Lecture Notes in Mathematics A collection of informal reports and.seminars Edited by A. Dold, Heidelberg and B. Eckmann, Z~irich
95 A. S. Troelstra University of Amsterdam, Amsterdam
Principles of Intuitionism Lectures presented at the summer conference on Intuitionism and Proof theory (1968) at SUNY at Buffalo, N.Y.
1969
Springer-Verlag Berlin. Heidelberg. New York
All rights reserved. N o part of this h o o k may be translated or reproduced in any form without written p e r m i s s i o n from Springer Verlag. 9 by Springer-Verlag Berlin 9 Heddelherg 1969 Library of C o n g r e s s Catalog Card N u m b e r 74-88182 Printed in Germany. Title No. 3701
Contents i 9 Introduction 2. Logic
2
o o o o o e o o o o e o e e t o o o o e e 6 o o o , o o t o o o o e J o o o o o o o o o o e D o o
.......................................................
5
3- Elementary arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
4. Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
5. Sequences and lawlike o b j e c t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
6. Elementary theory of real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
7, Ordering relations and order on the real llne . . . . . . . . . . . . . . . . .
26
8. Constructive or lawlike analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
9. Lawless sequences of natural numbers . . . . . . . . . . . . . . . . . . . . . . . . .
34
I0. Choice sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. Spreads and the theory of real numbers . . . . . . . . . . . . . . . . . . . . . . . .
57
12. Topology;
64
separable metric
13. Applications
spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .
of the continuity principles and the fan t h e o r e m v 71
14. W e l l - o r d e r i n g s
and ordlnals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15. Species revisited;
the role of the comprehension principle
16. Brouwer's theory of the creative subject
76
... 91
.....................
95
17. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
-
2
-
w 1. Introduction I.i This paper is intended as an introduction to the principles
of intultlonism.
notions,
results
not formal systems are emphasized;
proof theoretic
to illustrate relative power and formal consequences words,
as representing principles
fragments
of intuitionistic
in mathematics
mathematics.
important
Dialectica
and functionals
as illustrations intuitlonistlc
mathematics
from mathematics
only
(e.g. G6del's
completeness
problems of
introduction
As regards presentation
was determined by their usefulness
thus many well-developed
are not touched upon
For information
subjects of
(e.g. measure theory,
algebra,
pro-
regarding these subjects the reader is referred
~H 1966~.
of the subjects,
wish to mention especially (1966-1967)
it is not exhaustive;
of higher type,
and by personal preference;
geometry).
to Heyting's
of various
logic).
The selection of the examples
jective
The application
subjects are summarily treated or mentioned
interpretation
intuitionistic
In other
certain formal systems
is illustrated by suitable examples.
Although the paper has more or less a survey character, various
are mentioned
of various principles.
this paper presents the material needed to recognize
Basic
there is indebtedness
Professor Kreisel's
and Professor Heytlng's
lectures
lectures
to many sources.
I
on intuitionlsm ~n Stanford
on intuitionism
in Amsterdam
(1960-6i).
1.2 The subject of intuitionism might be described, mathematical
thought".
This succinct structive"?
description
immediately
raises a problem:
what do we mean by "con-
I might try to answer this question by a mathematical
would select
from the whole universe of possible mathematical
structive ones.
However,
there are two disadvantages
were to succeed in giving such a definition this would not lead to an autonomous pendent
at a first try, as "constructive
left with the
(if you want,
extra-mathematlcal,
whether there exists a basic intuitive
concept
satisfactory
even if I way. Firstly,
of constructive mathematics,
of any reference to "all possible mathematics";
which
arguments the con-
to this procedure,
in a formally
development
definition,
inde-
and secondly , we would be
but not irrelevant)
question of
of constructive mathematics
corre-
sponding to the formal definition.
On the contrary, tion between
I want to start from the idea that there
constructive
is a legitimate distinc-
and non-constructlve mathematical
in some cases we feel no doubt about
some argument
thought.
In other words:
being constructive,
whereas
in
-3-
other
cases we feel clearly that the argument
as presented
to us does not have the
form of a construction. Let us consider of a very
a simple example.
simple
kind.
If I express
then classically
we accept
m = 0
as d e t e r m i n i n g that
if
a natural
F
as well
point
out some of these
holds,
number
may be regarded
last t h e o r e m
number
various
like
a much
(i)
point
of view we cannot
some idealizations
concept
of a construction
them.
Every natural
It can be worthwhile,
of intuitionistic
of reality
(you
as we proceed.
clearer
between
F.
at all. We shall
3, than we have of a construction
so to speak.
parts
otherwise
in order to get anywhere
simplifications
we do not make any distinction to the same degree,
m = I
we shall admit
although we have
a small natural
integers
like
m. From a c o n s t r u c t i v e
say simplifications)
as constructions
by
m, since we have no way of deciding
In dealing with our subject, might
between
Fermat's
a statement
(i) determines
For example:
numbers
For all integers n > 2, and all positive x, y, z : x n + yn ~ z n
F
assert
Natural
mathematics
abstraction
and idealization
involved.
Two remarks
have to be added
in order to delimit
number
relative
the
representing
however,
I shall not attempt
say 999 ,
is constructive
to make
a distinction
to the degree
to do this
subject
representing
of
systematically.
of i n t u i t i o n i s m
more
accurately. First.
We may start with very
numbers,
and then gradually
simple
build up more
or "visualizable"
structures.
([Kr 1965],
In intuitionism,
3.4).
an intuitive structions
concept
which are
implicitely
principles
Finitist
constructions
the simplest outside.
Finitism
constructions,
complicated,
is concerned
with
by r e f l e c t i o n
involved
such as the natural
but n e v e r t h e l e s s
"concrete"
such constructions
we also want to exploit
of "constructive",
discover
represents,
concrete
the idea that
on the properties
in the concept".
only
there
is
of "con-
(I.e. we attempt
to
by introspection.) build up "from below";
so to speak,
an approach
(and in its kind rather
reflecting
on the general
notion
"from the outside",
"from above".
elementary)
of the approach
example
Logic
presents
from the
-
The approach (arbitrary)
"from the outside" constructions
4
-
leads us to considering
applied to constructions
intuitionism might be termed "abstract"
From the preceding remarks
constructions
applied to
... etc. For these reasons,
in constrast
to finitism as being "concrete".
it will be clear that we certainly cannot expect the
conceptual basis of intultionism
to be simpler than the conceptual basis of classical
mathematics. Second.
The
(mental)
constructions
we consider,
(idealized)
mathematician.
mind of an individual attempt
(necessarily nearly always
Talking about intuitlonistic logous mental constructions
inadequate)
mathematics makes
in the
The language of mathematics
to describe these mental
is therefore
to other people.
cesses of various human individuals
are thought of as to exist
is an
constructions.
a matter of suggesting ana-
Similarity
between the thought pro-
such communication
possible.
This mathematician,
who occupies
idealized
creature;
his ideas are supposed to be clear and distinct, not hazy and
confused,
as ours often are.
For the mathematicians
of real life, mathematical
keeping track of their thought essential
the formalist.
Finally,
but language
is not to be found in formal systems,
formalization
mathematics,
to suggest
help in is not
as it is for
is a very important tool in our research
for checking the principles
our constructions
formalization
more precise. separated
is an important
and in sorting out confusion;
mathematics
Nevertheless,
of intuitlonlstic
functions
language
is an
to the idealized mathematician.
For an intuitionist,
theorem,
himself with constructive mathematics,
to others,
used in the proof of a
and as a shorthand.
helps us in making our intuitive,
But if we want to study intultionism,
By these
languageless
insights
formal systems are not to be
from their interpretation. a word of warning.
Intultionlsm
is a complicated
and often tricky subject.
Many topics have not yet been investigated
sufficiently,
in a far from finished state. Accordingly,
some considerations
are in need of correction
or a more accurate
But the paper will have served its purpose program of Intultlonism.
therefore
the subject
is
in these lectures
formulation or a further development.
if the reader has got an idea of the
-5-
w 2. Logic 2.1 Logic represents
an example
general principles
about constructions
Logic is elementary the structure
of the approach
in
~e
The "elementary"
and constructive
reflection
on
proofs.
sense that it does not make use of deep insights about
of constructive
mentary as regards
"from the outside":
proofs;
interpretation,
on the other hand,
it is also very non-ele-
and in a sense, highly impredicatlve.
character of logic is illustrated by the fact that e.g. intultio-
nistic propositional
logic is capable of so many different
means that the insights about constructions are not specific,
used in the interpretation
since so many other interpretations
Let me first present
a rough description
interpretations
(which
of logic
are possible).
of the meaning of the logical constants,
which is all you need to understand the other sections.
A proof of A v B
is given by presenting a proof of
A
or a proof of
B.
A proof of A ^ B is given by presenting a proof of A and a proof of B. A proof of A r B A
into a proof of
-~A
is given by a construction which transforms B, together with a proof of this fact.
is proved by giving a proof of something
Vx Ax
is proved by exhibiting a construction
Ax Ax
is proved by giving a construction
any specific
any given proof of
llke
A + I = O.
c and a proof
d
which proves Ac.
scheme which yields a proof of Ax o for
x o , together with a proof of this fact.
In general,
the quantiflers
are supposed to range over a domain which has been "grasped as a whole" previously. 2.2 We shall try to make these explanations functions"
([Kr 1965]).
is only a rough sketch;
"meaning
The following introduction to the theory of constructions see subsection
We distinguish between constructions structions
more precise in terms of Kreisel's
2.3.
and general notions,
are the objects os our mathematical
reseaEch;
in short:
notions.
Con-
proofs are considered to
be constructions. Notions
are decidable properties
constructions,
of constructions.
this would lead us immediately
We cannot
into paradoxes
identify notions with of denotation and self-
-6-
reference
(for an example
see
[Kr 1965],
2.i52).
r61e of classes r e l a t i v e to sets in NBG-set is the i m p r e d i c a t l v e is not
One m a y compare this with the
theory.
Proofs may use notions;
c h a r a c t e r of our explanations.
"given" or "a priori bounded",
The domain of all c o n s t r u c t i o n s
does not "exist"
are p r o p e r t i e s w h i c h extend " a u t o m a t i c a l l y "
herein
as an entity;
and n o t i o n s
w i t h every e x t e n s i o n of the domain of
constructions.
In this section, d,
...
we use lower case letters
for c o n s t r u c t i o n s ,
lower case letters interpreted
a, 8,
x, y, z,
ac = 0
When a, b
also a construction. pair,
c I = a, c 2 = b
{O,I)
then
a w,
, b, b'
.."
(characteristic
c
c
function)
has the p r o p e r t y
,
c,
*'',
a notion
cl, c 2
by:
Greek
is
e.
ci = c2 = c
if
c
a,b is is not a
of a schema a p p l i c a b l e to other c o n s t r u c t i o n s .
Our first
is i n c l u d e d in the f o l l o w i n g assumption:
is a p p l i c a b l e to
Hence we may suppose an a p p l i Q a t l o n
yields
.*',
such that
b "
is a n o t i o n
(i.e. a decidable prQperty).
= a
m
the pair of c o n s t r u c t i o n s
=
,
9
if c = < a,b >.
" a
a(b)
c'
for c o n s t r u c t i o n s .
are used for notions;
C o n v e r s e l y we define
A c o n s t r u c t i o n may consist idealization
"'"
iff
are c o n s t r u c t i o n s ,
,
... as dummy v a r i a b l e s
"''' w' ~A'
as a function into
a, a'
say,
if
a
operator
is not a p p l i c a b l e to
.(.) b, and
(i)
to be defined
by s t i p u l a t i n g
a(b)
a
= c
if
applied to
c.
A n o t h e r basic a s s u m p t i o n
(and idealization)
" c
is a p r o o f of
is given by the a s s u m p t i o n
A "
any g i v e n a s s e r t i o n
is a n o t i o n
for
A.
(2)
More general
" c
is a p r o o f of A(ci,... , c n)
a notion
The n o t i o n a s s o c i a t e d w i t h
A
~A c = 0
" is
for any given p r e d i c a t e A(xi,...,
a c c o r d i n g to
iff
c
Xn).
(2) will be i n d i c a t e d by ~A:
is a p r o o f of
A.
(2')
b
-7-
Likewise
for
A(xl,... , x n)
~A(C; If we keep to our purely if we are in doubt prove
A
cl,...,
subjective
whether
c n) : 0 point
iff
c
of view,
a construction
c
is a proof of A(cl,...,
(2) and
proves
Cn).
(2') are rather natural:
A, then apparently
c
does not
for us!
Compare
this with the provability
whether
a certain
ordered
Logical
operators
are operations
functions
(of compound
predicate
sequence
in formal
of formulae
it is decidable
is a proof of a given
which t r a n s f o r m
assertions).
systems:
The simplest
meaning
functions
cases are
v
formula
or not.
into new m e a n i n g
and
~
Con~ unction :
WA^B(C) or more
: 0
iff
simply,
c : < cl,c 2 >
since
WA
cl, c 2
A
B(C)
and
~A(Cl)
are always
: 0
iff
: 0
and
~B(C2)
: 0
and
~B(C2)
= O.
defined:
~A(Cl)
: 0
Disjunction:
~AvB(C)
: ~A(C).
~B(C);
In the definition
i.e.
of ~A^B'
~AvB(C)
~AvB
is quite elementary,
applied
For the c o n s t r u c t i o n
of ~A§
: 0
iff
~A(C)
we meet the words
to decidable
: 0
"and" and "or",
In other words, fore a notion
for any notion
we are able to recognize
schema
such that
c
applied
but their use
principle:
" c is a proof with a free variable
applicable
: O.
relations.
we need a new basic
is a notion
or ~B(C)
d
of
= 0 "
(3)
if a c o n s t r u c t i o n to
~(d)
m.
d
proves
e(d)
c
is a universally
= 0. We introduce
there-
~:
(c, kd. ed = O)
: 0
iff
c
is a proof with
free variable
d
of
~d : 0.
-8-
Now we state Implication: WA§
= 0
iff
c = ~ ci,c 2 ~
and
~(cl, ~d. (IA ~Ad)~BC2d = O) = O. (~ denotes ordinary cut-off subtraction.) x ~y
(classical
implication)
will be used ~or
x =O§
= O.
Negatlqn: ~ A(C) = 0
iff
Universal Let
(I A ~A d) = O) = O.
quantification:
A(Xo,...,
~Ax0A(C;
~(c, ~d.
x k)
ci,...,
~(dl(Cl,..., Existential
be a predicate.
c k) = 0
iff
c = ( dl,d 2 ~
Ck), ~d. WA(d2(d,Cl,...,
and
Ck); d,cl,...,
quantification:
~VXoA(C;Cl,...,c k) = 0
The restriction
iff
c = ~ dl,d 2~
and
of logic to §
WA(di;d2,Cl,...,Ck)
on meaning functions
Logical validity
for the Intuitionistic
are conceivable,
propositional
F(PI, .... ,Pk ) is intuitionistically A~pl... A ~ P k V C
~F(C) = O.
are propositional
variables.)
Let F(RI,...,Rk) RI,...,Rk,
Ri
F
Pl
PI,...,Pk
is said to be valid if
As examples,
are taken to be predicate
in
calculus might be expressed as
valid Iff
logic with predicate
arguments;
suppose
Ax o ... AXnF
The validity of closed formulae of the predicate if
which are not definable
(4)
denote a formula of predicate
a predicate with
Xo,...,x n. Then
= O.
is more or less determined by tradition;
other operations terms of §
(PI,...,Pk
c k) = O) = O.
calculus
variables.
we discuss two logical theorems.
F
variables
contains dun~uy variables is valid. is also defined by (4),
-9-
2.2.1. Theorem.
A §
We have to show Proof.
Is intuitlonlstlcally
A~AVa(~A§
= 0).
Let
: 0
iff
(5)
(hypothesis).
~A(b) = 0 ~(c)
valid.
Hence by our hypothesis
~(c,kd.WAd = 1).
(6)
~.IA(C) = 1. The p r e c e d i n g argument represents a proof a'(b) with free variable w
A(a'(b))
C
of (6), i.e.
(7)
: O.
Therefore ~A(b) = O ~ A ( a ' ( b ) ) (5) - (8) represent a proof
2.2.2. Theorem.
c'
= 0
(ellmination hypothesis).
with free variable
~A§
< c', a' > : O.
~Vx
Ax §
b
is i n b u l t l o n l s t l c a l l y
Proof. We look for a construction
a
%~VxA§
(a) = O.
~
= 0
(8)
of (8). Therefore
valid.
such that
(9)
Let VxA(b)
(IO)
(hypothesis)
then ~(b,Ac.(l ~ ~ x A x ( C ) )
(II)
= O) = 0
By (11)
(12)
~ x A < d,d' > = i for arbitrary
d,d',
so
~A(d;d') (I0) - (13) is a proof
(13)
= i. a'< b,d'
>
with free variable
d
of (13) under hypothesis
(10), hence
~A(a'; (IO) - (14) represents a proof
(14)
d') = O. b'(b) with free variable
d'
of (14), hence if
- 10
we take
c'(b)
: < b'(b), a"(b)
~Ax~A(C'(b)) (10) - (15) is a proof ~nVxA(b)
-
> and a"(b)(d')
= a' < b,d'
> for all
(15)
: O.
b"
d' , we have
with free variable
: 0 :)~x,A(C'(b))
b
: 0
of (elimination hyp.).
(16)
Hence
=~VxA§
< b",c'
Rules are treated as follows.
If we want to show say
F,
G
F',
G' ~ H '
then we must show that to every c
such that
~F,^G,§
(17)
> = O.
~H
b
such that
WF^G§
: 0
we can find a
O.
2.3 Intentionally I refrained from a systematic development of the theory of constructions; the subject is far from being in a finished state. The p r e c e d i n g rough sketch of a general theory of constructions
in w h i c h we can e x p l a i n the m e a n i n g
of the logical constants is in need of refinement. the theory of constructions, [Gn 1968],
For a detailed development of
including intuitionistlc
arithmetic,
we refer to
[Gn].
The main refinements
in Goodman's development
the sketch given above are I ~ and most basic point of all: and likewise quantification
rules,
in [Gn 1968],
[Gn] as compared with
functions need not to be defined everywhere;
2 ~ ) proofs must be about objects alread Z cgnstructed, is supposed to be over domains which are already "under-
stood", had been "grasped" before. This automatically
introduces levels; the lowest
level consists of constructive rules operating on each other. For any given level
L
the proof-predicate:
d
proves cx = 0
for all
x ~ L
is
decidable and belongs to the next higher level. Each level is universally decidable in the sense of
5.3. The general idea might be summarized by saying:
"there is an
a priori bound on the possible proofs for any definite assertion". 2.4 Tec~ical
results about intuitionistic
logic are w e l l - ~ o w n
and easily available
-
in the literature.
For a Hilbert-type
11
-
system for intuitionistic
[K 1952], Ch. IV. A system with Gentzen-type rules, be proved is to be found in [K i952],
logic see e.g.
for which cut-elimination
can
w 77.
For future reference we mention a convenient
system essentially due to Spector
([sp 1962] >. Propositional and
axioms and rules:
P § Q, then Q;
(IV)
Q ^ P + P; (VI) P + P v Q (VIII) (X)
If R § P
If
Here~
P § (Q§
If and
and R § Q then
(I)
P § Q
P § P;
(VII)
then R § P ^Q;
stands for a contradiction,
If Q, then P § Q;
and Q § R, then
Q § PvQ;
(PAQ) § R;
(II)
(XI)
(III)
P + R; (V) P ~ @
§ P
If and
If P + R and Q § R, then P ~ Q
(IX)
9 ~
If
PAQ
P + R;
+ R, then P + (Q§
P.
"falsehood".
In accordance with our general
interpretation of the logical constants
P may be interpreted as P +-~. If we want
to avoid a separate symbol for falsehood,
we must replace
(XI)'
(AA-~A) § B; ( X I ) " ~ B
+ (B § ( A ^ ~ A ) ) ;
(XI) e.g. by:
(XI)"'(B § (A^-~A)) +~B.
