j(a
-
of
a
p
H
E
necessary a
Over X
over X, and la Ilp
- bllX
sufficient
bl
> 0.
X
0
--
is that for each
E
> 0
that b)(xk)
>
Ila
-
bllX
-
E
(1
< k
5
N+l).
0
49
Constructive approximation theory
For the proofs of these theorems see 15, section 51. From our point of view, it seems natural to look at best approximation over alternants: THEOREM 8 alternant
Let a - E C[O,ll, p (j,(xl,
approximant ~f
E
and E > and p,
H
..., X~+~))f x
a
Ixl,
a
0. Suppose that there exists an
0
0
E
C[O,ll.
Then there exist mappings
and if xl,
..., xN+l
6,w
of
R+
into
are -points of C0,ll with 0 -
I. Then gi+2(A) is gi+l(A) gi-l(A) v gi-2(A) (A) is gi(A) + fi-l(A) By Church's rule for HA , for some n gi+ I bVy3zT(g,y,z) and +
.
kCJZCT(Z,y,Z)
A
UZ=OI
+
gi+2(A)I
A
C3ZTT(g9y.Z)
A
UZZOI
+
, and
gi+l(A)I.
Therefore k[T(:,y,z)
Uz=O
A
+
[gi+l(A)
+
gi-l(A) A
gi-2(A)11
V
[T(g,y,z)
A
A
Us0
So, for some m , (1)
(2)
km gi+l(A) km T(n,y,z)
+
A
CT(g,y,z)
Uz*O
+
A
Uz=O
gi+l(A)
+
gi-l(A)
V
gi-2(A)1
.
Moreover, if we take m h 2 (3) (4)
I- ~(g,y,z) A uz=o T(;,y,z)
A
UzZO
-+
~hm~(r~(g,y,z)A u z = o l )
,
+
Thmm(rT(n,y,z)
.
A
From ( 2 ) and ( 4 ) we obtain
(5)
k T(n_,y,z)
A
Uz*O
r
-+
Thm( gi+](A)')
.
UztOl)
,
+
gi+l(A)l
D.H.J. DE JONGH
62
I-Ic
From
* gj(P)l
[gi+l(P)
tf
-
gj(P)
I- Thmm(rgi+l(A) * g j ( A ? )
(7)
,
j 5 i
for
Thmm( rgi-l
(A)1)
.
v Thmm( rgi-2(A)1)
i t follows that
T h m m ( r Ig . ( A p ) ,
Now by a p p l y i n g ( 7 ) and ( 4 ) t o ( 6 ) we o b t a i n +
,
I- T ( n , y , z )
1Thmm(‘gi(A)’)
Thus, s i n c e
t,,
i .
j
for
gi-*(A)
+
giM
Uz=o
A
*
,
Then by (8) and (5) and a g a i n p r o p o s i t i o n a l c a l c u l u s
Therefore V
gi(A)l)
k l l T h m m ( r g i + l (A)
k
and f i n a l l y
flection principle for
gi+l(A)
,
Thm,
duced i n t h e a s s u m p t i o n (*)
i
Now j u s t t h e c a s e g2(A)
+
g0(N
If
A
if
knf(A)
[A. HA
V
f,,W
is a sentence,
,
then
gi(A)l)
V
=
i
by
A
V
i s a f o r m u l a we u s e a r e -
T r o e l s t r a 1973, 1 . 5 . 6 ( a ) l and w e h a v e re-
see e . g .
1
[Because
gi(A)
V
I- Thmm(rgi+l(A)
and by Markov’s r u l e
.
1
r e m a i n s . But e x c e p t f o r t h e f a c t t h a t
t h e p r o o f r e m a i n s e x a c t l y t h e same.
g3(A)
is
0
t h e n t h i s p r o o f shows
t n l l A
or
I-,llA
+
A.
Visser o b s e r v e d t h a t t h i s c a n i n t u r n b e f o r m a l i z e d so a s t o y i e l d
I- Thm(rf(Ajl)
+
Thm(?lA
V
11A-tA)’)l
5. A PROBLEM OF G. KREISEL I n a l e t t e r t o t h e a u t h o r o f May 23, 1 9 7 0 , G . K r e i s e l
suggested e s s e n t i a l l y
t h e f o l l o w i n g problem. Does t h e r e e x i s t , f o r e a c h f o r m u l a
A
w i t h one a d d i t i o n a l f r e e v a r i a b l e
n
x
such t h a t , f o r each
of
,
HA, a f o r m u l a
I-B(n)
t+
B
gn(A).