Axioms and rules for quantifiers: Let in
x
be a variable not occurring free in
Q, and let
t
be a term free for
x
Px. Then
(XII)
If Q ~ Px, then
Q +AxPx;
(XIV)
A x P x + Pt;
Pt §
(XV)
(XIII)
If
Px + Q, then
VxPx
+ Q;
Axioms and rules for equality as usual. Intuitionistic p r o p o s i t i o n a l
logic is decidable
([K 1952],
predicate logic the interpolation theorem is provable completeness
w 80); for intuitionistic
(see IS 1962],
of intuitionistic predicate logic for the intuitionistic
is discussed in [Kr 1958],
[Kr 1958A],
for intuitionistic p r o p o s i t i o n a l
[N 1966]). The interpretation
[Kr i962]. We remark here that completeness
logic is proved fairly easily C[Kr 1958]).
-
w 3. Elementary
1 2 -
arithmetic
3.1 Arithmetic may be regarded as a piece of intuitionistic extent may be approached
"from below" 9
mathematics
which to a large
starting from very simple elementary
con-
struc tions. Natural numbers
are conceived as constructions
Juxtaposing units. The basis of this concept a unit, then another unit, this process
II
These very simple constructions of a certain natural number, the number itself,
In a picture
III
because
IllI,
obtained by
is the observation that we can conceive
look upon this two-ity
as often as we like. I
of a very simple kind,
(pair) as a new entity 9 and repeat
simply
....
are so to speak their own proof:
for the concept
the proof that it is a natural number is given by
its mode of generation
it has been obtained by this process
of generation
is at the same time "proof" that of natural numbers.
+ The notion of a successor
x
of a number
x
is clear 9 and also the properties
O~x+ +
§
x=y ~ ~ x Let
Q
= y
(I).
be any property of natural numbers.
variables
The induction principle
states
(x, y, z
for natural numbers) QO A ~x(Qx § Qx +)
The Justification
+
(2).
A X Qx
of this principle may be given as follows.
Take any natural number
y9 and let
Now we construct proofs of
QO,..
.9
QO, Ax Qy
(Qx § Qx+).
parallel to the generation
of O, O + 9 0 ++
9
QO9 QO,
QO ~ QI
~
QI,
QI9
QI ~ Q2
~
Q29
+ Qy ~
Qy.
,ee
Q(y-l) 9
Thus 9 by parallelling
Q(y-l)
the construction
of natural numbers,
We have been rather explicit about this point 9 in a much more complicated
situation later on.
The existence of primitive
recursive
construction
of function values.
functions
we prove
since an analogous
Ay Qy.
argument
occurs
is also seen by a "step by step"
-
13
-
3.2 A formal system for intuitlonlstlc
first order arithmetic may be found e.g. in [K 1952~, Ch. IV or in [Sp 1962~. The new axioms are the axioms for + (formulas (1)), recursion equations for
+, 9 ,
and induction
(2) with respect to
mulated in the language of first order predicate logic with equality and
Q +
for, +,~
For future reference, we state the following important result. 3.2.1. Theorem (Gddel; see [K 1952], w 81). L e t ~
denote intuitionistic proposi-
tlonal logic, predicate logic or first order arithmetic;
let
denote the corre-
sponding classical system. We define a translation as follows: (a)
P-
=,~P
(for prime formulae
(b)
(P§
:
P-§
(c)
(PaQ)-
:
P-AQ-
(d)
(-~P)-
:
"-'lP-
(e)
(PvQ)-
=
"~(~P-I~'-~Q-)
(f)
(AxPx)- =
AxP-x
(g)
(WxPx)- :
"-IAx-"~P-x.
(Instead of (b) we may also take Then
P~+
iff
5.2.2. Remark.
(P§
=
P)
"~(P~Q-)).
P-~ ~ ~
In the case of arithmetic
we can take
since the prime formulae of arithmetic are decidable.
P- = P
for prime formulae,
(In the formal system this
has to be proved by induction from 0 ~ x +, since we do not have the excluded third.) The theorem yields a consistency proof of classical arithmetic relative to intuitlonlstic
arithmetic. Moreover,
it shows t h a t ~ i s
not poorer than ~ *
3.2.5.Notatlon. We use the functions sg, :, etc. as defined in [K 1952]. N denotes the collection of natural numbers.
-
14-
w 4. Species 4.1 In this
section we introduce
intuitionistic
analogues
the notion
perties
which are in turn c o n s i d e r e d
Suppose
that we have any well defined
(such that the collection
Then w e l l - d e f i n e d (~ntuitionistic species
Px
an element
member
if
ly of)
of the theory
interpretation
~A
x
A
objects
species
of m a t h e m a t i c a l
principle~.
entities e.g.
A = N).
to be species
The extent
of the notion
of "well-deflned".
implies
as
are pro-
(entities).
to be an object,
are considered
P
may be regarded
speaking,
But
of
it is
that we know what
con-
xEA. x
of
A
can only be admitted
has been or might
have been defined
as an element before
or
(independent-
P.
In terms
selves
of
for a property
for an
It will be clear that (xEP)
A
by the interpretation
"well-defined"
a proof of
P
collection
of elements
determined
Species
Roughly
to be m a t h e m a t i c a l
form of the c o m p r e h e n s i o n
clear anyway that
of
sets.
itself may be c o n s i d e r e d
properties
is therefore
stitutes
of a species.
of the classical
of constructions
of the logical
are constructions.
is a species
(as used in w 2 to explain the intuitionistic
constants)
In other words,
if there
exists
species
correspond
a predicate
a construction
c
A
to notions
which them-
with a m e a n i n g
function
such that
~A(a;b) : 0 ~ ~ c < a,b > : 0 for all constructions a,b.
Essentially
impredicative
tifications
over all subspecies
universal
quantification
"subset") species)
there
of the c o m p r e h e n s i o n
of the basic
collection
over all subspecies
of
A
principle
A; and if we admit
("subspecies"
seems to be no reason not to accept ~
involve
e.g.
is defined
(A) = {X : XCA
quan-
like
} (the power-
as a species.
Typically
predicative
fications
over elements
The theory
of species
and Brouwer's this
applications
comments
stage mainly
applications
is a rather
collection
underdeveloped
are scarce. I introduce
for the purpose
in the sequel.
The discussion
we shall
open the question
leave
of the c o m p r e h e n s i o n
of the basic
principle
domain
of intuitionistic notion
formulation
is resumed
as to the extent
involve
quanti-
only.
the general
of a convenient
of the notion
A
mathematics,
of a species
of certain
at
results
in w 15. For the time being,
to which properties
are admitted
-
as species.
We remark,
all essentially we accept
e.g.
however,
predicative
essential
applications
are species.
Notable
are
principle
or a lawlike K
one might
objection
of the natural
to the a s s u m p t i o n
number
exceptions where
little
of the c o m p r e h e n s i o n
predicates
relative
of being a natural
any use of "species"
seems
of the comprehension
and predicative
llke those
-
that there
all arithmetical
Most of the applications
15
as well
principle;
numbers
that
WO
accepting therefore
as d e f i n i n g
in the sequel
species.
are non-
some basic properties
funtion
in w 9 and
Just
against
from
N
into
N
(see w 5)
in w 14. It is clear that
say "unary predicate"
is non-
essential.
4. 2o In classical
set theory,
sets are determined
this approach we may define
equality
x : Y 4--~Ax(xEX However,
it Is n e c e s s a r y
distinction
between
Our creation structions, structions indicates
of m a t h e m a t i c a l like natural
such entities properties an object
x,y
of natural
of type n + I
is itself an object
I. Likewise are species
we may
which defines of type zero.
Definitions.
is detachable
4.2.2.
in
Definition.
hess relation
of type zero;
of type
A species
X
~ y~-~x
(b)
x # y
(c)
x ~y--..x
n + I
such conx ~ y
Let us call zero,
the d e f i n i t i o n
might
e.g. of such
of the property",
be termed
... etc.;
an object
objects
with extensional
of type equality.
may itself be looked upon
is said to be inhabited
X, i f # ~ x ~ X ( x G Y
or secured
if V x(xEX).
vx 4 Y).
relation
X, if for all
(a) " ~ x
n, in relation of order
of order
"extension
2, 3,
between
"thought".
species;
equality,
given con-
is decidable;
of objects
but the
of type
immediately
of mappings;
(say ~) which
serve to introduce
a species
A binary
on
wlth concrete,
Properties
form objects
of objects
as an object
Y
equality
as related to extensional
The property
4.2.1.
starts
very relevant
equality.
or descriptions
of order zero. numbers,
with
by:
here the i n t u i t i o n i s t i c a l l y
objects
numbers,
X, Y
are given to us as the same object,
objects
i.e. the property
species
and extensional
we have a d e f i n i t i o n a l that
between
In agreement
@--~x E Y).
to stress
intensional
by their elements.
#
x,y,zg X
: y
~y ~ x
~ zvz §
on a species
X
is said to be an apart-
-
16
-
4.2.3. Remark. Equality on a species with an apartness relation is stable, i.e. --l--Ix : y § x = y, since x = y~-~-~x 4.2.4.
#y
and - I - I ~ 1 x
X x Y, X 2, X 3
X~Y,
~ y~---~-Ix # y.
X~Y,
X - Y, X ~ Y
are deflned in the usual way.
w 5. Sequences and constructive (lawlike) objects 5.1 5.1.1. Definition. A mapping
9, from a species
of process which assigns to any
x~X
an
X
y~Y,
(This stipulation is necessary whenever =
~(X)Y
X
into
Y
Y
is any kind
x= x' § ~x = ~x'.
does not denote basic definitional
equality, e.g. in the case where the elements of mapping ~ from
into a species and such that X
is said to be of type
are themselves species.) A (X)Y
(also ~ ( X ) Y
). A mapping
is said to be bi-unique (an injection) if
AxgxAx'eX
(Wx =~x' § x = x').
is said to be weakl~ bi-unlque (weak injection) if
Ax ~xAx'~x Remarks.
In case the equality in
injection
i s an i n j e c t i o n .
(~[X])X
(~[X]
~(X)Y
Ax xAx,
~Fx').
is stable, i . e . - T ~ y
~(X)Y
: y' § y : Y', a weak
possesses
an i n v e r s e
~-16
: {~x: x g X } ) . be species with apartness relations
#, #'
respect-
is said to be strongly bi-unique (a strong injection) if
x (x # x,
5.1.3. Remark. In case bi-Jection,
Y
An i n j e c t i o n
5.1.2. Definition. Let X, Y ively.
(x ~ x' ~ x
~(X)Y,
#, X
is a species with apartness relation
IX] = Y, then an apartness relation
~'
~, and ~ is a on
Y
is defined by
x ~' y -D ~ - I X ~ - I Y " 5.2 Integers and rationals are constructed from natural numbers in the same way as in classical mathematics. But in order to develop a theory of real numbers, we have to introduce the notion of a sequence. In general, a sequence is a mapping of type
(N)X, i.e~ a process which associates
with every natural number a mathematical object belonging to a certain species
X.
-
16
-
4.2.3. Remark. Equality on a species with an apartness relation is stable, i.e. --l--Ix : y § x = y, since x = y~-~-~x 4.2.4.
#y
and - I - I ~ 1 x
X x Y, X 2, X 3
X~Y,
~ y~---~-Ix # y.
X~Y,
X - Y, X ~ Y
are deflned in the usual way.
w 5. Sequences and constructive (lawlike) objects 5.1 5.1.1. Definition. A mapping
9, from a species
of process which assigns to any
x~X
an
X
y~Y,
(This stipulation is necessary whenever =
~(X)Y
X
into
Y
Y
is any kind
x= x' § ~x = ~x'.
does not denote basic definitional
equality, e.g. in the case where the elements of mapping ~ from
into a species and such that X
is said to be of type
are themselves species.) A (X)Y
(also ~ ( X ) Y
). A mapping
is said to be bi-unique (an injection) if
AxgxAx'eX
(Wx =~x' § x = x').
is said to be weakl~ bi-unlque (weak injection) if
Ax ~xAx'~x Remarks.
In case the equality in
injection
i s an i n j e c t i o n .
(~[X])X
(~[X]
~(X)Y
Ax xAx,
~Fx').
is stable, i . e . - T ~ y
~(X)Y
: y' § y : Y', a weak
possesses
an i n v e r s e
~-16
: {~x: x g X } ) . be species with apartness relations
#, #'
respect-
is said to be strongly bi-unique (a strong injection) if
x (x # x,
5.1.3. Remark. In case bi-Jection,
Y
An i n j e c t i o n
5.1.2. Definition. Let X, Y ively.
(x ~ x' ~ x
~(X)Y,
#, X
is a species with apartness relation
IX] = Y, then an apartness relation
~'
~, and ~ is a on
Y
is defined by
x ~' y -D ~ - I X ~ - I Y " 5.2 Integers and rationals are constructed from natural numbers in the same way as in classical mathematics. But in order to develop a theory of real numbers, we have to introduce the notion of a sequence. In general, a sequence is a mapping of type
(N)X, i.e~ a process which associates
with every natural number a mathematical object belonging to a certain species
X.
-
We shall use sequence example
x, •
leaves
x",
open various
which
(algorithm)
possibilities
is completely
which
As typical
A lawlike
sequence
for further
or lawlike
examples
applied
an element
which might
by a law, hEN
l.e.
the
~iven to us by an a l g o r i t h m
to any natural
of a species
X.
number produces
to be d e f i n i t l o n a l ~
equal
equality
imply
intensional
(x ~ •
for lawllke
for lawllke
sequences
context
otherwise).
We use
for the finite
x(x
- I);
~o
A hypothesis thesis"
relative
of
to lawllke
(N)N
sequences
Many
recursive
is recurslve,
formal
function
systems
or in Kleene's
to be consistent Another
(fairly
of a lawless values,
(I).
simple)
sequence
when
equal)
(N)N
equality
•
x'.
(unless the
sequence
•
...,
(of
almost
(N)N).
is known about
future
but what
future
sometimes
thesis"
called
states that
"Church's
every
lawllke
symbolism
(I)
n).
Compare
[Kr 1965~
of intuitionistlc
the opposite
Such a sequence
2.72.
mathematics
turn out
is conceived
sequence
at any moment,
sequence,
Is that
as a source
of values
•
•
of ...,
•
finitely many
casts are known,
can be said. sequences
as objects
to ensure that to every
one may create a lawless
of a lawlike
values.
of a die;
casts nothing
is needed
(N)N,
See also w 16.3.
One may also think of lawless
(E.g.
of
(n}x)
for fragments
notion,
thls wlth the casts
but about
anything
=
such that at any stage we know a finite
while n o t h i n g Compare
with
(ax
with GSdel number
proposed
of
form of Church's
AaVnAx ({n}
or more
: 0.
or "the intultlonlstlc
function
~x
number,
extensional
a, b, c, d
indicates
with a proof
applicability.)
From now on we shall use clearly
functions.
(intensionally
~x(xx
as
of the
that an algorithm
It wlll be clear that
does not
member
together
if they are given to us in the same way. = •
be described
a natural
(It may be m a i n t a i n e d
The simplest
recurslve
the proof of its own universal
are considered
idea of a
a prescription
n th
one may think of say primitive
in full also contains
sequences
The general
specialization.
sequence,
fixed in advance
is therefore
that the a l g o r i t h m described
In general.
tells us how to find for any
sequence.
generally,
-
for sequences
Is that of a constructive
a sequence
Lawlike
...
17
sequence
by t h i n k i n g
abtained x
by "abstracting"
a value
of say
•
~x.O ~ a
from
can be found. as a process
-
which generates
O, O, O,
18-
...; in applying at any stage any operation to this
sequence we do not use the law, only the initial segment at that stage.
O, O, O,
...
regarded as a lawless
O, O, ..., 0
sequence
stage we conceive all possible values at further arguments to lawless s e q u e n c ~ i n
Lawlike
sequences
complete objects,
obtained
is such that at any as possible.)
We return
w 9.
and natural numbers provide us with the simplest i.e. mathematical
objects which at a certain
examples
of
stage are completely
described.
Lawless sequences
are the simplest
are being generated;
The motivation
at no stage the process
for studying notions
by a law is abandoned, on sequences assumption
examples of "incomplete" is thought
objects,
objects which
of as being finished.
of sequence where the idea of determination
is provided by the circumstance that most of the operations
of rational numbers that play a r61e in analysis,
do not depend on the
of the sequences to be lawllke.
It will turn out, however, own account,
that lawless
sequences,
and useful in the discussion
well suited to the development
For a more satisfactory
although interesting
of intuitlonistic
of a theory of real numbers
logic,
are not very
and real-valued
theory of the continuum and the real-valued
need a more complex notion,
intermediate
of choice sequence will be discussed
on their
functions we
between lawless and lawlike;
this notion
in w iO.
We may generalize the notion of a lawlike sequence to lawlike operations ~(X)Y
is called lawlike if
For those readers
of "lawllke
~ ~(X)Y;
can be given completely by a description.
acquainted with the theory of creative
that the description to the creative
~
functions.
subject,
sequence" does not necessarily
I remark here
exclude reference
subject.
5.3 Let
X
be a (universally)
(By "universally
decidable
decidable"
species with an intenslonal
decided whether this construction
represents
an element
of
X
formulate the following A
"form of the axiom of choice"
is an (extensional)
predicate,
then
it can be
or not.
we shall suppose the species of natural numbers to be decidable.) When
equality relation.
we mean that for any given construction
In particular,
Then we may
or "selection principle":
-
Ax~X
~y~Y
A(x,y) §
19
-
V~@(X)Y
Ax
A(x,
~x)
(2).
For if we have a proof of Ax~X~y~Y
A(x,y)
then thls proof must contain a complete description occurring in When
A
(possibly involving parameters
A) of an operation which assigns to every
does not contain non-lawlike variables,
x
then
a
y~Y.
r may be supposed to be
lawllke. If
X
is a species wlth an equality relation
species
X'
of "definitions of elements of
the elements of element of
:, we may associate wlth X" (or In other words,
X, but with definitional equality
X"
X
a
consisting of
~). Such a "definition of an
may be thought of as a construction together wlth a proof that this
construction represents an element of
X. In general agreement wlth the principles
described in section 2, if we suppose the equality of proofs and constructions to be decidable, proof of
it may be assumed that
A x g X ~aygy A(x,y), V@~(X')YAx~X'
X'
Is universally decidable.
the only conclusion that is immediately evident,
A(~,
Ax~X~ygX' =
is decidable, Axe•
for we have
X
is such that there exists
a
a WE(X')X'
such that
and (2) may be asserted,
since it follows that
A(~,~x),
A('~,
determines
always can be extended
(x = y § wx ~ w y A wx = x)
CWx), and also
A ( x , y ) ~ x = x' § A(x',y), hence ~W
(X')Y
(X)Y.
In case the equality on
then
is
cx) (where ~ : {y : y ~ X ' A y = x}),
but there Is no reason to suppose that a m a p p i n g of to a mapping of
If we have a
~' ~ ( X ) Y
by
w-~ = ~; but
A
is extensional,
i.e.
A(~,r ~'~ = ~vx, hence
V~x~X
A(x,~'x).
Important special cases of (2) may be formulated as follows
(A
lawllke variables,
(N2)N):
{ } denoting a pairing function of type
B(n,•
...) + V x ' A n
B(n,km.•
AnVm
B(n,m,•
...) § V• A n
B(n,xn,•
AnVa
A(n,a)
+ VD
A n A(n,~m.b{n,m})
(5)
AnVm
A(n,m)
§ Va
An
(6).
...)
"'')
(3)
AnFx
A(n,an)
•
not containing non-
(4)
-
B
in
(3)
and
A
in
(5)
X = ~^B(n,•
are assumed
the proofs
•
of
to this point
i.e.
...)
§ A(n,b).
to a specific
AnVx
-
to be extensional,
...) § B(n,~,
a : b ^A(n,a) If we restrict
20
B, An~/m B
notion
always
of sequence,
provide
a
it remains
•
to be seen w h e t h e r
of the same kind.
We return
in w 10.
5.4 As will be clear like operator are usually
from the p r e c e d i n g
depends
on our notion
acceptable.
bar recursion
discussion,
More
[Ho 1968].
The principle
discussions
in sections
of constructive
controversial
of higher type.
the extent
in this
proof.
Definitions
respect
For formal work concerning
of bar r e c u r s i o n
of the notion
of lowest
of a law-
by recursion
is the discussion
this principle
of
see e.g.
type may be justified
by the
9, I0.
5.5 Let us return to the general native
explanations possibilities
For example,
interpretation
of the logical
of the logical
constants
we might
have looked upon the general
as a kind of " m e t a - e x p l a n a t i o n "
Take
a t h e o r e m of the general
tells
form
us that we have a m e t h o d which
an arbitrary
proof
of
the m a t h e m a t i c a l l y
described
A. The actual
precise
of using the
in w 2, there
outside
explanation
are alter-
mathematics.
A § B. Our m e t a - e x p l a n a t i o n
enables
us to construct
proof of the t h e o r e m
formulation;
of the logical
intuitionistic
of" ~-A
a proof of
"A + B" then
the precise m e a n i n g
case is "explained"
by the proof which
explicitely
Likewise,
law is interpreted
as a m e t a m a t h e m a t i c a l
a logical
Instead
which may be explored.
constants e.g.
constants.
presents
of
B
§ B" from
contains
~ in this
special
the required method. schema to be applied
in proofs.
In this manner,
logic
long way without Freudenthal,
using the explanation
in his paper
"Each theorem,
once
Intuitionistically liminary
is not properly
speaking,
in mathematics.
of the logical
has expressed
it has been correctly
orientation,
by the proof."
~F 193g]
included
a theorem
operations
this point
formulated,
(in the usual
a kind of summary,
whereas
Thus,
we may go a
in full generality.
of view as follows:
contains sense)
the t h e o r e m
its own proof.
is a short, proper
pre-
is given
-
GGdel's Dialectica interpretation
21
-
[G 1958~ is related to this approach.
If we take
a certain formal language suitable for describing a fragment of intultlonistic mathematics, operations
we are at liberty to eliminate some of the troublesome logical
(like implication)
in favor of defined notions which are conceptually
simpler and which "approximate" the operations which they replace. In G6del's paper, an essentially "logic free" interpretation of intuitionistic arithmetic is given, i.e. every arithmetical formula
A
the form
decidable.
that if
VsAt A'(x,t). A
s,t
lawlike operations,
A'
is a formula of intuitionlstlc arithmetic,
provable in a logic free theory of lawlike operations,
has an interpretation of then
It can be proved
VsAt A'(s,t)
i.e. we can construct
in the theory such that there is a free variable proof (computational) (w. r. t.
the free variable
is
of
s
A'(s,t)
t).
This result has been extended to analysis by C. Spector;
see [Ho 19681 for an
improved and smooth presentation. The essential steps in the interpretation of implication in GGdel's theory occurs when, after replacing
Vs4t A(s,t) §
Vs'At' B(s',t') by As(At A(s,t) § Vs'At' B(s',t')), ~sVs'(~t A(s,t) + A t '
this in turn is replaced by
B(s',t')),
and when, after replacing the previous formula by
AsVs'~t'(At
A(s,t) § B(s',t')),
we replace this by AsVs'At'Vt(A(s,t)
§ B(s',t')).
The final step yields V S ~ T A s A t ' ( A ( s , T s t ' )
§ B(S's,t')).
The resulting implication is essentially loglc-free.
This interpretation ef the
implication represents a strengthening relative to the original Intultlonlstic interpretation.
-
w 6. E l e m e n t a r y
22
-
theory of real numbers
6.1 In this
section we develop
a tiny part
We do not specify the notion think of all sequences
We shall introduce The methods
sequences
<Xn>n: i
are used for reals
theory
in mind;
lawllke.
to be definite,
or shortly
r, s
for lawless
section
sequences.
of rationals.
may also serve as a basis,
is technically
<Xn> n
in this section,
one may
in this
or Cauchy-sequences
or Dedekind-cuts
sequences
of real numbers.
The developments
and to a limited extent
by fundamental
intervals
via fundamental
We shall write
as being
real numbers
of nested
the approach
of sequence we have
involved
are also valid for choice
of the elementary
the simplest
for a sequence for rationals,
one.
xl, x2,
k, n, m,
but
i
....
x, y, z
for natural
numbers.
6.i.i.
Definition.
A sequence
of rationals
A k V n ~ m ( [ r n - rm+n[ Two real number
generators
n,
n
is a real number
generator
if
<2-k).
<Sn> n
are said to be equivalent
(notation
n ~ <Sn> n) if ~ k ~ n V m ( ] r n + m - Sn+ml Just as in the classical
theory,
equivalence
classes
generators,
the c o r r e s p o n d i n g
structive
one shows
~
are called real numbers. equivalence
to be an equivalence (If we think of lawlike
classes might be called
relation.
real number
lawlike
or con-
how to define
addition
and m u l t i p l i c a t i o n
for real number
generators.
Definition. n + <Sn>n :
< r n + Sn>n
n 9 <Sn> n = < r n ' S n > n One proves
easily
n exactly 6.1.3.
The
real numbers.)
It is evident 6.1.2.
<2-k).
n § n
as in the classical Definition.
+ <Sn>n ~ n + <Sn>n
theory.
Hence
etc.
for the reals we may stipulate
n ~ x a <Sn> n ~ y § n ~ x + y ~ < r n . Sn~ n ~ x . y .
-
Wlth the d e f i n i t i o n provides
of
x -I
we meet
us wlth the occasion
23
-
difficulties
in connection
to expand on the subject
with
zero; thls
of intultionlstic
counter-
"intuitionistlc"
counter-
examples,
6, 2. There
seems to exist
examples
constitute
Intultlonlsts wrong.
a widespread misunderstanding an essential
part
of Intuitlonistic
llke to give these examples
Thls popular b e l i e f
Is, I think,
of a t t e m p t i n g
to m l m l c k
mathematics,
for the fun of p r o v i n g
and also that
other people to be
wrong on both counts.
But It is a fact that these c o u n t e r e x a m p l e s the trouble
that
are often useful,
classical
arguments
If only to save us
when there
Is no reasonable
hope of succeeding.
Thls
is one reason
reason
for saying
something
is that they occur frequently
Let me first
state such an example
indicate
the assertion:
here to prove [Br 1920];
n
numerals
or take
It Is a fact that
e
of a circle
instead
that
of
~
Now we define
-iVn ~ m
generator
m
Let
Is a constructive
x0
denotes,
of
7.
Hm(n)
of the m th
sequence
(I wlll not bother
such a decimal
[H 1966],
Let
as usual,
expansion;
2.3.)
whether
question.
m by
= 2 -n
J
real number generator,
since the decimal
expansion
of
lawlike.
denote the real n u m b e r
We have no m e t h o d
of d e c i d i n g
(x 0 = O v x O ~ O) ~-~ (VnHlnr relative
The other
( Hin) § r m = 0
n ln ~ n ~ m § r m
is evidently
expansion
7, and look at
up tlll now it is u n k n o w n
a real number
~
and its diameter.
possesses
V n Hin v i V n Hln and we do not have a m e t h o d to settle thls
way.
of the last decimal
In the decimal
intultlonlstically
counterexamples.
In the literature.
is the number 7
these
in the c o n v e n t i o n a l
the ratio between the circumference of 10 consecutive
about
to zero.
defined
by
m.
x 0 = 0 v x O ~ O, since x 0 = 0 ~-~-IVn(l~In), hence nln).
x0
might
be called a floating number
see
-
We can state thls Ax(x
= Ovx
in a more
~ O) would
general
imply
an
xX
for any predicate
they reduce
likely that we shall hope of proving V n XnviVn reals,
Xn
VsAn
= O#x
~ 0)
for decidable
unless
X. If
x
numbers:
Xn) X, like we did for
NI"
of a certain solution.
kind,
for which
Specifically,
we have a m e t h o d
refutations;
we have no
of solving
Is supposed to be r e s t r i c t e d
it is un-
all problems
to lawllke
then ~ Ovx
for any decidable (an = o*-*Xn)
In the literature a specific like
problems
of natural
do not provide m a t h e m a t i c a l
ever flnd a constructive
Ax(x
Ax(x since
decidable
counterexamples
a p r o b l e m to unsolved
X
§ (Vn Xn v ~ V n
wlth every
It will be clear that these
-
way:
A n(Xnv-iXn) for we can associate
24
= O) + A a ( V n ( a n X
not containing
~ o)v1~n(an
= o))
free n o n - l a w l l k e
(I)
variables
(w
the counterexamples
problem,
but
mostly
occur in the form of a r e d u c t i o n
it is often worthwhile
to bring them in a general
to
form
(i).
For example,
this
can be useful
would read,
if we translate
recurslvely
decidable
ing a characteristic
Reductions variables
like
"lawllke"
if a recurslve
(I) are especially
version
by "recursive":
counterexamples
So
sets
(a Is viewed
(I)
it is
as determin-
relevant
funtlons
for which we want to investigate
of Church's
of assertions
If we want to study formal
thesis"
may be added
discussed
If the
consistently.
is, that we cannot
for which we can give
systems wlth
be sure that
Intultlonlstlc
(weak-)
are
Ax(x
Is rational
Ax(x
possesses
AxAy(xy The actual
theory.
for recurslve
set Is empty or not
The consequence of the counterexample -i x is everywhere defined.
A few other examples
wlth r e c u r s i o n
function).
for constructive
"intuitionistic
for a comparison
or
is not rational),
a decimal
= 0 § x = O vy
construction
x
representation),
= @.
of such counterexamples
may be left as an exercise.
-
25
-
6.3.
6.3.1.
Definition.
n 9 n < <Sn>n ~D Y k V n ~ m
(Sn+m - rn+m 9 2-k),
n ~ <Sn>n ~D Vk%/n~m (ISn+m - rn+m]
> 2-k)"
It is easy to prove: n < <Sn>n m n ~ n A
<Sn> n
n ~ <sn 9 n A n ~ n ~ < Sn>n This justifies 6.3.2.
<
<S'n>
n>n
~
<S !
n
'
n>n"
~ x V<Sn> n ~ y
x ~ y ~D~n
~ x W < S n > n ~ y ( n ~ <Sn>n).
~
relation
satisfies ~
[H 1966],
for real numbers
for an apartness
is an apartness
2.2.3,
x ~ y § x @ z ~ z @ y
relation,
+ z;
2.2.5. present
z
m
x.z
y.z.
x : y § ~y, x ~y§ ~x, we shall prove -Ix @ y § x = y.
of
(x ~ y).
(It n - rn+ml 9 2-k-2), A m
Isn+ m - rn+ml
This contradicts A m l r n + m - Sn+ml The proofs of
=> 2 -k-i,
our assumptions,
hence
< 2 -k+l. This proves
i.e.
(Is n - Sn+ml
< 2-k-2)
x ~ y.
Irn-Snl
verify that
is clearly defined
In Intultlonlstic
x ~ y ~-* x < y v x if
so
x = y.
~ y ~ x.z ~ y.z are straightforward. > y.
x ~ O.
formal systems partially
classically
<2 -k, and thus
n ~ <Sn>n,
x ~ y + x + z ~ y + z, z # 0 ^ x
6.3.4. We readily
difficulty;
as given in w 4.
Irn-Snl L 2-k.
Then for all
x -I
relation
and
o^x
The proofs
a kind of
~. It remains to be shown
such that Am
and suppose
and represents
little difficulties;
Let n ~ x, < S n > n g y , and suppose n, k
( n < <Sn>n),
defined inequality
the properties
x +
6.3.5.
-~ < r !
n n>
x < y ~DVn
of the negatively
6.3.3. Theorem.
Choose
<S !
n>n
Definition.
Ss called the apartness
Proof.
§
the following definition:
poslti~analogue that
~ s ' n n>
defined operations
we can always make an operation
present
everywhere
a special
defined,
since
-
the "excluded third" is available.
26
-
I n t u i t i o n i s t i c a l l y this does not work; one has
to introduce variables for elements b e l o n g i n g to a domain of d e f i n i t i o n which is undecidable relative to the basic domain of our theory. need in a formal theory special variables
In the case of
for the domain of reals
x -i, we
~ O.
w 7. Orderin ~ relations and order on the real line 7.1 x < y
has been defined for real numbers
7.1.1. Definition.
in the previous section.
x 9 y =D y < x; x ~ y :D-~X 9 y, x ~ y =D-IX < y.
Now we may prove 7.1.2. Theorem~
(z) (Ii) (iii) (iv) (v) (vi) (vii)
x @ y * x < yvx
(viii)
x ~ y ^ y ~ z + x ~ z.
9 y,
x
y # x§
x < y § x < zvz x
< z ~x
=y < y, < z,
y 9 z§
z,
~ z§
> z,
x 9 y^y
Proofs. Most of the assertions are easily proved; We prove
Vl
(supposing
as an example.
Let
< r n > n g x, < r ' n 9
and a number
k, n
such that
A i ( r ' n + i - r"n+ i 9 2 -k) m O 9 n such that
~i 9 m O AJ
(Ir'i- r'i+jl
i > mo, ri-r" i < 2 -k-2 9 Then r'i+ j > r' i
for every
[H i96612.2.6.
x ~ y, y 9 z. There exist
A i 9 m o A j (Ir i - ri+jl
Suppose
see
J, hence
_ 2-k-2
< 2-k-2).
r'i~ r"i+2-k > r i + 3.2 -k'2 , and 2-k-2
9 ri+ j +
2-k-2
y 9 x. This contradicts our suppositions,
r i - r" i >_ 2-k-2; so 7.1.3. Definition.
9 r i + 2.
< 2-k-2),
Let
r i - r" i 9 2 -k-3 n ~ x ,
for every
<Sn> n & y .
hence
i 9 mo, hence
x 9 z.
Ixl is the equivalence class of
n, max (x,y) is the equivalence class of < max Likewise for min (x,y) ( = inf (x,y)).
(rn, sn) >n"
-
the "excluded third" is available.
26
-
I n t u i t i o n i s t i c a l l y this does not work; one has
to introduce variables for elements b e l o n g i n g to a domain of d e f i n i t i o n which is undecidable relative to the basic domain of our theory. need in a formal theory special variables
In the case of
for the domain of reals
x -i, we
~ O.
w 7. Orderin ~ relations and order on the real line 7.1 x < y
has been defined for real numbers
7.1.1. Definition.
in the previous section.
x 9 y =D y < x; x ~ y :D-~X 9 y, x ~ y =D-IX < y.
Now we may prove 7.1.2. Theorem~
(z) (Ii) (iii) (iv) (v) (vi) (vii)
x @ y * x < yvx
(viii)
x ~ y ^ y ~ z + x ~ z.
9 y,
x
y # x§
x < y § x < zvz x
< z ~x
=y < y, < z,
y 9 z§
z,
~ z§
> z,
x 9 y^y
Proofs. Most of the assertions are easily proved; We prove
Vl
(supposing
as an example.
Let
< r n > n g x, < r ' n 9
and a number
k, n
such that
A i ( r ' n + i - r"n+ i 9 2 -k) m O 9 n such that
~i 9 m O AJ
(Ir'i- r'i+jl
i > mo, ri-r" i < 2 -k-2 9 Then r'i+ j > r' i
for every
[H i96612.2.6.
x ~ y, y 9 z. There exist
A i 9 m o A j (Ir i - ri+jl
Suppose
see
J, hence
_ 2-k-2
< 2-k-2).
r'i~ r"i+2-k > r i + 3.2 -k'2 , and 2-k-2
9 ri+ j +
2-k-2
y 9 x. This contradicts our suppositions,
r i - r" i >_ 2-k-2; so 7.1.3. Definition.
9 r i + 2.
< 2-k-2),
Let
r i - r" i 9 2 -k-3 n ~ x ,
for every
<Sn> n & y .
hence
i 9 mo, hence
x 9 z.
Ixl is the equivalence class of
n, max (x,y) is the equivalence class of < max Likewise for min (x,y) ( = inf (x,y)).
(rn, sn) >n"
-
7.1.4.
Theorem.
(1)
max(x,y)
(~)
Ixl.
7.1.5.
+ min(x,y)
lyl : Ixyl,
Definition.
: (z :-~(z
The d e f i n i t i o n or
2.2.7,
Theorem.
(I)
Ix,y]
(If)
~ llxl - lyll.
: Ixl.
in thls
[H 1966]
= [min
[x,y]
i-xl
definition,
= (z
x, y:
< x^z
< y)}.
form b e c a u s e Ix,y]
we do not
is a l w a y s
always
x~y
know whether
inhabited.
3.3.2.
(x,y),
x ~ y ~ Ix,y]
(III)
Ix - yl
> y)^-~(z
is g i v e n
x ~ y. By this
7.1.6.
2.2.8.
For real n u m b e r s
> x^z
-
= x + y.
Ixl + lyl ~Ix + yl,
(~)
Ix,y]
[H 1966]
27
max
(x,y)]
: (z : y ~ z ~
: max(x,y)
x}.
~ z ~ min(x,y)}.
7.2 7.2.1.
Definition.
on a s u b s p e c i e s the
following
<
X
be a species,
>, ~, ~
properties
x < y § x ~ yA
x : y ^ y < z § x < z
(c)
x < y a y = z § x < z
(d)
x < y ^ y < z § x < z x < y V x > y
(e 2) x ~ y A
x ~ y § x = y
(e 3) x ~ y A
x ~ y § x > y (x < z v z
(a) - (d),
called an
order relation
satisfies
(a) - (d),
(e2) , (f)
satisfies
(a) - (d),
(el) , (e 2)
7.2.2. being
Remark.
7.2.3. defined
Theorem. by
AxgX
then
Let
X
Ay~X(x
<
< y)
to
<
(relation)
as in 7.1.1.
defined Consider
(b),
be a species
a virtual
on a species be o m i t t e d
with
~ y -D x ~ y ^ x
ordenin~
satisfied.
a pseudo-orderlng
It Is c a l l e d
may
a partial
(e i) are
is c a l l e d
defined (c)
.
is c a l l e d
if (a) - (d),
If a r e l a t i o n
extensional,
predicate
relative
x ~ y
(e 1) x = y V
satisfle~
a binary
(a) - (f):
(b)
x < y ~Az
<
are d e f i n e d
(a)
(f)
When
Let
of X 2,
relation.
ordering
Is a u t o m a t i c a l l y everywhere
pseudo-orderlng ~ y)
relation.
A relation
< When
<
is
which <
relation. understood
as
in 7.2.1.
<. T h e n the r e l a t i o n
is a v i r t u a l
ordering
(KH 1966~
7.3).
-
<
for reals is a pseudo-order,
The usual relation
<
28
hence the corresponding
for rationals
Let
<'
due to Brouwer,
are interesting because of
be a partial order on a species
species of valid a s s e r t l o n s o f the form (a)
is a virtual order.
in terms of deducibility.
7.2.4. Definition. unextensible
~
is an ordering.
The following definition and theorems, their formulation
-
x = y
or
x r
y
X, and let
with x , y ~ X .
Y
<'
be the is called
if
x,y~X,-ix
<' y
not deducible
from
Y
by means of
(a) - (d)
of 7.2.1., then
x <' y ~ Y .
(b)
x,y~X,
x $ y
not deducible from Y
by means of
(a) - (d)
of 7.2.1., then
x = y ~Y.
7.2.5. Theorem. ([Br 1927], [H 1966] 7.3.1). Every virtual ordering relation is unextensible. Proof. Let suppose
X
be a species with virtual order
x ; yGY.
deduced from
Y
Then
x < y
~. Let
according to 7 . 2 . 1 ( a ) ,
by (a) - (d), then -~(x ; y ~ Y ) ,
by (e 3) x } y ~ Y .
Likewise for
be as in 7.2.4, and ~ y
and likewise-1(x
cannot be
= y~Y),
hence
=.
The same kind of r e a s o n i n g has been used extensively 7.2.6~ T h e o r e m
Y
so if ~ x
in the proof of
([Br 1927]). Every unextensible partial ordering relation is a
virtual ordering relation. 7.2.7. Remark. Definition 7.~.4.
and the proofs of 7.2.5,
in terms of g e n e r a l i z e d inductive definitions here.
7.2.6. may be r e p h r a s e d
(w 15); we shall not execute this
Presumably, the resulting r e f o r m u l a t i o n represents an a p p l i c a t i o n of g.i.d'.s
which is weaker than the introduction of
K (w 9)
or the introduction of WO (w i4).