We w i l l p r o v e h e r e t h a t n o t e v e n f o r e v e r y s e n t e n c e d o e s such a f o r m u l a e x i s t and i n f a c t t h a t no s u c h f o r m u l a e x i s t s f o r any words f o r any f o r m u l a
A
A
such t h a t e x a c t l y t A + A
,
i n other
f o r which no p r o p o s i t i o n a l f o r m u l a i s p r o v a b l e e x c e p t
when it i s p r o v a b l e a l r e a d y i n
.
IC
a n i n f i n i t e number o f n o n - e q u i v a l e n t
So t h i n g s go wrong a s soon a s t h e r e r e a l l y are f o r m u l a s t h a t c a n b e o b t a i n e d from
A
with
p r o p o s i t i o n a l c o n n e c t i v e s . Kreisel s u g g e s t e d t h i s problem b e c a u s e i t s n e g a t i v e sol u t i o n shows v e r y c l e a r l y ’ w h y i n
HA
i t i s n e c e s s a r y t o i n c l u d e t h e number o f im-
Formulas of one propositionalvariable
63
plications when assigning a degree of complexity to a formula (cf. Kreisel L6vy 1968). Thm 5.1. If, for a sentence A
HA , exactly
of
I-A
A , then for no formula B
+
of HA , for each n , I-gn(A) ++ B(n) . Proof. We form the same Kripke-models KO, K , and K as in the proof of Thm 2.,1. It was proved there that, for each q E K and each 1 2 3 , qll-gi(A) iff pi 5 q . Let us say that a sentence C has degree d
in the Kripke-model
the smallest natural number with the property that, for each pkll-C.
K
,
if
d
is
j,k t d, p.II-C iff J
For our purpose it will be sufficient to prove that, for any formula
...,xn)
C(xl,
the degrees of the sentences C(g,
will be proved by induction on the length of
,...,%)
C
in K
are bounded. It
that there is a number nC which
is such a bound.
(i)
If
C
is a prime formula, then nC = 0 , since K
in HA (ii)
If
C
is A
(iii)
If
C
is A v B
, ,
(iv)
If
C
is A
, then for each q
qll-A
+
is a model for HA
and
all prime formulas are decidable.
B
A
+
B B
then we take nC
=
again take n C
max(nA,nB).
iff, for each
r
2
q
,
=
max(nA,nB).
K ,
E
rll-A , then rlkB.
if
One sees easily from the picture in the proof of Thm 2.1 that this implies (v)
that we can take nC = max(n A ,nB ) + 1 . C is V x D , then, since the domains of all the p.
If
the same we can take nC (vi)
If
C
is
=
.
max(%,2)
3 x D , then we can take n
=
C
n D ’
for
i
t
2
are
’
[Remark. A stronger result, obtained by proof-theoretical methods, is found in Theorem I of Leivant 1981.1
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(1960),
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intuitionistic formal systems.
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25, p.27-32.
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V
C, A
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propositional calcuLus.
JSL 25, p.327-331.
Smorynski, C.A. (1973), Investigations of i n t u i t i o n i s t i c formal systems by means of f i i p k e models. Ph.D.Thesis, University of Illinois at Chicago Circle.
-------
(1973A),
Spector, C. (1962),
Applications of Kripke models. In Troelstra(l973).
p.324-391.
Provably recursive functionals of analysis, a consistency
proof of analys,is by an extension of principles formulated in current in-
t u i t i o n i s t i c mathematics, in: J.C.E. Dekker(editor), Proceedings of the symposia in pure mathematics V. American Math.Soc., Providence, RI, p. 1-27.
Troelstra, A.S. (1971),
Notions of r e a z i z a b i l i t y f o r i n t u i t i o n i s t i c arithmetic i n
a l l f i n i t e types, in: J.E. Fenstad(editor), Proceedings of the second Scandinavian logic symposium. North-Holland Publ.Co., Amsterdam, p.369-425.
-------
-------
(1973)(editor),
Metamathematical investigation of intuitionistic
arithmetic and analysis. Springer Verlag, Berlin. (1975), Markov )s principle and Markov ) s rule f o r theories of choice sequences, in: J. Diller, G.H. Mcller(editors), Proof theory symposium Kiel 1974. Springer Verlag, Berlin, p.370-383.
Visser, A. (1981).
Aspects of diagonalization and provability. Akademisch Proef-
schrift, Rijksuniversiteit Utrecht.