-
w 8. Constructive
29
or lawlike analysis
8.1. In this section we suppose all sequences that the results sequences
obtained
by a lawlike
sequence
<m>n
<Xn> n
of reals
~kVn~m(Ix
< 2 -k)
it will turn out based on choice
- Xn+ml
Every Cauchy-sequence
Since <Xn> n
<m>n
for every
4k AnVmAi
if
.
then it is clear that also §
n - Xml
<Xn> n
is a Cauchy-sequence,
, m 6 x n
n.
< 2-k).
A k V k ' A n S m ( n , m ~ k' 8.1.2. Theorem.
for every
supposed to be given
(with limit x) if
is a Cauchy-sequence,
<Xn> n
is always
is said to be a Cauchy-sequence
is said to be convergent
If
<Xn> n
, m 6 x n
A k V n 4 m ( I X n + m - Xnl
Proof.
although
in 8.1. - 8.3. are also valid in analysis
or lawlike sequence
8.1.i. Definition.
Remark.
to be lawllke,
(w iO, II).
A constructive
<Xn> n
-
< 2-k).
is convergent.
there is a lawllke sequence
of rationals
n, such that
(Irm+i, n - rm,nl
< 2-k),
hence Va ~n4kAm
(Ira{n,k} + m,n - ra{n,k},nl
< 2-k).
We put an <Sn> n
ra(n,n},n
is a lawlike real number generator.
Am(IXn+ m - Xnl
This is seen as follows.
< 2-k), n ~ k, and remark that
m > i~Xm
and
Ixm - ra{m,m},ml
IXm-Sml
~ 2 -m, since
~ 2 -m,
Then ISn+ i - s nl ~ ISn+ i - Xn+ il + IXn+ i - x nl + Is n- x nl < 2 -n-i + 2 -k + 2 -n < 3.2 -k .
Let
-
30
Hence there exists a lawllke real number Ix - Xn+il
~ Ix - Sn+il
+ ISn+ i - Xn+il
-
x
such that
~Sn~ n 6 x.
~ 3.2 -k + 2 -n-i < 4.2 -k
for
n ~.k.
8.2. 8.2.1. Definition. like functional
A real-valued
u
Definition.
~
v §
v(u)
A real-valued
AkMmAxAy f
h k Ax~IAy
8.2.3. Definition.
~(v).
function
continuity x
of real numbers,
if
a l.u.b,
f
( Ix - Y l < 2-k § Ifx
A real number
AygX(y Clearly,
~
u, v:
is said to be uniformly
continuous
< 2-k).
If
It is evident how to formulate
X
is be supposed to be given by a law-
(Ix - yl < 2 -m § Ifx -fyl
is said to be continuous
of a species
f
such that for real number generators .
8.2.2.
function
} x^AkVy~X(y
of a species
-fyl
< 2-m)
9
and uniform continuity
for an Interval.
is said to be a lowest upper bound
(l.u.b.)
> x - 2-k)).
X, if it exists,
is uniquely determined.
Not every species of reals can be proved to have an l.u.b. 8.2.4. Definition.
x
l.u.b, for {fy : y ~ [ O ,
is said to be a l.u.b,
8.2.5. Theorem. If f is a uniformly possesses a 1.u.b. on [O, I]. Proof.
Consider
for every
Xn, k = n.2 -k, and put (sup
Xk
k
f
on [O, 1], Iff
x
is
a
and every
continuous
function on [O, I], then
f
n
such that O _< n < 2 k
the number
X k = {fXn, k : O =< n <= 2k}, sup X k = x k.
is clearly defined,
and nn E Y i generator for
for
I]).
for if { Y o " ' ' '
for 0 ~_ i < n, then sup {Yo''''' Yn })"
Yn } is a finite species of reals,
O 1 <sup {rm, rm,...,
n rm}> m
is a real number
if
-
We shall prove
<Xk> k
31
-
to be a Cauchy-sequence.
An Ax AY(Ix-Yl
Let
~ 2 -an § Ifx-fYl
a g(N)N
be a funtion such that
< 2-n).
Then An Ai(Xan+i Let
(Xan+i - fXan+i,k) IXan+i,k - Xan,jl
Then
~ Xan).
< 2 -n, and let ~ 2 -an.
IfXan+i,k - fXan,j I < 2-n;
Xan - fXan,J
Xan+i - Xan = (Xan+i - fXan+i,k)
~ O. Therefore
+ (fXan+i,k - fXan,J)
+ (fXan,j
- Xan)
< 2 -n + 2 -n = 2 -n+i.
Hence
IXan+i - Xan I ~ 2 -n+l.
By the previous
theorem
proved to be an l.u.b, For suppose so
contradiction. since
for
f
converges
2 -n, and since
Hence
x
can easily
inequalities.
IY - k. 2-anl x ~ xn #
x. Thls limit
< 2 -an, then
[email protected]~
I f y - f ~ . 2 - a n ) l < 2 -n,
it follows that
fy - x < 2-n;
AY(fY ~ x).
x - Xa(n+2)
it follows that
to a limit
by manipulating
fy - x > 2 -n, and let
fy- f ~ . 2 - a n ) <
Also,
<Xn, n
~ 2 -n-l, and for some
x - f(Xa(n+2),k)
k
Xa(n+2)
- f(Xa(n+2),k)
< 2 -n-l,
< 2 -n.
8.3. 8.3.1. f
Definition.
assumes
Let
its 1.u.b.
8.3.2. We cannot assert possesses Let
a, b
a maximum.
f
be a real-valued
(possesses
be a lawlike
a maximum)
that every function
We illustrate sequences
if for some real number uniformly
continuous
such that
a real number generator
: 0)). n
as follows.
x, fx = y.
on [0, i]
this with a "weak counterexample".
~l(Vn(an = o) a Vn(bn Then we define
function on [0, I], with 1 . u . b . y .
- 32
-1Vm
Let
<= n (am : O ~ b m < re(am'
# O ) ^ m __< n § r n = 2 -m
bm = O^Am'
< m(bm'
~ O)^m
If
= 0 v bn
Let us c o n s i d e r
Then If
Vn(an
: O), t h e n
= O), t h e n
& n § r n = -2 -m.
x O > O; if ~ n ( b n
: O), t h e n
x O < O, and if
x O = O.
the f u n c t i o n
f(x) and s u p p o s e
: O) § r n : O
am : O A A m '
n ~ x O.
-1Vn(an
-
f
g i v e n by
: I + Xo.X
that
f
assumes
Its 1 . u . b . ,
say In
xi9
0 < x l v x I < I. O < xl, t h e n ~ V n ( b n
every
uniformly
Aa A b ( ~ ( V n ( a n Hence
: O);
continuous
if
x i < I, t h e n ~ V n ( a n
function
= o)^Vn(bn
assumes
Its
: O).
l.u.b,
= O)) § ( I V n ( a n
So t h e a s s u m p t i o n
that
implies
= O)v~Vn(bn
= O))).
also Aa AD((An~(Vm An V n ' ( ( V m
and t h e r e f o r e
a{n, m } = O ~ V m
a{n,
VcAn((Vm
disjoint
= 0 § n'
b{n, m} : O)A
= O) §
(Vm b{n, m}
= O + n'
~ O))),
(5.3,(6))
~a Ab((An1(Vm
Interpreting
m}
a{n,
a{n, m} lawlike
r.e.
sets
m}
: O~Vm
b{n,
= O § cn = 0 ) A
functions
m}
: O)
(Vm b{n,
as r e c u r s l v e
can be s e p a r a t e d
§
m}
: O § cn ~ O))).
functions,
by a r e c u r s l v e
we o b t a i n :
"Every palr
of
set".
8.4. In l n t u l t i o n l s t l c intultionlstic The
the
ones
splltting-up
classically
(i.e.
A simple is the see a l s o
reflects
possible
a property
which
notion
point
notion,
split
are two
constructive
to Its d o u b l e
up into m a n y
w h e n we use
in c o n s t r u c t i v e
Often there
classical
content
between
lnterestlng content,
negation).
cases
inequlvalent logic). various only:
and a s t a b l e
Stable
notion
notions may
be
of view.
by the n o t i o n s
but
notions
equivalent
with maximal
is e q u i v a l e n t
was p r o v i d e d
strongest w 16.6,)
classical
the d i f f e r e n c e s
definitions.
from a formal
example
most
( w h i c h are p r o v a b l y
equivalent
strongest
interesting
mathematics,
~
is stable.
~
and
~
for r e a l s
(For t h e r e l a t i o n
(section
between
6);
~ a n d ~,
-
33
-
Another example is provided by convergence of sequences of reals:
in 8.1.I. the
positive notion is defined; a weaker but stable notion is: 8.4.1. Definition. (with limit
A sequence of reals
<Xn> n
Is said to be negatively convergent
x) if
A n w ~ V m A k ( I x - Xm+kl < 2 - n ) . The stability of the notion is an immediate consequence of the logical rules ~Ax
Ax + Ax~-~Ax, ~ A
For some more examples,
<-~A. see e.g. [H 1966], 7.3.2,
~R 1960],
[R 1963].
8.5. The difficulties
inherent to the general interpretation of e.g. implication often
can be avoided by making explicit the extra information required to exist by the premiss of the implication.
Thls Is illustrated by theorem of the form
Af(UC(f) § A(f)) f
(i)
a variable for arbitrary real-valued functions on the species of lawllke reals,
UC(f) -D f is uniformly continuous. The validity of UC(f) requires the existence of a modulus of continuity a such that
Ax Ay An(Ix-yl
and the existence of a sequence bo, bl, b2,
..., b m
< 2 -an § [fx-fyl
< 2-n),
defined on all rationals
k.2 -am, and taking rational values such that If(k. 2 -am) - bm(k.2-am) I < 2 -n. Thls information may be coded into one lawllke c ~(N)N,
and the coding can be arranged in such a way that conversely every
defines a uniformly continuous instead of (I) we may prove
fc(l.e. Ac UC(f c) and
Af(UC(f) §
Aa(A(fa)) which intuitively expresses the same theorem as (I). Compare also Krelsel's remarks in [Kr 1968A], appendix.
c Now
-
54
-
w 9. Lawless sequences of natural numbers 9.1. The material p r e s e n t e d in thls section is based on [Kr 1968], behavlour of lawless sequences quences.
is radically different
[Kr 1958]. The
from that of lawlike se-
In thls section we shall discuss some important properties of lawless
sequences of natural numbers. Throughout thls section
~, 8, ~
denote lawless
sequences. To begin with, we introduce a standard e n u m e r a t i o n of all finite sequences of natural numbers, Let
Xx.~y.
including the empty sequence.
{x,y} denote a p a i r i n g function with inverses
AzVx~y({x,y}=
z); A x { J l x , J2x}
= x, {x,y} = {x',y'} § x = x ' ^
We define inductively enumerations
~o(Xo)
=
~i(Xo,
x i)
Jl' J2' such that
from
Nk
onto
Xu)= {X O, VU_l(Xl,...,
Xu)}
y = y'.
N:
x0
Vu(XO,...,
= {x 0,
xl}l
An enumeration for all finite sequences Is obtained by assigning zero to the empty sequence,
and assigning Xu > = (u, v u (Xo,..., Xu)} + 1
<Xo''"' to the sequence
Xo,...
x u. The empty sequence is also written as
ith (n) indicates the length of the sequence wlth number function
@
< >.
n. The concatenation
is introduced by <Xo,..., Xu> 9
<Xu+ I, ... , Xu+v > = < X o , . . .
, X u + v >.
We may define ~ also for Concatenation of finite sequences wlth elements of (N)N:
<Xo,.-.,
,4v((v
If
y
Xu_l>@X
< u § x'v
= X'
x v ) A ( v ,~ u -* x ' v
=
is an initial segment of
xEY
~0 = < > ,
"'~ V x ' ( x
~x = < x O , . . . ,
*-~
x(x6(N)N),
=
x(v.u))).
we write
x6Y
= y~x')
X(X-'l)
>
for
X > O.
A
In formulae we shall often write
x
for
<x>,
to
save
space,
or formally
-
Throughout arguments
this
section,
X, Y, Z
55
-
denote p r e d i c a t e s
for w h i c h all n o n - l a w l i k e
are s u p p o s e d to be e x h i b i t e d when they occur in formulae.
9- 2. As a first p r i n c i p l e
for lawless
sequence with a r b i t r a r y
LSI
Ax~(e
As a consequence, prescribed
sequences we state that we can find a lawless
p r e s c r i b e d initial
segment:
gx).
we can find i n f i n i t e l y many different
initial
segment
AxA~(~n4~
lawless
sequences with a
n, since
<x> § a ~ n ) .
9-3. If
-
is used to denote " i d e n t i t y
if
~ - 8
expresses
LS2
that
a,8
by d e f i n i t i o n "
are to denote the
(or " i n t e n s i o n a l equality") same object
i.e.
(thought-object),
then
s - B v ~ I 8.
For either we start t h i n k i n g of
a,S
as the same object,
or we do not.
We also have ~
(or abbreviated:
For if
S
~*
m ~ 8
~-*
~ ~ S, it is absurd
Ax(~x
(contradictory)
to be equal,
e,8.
So if we have a proof of ~ ~ 8
since at any moment
only.
(1)
~x)
~ = 8).
~,S
that
:
Ax(ex
that we could ever prove all values of
(stage) we only know initial = 8x), this
The c o n v e r s e i m p l i c a t i o n
can be on account
segments of of the fact
from the left to the right
is evident.
9.4. For any p r e d i c a t e
X:
Xa ~-~ %'y(a~y
^ AS&Y X8).
For what
is given about
consists
of I ~ ) its " i n d i v i d u a l i t y "
finite
initial
segment
only n o n - l a w l l k e
a certain
of values.
p a r a m e t e r of
equal to another n o n - l a w l l k e
individual
(expressed by
X, r e f e r e n c e to i n d i v i d u a l i t y parameter)
is ezcluded,
therefore X8
segment.
s e q u e n c e at a c e r t a i n stage
if we have a p r o p e r t y
cannot depend on such a reference; only.
lawless
or "identity"
Hence
of an initial
segment
(2)
Thus
Xm
LS2) and 2 ~ ) X~,
~
(by setting
and hence a p r o o f of
must be a s s e r t a b l e
must hold for all
8
a
b e i n g the
X~
on account
with the same initial
-
More generally , we have for any predicate
~
(S'~O'''''
LS3 where
Vn(~n ~(m'mO'''''
a ~ aOA~ The clause
-
X:
~p) ^ X (a'aO'''''
AAf~n(~
~p)
36
ap) §
(8,~0,... , ~p) + X(8,al,...,
mp)))
is an abbreviation for ~ ~i^...^ ~
~ (a,=O,... , ap)
~
in LS3
=p. serves to exclude reference to "individuality".
That this is essential is revealed by the following example. Take for Then the assertion
X(a'mO):
~=~0"
AA8 E n (8 : ~O))
X(a,~ O) § is evidently false.
(I) and (2) are derivable with the help of LS3. For (2) this is immediate,
since
it is a special case. For (1) we argue as follows. a = B A a ~ B § Vn(a~n
^Aa' ~n(~'
~ B § ~' = 8))
is a consequence of LS3. The conclusion is evidently false, hence
a = S § ~ a ~ S,
and so by LS2 we obtain (1). 9.5. Let us use "LS" to denote the lawless sequences of (N)N, and let
F, A
for lawlike functionals from LS into a species of lawlike objects
(natural numbers,
constructive functions,
species
be used
or relations).
It is a consequence of (2) that such functionals are continuous: r~ = x ~ - ~ V n ( ~ 6 n (Likewise for If we combine
r's
^ AS~n
(r8 = x))
(3)
of other type).
(3) with the selection principle:
or directly from (2), we obtain a weak form of continuity: Am
Vx x(~,x)
§ ~xV~(~x
= Nx §
x(B,y)).
(4)
We obtain stronger forms of continuity principle if we make stronger assumptions about the class
(LS)N
(functionals
from LS into the natural numbers). For example,
it is reasonable to assume that we actually know, given
m,
if
ra
can be
-
computed from Essentially segments species
37
~x, or not.
stronger, but also plausible,
~x
-
needed to compute
rm
is the assumption that these initial
can be taken from an a priori given decidable
(to be obtained from a proof of
Am Vx
X(m,x)).
Ar ~ (LS)N Va(Aa Vx a~x ~ 0 a /~n(an ~ 0
and likewise for functionals
r
Formally:
+ Vx A,, 6 n ( r a
= x))
(5)
of other types.
It is easy to verify that we may without
loss of generality
suppose the
a
in (5)
to satisfy the further requirement: An /Ira(an ~ O + m = O v a ( n ~ m )
= O).
A further assumption about the lawlike elements of (LS)N (essentially an assumption about the constructive functions a occurring in (5)) is discussed extensively
in
the next two subsections.
We introduce an inductively defined class
K
of constructive functions of
(which may be interpreted intuitively as n e l g h b o u r h o o d Think of a class
P
of constrnctive
(N)N
functions on Baire space).
functions which satisfies two closure
conditions:
I~ )
P
contains all non-zero constant
functions:
Vx(a = ~n. x+l) § a ~ P 2~ )
If aO = O, and for every aGP,
y
~n.a(y@n) ~P,
then
or in a formula:
aO = O a ~ y ( ~ n . a ( y @ n ) 6 P
§ a6P)
9
If we write AK(P'a)
~D V x ( a = ~n.x+l) ~ ( A y ( k n . a ( y @ n ) 6 P )
then the closure conditions
for
P
AK(P,a) § Pa .
^aO
= O)
may be expressed by (6)
-
The minimal class
K
38
-
satisfying (6) is exactly the class such that
agK
is proved using (6) only, or in other words, the proof conditlons for a ~K
is proved using
I~
and
2~
only. The minimality of
K
(or Ka)
aG K
are:
is formally ex-
pressed by a schema (P a predicate letter)
Aa[AK(P,a) § Pal + [KfmP] Let us look at the structure of A natural "direct" proof of
K
(7)
9
also from a slightly different angle.
a 6K
may be visualized as a (in general infinite)
well-founded tree with the topmost node corresponding to the conclusion Ka. Terminal nodes correspond to inferences on account to closure cohdition I ~ Passing from a row of immediate descendants of a node
v
to
ponds to an inference on account of (2o). The elements of proof of
aEK
In this way
are all of the form a
v
K
itself corres-
occurlng in such a
kn. a ( m ~ n ) .
itself represents its own "natural" proof of
a 6K
(somewhat
analogous to the situation for natural numbers). I n d u c t i o n for natural numbers was Justified by a step by step parallelling of the construction of
n
for every natural number
one can Justify (7). Suppose
AK(P,a) ,
With every inference of type
i~
inferences of type 3~ ) 4~ )
3~
2~
n
(see w 3). By a similar argument
aGK. in the natural proof of
a gK
we associate
4 ~ respectively:
Vx(b : An.x+1) § b @ K ^ b ~ P bO : 0 A ~ y ( k n . b ( y @ n ) g P ) A
^t~v(In.b(y~n)6K)
§ b~K~P.
In this way we obtain by replacing in the natural proof of type
l~
a~K^a6Q.
2~
by the correspending inference of type
K
3 ~ or
every inference of 4~
a proof of
This Justifies (7).
The idea of a "natural" or "direct" proof of proofs of proof theory.
a gK
is analogous to the cut-free
- 59 -
9-7The elements of
K
may serve to define continuous functionals of types
((N)N)
and ((N)N)(N)N. We shall, throughout the remainder of this paper, use of
to denote elements
e, f
K.
One easily proves for elements of
K:
(8)
Ae AX Vx(e~x ~ O) Ae An Am (en # 0 § e ( n g m ) Let us give the proof for (8). Take
(9)
= en) .
Pa
to be
AxVx(a~x # o).
P(~n.x+1) holds. Suppose
Take any
•
P(An.a(<• Therefore proved
aO = O,
AY P(ln.a(yJn)).
• = <xO>~•
for a suitable
~n),
it follows that
Vy
x'. Hence, since
(a(<xO> @ ~'y) ~ o).
a ~(y + I) ~ O, and thus AxVx(a~x # 0), so
Pa
holds. Thus we have
Aa[AK(P,a) § Pal.
By (7) this implies (8). Likewise one proves
(9).
From (8), (9) it follows that we may unambiguously define functionals
~e' Ve by
~e • = y ~-~ Yx(e~x : y+l) (~e•
x = y ~Vz(e(~z)
: y+i).
Now we put K@={| e : e ~ K } , The functlonals of
K e, K e O
K~
e : e~K}.
are everywhere defined on
(N)N
follows from their mode of generation, the functlonals of
and continuous.
As
K @ satisfy (5), since
we have evidently: ArGKeVe~K(Am~x
e(~x) ~ O a A n ( e n
~ 0 §
Am~n(ra
= x)).
Now a basic assumption on the structure of the continuous functlonals one can make is: LS4
K e= (LS)N
LS4 is the basic assumption underlying Brouwers argument for the bar theorem and the fan theorem (for choice sequences,
[Br 1926 A~), and turns out to be quite
- 40-
strong proof theoretically. The plausibility of
LS4
lies in the fact that in order to prove
have to prove
ra
segment of
only; no "intensional" information about
proof of
a
to be defined for an arbitrary
r ~ (LS)N
such that
a
r ~(LS)N,
on accmunt of an initial a
can be used. So a
must start from a given collection of initial segments
~.I(nm~)
is constant,,and then proceed inductively to show
to be defined for every
we
a; and this leads naturally to a functional of
n ra
K @.
Now our strongest form of the continuity principle for lawless sequences becomes:
(AaOAal...Aap(~(ao,
.. , ap) § V x x ( x , a o , . . . ,
ap)) §
LS5. ~ V e ~ K ~ a o ' ' ' A a P (~(aO''''' ap) § X(e(~p(aO,..., where
~(aO,...,a p)
0 ~ S ~ p,
a i ~ aj
is an abbreviation for
and where
e(x)
ap~, aO,..., ap)), for all
i ~ J, O ~ i ~ p,
is defined by
e(x)
= x
~
~e(X)
= x.
e[x
= X '4"* Ye(X) = X'.
Likewise
A special case of LS5
is
As Vx
x(~,x)
+
VeAaX (~,
e(~))
or equilently Aa Vx X(a,x) § VeAn (en # 0 § Vx~a ~nX(a,x)). The corresponding principle for constructive functions is AaVa X(a,a) § Ve An(en ~ 0 § V a Aa6n x(a,a)). $
Likewise for species etc. 9.8. Bar theorem.
The hypothesis
LS4
may also be expressed in a completely different
way, namely in the form of the "bar theorem" accident).
(the name is more or less a historical
The bar theorem may be expressed as a schema:
Aa Vx X(ax) ^ An(Xn § Yn) ~ An Am(Xn § X ( n @ m ) ) An(AxX(nmx)
§ Yn) § An Yn.
^
-
-
Without use of the axioms of the theory of lawless sequences
Sketch of the proof.
one proves by induction over Ze ~ D A m
41
K
with respect to
{[An(en ~ 0 + Y ( m @ n ) )
^ An(AyY(m@n~y)
e : AeZe, with § Y ( m @ n ) ) I + Y(m)}.
Then one easily proves An(en
~ O § Y~ A An(AyY(n~y)
§ Yn) + A m
Ym
(10)
using AeZe. Now suppose
A~ ~x X ~x, A n ( X n § Yn), A n A m ( X n § X ( n @ m ) ) ,
From ~a Wx X(~x)
we conclude to the exlstence of an
~n(en
X(~(en ~ I)).
~ 0 § Aa~n
e
An(Ax Y(n@x)
+ Yn).
such that
Then we prove by ordinary induction: ~n(en
~ 0 + Yn)
(11)
.
This is done by induction on the length of For
n.
ith(n) ~ en : i, (li) is irmnediate. For the remaining cases, our induction
hypothesis
is: 4n(lth(n)
and this is proved for all Am(Ay Y ( m @ y ) Essentially,
= (en A I) + k ^ en ~ 0 + Yn) k
by induction with respect to
+ Ym) we can infer, using
K. From
the bar theorem expresses an induction principle for w e l l - f o u n d e d
species of finite sequences: m
= m@m'^m'
if we take
R = {n : A m ( m
~ 0)), then
An(en ~ O a R n § Y'n) ^ A n ( ~ x
Y'(n~x)
§ Y'n) § A n ( R n + Y'n).
This is seen by taking in the formulation of the bar t h e o r e m Xn ~D (en ~ 0), Yn ~D (Rn + Y'n). On the other hand it is worthwhile remarking that if we define aLE
(il) and
(iO), A m Ym.
~D A~ Vx(a~x
~ o)^AnAm(an
then the axioms for Sketch of the proof:
K
~ 0 § a(n~m)
K
by
: an),
are provable from the bar theorem. Aa[AK(K,a)
+ Ka]
is proved straightforwardly.
-
In order to prove and take
Aa[AK(R,a)
42-
+ na] + K g R
+Hal, ~a
we suppose Aa[AK(R,a)
Xn ~D an ~ O, Yn ~D R ( X m . e ( n ~ m ) ) ;
then we apply the bar t h e o r e m
for X,Y.
9-9. Functional
relationships
between
lawless
sequences
are given by relations
X
such
that Aa V!8 Let us consider
X(a,B,yO,...,
the simplest
Aa V!~
yn ).
case:
X(a,S,~).
By LS3 we obtain XC~,s,v)
+ a = S v
Vn(S&n
=
V
8 ~ {
r
(S,a,v) a
^ AS' ~ n(r
+ X(a,S',y)))}.
Since
Vn(sen^AB' E n( r (s',~,y) + X(a,B',~)) conflicts
with
As we shall
X(a,8,7),
Aa rIB
xCa,B,~) §176
see, this behaviour
and r e a l - v a l u e d
The rigid
Aa VlS
functions
"lawlessness"
identity-mapping
Even
of lawless
another In fact,
lawless
sequences
unfit
for a theory
into lawless
way,
sequence
involving
a, which
suppose
§
r
sequences
given by
implies
that
LS
ra
infinitely
many
be true
LS3.
is closed under the
is not
lawless
function
for lawless
does not t r a n s f o r m since
values,
it is in a
related
to
sequences.
it follows that
8 ~ avVn[86n^A~n(~
Since the second m e m b e r
ra = ~x.a(x+l)
sequences;
cannot
S = ra. From
8 : ra
~ a § ~ : ra)]
of the d i s j u n c t i o n
is evidently
.
false,
it follows that
8=ra+B~a.
But in this
case,
implies,
(7):
by
a : 8 = ~x.aO,
~nKa6n which
is evidently
In fact,
of reals
II).
only.
sequences
well-determlned
(12)
makes
(section
a simple t r a n s f o r m a t i o n
lawless
we have
analogous
AAB~n(8
false. to
Hence
i.e.
a constant
function.
But
a : ~x.y
= Xx.y)] for no lawless
(12) one can prove:
a, 8 8 = ra .
Aa VIB x(a,s)
+ Aa x(a,a).
-
43
-
9.10. Lawless
sequences
intuitionlstlc
are a useful tool in dealing wlth questions
logic
(see e.g.
~K 1958~).Speclflcally, valid,
of completeness
they provide
but intuitlonlstlcally
for
us wlth simple
refutations
of certain classically
invalid logical
principles.
In thls respect they are easier to manage than the choice sequences
of section I0. As an example,
we glve the following theorem:
Theorem. (I)
~ As V x ( s x
(II)
= O)
A~ ~ V X ( a X
= O)
(III) n A s ( V x ( s x
= O)v1~x(~x
(IV) ~ ( A s ~ V x ( a x (V) Proof.
= 0))
= O) §
1~s(-,~Vx(~x
= 0))
= o) § V x ( s x
(I) Suppose Aa V x ( s x
= O). It follows
An(fn ~ O § A s ~ n ( a ( f n and If we determine
u, m
= 0)) from LS5
~ I) = O)
~ O
then ~ s ~ m ' ( m ( f m
: I) = O) for m' = (~-y~.i)(u + (fm : I))
which Is plainly
contradictory.
~Vx(=x
~Vx(sx
(III)
-lVx(~x
follows that
= O). By (7),
= O) ~ * V n ( s
This Is evidently = O)
f
such that
m = ~y-J~U~n
(II) Suppose
that for some
e n ^ ~B 6 n ~ V x ( B x
false,
hence ~ I V x ( s x
is contradictory,
As Vx(sx
= O)).
= O). so if we suppose ~s(Vx(sx
= O), which conflicts
(V) as simple consequences
of (I),
wlth
(I). Likewise
we obtain
= O) it (IV),
(II).
Thls theorem therefore refutes the logical principles A v~A, and "Markov's principle" for lawless sequences, by II, IV, V. In its proper sence only states
= O)v~Vx(ax
something
for mechanically
Axm~A § A, (Markov's principle
computable
functions).
-
44
-
w iO. Choice Sequences iO.i. Choice sequences
are intermediate
in character
between lawless and lawlike
sequences.
Lawless
sequences have shown themselves
to possess
that they cannot be linked by a non-trivlal
a strong "spirit of freedom"
relationship
involving infinitely
many values. Lawlike
sequences produced decent analysis,
exploit
the fact that most operations
made us suspect
but there we had no opportunity
on sequences
that every well-defined
are continuous.
real-valued
to
Lawlike analysis
function is continuous,
but
we were unable to prove it.
Choice sequences possess restrict
their freedom,
be (extensionally) sequences,
some freedom, or even bring
but they are "on leash" ~em
completely to reason and force them to
equal to a lawlike sequence.
real-valued
functions
so to speak; we can
In analysis based on choice
are continuous.
Brouwer does not spend much words on telling us what exactly a choice sequence ([Br 1924],
footnote
3 on page 245,
His notion of sequence and restrictions
[M 1967];
The concept
see also
1952]
footnote on page 142).
described by saying that we choose values
in the form of a spread-law
future choices of values; paper
[Br 1942],[Br
is approximately
(see the end of this section)
for a recent discussion
of this notion see Myhill's
of a choice sequence as it is discussed here is probably neither the one, but among the alternatives
it seems to me to be the most fruitful notion for analysis. in
on
IT 1968], page 220.
only possible nor the simplest discussed
is
IT 19681
and the system of the monograph
~K, V 1965~ may be interpreted by means For lack of a better name, here, GC-sequences "choice sequence"
discussed so far,
The formal system
CS
of Kleene and Vesley
of this notion.
I have baptized the notion of choice sequence discussed
(from: Generated by Continuous and "GC-sequence"
In the sequel,
will be regarded as synonyms.
There is a definite need for a formal development occur in the intuitive Justification
operations).
in which the basic notions which
occur explicitly.
It may well be that a
detailed formal analysis would reveal some hidden assumptions,
but I believe never-
theless that the notion of choice sequence as discussed here is a good approximation of what we need. available.
Besides,
at present
I do not have a more satisfactory
alternative
-
45
-
I0.2. By associating number
with the choice
generators,
we shall obtain
Working
in a theory
x, y 6 ~ +
x, y 6 ~ .
On the other hand,
when
speaking
in the course
such that
Vz(x
This
u"
choice
= f(z))",
A choice
where
s0
for computing
values
this
newly
of
values
SO; at a later stage,
a continuous sequence,
s
this d e s c r i p t i o n
as a sequence
successively
(extensional) process
(numerical taken
r0
real number
x, it
llke "x
is such that
(We use
is
g(u)
s, B, ~, ~
= z
for
Xo, Xl,... ; we may decide
of values by r e q u i r i n g
choice
sequence
a continuous we compute
(that
operation
further
we may again decide
s = rOS 0 ,
is, a new process
(one may think of a
values
of
s
to restrict
s 0 = rlSl,S 1
values an
R
in a more
a newly
by selecting
a0
by selecting
started
choice
schematic
<x I, RI>,
on the
the
<x 2, R2>, xi
sequence
of pairs must
must
satisfy classy),
<Xn> n
satisfy
a certain but
Extensional
equality
of choice
with
already
relation
sequences
An(x n = yn ) *-*
of numerical
s
=
8.
a
-
the
values
a priori chosen,
are
of
s. The
an
Rx+ 1 Rx
must
be
know at any stage only an
sequence, the
<<Xn,
Ri
conditions
C ( R x + I , R x ) , the
for the rest we
as
numbers,
some general
initial segment of a. For our notion of choice r e l a t i o n s h i p s to other choice sequences.
may be expressed
...
are natural
chosen must be consistent
from a certain
way. We think of a choice
of pairs:
generated;
conditions
of generation
following
values
choices
and requiring
~ <Xo, RO>, which are
e.g.
and so on.
We now present sequence
rI
z
sequence.
by chosing
for some time,
operator
(see w II).
closure p r o p e r t i e s
specifications
"x = f(z);
real
section.)
is started),
K~);
real numbers
individual
to require
introduced
functional of
an arbitrary
of choice
stage to r e s t r i c t future
is a "fresh",
certain
below,
functions).
is obtained
s
of (choice-)
and later on e.g.
throughout
sequence
at a certain
about
the following n o t i o n
sequences
(N)N, to be introduced
we require
of our argument
(f, g lawlike
suggests
of
a theory
of real numbers,
is natural
for some
sequences
Rn>> n
and
Ri
8
stipulate
~
<
Rn~>n
-
For any
a ~ <<Xn, Rn>> n
The conditions
Ri
we write
~n
46
-
Xn, Rna for R n
for
are taken from a special c l a s s 2 ; ~ c o n s i s t s
of all conditions
of the form:
R ~ ~a.(a where
the
to,...,
Furthermore condition:
rn
= r o a 0 ^ ~0 = r 1 " 1 ^ " ' ' ~ n - I are
continuous
we require that the if
Ri
as presented
depends essentially possibility
Ri
of letting
Ri+ i
an-i = rnmnAmn
~
of
a
satisfy the
must have the form
= rn+i ~ n + I A " ' ^ a n + p - i
above is too simple,
on one other choice
(1)
on s e q u e n c e s .
used in the description
has the form (I), then
~a.(m = ro~ 0 ^... Now the picture
operations
=rnan)
= Fn+p ~n+p )"
in the sense that at any stage
sequence
only. There is also the
depend on more than one other sequence
in a given stage.
Let us put
~u (~0'''''
~u) = ~x. ~u(~oX,...,
~uX).
Now an example of the more general type of condition
R ~ ~a. Cm = roa 0 ^ a 0 =
ZO v2(~l'a2) ^ al =
is:
rl v2(a3'
a4 ) ^ a 2
= F2 v2(a5'a6)]
or in general m = r~m(~1,... , am) ^i=IA ai =
R ~ ~.[~
A
m m A Ai i=l J =I
But this extension principles
~iJ
=
rlJ ~m
lJ
of the c l a s s ~ d o e s
valid for choice sequences,
to the simplified
ri(ai1''''' (~lJ1'""
given below,
decide
8 : kx.J2Yx
later on
"'']
"
alter the discussion
of the
hence we restrict the discussion
version.
a
Fry
aiJ mlj) ~
not essentially
Note that if at a given stage =
al m i) A
a = ~x.Jlyx ,
:
r~2(m,8) , then it is always possible
that we
(or in short ~ = Jl Y, 8 = J2y),
so then
.
From the preceding description form choice sequences
it will be clear that continuous
into choice sequences.
operations
trans-
-
For if
~ - <<sx, R x
and if we start
with
>x'
8 = ra
as
say
8-<
47
-
R 0s - ks [s = roSo]
,R s 1 -- xa Is = row 0 ^ s O = r la 1] ,
80, ROB > ,... with
ROB
- k S . [B : r s ^ ( ~
in
R6
:
DmO]
then the necessary g
~nformation segment
to insure
of pairs
conditions)
so
It is perhaps generation choice
8
conceived
contained
appropriate choice
as a choice
to describe sequence,
B
(hence in an initial
(within the general
sequence
the process
of generation
by conditions
are then to be interpreted
a priori
too.
but as the g e n e r a t i o n
which are i n t e r c o n n e c t e d
AsAs
As"
Remark.
may
more
is completely
8), and we are free to extend
of a single
sequences
statemen~ works
of
8 = ra
of the form
as "for all
s
not as the
of a "network"
of
(i). Universal
in all possible
net-
etc.
On a formal
a wide variety
scheme
of notions
of description
as used above
it is possible
to construct
of sequence.
10.3. The principle section
of intenslonal
for extensional
predicates
they occur in formulae.
X(so''''' A first principle an extensional
that
The implication
From now on, X, Y, Z
with all n o n - l a w l l k e
Extenslonality
~I ' ' ' ' ) ^ ~ i
variables
exhibited
when
: S § X(so,...,
~i-l'
sequences
8, ~i+l,-..).
is the following.
Let
X
be
Then
Vr(VB(~
from the
are used in this
means
is valid for choice
predicate. x~ ~
continuity.
right
= PB)^AB
to
Xs. Then this must be possible
the
x(rs))
.
is
trivial.
left
on account
(2)
Suppose
of an initial
segment
that of
we c a n a s s e r t s:<Xo,Ro>,...,
<x n, Rn>. Suppose Xa.
e.g.
Rn~
to have the form
a" is as yet completely
s : P'm'Aa'
undetermined,
then
: r"a". a"
If at the time we assert
is entirely
arbitrary,
hence
A~ x ( r ' r " ~ ) . If
s"
always
is d e t e r m i n e d be conceived
operation
r (n)
up to
n =
as beinB obtained
from
s"
=
Vx>
by a special
say, then
a"
continuous
such that
AxAy{(y s x + (r(n)x)(y)
: Vy)^
(y> x § ( r ( n ) x ) y
: XY)}.
may
-
So
a
may be conceived
48
as being obtained
-
as
~ = r'r"r(n)~ m , hence
A~x(r,r,,r(n)~). IO.4. Extensional
continuity
The next principle continuity
for choice
(called Brouwer's
Ac, Vx x ( ~ , , x )
Suppose which
principle
~ A~, V x V y
So Vx X(a,x).
computes
sequences
in
AB(gx
A proof of
a natural
to be discussed
number
~K, V 1965~):
= ~ x -, x ( B , y ) )
~a Vx X(e,x) We
is the axiom of extensional
.
must
(3)
contain
for an a r b i t r a r y
~
a procedure
such that
X(~,Wa).
Thus we have j~ X ( a , ~ ) .
In general
W
is an intensional
as restrictive
conditions
introduced
Now we introduce
a certain
which transforms
a choice
If
m ~ <<Xn,
mode
U
indicates
restrictions
operator,
~
possible
~. That
for
tically the restrictions. the "thought
object"
restrictions
possible.
makes use ef values of the choice
an abstraction choice
as well
process
sequence
sequences.
Ab_~s~E
Abstr ~.
(~)
as a choice
sequence
k~.
e = a, and where we forget the
<xl, U>,...
the universal from
w
into another
Rn>> n , then we think of ~ E
of g e n e r a t i o n
i.e.
during the generation
conceptual sequence
<Xo, U>, where
operation,
condition
is, at any moment
e, because we abstract
we consider from
a
all future
by omitting
Remark that Abstr (~) is not a lawless
Abstr (~)
is such that we think
systema-
sequence,
at any stage
since
future
m ~ w - - m
Maybe
it is more accurate
but an object operations,
generation xgN
Abstr (~)
which is i n d i s t i n g u i s h a b l e
or yet more precise:
automatically
We cannot
to say that
defined
assert
~ = Abw
of Abstr
(a) from
it is effectively
simply a choice
from a choice sequence
each operation
for objects
is not
defined
mathematical
sequences
is
llke Abstr (~) too.
since we must n e c e s s a r i l y ~
w.r.t,
on Choice
sequence,
for this.
verifiable
that
~x : Abstr
(~) x
know the mode
But on the other hand,
o
of
for any given
-
Formulating
this in yet another way,
Abstr a means that
in applying
we cannot make use of ~x : Abstr
49
-
abstracting
from the mode
any kind of m a t h e m a t i c a l
a = Abstr
(~) x for any initial
segment
Abstr
(a)x
to one of the specific
attempt
.
Although
Abstr
it is permissible ~t = Abstr
for any constant
...,
~ = Abstr
w Abstr
<Xn, Us. Hence
(s)
W(Abstr
to be determined 8
implies
(a))
for any
depends
a
beginning
A=
Vx
<Xo, U>,...,
a : 8, it follows
<Xo, U>,
<Xn, U>,
that
X(a,x)
must
hold i.e.
(a))).
on initial
segments
(3). Expressed
x(=,x)
+
VrA=
of numerical
values
only,
we
otherwise
(4)
x(=,r=),
operator
from choice
as in the case of lawless
on all sequences
K':
(GC)N
As in the case of lawless strengthening
with
segment
= ~x + x ( ~ , x ) ) ,
continuous
For the same reasons has to be defined
from an initial
to
and thus we have Justified
being a lawllke
to obtain
must hold.