THE L.E.J. BROUWER CENTENARY SYWOSIlM A S . Troelstra and D. van Dalen (editors) 0 North-Holland Publishing Company, 1982
65
COATINDOUS SUMS OF SQUARES OF F O R M
Charles N. Delzell Department of Mathematics Louisiana State University Baton Rouge, Louisiana 70803 USA
We give a continuous representation of positive semidefinite (psd) n-ary quadratic forms over an ordered field as sums of (almost n!e) nonnegatively-weighted squares of linear forms. This answers a question of Kreisel, who noticed in 1980 that (already for n = 2) the usual "completion-of-square" process gives a discontinuous representation. For n = 2 J.F. Adam has recently reduced the required number of continuous summands to 2, but only over Euclidean ordered fields. We also show that any universal representation of psd quartic forms as sums of squares of quadratic forms must be discontinuous at (X2 + Y2)'. Contents:
0. Introduction 1. Continuous Versions of Some Classical Results (a) Lagrange's and Siegel's Theorems on Sums of Squares (b) The "Weak" Hilbert Nullstellensatz (c) Hilbert's 17th Problem 2 . Sums of Squares of Forms
3. Statement of Continuity Results 4 . Significance of Continuity (a) Geometric Significance (b) The Case of Recursive Ordered Ground Fields (c) The Case of IR 5. The Proof for Quadratics 6 . The Proof for Ternary Quartics
0. Introduction We construct (5.1) a representation of psd (positive semidefinite) quadratic forms in X1, Xn over an ordered field K as sums of (almost n!e) nonnegat ively-weighted squares of linear forms such that each summand, regarded as a function of the Xi and their coefficients, is rational with integer coefficients and (extensible to a function which is) continuous with respect t o the usual interval topology on K, hence also with respect to finer "computational" topologies on enrichments of IR by certain kinds of representations. This answers a question of Kreisel, who noticed in 1980 that (alread for n = 2) the usual SOS (sum of squares) representation of aX2 + 2bXY + CYJ (writing (X,Y) for (Xl,X2); a > 0, c 0, ac - b2 > 0) as
...,
(0.1)
a-'[
(ax +
+ dY2], o r
Research supported in part by NSF grant No. MCS8102744. 1980 Mathematics Subject Classification: Primary 03355. 10C05, 10C10, 10505, 54C05; Secondary 10C04, 54H13.
66
C.N. DELZELL + (bX + CY)']
c-l[dX2
(0.1')
(d = a c - b 2 ) , o b t a i n e d by " c o m p l e t i n g t h e s q u a r e , " i s d i s c o n t i n u o u s n e a r t h e ray ( a , b , c ) = (O,O,c) ( c > 0 ) , s i n c e a s a + 0 , b 2 / 4 a v a r i e s between 0 a n d c. Our p r o o f o f 5 . 1 c o n s i s t s , f o r n = 2 , o f t a k i n g t h e convex combinat i o n s (by a / s and c / s , where s = a + c ) o f 0 . 1 and 0 . 1 ' t o g e t t h e c o n t i n uous r e p r e s e n t a t i o n
+ by)' + dY2 + dX2 + (bX + cY)'].
(ax
s-'[
(0.2)
W e a l s o show ( 6 . 1 ) t h a t any map o f psd t e r n a r y q u a r t i c forms i n t o e q u i v a l e n t SOS o f q u a d r a t i c forms must have a jump d i s c o n t i n u i t y a t ( X 2 + Y2)' ( H i l b e r t [1888] gave t h e f i r s t d i s c o n t i n u o u s map of psd t e r n a r y q u a r t i c s i n t o e q u i v a l e n t SOS o f q u a d r a t i c s ) .
1. C o n t i n u o u s V e r s i o n s of Some Other C l a s s i c a l R e s u l t s S i n c e t h e e a r l y s i x t i e s Kreisel h a s asked whether "continuous" v e r s i o n s of v a r i o u s c l a s s i c a l r e s u l t s a r e p o s s i b l e . L e t u s l i s t some o f t h e a n s w e r s : ( a ) L a g r a n g e ' s and S i e g e l ' s Theorems on Sums o f S q u a r e s I n t h e c a s e o f L a g r a n g e ' s theorem, H e i l b r o n n [1964] c o n s t r u c t e d i n t e g r a l functions f l , f 2 , f 3 , f 4 which s a t i s f y ( w r i t i n g m = 4 ) (1.1) and s u c h t h a t
&fi(z)
I n [Delzell, follows: let F
m
such t h a t
9 . for
0
0,
vn,m
> u ( k ) , [ lrn-rm\< l / k ] .