VxVyaB(ax
r
in an
(a).
= x
for any
Rn+lS^Rn+la
conclude
one encounters
stage.
(a) x)
Ha X ( a , W ( A b s t r Since
at the given
(a)x
= Abstr
< a n + 1 , R n + l >, X(B,x) Since
problems
use of
term t, we may not assert
implies
Now suppose
computed
(~),
to assert
and then use the rule of g e n e r a l i z a t i o n
since this
on Abstr
(a)(t)
~x = Abstr
~x(~x
operation
(a), but at any stage we can make
It is useful here to point to formalize
of g e n e r a t i n g
Abstr
sequences
sequences
(the species
(specifically:
(a)) we may introduce
GC)
since
(5) there
is an intermediate
possibility
of
(3):
A r ~ ( G C ) N Ya(A~ Vx
a~x ~ OaAn(an
~ O * Vx A a ~ n ( r a
r
here the a s s u m p t i o n
. sequences,
into N.
: x))
.
(6)
-
50
-
IO.5. Formal consequences
of the principle discussed.
We remark that a lawlike
r
of type
(GC)GC
Ax Ve Aa((r~)x
belongs to
Ax(xa.(rm)x~K@),
i.e.
= e(m)), hence
= xn.f(x@n)(~)).
Therefore the axiom of intensional
K~@, since VfAx
Am(e(~)
continuity
=
combined with
(5) reads:
x~ § An important
= ~)^A~ X(elY)].
consequence
(7)
of (7) is
Aa[Xa § Ym I *-~ Ae[Aa X(eJ~) § AmY(eJa)~ Proof of (8).
The implication
conversely
4e~a
m = fJS^A~
X(fJy).
Hence 4y Y(fJy),
X~. Then for some and thus
continuity
A~Vx x(~,X,~o,... ,) § Proof of (9).
(8)
from the left to the right is immediate.
X(eJa) § A~Y(e[m)],
The principle of extensional X with parameters:
.
Ym.
(3) can be strengthened
= ~x §
Suppose
f, $
to the case of an
x(B,y,~o,...)).
(9)
Let us take the case with a single extra parameter:
AB~4~Vxx ( ~ , x , B )
§ A~VxVya~'(~'y
By (8) this is equivalent
= ~y § x ( ~ ' , x , B ) ) ] .
to
Ae~BAaVx X ( a , x , e 18) § Asa~VxVya~'(~'y = ~y § X ( a ' , x , e J B ) ) J .
x(~,x,els).
Suppose
48AeVx
Since
AYV~VS(JlY
= ~ ^ J 2 Y = 8) and
A~AS~({a,8}=
y)
(closure of the notion of choice sequence under continuous ~mA8%IX X(~,x,eJs) Therefore
e-~ AyVx X(JIy,x,eJJ2y).
from our supposition i%yVx X(JlY,x,eJJ2y)
operations)
we have
-
51
-
hence, using (5):
AyVxVy~,' J(-~-~i~y : JIy'y § X(JlY',x,elJ2y')) and thus
A~ABVxWA~'AB'(~y
: ~'y §
x(~',x,elB'))
so
AS'AmVxVyAs'(~y
= ~'y
§ X(a',y,e[89)
and thus we have proved the case with a single extra parameter. Likewise In the case of more parameters;
ABA~[A~Vx
x(a,x,s,~)
e.g.
§
= ;y § x(=',x,s,y))]
Is reduced to the previous case by first applying
ABAy Y(8,~) +~ A6Y(JI6, J26). Remark. The method used to prove
(9) ls applicable to many cases where a result
without parameters has to be generalized to the corresponding result with extra choice parameters. We mention especially: A~Vx X(~,X,~o,...,~u_ 1) § The bar theorem for
X, Y
VeA~
X(~,e(Vu(~,~O,...,~u_1)),
~O,...,au_l).
(lO)
not containing choice variables is proved in exactly
the same manner as in the case of lawless sequences. The bar theorem for arbitrary X, Y
Is contained from the special case by a reasoning analogous to the argument
In the proof of (9). The derivation of the following formula deserves special attention
AxVy
X(x,Y,~o,...)
+ VB4x X ( x , B X , ~ o , . . . )
(117
9
(11) ls obtained from the special case (Justified In w 5)
Axe9 X(x,y) § VbAx X(x,bx). In order to prove (11), we do not use (8), but (7) instead. Proof of (II).
Let A x Y y X(x,y,a). Then there is an
fly = a, A6AxVy X(x,y,fI6). therefore also
We use (3) and obtain
AxA6 X ( x , ~ n . e ' ( x e n ) ( 6 ) ,
f16)
f
and a
AxVeA6
for some
y
such that
X(x, e(6),f16), and e'~K.
- 52 -
Define
S : kx.(kn.e'(x@n)(6))
Ax X(x,Bx,m)
and this proves
: e'l~ and take
6 : y, then it follows that
(li).
IO.6. Derivation of the fan theorem. iO.6.1. Definition. (notation:
A lawlike function
a
is said to represent
a spread-law
~RE (a) or a ~ S~E) if
(a)
aO ~ 0
(b)
AnAm(a(n@m)
(c)
AnVx(an
~ O § an ~ O)
~ o § a(n@x)
~ O).
The spread-law is said to be finitary if in addition (d)AnVz~x(a(n~x)
~ O § x 9 z).
A finitary spread-law is also called a fan-law. Intuitively,
a spread-law represents a set of nodes
finite sequences of natural numbers,
{n : an ~ O} of a "tree" of
with branches directed downwards; the topmost
node corresponds always with the empty sequence.
A fan-law corresponds to a finitely
branched tree. IO.6.2. Definition.
For any
x~a I0.6.3. Definition.
a 6 S~E,
~DAn(a(xn)
AnAx(a(n@x)
ea ~ K ,
c
a
Let
b
be a function such that
~ 0 ^ bn ~ x).
such that
~ o § c(n, x} = x ) ^ ( a ( n @ x )
d, d I
we may associate a continuous operation
as follows.
~ 0 § a(n@)
Then there exists a unique
and unique
~ O).
With any spread-law
ra, r e p r e s e n t e d by an element
SnAxK(a(nmx)
X ~(N)N:
= o § c{n, x} = bn) S
such that
dO = O,
dlO= 0
di(n@x)
= c{dn, x}, d ( n ~ x )
= dnOc{dn,
x}.
-
rax
Now
: Xx.d(~(x
+ I)).
ea
ea ea
raa
: a.
10.6.5.
If
For all
Lemma.
a
a
case,
raa6a,
not
ra is d e s c r i b e d
Let us put Let
r = ra
hence
for
:
+ I
sequence
raraa
the a
~-* A i
1
the
-
!1
sequel.
structure
the ~<
of the
~tnary
v(x i
spread
~ 1),
proof,
we g i v e
then
the
proof
in
law:
An(an~{O,1})
9
(so
hx = O
We p r o v e
if
x = O
by induction
over
x > I, hx = I
or K
with respect
s z).
haO = O v h a O
= 1,
s x).
Suppose
+ i)) s
x 0)
A~(e1(r~x.a(x
+ i)) s
x i)
e 0 : Xx.e(6@x),
eI
: ~x.e(i~x).
+ 1)) &
hence A~(e(ra) Vzh~(e(ra)
s
sup ~ z).
(Xo,
xl))
sup
now: s x].
a n d we c a n f i n d
Aa(e0(r~x.a(x
Aa(ehao(r~x.a(x
SO
9 a, a G ~ E ,
such that
= ram.
eO : O, A x V z A a [ x n . e ( x ~ n ) ( r a )
whe2e
z =< v.
x - 1]).
e = ~n.x + I, t h e n ~ a ( e ( r a )
We h a v e
if
by
= ~x.(1A]x
in
v < z
a:
obsoure
h = xx. l i l x
AeYzAa(e(ra) If
to
xv >
if
by
z).
by taking
a<XO,...,
rax
= 0
F o r any f a n - l a w
In order
a simple
may b e d e f i n e d
is any c h o i c e
AeVzA~(e(raa) s Proof.
-
x v > : d<xo,. -., Xz>
Xo,...,
I0.6.4. Remark.
53
Xo, x 1
Therefore
(x0, x I)
such that
if X = 1). to
e."
- 54 -
In the general advantageous
VzAa(e(ra~) If
d
case the argument
to apply induction
is slightly more complicated; over
K
w.r.t,
e
to
is the function as indlca~@~
Remark.
10.6.7. A~Vx Proof,
Theorem X(ra~,
x, aO,...)
e
in the definition
obtain a
y
such
of r a we use is the fan-law are left to the reader.
f
e
~ z). Let
a
be a fan-law.
Then
= flaky § X(raS , x, ~o,...)).
such that
variable
Aa X(ra~,
such that Aa(e(f(a)) Aa(f(ra~)
in
X, and suppose A a V x X(ra~,X).
e(a)),
hence Aa X(raa,
e(raa)).
~ 0). We apply the lemma 10.6.5
Then
and
~ y). Hence for any
: raSY § X(raS , e(ra~))).
to the general
Remark.
For any
case with parameters r
such that
x) §
as in the proof of (9).
AS@aVe(rs
= a), we have
:.-~y § x(rs, y)). sequences,
corresponding
combinations
to quantifier
"signature"
aO' ~1'''"
Aa(raea),
As in the theory of lawless a specified
~ a ( r a e a) and
that
§
such that
~S(ra~Y
A~Vx x(r~,
r
be the only non-lawlike
there exists an
10.6.8.
and any
(fan-theorem).
Then we can find an
Extension
a
= ~): AeVzAm(e(ra)
Let
is a fan-law §
It is easy to see that once we have proved this lemma, we also
have for any fan-law AS G a V a ( r B
Ae~a(a
~ z)).
e(raa) = k n . e ( d < a O > m n ) ~ b k X . u ( x + I)), where b k n . a ( d < m O > @ n ) , in the induction step. The details 10.6.6.
here it is more
we may also obtain continuity
AaVa,~aVX a (X a
a
of arguments).
A~Va x(,. a. , o . . . . )
§
A,VxVaAB(~x
A~FXaY(~,
+ A~VxFXaAB(~x
principles
a species variable with
Thus we obtain a.o.
X a, ~O,...)
A a V a X(~, a) + V e A n ( e n etc. etc.
~ O § VaAa~n
= ~x
§ x(8.
= ~x ~ X(a, a))
a.
~o,...))
Y(B, X a, aO,...))
(12)
(13) (14)
-
One would expect for
AeVB
55
-
the following form of continuity:
A~VB x(~, B) §
x(e, ela).
(15)
But if one tries to reproduce the informal argument given in I0.4 continuity, we would have to use Abstr (~) = m
for AaVx-
to Justify this principle.
See
[M 1968], page 21B, and this paper, 16.4. The only (weak) argument in favour of (15) is perhaps that continuous relationships are the only ones made possible explicitly,
as is seen from the descriptlon
of GC-sequences. 10.8. Formal systems for IntultionIstic In IT 1968S a system described.
CS
CS
analysis.
for intultionistic
analysis with choice sequences is
is based on four sorted Intuitlonistlc predicate logic with equality,
with numerical variables, vars. for constructive functions, vats. for elements of
choice sequences and
K, and contains the usual arithmetical axioms, axioms for
K, abstraction operators with rules of x-conversion,
and the axiom of choice in
~ e form A x V a F(x, a) + V b / t x
F(x, ~y.b{x, y})
and the following axioms for choice sequences F~ §
Ve[YS(eIB
=
c~)^
A ~ V a F(~, a) §
/~'y
F(ely)]
# 0 +VbA~
6 n F(~, b))
A(~VS F(~, B) +VeAc~ F(cL, elot). For a more detailed description see
IT 1968~; the axioms mentioned there are
slightly different but equivalent. CS
contains a subsystem
IDK
(essentially consisting of the part of
CS
without
choice variables). For future reference we describe IDK
IDK
here separately:
is based on two sorted intultlonistlc predicate logic with equality, with
variables for numbers and constructive functions, of ~-converslon,
the usual arithmetical axioms
abstraction operator and rule
(induction,
defining axioms for some primitive recurslve functions like pairing function), predicate constant
the axiom of choice in AxVa-form, K:
successor-axloms, +, "' @ 9 J1' J2'
and the axioms for a
-
56
-
Aa[AK(K,
a) § Ka]
~a[AK(Q,
a) § Qa] § Aa[Ka § Qa].
(Alternatively,
instead of introducing a predicate constant
variables for elements of
K, one may introduce
K).
The main result stated in KT 1968] is essentially this: thereexlsts
a translation
(more accurately:
F
T
of closed formulae of
some conservative extension of
CS
into formulae of
IDK) such
IDK
that
A *-+ T(A)
for every closed formula
~-CS
CS
A
A iff
of
CS. We can prove much more however:
hDK
T(A)
can be proved finitistlcally. This result establishes choice sequences.
consistency for a considerable part of the theory of
It enables us to enterpret the continuity axioms as defining
a special interpretation of certain quantifier combinations. Take e.g.
(15): this may be read as an explanation of the quantifier combination
AaVB. Another way of reading (15) is, that it expresses a restriction on possible proofs of AmVB X(a,B). The unsatisfactory aspect of this interpretation of the theory of choice sequences is, that although we know it to be successful for "interpretation"
CS, we do not know if this
or "explanation" can be maintained for arbitrary intultlonlstlc-
ally acceptable extensions of
CS.
-
w 11.
57
-
Spreads and a t h e o T j of r~al numbers
11.1. In this section we introduce the notion of a spread, well-known literature.
is often very convenient.
Moreover,
it is necessary to understand the notion in
order to be able to read literature on "traditional
intuitionism".
a spread-law has already been defined in the previous of the definition 10.6.1 11.1.1. Definition. associates
is obtained if we replace
A complementary
objects of a species
ly, to {n : an ~ O A n
~ O} if
S a
section. a
such that
A generalization
(lawllke) mapping
~
of a tree or spread-law
to the positive natural numbers is the spread-law considered).
SpT.(a),
~
a
The notion of
by another kind of sequence.
spread may be generalized by a d m i t t i n g non-lawllke mappings A pair
from intultionistlc
The general notion of a spread, although not absolutely indispensable,
for
(or, equivalent-
The notion of a a
and/o~ for
complementary m a p p i n g into a species
is called a dressed spread. When we take for
~
a mapping
into
N
6. S
such that
is a naked or u n d r e s s e d spread. When we talk ~<Xo,..., XU> = X u, then about a spread a(wlth Sp~(a)) we mean the u n d r e s s e d spread associated with a.
is said to be a subspread of
and if
~I(X - {0}) = ~'I(X - {0}).
A sequence
•
• ~
if
~>)
~ O) ^ k x .
With a complementary m a p p i n g
~} : < ~ ( n 11.1.2. Definition. equivalence relation
~'(n
a mapping
+ 1)> n : k x . ~ ( n
A species ~
~
X
X = {n : an ~ O } ~ { n
~
+ 1)
: a'n ~ O}
(notation
: •
is associated defined by (x e ( N ) N ) .
+ 1)
is said to be represented by a spread with an
(represented by )
S ~, the speeles of equivalence
onto
if
is said to be an element of the spread
V•
If
classes w.r.t.
X. If this m a p p i n g is lawllke,
if ~
the r e p r e s e n t a t i o n
11.1.3. Definition. A spread with a flnltary
is a spread, and
can be m a p p e d b l - u n l q u e l y is called lawlike.
spread-law is called a fan or
flnltary spread. 11.2.
Let
n
be any given lawllke e n u m e r a t i o n of the rational numbers,
be an extensional species of sequences of
(N)N.
and let
-
Let
Rng X
-
denote the collection of real number generators
equivalence classes of generators
58
Rng X
n, x ~ X -
with respect to the relation
~
(as defined in w 6) constitute a speeies of reals relative to
n
: Re X.
If
is closed under composition with lawlike functions, i.e. if
X
AxAa(• then
Rng X
The
for real number X,
(I)
§ ~x.a•
is independent of the particular lawlike enumeration of the rationals
chosen. For let
n
be another such enumeration. Then there is a lawlike bl-unique
mapping
a
N
of
onto
N (supposing both enumerations to be without repetitions)
such that An(r~ = ran). If n = n, b x ~ X . n =
, a•
If we take for
X
worse:
n
n
is a real number generator,
Conversely, if n
x6X,
then for
is a r.n.g., x ~ X ,
then
the lawless sequences, condition (I) is not fulfilled. But even
can never be a real number generator for lawless
Ak~d~m(Irxn
X, since
- rx(n+m) l< 2 -k)
is an a priori condition on all values of
x
which makes
x
non-lawless.
(This
is easily proved by the methods of section 9, and may be left as an exercise to the reader.) On the other hand, For
(1) is fulfilled for lawlike and choice sequences.
Ro(X) not empty, real-valued (lawlike) functions f:'om
determined by lawlike operations ~, ~ ~ Rng X ^ ~
w ~ ( R n g X) Eng X
Re X into
Re X
are
such that
~ ~ + w~ * W~
11.3. There is a more general way of defining analysis of reals and real-valued ~unctions, namely relative to a spread Let
S =
every
x6a.
S
and a species of sequences
be a spread, such that
Rng(X, S)
Cmx
denotes the r.n.g's
X~__(N)N.
is a real number generator for ~x
for
x~X.
The species of equivalence classes w.r.t, equivalence between r.n.g's is denoted by
Re(X, S). Real-valued functions relative to
Re(X, S)
are defined as in the
-
case of
59
-
Re(X).
We describe
a spread
enumeration
of rationals,
SO
which can be widely used. as before,
all indexes of rational numbers We define
S O =
< ko > = rk 0
r
< ko'''"
km-l>
represented,
n
< ko'"''
be a fixed lawllke
be an enumeration
such that
a 0 = ~x.l, and taking for
= rn + r
SO
Let
<s(n, k, m)> m
(without repetitions)
by taking
r
Pictorially
s
and let
of
Irn-S I < 2 -k.
r
km-l' k> = rs(n,m,k)
.
looks like
ro ri;/73 rs(3,1.0~/rs(3,1,1)
rs(3,1,2)
rs(s(3,1,1),2,0)
If
<Sn> n ~
r >, then
<Sn> n
...
O00
is a real number generator,
ISn+ m - Snl ~ ISn+ m - Sn+m_ll@...+ISn+ 1 - Snl<2-n-m+1+...+ SO
does not contain all possible
It is possible
to construct
We leave the construction
of
a spread Si
2-n-I+2 -n <2.2 -n.
real number generators,
a priori bound on the rate of convergence, clear from (2). Si
(2)
since we can find an
valid for every
which contains
since
x~S O ,
as will be
all real number generators.
as an excercise.
With respect to S O we can talk about Bng(LS, S O ) again lawless sequences do not produce a satisfactory
and Re(LS, SO). But once theory of reals and real-
valued functions; for example, O ~ R e ( L S , SO)! And if we look for real-valued functions, matters become (If possible) worse. Because of the lawlessness,
-
if
n,
<Sn>n&Rng(LS,
60
-
SO) , then
n ~ <Sn> n *-~An(r n = s n) since If n ~ <Sn>n~ n ~ <Sn> would impose a condition of relatlve dependence on the underlying lawless sequences of (N)N, contradicting their lawlessness. Hence a lawllke real valued function must be representable X(a,8)
(s,B~LS)
such that
AaVI8
X(m,B)
which implies
ponding real valued function must be the identity: If we admit extra parameters need not be lawllke), (~,8,y LS-varlables)
In
X
fx = x
by a predicate AmX(m,m), for all
(so the real valued function
the situation does not improve:
f
so the corresx6Re(LS,
SO).
defined by
in 9.9 we derived
Aa%aBX(a,B,~) § hence if fx_ Is the function defined by to y In R~(LS, SO) , then AX(fxoX
=
XVfxoX
=
x O)
X, and
x0
the real number corresponding
9
11.4.
If we restrict the spread
SO
described
In 11.3
<Sn> n G S 2 ~-+An(ISnl < I), then S 2 represents • 2 Is a r.n.g, for a number of [-I,I]. For a3
[-i,l]
we define yet another representation
a spread S 2 such that ~-1,I~ in the sense that any
by a fan
S 3 = 9
is given by
~
50
~
0
3 < Xo'''''
r
to
Xv> ~ 0 ~-~Ai < v(x i < 2).