Our c o n t i n u o u s v e r s i o n s o f A r t i n ' s theorem ( § l ( c ) ) and o f (d = 2 i m p l i e s 2 . 2 ' ) f i l l i n t h i s g a p , and n o t o n l y f o r K = R = R , b u t more g e n e r a l l y whenever R c o n t a i n s a dense r e c u r s i v e s u b f i e l d F: t h e n i f t h e c o e f f i c i e n t s o f a psd form a r e g i v e n by a p p r o x i m a t i o n s from F, w e c a n compute a p p r o x i m a t i o n s o f t h e SOS r e p r e s e n t a t i o n t o c o m p a r a b l e a c c u r a c y ; t h e p r o o f s a l s o a r e i n t u i t i o n i s t i c . On t h e o t h e r hand, o u r d i s c o n t i n u i t y r e s u l t f o r (n,d) = (2,4) does n o t n e c e s s a r i l y imply t h a t 2.2 c a n n o t be proved i n t u i t i o n i s t i c a l l y f o r (n,d) = (2,4),even when R = R . 'Ihe r e a s o n i s t h a t t h i s d i s c o n t i n u i t y r e s u l t i s w i t h r e s p e c t t o t h e u s u a l i n t e r v a l t o p o l o g y on R , and n o t w i t h r e s p e c t t o f i n e r " c o m p u t a t i o n a l " t o p o l o g i e s on v a r i o u s "enrichments" o f R by s p e c i f i c r e p r e s e n t a t i o n s , say o s c i l l a t i n g b i n a r y expansions with t h e corresponding Baire s p a c e o r "weak"
t o p o l o g y on
y,
o r Cauchy s e q u e n c e s o f r a t i o n a l s w i t h t h e to-
&.
While any f : R + R conp o l o g y i n h e r i t e d from t h e p r o d u c t t o p o l o g y on t i n u o u s w i t h r e s p e c t t o t h e u s u a l topology i s o b v i o u s l y c o n t i n u o u s w i t h res p e c t t o t h e c o m p u t a t i o n a l t o p o l o g y on Cauchy s e q u e n c e s , t h e c o n v e r s e i s d i s proved by e a s y examples; i n f a c t , f u n c t i o n s c o n t i n u o u s f o r t h i s c o m p u t a t i o n a l t o p o l o g y need n o t r e s p e c t e q u i v a l e n c e between Cauchy s e q u e n c e s a t a l l . Brouwer had o b s e r v e d t h a t e v e r y c o n s t r u c t i v e l y d e f i n e d f u n c t i o n from R t o P (whose e l e m e n t s h e d e s c r i b e d by " f r e e c h o i c e sequences" u o u s r e l a t i v e t o t h e p r o d u c t t o p o l o g y on
value
sn
h a s been established, then
o f information about
(rn).
f
&,
for i f
(rn)
E
and a
c a n h a v e used o n l y a f i n i t e amount
'Ihus i f 3.1 c a n n o t b e proved f o r
with respect t o t h i s particular proved i n t u i t i o n i s t i c a l l y .
e) i s c o n t i n -
f ( ( r n ) ) = (sn),
computational
topology,
then
(n,d) = (2,4)
it cannot be
5 . The Proof for Quadratics Let let
C = (C. .) IJ
be i n d e t e r m i n a t e s ,
c = ( c i j ) E Sym(n+l,R),
for
0
< i , j < n,
with
t h e t o p o l o g i c a l (R-vector)
Cij
= Cji,
and
s p a c e o f symmetric
71
Continuous sums of squares
( n + l ) x ( n + l ) m a t r i c e s o v e r R. i n X = (X,,.. ,Xn) and write PA,? = { c
1J
R("+l)'
be t h e g e n e r a l q u a d r a t i c form
I
1
i s psd o v e r
Sym(n+l,R)lZc;jXiXj
E
i s an i s o m o r p h i c image i n
PA,2
Z i j C..X.X.
Let
..
of
N(n) = ( n + l ) !
Pn,2.
k=O
R
Set
&.
0 < < n, such t h a t
Theorem 5.1: For f i x e d n and f o r c o n s t r u c t r a t i o n a l f u n c t i o n s pk, a,cC)
XI;
in
1 < k < N ( n ) , we can
(5.2) throughout
Rn+l,
x
and
> 0,
pk(c)
for
1 < k < N(n)
,
p k ( c ) ( ~ a .akE(c)xa)'
continuous
(c;x)
B e f o r e g o i n g t h r o u g h t h e p r o o f , i t i s i n s t r u c t i v e t o c o n s i d e r t h e simp l e s t case, n = 1 , and v e r i f y c o n t i n u i t y of 0.2. The o n l y p o i n t i n P1,* where t h o s e c o e f f i c i e n t s c o u l d be d i s c o n t i n u o u s i s where s = 0 , h e n c e a = c = b = O ( t h e t r i v i a l f o r m ) , and even h e r e e a c h c o e f f i c i e n t e x t e n d s c o n t i n u o u s l y (by 0 ) : namely, t h e i n e q u a l i t i e s 0 < a 2 / s < a , 0 < c 2 / s < c , and 0 < max{d/s,b2/s} < a c / s < a show t h a t t h e c o e f f i c i e n t s a2/s, c2/s, d/s, b2/s, and h e n c e a l s o 2 a b / s and Lbc/s must approach 0 a s a and c d o . Proof o f 5.1: For
n
I n d u c t i o n on
>
0
n.