Is given by
Ir
3 < ko>
: (ko-l)2"l
3 < ko'''''
kn> = r
+ (kn-l)2-n-1
-61 -
In a picture:
I
I
l
l
It is not hard to see that every We remark that sequences
or
operations
Re(X),
Re(So,X)
X = species V, w'
Ax~Rng w'
are equivalent
of choice sequences
for
for an
x ~ [ - i , l I.
X = species of lawlike
in the sense that there are lawlike
X), w' ~ ( R n g ( S O, X)) Rng X, and
X(~x ~ x), 4 x ~ R n g ( S O, X)(~'• ~ x)-
of
~
is proved by showing
~•
W x ' ~ R n g ( S O, X)(• ~ X').
(By a selection principle necessarily
continuous.)
x e <Sn> n ~ R n g
X
this implies the existence We take the case of lawllke
Vo~kAnAm(ISbk+n c
of a lawllke
operator,
sequences.
implies
A k V i A n A m ( I s i + n - Si+ml
Define
is a r.n.g,
I
we may take the identity.
The existence
Hence
3
!
such that
E ( R n g X) Rng(So,
For
•
I
< 2-k).
- Sbk+m I < 2-k).
by O
I~ It follows
:
bO
(k+1)
= c(k) + b(k+l).
that
<Sck~k G Rng(S,
ISb(k)+i - Sb(k)+j I < 2 -k
X), since
for certain
ISc(k~l)
- Sck I =
i, J, h e n c e A k l S c ( k + i )
- Sckl
< 2-ko
not
-
Likewise one easily proves or
62
an equivalence
-
for
Re(S2, X),
Re(S3, X), for
X = GC
X : species of lawlike sequences.
Remark.
It follows from our experiences
with lawless
entirely
trivial to prove such equivalences!
sequences
that it is not
11.5. The homogeneous Theorem.
character of
The subspread
am ~ 0 ~-~ (m <_o
r176 Then
S
of
$5, defined by:
kn > v
[r - 2 -n-i, r
is expressed by:
S :
obtained by restricting
Proof. Let us extend
S3
r
to {m : am ~ 0 a m
+ I) = (~r
+ I)
contains exactly the
X ~S 3
choice reals.
by stipulating
for all
b, n.
with an initial
We give the proof of our theorem for the case be straightforwardly
~ 0} represents
r + 2-n-lsf) to finite sequences
~ O,
for simplicity
of choice reals the proof is completely number in [-1,1S. Then /~kVn(Ix - n.2-ki
r
analogous.
S
determined by .
in notation; Let
x
but in the case
be any lawllke real
< 5.2 -k-3 ^ inl <_ 2k).
Hence VbAk(ix-b(k).2-kl i2-k.b(k)
- 2 -k-1. b(k+l)i
< 5.2 -k-3 ^ [b(k)[ ~ Ix-
b(k). 2-ki
< 2k),
+ Ix-
so
b(k+l).2 -k-l]
5.2 -k-3 + 5.2 -k-4 : 15.2 -k-4 < 2 -k. Therefore
12b(k) - b(k+i) I < 2, so
1 2 b ( k ) - b(k+l) I ~ 1. Now we define
c
by
kn>-
S = $3, in such a way that it can
adapted to obtain a proof for any
We give the proof for lawlike reals,
segment
I b(k)i < 2k i c(k) = b(k) b(k) = 2-k c(k) = b ( k ) - i b(k) -2 k c(k) = b(k)+i.
-
If
Ib(k) l < 2 k,
If b(k+l) If b(k)
Ib(k+l)l
= 2k+l,
then
< 2k+l b(k)
= 2 k, then b(k+l)
we h a v e
12 c ( k )
= 2 k+l or b(k+l)
12 c ( k ) x
-
= 2k; hence again
12c(k) - c(k+l) I ~ I. Likewise for b(k)
Now
63
is represented by
- c(k+l)l
- c(k+l)l 12 c ( k )
< 1.
- c ( k + l ) I ~ 1.
= 2 k+l - I. In both cases,
= -2 k or b(k+1)
= -2 k+l. Therefore
~ 1. 2-k-I> k ~
S 3.
11.6. Let
~,6,y
~0 = O,
denote choice sequences.
Ax(~(x+1)
We define
S
= mx), m
Let
by stipulating
r
.
@ <x>) : J2 x S"
= <~x.l,~>, where
~
ls defined by
an arbitrary but fixed choice sequence 9
S" = <~x.I,r
is a subspread of
S
In the sense described below.
We define
Si = I,
B <Xo,...,
Xu>
= I ~-*
<JlXo > = J 2 X o A m < J l X o , ~<JlXo,..., is obtained from B
m
Jlxl>
= J2x2~...
JlXu > = J2Xu , Bn
0
otherwise 9
by a continuous operation; hence
has all the properties of a spread-law,
immediately from the definition,
and if
for
B(n@m)
B
Is a choice sequence.
~ 0 § Bn ~ 0
follows
B<Xo,... , Xu> ~ O, and we take any
x,
then B<Xo,... , Xu, y> ~ 0 for y = {x, ~<Jlxo,..., JlXu , x>}. Now If X = {n : Bn ~ 0}, then
To every
a spread
S
such that
S
:
a
an arbitrary lawlike spread law,
is treated likewise, with as a result:
S a =
wlth
~0 = O, Ax(~(x+l)
= ~x)
~ a
choice sequence,
wlth a cholce-spread law and a lawlike complementary law may be found {8 : S ~ S
} = (B : B G S } 9
Remark. Thls theorem expresses the essential content of Brouwer's stood ~aper
S
EBr 1942 A S .
often m i s u n d e r -
-
Conversely,
if one has a spread
64
-
<s,a>, ~ a
spread law, a g (N)N
plementary law, then the species of choice sequences contained in represented as defined like
ra
w 12. Topology~
such that for all
y
am(ray)
a lawllke com<m,a>
= 84(y), where
may be r
is
in I0.6.3.
separa@le metric spaces
i2.1. In order to be able to present some mathematical applications of the principles for choice sequences, we discuss in this section the introduction of separable metric spaces in intultionlstlc mathematics,
a,S,y
are variables for choice
sequences. 12.1.I. Definition. A metric space is a pair on
<X,p>, X
X, i.e. a (lawllke) real valued function on
a)
p(x,y)
~ o, ~(x,y)
b)
~(x,y)
=
c)
~(x,y)
} ~(x,z)
The values of
p
XxX
a species,
p
a metric
such that for all
x,y,z~X
= o -- x : y,
r
+ ~(z,x).
are supposed to be choice reals.
A metric automatically induces an apartness relation given by
p(x,y) > 0 *-~ x @ y.
U(c,p) = {q : p(p,q) < r 12.i.2. Definition. <X,p> if
A sequence of points
,~X~kVn(p(p,p A sequence of points
n
n
is said to be dense in a space
n) < 2-5. is called fundamental if
AkVnAm(p(pn,Pn+ m) 9 2 -k. Two fundamental sequences are said to be equivalent
AkVnAm(p(pn+m,qn+ m) 9 p
2-k
is said to be the limit of a sequence
(notation
l~m qn : p
or
~m
qn : p)
AkVmAn(p(p,qn+ m) < 2-5.
n if
(notation
n ~ n )
if
-
Conversely,
if one has a spread
64
-
<s,a>, ~ a
spread law, a g (N)N
plementary law, then the species of choice sequences contained in represented as defined like
ra
w 12. Topology~
such that for all
y
am(ray)
a lawllke com<m,a>
= 84(y), where
may be r
is
in I0.6.3.
separa@le metric spaces
i2.1. In order to be able to present some mathematical applications of the principles for choice sequences, we discuss in this section the introduction of separable metric spaces in intultionlstlc mathematics,
a,S,y
are variables for choice
sequences. 12.1.I. Definition. A metric space is a pair on
<X,p>, X
X, i.e. a (lawllke) real valued function on
a)
p(x,y)
~ o, ~(x,y)
b)
~(x,y)
=
c)
~(x,y)
} ~(x,z)
The values of
p
XxX
a species,
p
a metric
such that for all
x,y,z~X
= o -- x : y,
r
+ ~(z,x).
are supposed to be choice reals.
A metric automatically induces an apartness relation given by
p(x,y) > 0 *-~ x @ y.
U(c,p) = {q : p(p,q) < r 12.i.2. Definition. <X,p> if
A sequence of points
,~X~kVn(p(p,p A sequence of points
n
n
is said to be dense in a space
n) < 2-5. is called fundamental if
AkVnAm(p(pn,Pn+ m) 9 2 -k. Two fundamental sequences are said to be equivalent
AkVnAm(p(pn+m,qn+ m) 9 p
2-k
is said to be the limit of a sequence
(notation
l~m qn : p
or
~m
qn : p)
AkVmAn(p(p,qn+ m) < 2-5.
n if
(notation
n ~ n )
if
-
12.1.3. Definition.
A space
<X,p>
respect to choice sequences) called a basis for
65
-
is said to be a separable metric space (with
if there is a lawllke sequence
<X,p>)
n~X
(n
is
such that
ApgXVa(~i ~ Pan = p)" 12.1.4. Remark.
We might have defined "separable" with respect to other notions
of sequence also; but in the sequel we shall consistently assume "separable" to be defined relative to choice sequences. 12.1.5. Definition. choice sequences) fundamental
A separable metric is said to be complete
if for the lawlike
<pan>n
12.1.6. Remark. Let
n
converges to a point
n
and
n
P(q~n,Pa{an,n})
n
every
p~X.
be lawllke sequences, both dense in
Then J~k~nVm(p(pm, q n) < 2-k), hence V a ~ ( p ( p a { n , k } , a fundamental sequence; then
(with respect to
of the previous definition,
qn ) < 2-k). Let
n
be
.> since a(~n,n# n' ~n.a{en,n} is a choice sequence, say S, we have
< 2 -n. Since
n ~ n . The correspondence
is equivalent with
X.
is lawllke.
In case the sequence n is lawllke relative to a choice parameter, correspondence is lawllke relative to the same choice oarameter.
the
12.2. A basis
n
for a separable metric space
(n is called discrete if theorem however.
Theorem. Let
<X,p>
find a sub-sequence Proof. Let
b
<X,p>
~nAm(Pn ~ P m V P n
<Pan>n
which is a basis for
be a lawlike function such that
a
by recursion,
and we put
aO : O. We want to achieve
= Pm ))" We can prove the following
be a separable metric space with a basis
Jb(n,m,k).2 -k - p(pn,Pm) J < 2 -k. Now we define
is not necessarily discrete
~
(Po,Pl ) < 2 -2 § al = O, (Po,Pl)
~ 2 -1 § al = 1.
<X,p>
n . Then we can
and which is discrete.
-
66
-
This can be done by stipulating b(O, I, 2) < 2 § al = 0 al = I otherwise. For
Ib(O, 1, 2)2 -2 - 0(po , pl)l < 2 -2 , hence
b(O, 1, 2) < I + 4p(po , pl ) < 2. Conversely,
(b(O, I , 2) >1 4P(PO, Pl ) - 1 1 { 1, i . e . Now suppose aO,..., ac(k - I) ~d(i A cCk-1) § al & k-l). Let that
<Jo,...,Jn >
<J~,..., J~> !
JO''''' Jm-I (a k)
such that
Jk >
I I< m - 1
so
as follows. Take J~
~_ i
such that p(pj~ , pjm ) < 2 -k-1
i ~ m-i o(Pjl , pjm ) { 2-k, then
Jm = Sm "
: 0 & I & m} by d.
Jm = Jm-1
p(Pj~ , pj ) < 2 -k-1
for an I & m-1. Then
Ib(Jm, J~, k+l) 2 -k~l - 0(pj~ , pjm) I < 2 -k-i, therefore b(Jm, J~, k + 1) < 1 + 1, so
Then
suppose
b(Jm, J~, k+l) & I, hence
p(pj~, pjm ) { 2 -k
b(Jm, J~, k + 1) 9 I
for all
This shows that our construction of
for every
i & m-l, hence J~
for
such that
Jm = Jm-1"
I f for every
such
= k~
J~ = Ji
Then we put
Conversely,
Ok
!
Denote Inf {b(Jm , J~, k+l)
Suppose
step, such that
be an ordering of the elements of
n I < n 2 § Jn+nl < Jn+n2
from <Jo''''' Jk >
If there exists an
k th
{aO,..., ac(k-1)}
have been chosen, we want to choose
then
(b k)
implies
2) 9 1 .
be an orderlng of the elements of
{0,..., k} - {Jl,.., Jn }
If
b(O, l s
<Jn+l'''''
implies
P(Po' Pl ) ~ 2"I
to be constructed after the
hi< n 2 * in1 < Jn2 . Let
Construct
0(po , pl ) < 2 -2
satisfies
d
t <m-1. d > 1.
(ak), (~).
i An.
-
67
-
Finally we take ' a(c(k-l) + I) = Jn+l Apparently
for
1 A i
k-n.
c(k) = c(k-1) + (k-n).
Now we prove the discreteness
of
<PaO''''' Pac(k) >
k. Suppose the discreteness
for every
to be proved. From the fact that follows that PJm
in case
a
Jm(Jm ~
<Pan>n
by proving the discreteness
(ak) , (b k)
{Jo''''' Jm-1 })
<Pao''''' Pac(k-l) >
are satisfied at every step, it
is included in
lies apart from every element of
<PaO''''' Pac(k) >
of
of
{J~,..., J~} only PJm-1 }. Therefore
(PJ0 ' ' ' "
is discrete.
Thus we obtain the desired conslusion by ordinary induction. There remains to be shown that <Pan>n is a basis. In view of 12.1.6 m it is sufficient to show <Pan>n to be dense in the space. To see this, we remark that for every
I < k
~(0 & ~ & c(k)) at the
k th
So if we take
either
PI~{Pao'''''
such that
Pac(k) }p or there exists an
~(Pi' PaL ) < 2"k+l
step, there exists a
J~
(for if
such that
i ~ {J~,..., J~}, i ~ k
o(Pj~ , Pl ) < 2 "k+l, m ~ k).
p(p, pn ) < 2 -k-l, then we can effectively find a
~ c(k+2), such that
0(pn , paL) < 2 -k-l, hence
PaL'
o(P, PaL) < 2-ko Q.e.d.
12.3. 12.3.1. Theorem. Every complete sepaFable metric space represented by a spread.
r
=
(IT 1966], 2.2.1)
<X,
can
o>
be
Proof. Let n be a discrete basis for r, and let <s(k, n, m)> m be a lawlike enumeration of all indexes i (possibly with repetitions) such that 0(pn, pi ) < 2 -k. We may suppose s to be chosen such @hat Pn = Pm - * A k A I
(s(k, n, i) = s(k, m, I)).
~hls is not essential but technically convenient.) taking a = Xx.l, and fixing ~ by
~
Now we define
5
=
= Pko
= Pm § r
Then Just as in 11.3, S
represents
r.
kn-l' k> = Ps(n, m, k)
by
-
68
-
12.3.2. Lemma. Let r = 9 p9 be a complete metric space, then the spread representation indicated in the proof of 12.3.1 has the following property: AxgxV~(lim
r
: x ^ A k ( U ( 2 -k, x ) ~ W ~ k ) )
where W n = {y : V S ~ n ( l i m Proof.
Take a certain
9qn~n ~_ 9
such that An(p(qn'
so
x~ X
$~8 = y)}.
and let
y
be such that
p(x, y) 9 2 -k-2. Construct
~n(e(qn , x) 9 2-n-2). Then apparently
qn+l ) ~ P(qn' x) + P(qn+l' x) 9 2 -n)
r
Vk-l> = qk-1 " Clearly,
p(y, qk.1) ~ ~(x, y) + o(x, qk.1) < 2 -k-2 + 2"k-i 9 3 .2"k-2 9 Then we can find An(p(q~,
(q~>n ~
9
y) 9 2-n-2), hence
Then the sequence
9
such that
= y'
~(q~, qk_i ) 9 2 -k.
qi''''' qk-i' qk' qk+i" q k + 2 ' ' "
verified. This sequence is equal to Therefore
lim 9
~$8
9 E S, as is easily
for a suitable
8, with
~k = 9
y&W 9 "'''Vk-l>
Q.e.d.
12.4. 12.4.1. Definitlon. A topological space is a pair species of V such that
(b)
Finite intersections to
7
,
~a
species of sub-
and arbitrary unions of elements of ~
again belong
"
This definition is exactly the same as one of the well known classical definitions. 12.4.2. Definition. f is called a continuous mapping from a topological space into a space if f is a mapping (V)V' such that
The definitions of homeomorphlsm, given accordingly.
bamis of open species, and nei~hbourhood ' are
A point p is said to be a closure point of X ~ V , if (every neighbourhood of p contains a point of X).
AW6 ~(p~W
* Vq(q6X~W))
9
-
X'~the closure of
X
69
-
consists of the species of closure points of
X.
So far, all things look very much the same as in classical topology. But we must be aware of the fact that classically equivalent definitions are not necessarily intuitionistically equivalent. For example, the notion of weak continuity, characterized by:
Axgv,
(r'1[x'] - : f-l[x-])
(the complete original or counterimage of a closed set is always closed) is classically equivalent with, but intuitionistically weaker than continuity (IT 1966], 2.1.8). With a metric space
~
we may associate in the usual way a topological space
Ap6xV,,O
^qev
(p(p, q) 9 c ~ q e X ) .
Or if we put U(r
p)
= {q : oCP,
q) 9 r
then
The r
U(r
p)
constitute a basis for ~ ( ~ ) .
Usually we shall not distinguish between
and
when the meaning
is clear from the context. 12.5. The results mentioned in this subsection may serve as an illustration of the more typical questions of intuitionistic topology. i2.5.1. Definition. A space r = ~ V , ~ > complete, metrizable (i.e. ~ = ~ ( p ) , 9 representable by a finitary spread.
is said to be located compact if complete, for a suitable e)
r is and
12.5.2. Definition. A space r = ( V , ~ , is said to be locall~ compact, if r is complete, metrizable, and if every p ~ V possesses a located compact neighbourhood U(i.e. U is located compact in the relative topology ~' = { X ~ U : X e ~ } induced by ~ ). Closed intervals
[a, b]~__ ~ ,
a ~ b
are examples of located compact spaces;
-
70
-
, the species of (choice-) reals is an example of a locally compact space. Arbitrary pointspecies in a topological space may be defined very nonconstructively; so one feels the need for a subclass of pointspecies for which some extra information is available. The notion of a located pointspecies turns out to be useful: 12.5.3. Definition. Let r = < V j y > X~V is said to be located, if Ap&VAU~
be a topological space.
q'~(p~U § {Vq(q~U,',x) ~, V w 6 ~ C p G W
i2.5.4. Definitlon. A species metrically located if
X~V,
^wr~x = ~
)}).
a metric space, is said to be
p(x, X) = inf (p(x, y) : y ~ X }
is defined for every
x~V.
The significance of the notion of locatedness becomes clear from the following result: 12.5.5. Theorem. Let r = < V , ~ ( p } ~ be a located compact space. Then an inhabited Xm.V is located iff X is metrically located. (Proof e.g. in IT 1988 A], 3.14(a)). So metrical locatedness is a topological notion. In general, this is not the case; e.g. ~ can be metrized in such a way that a certain located pointspecies cannot be proved to be metrically located. But the following theorem holds: i2.5.6. Theorem. To every locally compact space < V , ~ > we can find a metric p such that ~ = ~ ( p ) j complete, and every located inhabited species X ~ V is metricall~ located with respect to ~ (the converse is trivial). A proof may be found in IT 1968 A], 4.6. 12.5.7. Definitions.
Let
be a topological space, X ~ V ,
Interior X = ~ t
X = {p : Vr
U(r p ) ~ X } .
X ~ Y ~m AP ~ V ( p ~ Y v p ~ X ) is the analogue of the classical relation X" ~ 12.5.8. Theorem. Let located, X~Y
iff
~V,0>
Y~V.
Int Y.
be a complete separable metric space. Then if
X" ~ I n t
X
Y.
Proof. The proof is contained in IT 1968 B], and uses continuity principles for choice sequences in an essential way.