For
ccijxixj (% ,...,Xa.-l,Xa.+l
=
so have t h a t
(5.3)
(cij)
p a
p1 = Coo
a10 = 1.
and
we make u s e o f t h e r e p r e s e n t a t i o n c o n s t r u c t e d f o r
t o construct the representation for
where
n = 0, t a k e
€
PA,2
0, and b o t h
P h , 2 , a s follows.
+ 2Xa.Zifa.
= CLa.X$
ciaxi
,...,X,,),
and where
i f f , for
0
1,
C.N. DELZELL
12 i
then given
qa(O i s psd f o r
c
: V + V. We normally call the arrows morphisms. By introspection, we may recognize a truth whose verification seemingly requires more data bv observina that. no matter which of an exhaustive coilection of possibilities fo; extra data transpires, the verification will occur. We call such exhaustive collections of possibilities for future data covering families. Formally we require that these satisfy the axioms for a Grothendieck pretopology: 1) I < > ) covers U for each state U, f: Vf + U I fEK 1 covers U, and e: W -t U 2) If K = then I g 1 e*g factors through some feK } covers U. These are clearly valid for the intuitive notion to hand. 1.2 Remarks. Our models are similar to the familiar Beth and Kripke models. Formally, the forcing definition for sites is that for Beth models with bars replaced by abstract covers, and passage to a later stage, 5 , replaced by arrows representing incoming data, +. We make a distinction between a constructive explanation of meaning, given in terms of a notion of proof, and an intuitionistic explanation of meaning which we shall give in terms of data. Unlike Dummett, (Elements p. 403) we do not view these as rival accounts. We assume a basic conception of the mathematics of lawlike or constructive objects. (We discuss later the minimum demands we make on this metatheory - which may be classical.) We introduce various notions of data and representations for non-contructive objects based on these notions. Our analysis of the meaning of predicates involving non-constructive parameters leads us to the justification of various intuitionistic principles; it does not affect the mathematics of lawlike objects. Technically the theory of non-constructive objects is a conservative extension of our metatheory. Our-project is not novel: Beth models were introduced to formalize iust such an explanation of meanins. Our use of catesories in place of posets arises from a basic philosophical difference. We aim; not to qive a model of the activities of a sinqle idealized mathematician, but-rather, to analyse the objective mathematical truths which may be justified on the basis of a particular conception of data. Thus, for us, the information to hand at a particular time is not, in general, part of the state, but part of the representation of a particular nonconstructive object.
Notions of choice sequence
93
Two extreme examples of categories are posets and monoids. In a poset, there is, for each pair of objects , at most one morphism p5q from p to q. Posets represent a totally subjective conception of data, which identifies the state with the information to hand. A monoid is a category with only one object, only the morphisms and the (total) operation of composition matters. We use them to represent objectively conceptions of data for which we can recognize that future possibilities for data are independent of the information to hand. Some types of restriction on future data compel us to consider separate states to represent the differing data which is acceptable. In these cases an objective viewpoint amounts to being able to juxtapose two states given independently to give a single process. Formally, given two states A and B, there is a state A X B accessible from A and B by morphisms TA nB A -A X BB which are covers and for each pair e: CA and f: DB there is a unique e x f making the diagram commute A
k
f A
1 C-B
X
3B
C
tx D
e x f
b
I D
Note that, A x B is not a categorical product since we have no pairing in general. The requirement that the “projection“ maps cover says that, we can always introduce a new process independent of that under consideration. 1.3 Definitions. A spread is a subtree S c N4N with every branch N, we say ~ E S infinite: if aES then a*ns.S for some nsN. For a:N+ iff Va. ( m a 4 aES). If S and T are spreads, a neighbourhood function F: S + T is a monotone function such that for each nsN, the set of nodes aaS such that Lth (F(a))zn is an inductive bar of S. Given a s s and F: ST we define F(a) by asa + F(a) E F(a). Spreads represent subsets of 7 8 , neighbourhood functions represent continuous functions. Composition of neighbourhood functions gives the composition of the associated functions. Some functions are canonically represented: in particular, the open inclusion corresponding to a finite sequence e is represented canonically by Xa.e*a (and in many other ways by merely deferring the information); more generally, if f is an open map it has a canonical representation F(a) = A { b I aEf-’(b) 1 . Countable dependent choice and a suitable form of Bar Induction imply that every continuous function has a neighbourhood function. We write B for the universal spread of all sequences. 12 TYPES
OF DATA
The examples which follow should make clearer the translation from the informally rigorous description of a notion of choice sequence to the appropriate site.
M.P. FOURMAN
94
2.1 Open Data. The simplest data we shall consider, open data, consists of finite sequences a of natural numbers. We can construct a sequence a from such data in many ways, the simplest of which is to consider the information thus far received as an initial segment aEa. We have decided in advance that, having received the information a we will treat subsequent data (another finite sequence, b) in a particular way: we concatenate a*b. Thus there are various states which in this example may be identified with the information to hand. Incoming information takes us from one state to another.
Abstractly, we have a category whose objects are the states and whose morphisms represent finite amounts of information. In our present example this structure is represented abstractly as the tree of finite sequences or, more concretely, as the category of basic opens of Baire space and open inclusions between them. Since, to recognize that aEa*n for some nEN it suffices to recognize that aca, we must let { a*n I nEN 3 cover a for each finite sequence a. These covers generate the open cover topology for formal Baire space (see Fourman & Grayson (this volume) ) . Our analysis has merely served to reconstruct the Scott (1968) - Moschovakis (1973) model. From a more objective view of mathematics the distinction between various states seems unjustified: it portrays the activity of a particular idealized mathematician rather than the mathematics which results from reflection on the general nature of such activity. A more satisfactory model (from this point of view) is given by the monoid of finite sequences. We can picture this concretely as the monoid of neighbourhood functions canonically representing open inclusions. Again the appropriate topology is the open cover topology. This model corresponds to the liberation of the idealized mathematician: realizing her situation, she can transcend it and is free b?
.
-
This conception of open data allows that all possible data can be coded up in a single choice sequence. 2.2 Independent Open Data. We modify our model to consider not a single generating process but a potentially infinite collection. It is essential to distinguish this from a potentially infinite sequence of processes, which we could code as a single choice sequence. The force of this distinction is that, at any stage, the information we have is just the collection of initial segments to hand. They are not taken in any particular order.
The subjective states for this notion consist of finitely much information about finitely many sequences. Concretely we represent such a state by a basic open U c B n modulo any action by a permutation of n.
Notions of choice sequence
95
Actually it is more convenient to consider basic opens as states. At any stage we may introduce finitely many independent generating processes as well as obtaining more information about those already considered. We represent such information by a map
u-v which is induced by the projection corresponding to some injection n r+ m. The morphisms induced by permutations of n have the effect of identifying states which represent different orderings of the same collection. In addition to allowing open covers as before, we stipulate that the projection
u
ni
-
Bm-
T(U)
ni
Bn
is a cover. This reflects the possibility of adding finitely many independent processes to any discussion. In general, a family of morphisms covers iff the union of the images is an (open) cover of V. All the morphisms here are open maps and we shall later consider them as represented canonically by neighbourhood functions. Again, we have arrived at a well-known model: according to Hyland (personal communication), the forcing definition for this model corresponds to the Kreisel-Troelstra elimination of lawless sequences. more objective representation of this type of data is obtained by identifying states in which the same number of generating processes are considered. The site we use to represent this type of data has as objects, the various Bn, and as morphisms, compositions of projections Bm-Bn induced by n-m and open inclusions Bn-Bn induced by n finite sequences. Once more, all the morphisms are open maps canonically represented by neighbourhood functions and we use the topology in which a family of morphisms covers iff its images cover.
A
2.3 Lawless Data. This conception of*data was motivated by the following passage from Troelstra (CS p. 16) : "Suppose we have started two lawless sequences a and 0, alternately selecting values: a0, 00, al, 01, 1x2, 02, Now we may also regard this as a single process y , with y(2n) = an, y(2n+l) = On. However, we cannot regard cx,B,y as all being lawless within the same context: either we have to decide a and 0 to be lawless, and then y is a sequence which is not itself lawless .; or we consider y as lawless, inwhich case a,B are sequences (not lawless ones) constructed from y."
... .
.. .
..
This discussion cannot be expressed in the Kreisel-Troelstra formalisation of choice sequences. This is because, their notion of lawlessness is not an objective one. In our previous example, all the states are, in fact, remarkably similar: Bn is homeomorphic to B. Essentially, the differences between the states arise because we have chosen a coding B Bn in term of which we choose which maps to put in our category. We now take the point of view that different codings
MP. FOURMAN
96
simply reflect different ways of considering the same reality. Thus the controversy as to which of a, B and y are lawless, in our example above, arises from the difference in viewpoint formalized by the pairing B = B x B. We now consider an abstract view of the same kind of data, which is independent of such codings. We call this lawless data, it consists of those endomorphisms e: B + B which can be decomposed as "projections modulo some coding": e = j-1.r,i for some homeomorphisms i and j.
Bm& UI
Bn UI
u-v i\ B
4
A s usual, all our morphisms are open maps and we use the topology in which surjective families cover.
We shall see that lawless sequences for this conception of data behave more sociably than is traditional. For example, two views of the world may at some stage turn out to be the same so equality is not decidable. To formalize our discussion of this type of lawlessness, we shall introduce a notion of independence: basically, a and B are independently lawless iff y = < a , % >is lawless. Returning to Troelstra's example, a, B and y are lawless a and 4 are independent and y is independent of neither of them. 2.4 Spread Data. Here we attempt to formalize Brouwer's description of the generation of a free choice sequence.
...
" the freedom of proceeding, without being completely abolished, may at some time p, undergo some restriction, and later on further restrictions. " Brouwer (Cambridge p. 13)
The restrictions discussed by Brouwer demand that future choices belong to some spread. Spreads correspond to certain sublocales of B. We consider such sublocales F s Bn and morphisms between them induced by projections. We take as covers projections and open covers. This gives us (in this example) the topology in which a family covers iff the interiors of its images cover. This topology involves no new insights, many stronger topologies (more covers) are conceivable: It is certainly plausible that we might justify the conclusion that every member of a spread S belongs to one of the spreads Ti without showing that the interiors of the Ti cover S , by appealing to particular properties of S. This would be reflected in our models by adopting a stronger topology. What we will show is that it is consistent to assume that the only covers are those we have built into the definition of the topology. The main insights justified by this conception of data are the relativisation of V a 3 % choice and continuity for lawless a to lawless elements of some spread and the extension of Bar Induction to give induction over arbitrary spreads. Brouwer's conception of choice sequence has been criticizad for not
Notions of choice sequence
97
b e i n g c l o s e d u n d e r c o n t i n u o u s o p e r a t i o n s . The s p r e a d s w e have i n t r o duced a r e b l a n k o r naked s p r e a d s , which, f o r Brouwer, s i m p l y p r o v i d e a framework f o r t h e g e n e r a t i o n o f m a t h e m a t i c a l e n t i t i e s . By a t t a c h i n g In " f i g u r e s " t o some nodes of a s p r e a d S w e p r o d u c e new o b j e c t s . p a r t i c u l a r , any neighbourhood f u n c t i o n F: S + T p r o d u c e s f o r e a c h c h o i c e s e q u e n c e a€S a s e q u e n c e F ( ~ ) E T . The i n f i n i t e s e q u e n c e s gene r a t e d i n t h i s way a r e c l e a r l y c l o s e d u n d e r t h o s e c o n t i n u o u s o p e r a t i o n s which have neighbourhood f u n c t i o n s . We s h a l l s e e t h a t ( i n o u r models) a l l l a w l i k e f u n c t i o n s have neighbourhood f u n c t i o n s . F u r t h e r more u s i n g s u c h d r e s s e d s p r e a d s ( w i t h S E < S , F > i n t e r p r e t e d a s , f o r some ~ E S , 5 = F ( a ) ) , we s h a l l see t h a t an axiom of " s p r e a d d a t a " i s v a l i d f o r t h e s e sequences. 2 . 5 Continuous d a t a . W e s t a r t from B r o u w e r ' s 1933 d e s c r i p t i o n of a d r e s s e d s p r e a d a s r e p o r t e d by van Dalen (Cambridge p . 1 7 1 . Here A g e n e r a t e s a l a w l e s s sequence and B a p p l i e s t o it a neighbourhood f u n c t i o n t o o b t a i n a s e q u e n c e F ( a ) a s d e s c r i b e d e a r l i e r . W e modify t h i s p i c t u r e by no l o n g e r r e q u i r i n g t h a t A ' s sequence b e l a w l e s s : it may i n f a c t b e g e n e r a t e d a s a c o n t i n u o u s f u n c t i o n of some sequence g e n e r a t e d by X who, i n t u r n , r e f e r s t o Y , and s o on. W e r e q u i r e t h a t a l t h o u g h t h i s c h a i n of dependence may b e p o t e n t i a l l y i n f i n i t e , a l l t h a t B c a n be aware of a t any g i v e n s t a g e i s a f i n i t e c h a i n of dep e n d e n c i e s , r e s u l t i n g i n t h e knowledge t h a t B = r ( a ) f o r some n g e n e r a t e d by someone down t h e l i n e , and some neighbourhood f u n c t i o n
r.
We r e p r e s e n t s u c h d a t a by a neighbourhood f u n c t i o n r : S + T between s p r e a a s . Note t h a t , a l t h o u g h i n p r i n c i p l e w e s h o u l d want t o c o n s i d e r dependence on more t h a n one s e q u e n c e , s u c h d a t a r e d u c e s t o dependence We g i v e t h i s on a s i n g l e s e q u e n c e by means o f t h e p a i r i n g B x B c. B. c a t e g o r y t h e "open c o v e r t o p o l o g y " i n which t h e c a n o n i c a l r e p r e s e n t a t i v e s of a c o v e r i n g f a m i l y of open i n c l u s i o n s form a c o v e r . Of a l l our models w e b e l i e v e t h a t t h i s one b e s t r e p r e s e n t s t h e n o t i o n of c h o i c e s e q u e n c e . N e v e r t h e l e s s , w e d i s c u s s two v a r i a n t s . F i r s t l y , i f w e a r e concerned o n l y w i t h e x t e n s i o n a l p r o p e r t i e s , w e can use c o n t i n u o u s f u n c t i o n s i n p l a c e of neighbourhood f u n c t i o n s . Secondly, i f i n s t e a d of u s i n g a r b i t r a r y s p r e a d s w e c o n s i d e r t h e monoid of c o n t i n u o u s f u n c t i o n s B + B , w i t h t h e open c o v e r t o p o l o g y , w e o b t a i n a model f o r Kreisel and T r o e l s t r a ' s t h e o r y C S . ( T h i s was observed i n d e p e n d e n t l y by Moerdijk & van d e r Hoeven ( 1 9 8 1 ) , Grayson ( 1 9 8 1 ) and t h e a u t h o r ( 1 9 8 1 ) ) . The f o r c i n g d e f i n i t i o n f o r t h i s model c o r r e s p o n d s t o t h e e l i m i n a t i o n mapping f o r c h o i c e s e q u e n c e s of Kreisel and T r o e l s t r a ( 1 9 7 0 ) . I n t h e s e models w e v e r i f y f u l l V a 3 B c h o i c e and c o n t i n u i t y p r i n c i ples. The a d v a n t a g e of t h e e x t e n d e d model i n which w e a l l o w a r b i t r a r y s p r e a d s as domains i s t o j u s t i f y r e s t r i c t e d v e r s i o n s of t h e s e and e x t e n d e d Bar I n d u c t i o n a s f o r s p r e a d d a t a . 2.6. O t h e r t y p e s of d a t a . I n o u r p a p e r C o n t i n u o u s T r u t h (1982) we c o n s i d e r more g e n e r a l t y p e s of d a t a ; i n p a r t i c u l a r , d a t a r e p r e s e n t e d by c o n t i n u o u s maps between opens of Rn. W e a l s o g i v e a g e n e r a l t r e a t ment of t h e " e l i m i n a t i o n mappings" a s s o c i a t e d w i t h e a c h t y p e of d a t a and t h e r e l a t i o n s h i p s between v a r i o u s t y p e s of d a t a m e d i a t e d by geometric morphisms between t h e c o r r e s p o n d i n g t o p o i .
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13 NON-CONSTRUCTIVE OBJECTS
We now embark on the analysis promised in 2.1. An understanding of a collection of objects is merely an understanding of what it is to be presented with such an object and of what it is to show that two such objects are equal. This does not automatically give rise to a determinate collection of predicaterrather we must introduce predicates by explicitly giving their meanings. Other predicates may, of course, be compounded from ones previously understood using the logical connectives. We suppose the meanings of statements involving lawlike parameters, quantification over lawlike objects and the meanings of the logical connectives applied to such statements, to be understood. Traditionally, an explanation is given in terms of an informal notion of construction (for example, Dummett (Elements p . 12ff.)). Our explaination of the meaning of statements involving non-constructive objects is independent of this (and, to a large extent, of its results), similar to it in form, and different from it in content. The meaning of a statement involving non-constructive objects is given in terms of a constructive understanding of which items of data justify a given assertion. 3.1 Non-constructive Objects. Our archetype is given by Brouwer's notion of a dressed spread: A partial function $ assigning lawlike objects to the nodes of some spread S. The idea is that any choice sequence a of the spread S generates successive approximations, $(a) for aca, to a non-constructive object $ ( a ) . Abstractlv, we assume that the constructive objects @(a) have a preorder, x < y if x contains "more information" than y, and that $ is monotone, a