Is
-
w 13. App!Ication~of
71
-
the c ontinulty principles
and the fan theorem
13.1. We start wlth some applications In its simplest form
A~Vx x(~, x, % , . . . ) Proof-theoretlcally,
Def!nltlon.
mapping from
V
Let
n~___V
wlth a limit
fp.
13.1.2.
Theorem.
r
metric space,
f
Let
~'>
consequences
and let
Is sald to be sequentlally wlth a limit
be a separable
a mapping from
r
.
Is not an essential strengthening
r = , r' =
V'. f
n
for choice sequences
= j y § x(B, x, % , . . . ) )
[Ho,K 1967]) but Its mathematical
Into
verging sequence
§
thls principle
(as follows e.g. from and rather elegant. 13.1.1.
of the continuity principle
p
f
are interesting
be a (lawllke)
continuous
If every con-
Is mapped onto a converging
complete metric space,
Into
of analysls
r'. Then
f
r'
sequence
a separable
Is sequentially
continuous.
Proof. See IT 1967]. This theorem Is contained In the stronger theorem 13.1.6 given below. 13.1.5. space.
Theorem. If
f
continuous, Proof.
Let
r'
be a separable metric space,
Is a (lawllke) mapping from then
f
([T 1967]).
r'
Into
and let
r"
and
F" f
be a metric
Is sequentially
Is continuous. Let
n be a lawllke sequence of points dense in r' = . Let r" = <W,0'>. Let x be an arbitrary point of r' and suppose llm n = x. Now we want to construct i such that p(pB{n,l},X)
O(pl, The construction From
n
< 2.2 -n
for every
II
(1)
x) ~ 2 -n § Pl ~j
proceeds
as follows.
we obtain effectively
n , llm n
= x, such that
An(p(pyn J x) < 2-n). Let
b(n, m, k)
be a (lawlike)
An~r~k
enumeration
of rationals
Ip(pn , pm ) - b(n, m, k) I < 2 -k.
such that
-
?2
-
We have b(i, v(n+2), n+2) 9 6.2 -n-2 v b(i, v(n§ If
0(pi , x) ~ 2 -n, then
Ip(pi, py(n+2))
P(Pi' Py(n+2) )
n+2) ~ 6.2 -n-2.
5'2"n-2; also
9
- b(i, v(n+2), n+2) I 9 2 "n-2, therefore
b(i, v(n+2), n+2) 9 6.2"n'20 If
b(i, y(n+2), n+2)
9
6.2 "n'2, then
0(pi , pv(n+2))
7.2 "n-2, so
<
P(Pi' x) 9 8.2 -n-2 = 2.2 -n. Therefore we may take for
i
b(J, y(n+2), n+2) 9 6.2"n-2; then tinuous operation from
~)).
Now we define a spread
<~x.i,+>
an enumeration of the
(I)
is satisfied.
for every
Furthermore,
a'@*s
Ps(n,in}"
is obtained by a con-
9
is a fundamental sequence in
for any fixed natural number
A6VnAm(P'(fPs{n+m,
such that
by stipulating
$ =
(6
J
P'
converging towards
~ :
6(n+m)}' fx) 9 2-v),
hence with an application of the continuity principle we find, for any n, n' such that A6(~'n'
~n'
=
+
Am(p'(fps(n+m ' ~(n+m)}'
Without restriction we may suppose Ai(p(pi,
2-v)).
<
n' = n. From this it follows that
x) < 2 -n * o'(fPi' fx) < 2-v).
For whenever
P(Pi' x) 9 2 -n, there is a
Now take any
y
n_~
fx)
6',
such that such that
f is sequentially Therefore.
6
with
~'n = ~n,
6{n, 6n} = i.
~(x,y) < 2 -n. Then there is a sequence Am(~(p6m , x) 9 2"n), l~m P6n = y'
continuousj hence for a suitable
k
P'(fP~k'
fy) < 2"v"
p'(fy, fx) $ D'(fp6k, fy) + D'(fp~k, fx) 9 2 "~ + 2 -~ = 2.2 -~ and thus Ay(p(y,
x) 9 2 -n * ~'(fy, fx) < 2.2 -v)
.
q.e.d.
x.
- 73
13.1.4. Theorem. Let Let
r
:
r
be covered by a species
-
be a separable, complete, metrizable space.
O~
of pointspecles; then
r
Is also covered by
the species of interiors of these pointspecies. Proof. Let us use Then we have By 12.3.1
W
hxgV
as a variable for polntspecles. VW(W60~^ x@W).
there is a representation
~x.l,r
for
r.
Hence we also have:
Now we apply (13) of 10.7
and we obtain
A~VnVWAs(~n : ~n § llm r We then use lemma 12.3.2. Take any
x, and find lim r
We have an
n
Hence
W6~
: ~n
U(2 -n, x)_~W, so
This proves
AxEV
such that
: x ^ A k ( U ( 2 -k, x ) ~ W ~ k ) .
and a
As(~n
~
such that
§ lim
~
86W).
x &Int W.
VW(W~^x~Int
13.1.5. Remark~ In case O ~ w a s
W). a subspecies of
N, it would have been sufficient
to use (3) of w 10 instead of (13). We remark that the lawllkeness of ~
and the elements of ~
is essential.
For consider the covering by species of one element; then the interiors do not cover
rl The reason is
that arbitrary points are not lawlike. The theorem may be
generalized however to the case where the elements of O ~ contain choice parameters from a fixed finite collection, From 13.1.4
one also obtains
13.1.6. Theorem. let
r
be a complete, s e p a r a b l % m e t r i c
separable metric space; then any mapping from Progf. Either we may combine 13.1.2
r
into
i : i~N},
r"
r"
be a
is continuous.
and 13.1.3, or we deduce the theorem directly
form 13.1.4, by looking for any fixed natural number (W
space, and let
defined as follows. Let
n
~
at the coverings
be a basis for
r" : ; then
- 74 -
W , i = {y : y ~ r ^ P(Pi, fy) <2-~} " Then an application of i3.1.4
readily yields continuity of
f.
i3.2. Intermediate in strength between (i3) of w iO and the correspom44ing stronger form A~ VXa~(~,X a) § VeAn(en ~ 0 §
VxeAs&nY(s,Xa))
we have As VXaY(s,X a) + V a [ A n ( a n
~ 0 §
VX~As6nY(m,X
a))^A~vn(a~n
~ 0)] .
I f we apply this principle instead of the weak form of continuity, one obtains an intuitionistic version of Lindel6fs theorem: 13.2.1 Theorem. Let r = be a separable, complete, metric space, and let r be covered by a species ~ of pointspecies. Then there exists an enumerable subspecies ~ ~ such that {Int W : W G ~ } is a covering for r. Proof. The proof can be given by remarking that if As~q~lim and if a is such that then there is a species
~sG
AsVn(a~n ~ 0), An(an ~ o § We[nJ with parameter n such that
An(an ~ o § W ~ [ n ] & ~ The collection
~=
w
{W~[n]
^A~6n(lim
: an ~ O}
6 n(lim Ct= ~ W)),
Cea6W*[n])).
covers
r; then apply 13.1.4.
13.3. We finish this section with applications of the fan theorem. Very well known is i3.3.1. Theorem ([H 1966], 3.4.3). A real-valued (lawlike) function on [-I, I] is uniformly continuous. Proof. Let us use the spread 11.5.
S 3 =
, f
S3
described in 11.4, with the theorem proved in
is a real-valued lawlike function on
Let ~ be a fixed natural number, and let any fixed v 6 N : A=6a3Vmlf
llm ~
-m.2-vl
S
be the undressed spread
< 2-v
or
A=Vmlf lira r
c= - m.2-vl
I-i, 1].
<2 -v.
a3 Applying the fan theorem yields:
VnA~VmAB(ra3~ n = ra3's n § If llm Cmra38 - m.2-ul
< 2 -v)
a 3. For
-
75
-
or equivalently VnAa6a3VmAsga3(~n Suppose
n
and let
x,y
= ~n § If lim ~es - m.2"vl
to be such that
Aa~a3VmAs~a3(~n
be such that
lim Cea = x, lim r
Ix-yl
= y,
{W~n
: ~a
{[(k-l)
Let
m
3}
2 -n-l,
= ~n * If lim ceB - m.2-vJ
Then we can flnd
~n = ~n, since It follows
W~n = {lim r then
<2 -n-i.
< 2-v).
: ~n = ~n ^ B 6 a
Is exactly (k+l)2-n-1]
a,S@ a 3
from 11.5
< 2-v),
such that
that if we define
3}
the collection : -2 n*l < k < 2 n*l}
.
be such that
A~6as(~n
=
~n
§
If llm ~*~ - m.a-vl
<
2 -v)
then
Ifx If
13.3.2.
- fYl nm
~
=
If
llm
- m.2-~l
~em - f l l m *
If
nm
~sl
,~B
If we adapt the proof of 13.1.4
form of Heine-Borel
for
~I,
form of Helne-Borel
for every
- =.2-~1
compact
Q.e.d.
9
we obtain a proof
1] . More generally, located
< 2-~.1
of the int~Itlonlstlc
we can prove an intultlonlstlc
space.
See
EH 1966]
5.2.2.
-
w 14. Well-orderings.and
76
-
ordinals
14.1. In this section we dSscuss a simplified version of Brouwers theory of well-orderings and ordinals
([Br 1926],
[H 1959]).In Brouwers original definition,
the type of a
well-ordered species may become arbitrarily high, and is in fact itself characterized by an arbitrary well-ordering.
Therefore we shall use a standard coding of well-
orderings by means of species of natural numbers, in order to avoid such conceptual difficulties. 14.1.I. Definition.
Let
XO,... , Xn~__N. We define operations
+, Z, 9 :
X 0 +...e X n = { ~ x : 0 .< i .< n ^ x E X i} ; as a special case we have Xo+
=
{<0>
~
x
x 6 X O}
:
.
We write 9 X
or < x ~ X
for { < x ~ y
: y~X}
.
The infinite sum is defined by X 0 + X I +... = ~X I = { < l > ~ x XI.X 0 = XIX 0 =
{n@m
: I~NAxEX
: n~X OAmEX
I} 9
I} 9
Now we introduce a class of species, WO, by a generalized inductive definition (analogous to the introduction of 14.1.2. Definition.
Every
F@WO
K
is section 9).
(WO = the class (species) of well ordered species)
is constructed by means of the following principles: (a)
(b) (c)
(o}
~
wo.
FO,... , F n ~ W O n
§
Fo+~WOAPo
lawllke, A n ( F n ~ W O ) ,
14.1.3. Convention. elements of WO.
+'''+ F n ~ W O "
then
~Fi6WO.
In this section, we reserve the capitals
Corresponding to the generalized inductive definition of 14.1.4.
Principle of proof by ~nduction over
If the following hypotheses
(a), (b), (c)
WO.
are satisfied
W0
F, G, H, J
we have the
for
- 77
(a)
{0}~
-
X,
(b)
FO,... , F n ~ X
(c)
An(F n6x),
§
Fo+6XAFo+...+
F n ~ X,
n lawlike § Z F n ~ X . n
then WO~X. Just as in the case of ordinary induction, we may Justify definition by induction over
WO
14.1.5.
by the principle of proof by induction over Definition.
Let
n = 9
n~m
~-+ Vi(i ~ r ^ i
WO.
~ is an ordering on the natural numbers defined as follows.
, Xr> , m = 9
Ys ~ "
~ s ^x i 9
^Aj
9 i(x i = y~)) v (r , s A A J
~ s(xj = yj)).
This ordering figures in the literature as the so-called B r o u w e r - K l e e n e between
finite sequences
is a refinement n
9
m ~-+Vn'(n'
of natural numbers.
of the partial ordering on sequences defined by ~ OAn~n'
= m).
is a total ordering, hence that
FO+...+ Fn, ZF n
sense.
In the sequB1,
standard ordering ~IF 14.1.6. X ~ Y
Definltlon.
correspond
induces a total ordering on any
(restriction of
If
X, Y
r
i4.1.7.
~
to
are order preserving. WO
Remark
FGWO
refers to the
F).
are two (partially)
of
FGWO.
to the formation of ordered sums in the usual
any t e r m i n o l o g y i n v o l v i n g order on
The following properties
(I)
~
ordered species, we write
to indicate the existence of a one-to-one m a p p i n g
such that
ordering
(In short:
r
r
from
X
onto
Y
is an order isomorphism.)
are easily proved by induction over
WO:
Theorem.
Every
F
is decidable.
(II) Every
F
possesses
(III)For all
x~F
a first element.
(FgWO)
is a last element
of
(IV) ~ F A n ~ m ( n ~ F ^ n @ m 6 F AF~aAxAy (~x~FA~y~F
either there is an immediate successor in
F
or
F. § m = 0). This property is more simply expressed as § x = y).
The proofs are left as an exercise to the reader.
-
14.1.8. Theorem For any species
78
-
(Transfinite induction with respect to X~_~F:
41F).
A x ~ F [ A y ~ F ( y ~ x § Xy) § Xx] * A x ~ F ( X x ) . Proof. By i n d u c t i o n over
W0 with r e s p e c t to
F. For
F = {0}
the theorem i s
evident. Suppose now the assertion of the theorem to be valid for From this hypothesis we prove for any n (by ordinary induction)
F1, F2, F3, . . . .
A X 6 F o + . . . + Fn[AY6Fo+...+ Fn(Y ~x -* Xy) * Xx] § A X e F o + , . . + Fn(XX). For
n = 0, this is immediate. Ax6F0+...+
Suppose
Fk+I[~Y6F0+...+
(1) to be proved up to
(1)
n = k, and let
Fk+l(y ~ x § Xy) ~ Xx].
(2)
This implies AXEFo+...§
Fk[~Y~Fo§247
and hence by our ~ d u c t i o n AXs
Fk(YlX
~ Xy) ~ Xx]
hypothesis
Fk(XX).
Now take an arbitrary As a consequence
(3) z ~r
+ I) @ Fk+ I .
of (3) we have
A Y e F o + . . . + Fk+l(Y.~Z * Xy)'~, A y 6 , k
+ 1 , @ F k + l ( y . ~ z -* Xz),
therefore from (2)
Az ~ ~ F k + l [ ~ y ~ . ~ k + 1 , ~ F k + i ( y ~ z
-, Xy) § Xzl,
and this yields, since the theorem was supposed
to hold for
Fk+i,
Az 6
Fk+l(Xx).
Now if F = ZFi, then any x ~ F the previo-siargument Xx.
belongs to
FO+...+ F k
for some
k, hence by
14.1.9. Theorem. Let F be any well-ordered species, Every strictly descending (w.r.t ~) sequence contains an element outside F.
- 79
-
Proof. Take xx to be: every strictly descending contains an element outside F. Clearly Ax E F~y hence
beginning
for an arbitrary
<Xn, n
with
x0 = x
6 F(y ~ x § Xy) § Xx],
(by the previous
sequence
sequence
theorem)
Ax ~ F(Xx),
with an element
strictly
of
descending
F
i.e.
contains
sequence
every strictly an element
descending
outside
F. Since
x 0 ~ F v x 0 6 F, this proves
our
theorem. 14.2. 14.2.1.
Theorem.
G(Z Fi)= z GFi~ G(Fo+...+F n) = GFo+...+GF n i i
Proof.. immediate. 14.2.2.
Theorem
~ F ~G(.WG ~ WO).
Proof.
by induction
14.2.3.
Defl nition. F[u)
= (x
on
WO
with respect
Fer any
u, v E F
to
F, using the previous
theorem.
we define
: uix^x~F),
Flu,v) = : u i x Iv ^ x E F(V) = (x : x ~ V ^ X ~ F}, where F(v),
u ~ v
FKu,v) , FKu)
or an element For
u ~v,
14.2.4. Let (a)
Is an abbreviation
of
FKu,v)
for
are detachable
subspecies
= FKu) F) F(v).
Definition
§ ~u v
= v. F. For any
of
v, F(v)
is empty
WO.
~u
F = F(u) %2 Flu).
is a mapping
u = <Xo,... , xt, Xt+l,... , Xt+s> v~u
u ~vvu
defined
as follows.
= <0,...,
Xt+l,... ; Xt+s >.
= v
(b) u_~, for some
i .< t
Yi > 0 § Su
yr > =
Yr >"
(c) u ~ < Y o , . . . , Yr'' t < r < t+s, Y0 = ''" = Yt = 0 § ~u
(d) If ~u 9
u ~
Yr>' t+s
''''' Yr > =
Yt' Yt+l - Xt+l''''J
Yr - Xr>'
9 r § Yt' Yt+l - xt+IJ'''s
Yt+s - Xt+s*
Yt+s+l'''''
Yr >"
- 80-
It may be helpful to the reader to verify by representing
a simple finite
a construction
Cu
tree
(see the picture below) what exactly
0
F
by
does.
F = F0 + F I + F2 F 0 = FO, 0 + FO, I
<010~.... <0,0}0>
<0,i~... <0,0,I>
<1,1><2,0>
<2,1>
<2,2>
FO, 0 = FO00 + FO01
<0,1,0>
FO00 = (0}
(The terminal nodes of the tree represent
the elements
of
We use in the remainder of thls section S,T as letters and species Flu), F(u), F [ u , v ) ( u , v 6 F , F E W O ) .
F.) for species
We introduce a mapping 0 from species Into specles;, Is defined If u is the first element of S, then *S = {r x : x ~ S}. Clearly
AS(@S g WO))by induction),
and
... ~ F(nl)
+
of
Definition.
(Equivalently: 14.2.7. (I)
F < G
for any
S.
of
~nl> I b e a strictly
~F[n2,n3)+.. .
F, hence
F(n 1)
Is
~D Vx(x 6 G A F ~ G(x)).
F < G ~D VH(F+H ~ G).
Theorem.
If r is an order-lsomorphlsm and r Is the identity.
(II) AF A G A x , Proof.
WO, and let
@F[nl,n2 ) +
Proof. Since n o < nl,n I Is not the first element inhabited. Otherwise the proof Is routine. 14.2.6.
WO
AS(S ~ 0S).
14.2.5. Theorem. Let F be an arbitrary element increasing sequence of elements of F. Then F = F(ni) v F[nl,n 2) ~ F[n2,n3)
of
between
F
and a subset
y 6 G(F ~ G(x) a F ~ O(y) ~ x = y).
(I). It Is possible
to prove
A x E F(Ay E F(y ~ x § Cy = y) § r
= x).
X~
F, then
X = F
-
For suppose
Cx ~ x A A y
r 1 6 2 1 6 2 = r162 Then
r
and o b t a i n
Let
to the wlth
conflicts
Ax ~ F(r
(I).
for
Deflnltlon.
14.2.9.
x
~ y; h e n c Z x
theorem
F 9 G
r
n
r162 = Cx, h e n c e
~ x. S u p p o s e
would
Therefore
from
from
x ~ Cx.
be an i n f i n i t e
descending
Cx = x. Now we a p p l y
F ~ G(x), G(y)
14.1.8
onto
F ~ G(y) G(x),
we c o n c l u d e
In c o n t r a d i c t i o n
= y.
we d e f i n e :
It f o l l o w s
that
If
thls
F ~ G(u),
definition
then
G-F
~D CG[u).
Is unique.
Theorem. F ~ F',
(II)
F < G § F + (G - F)
(III)
F < G r H.F
(IV)
F 9 G § H.(G
G ~ G' § F . G ~ F'.G'.
Proof.
(I) Is e v i d e n t ,
proved
simultaneously
H.F HF
14.2.10. Then
then
isomorphism
(I)
Hence
or
14.1.9.
F ~ G(y),
F o r any
F r o m the p r e c e d i n g
n, and so
wlth
of an o r d e r
Likewise
Cx = x
: x).
x ~ y, F ~ G(x),
existence
14.2.8.
for e v e r y
and thls
-
g F(y < x + Cy = y); t h e n a p p a r e n t l y
Cx = x. T h e r e f o r e
< r
sequence;
(II).
l.e.
81
< HG,
< H.G. - ~) ~ H . G
(II)
f r o m the
definitions,
(III)
and
(IV)
are
as f o l l o w s .
H(G-F)
lim H n = z H' n n n
- H.F.
follows
+ H.(G-F)
Deflnltlon.
~ G.
~ H.(F
~ HG-HF
Let with
(with
n H6
+ (G-F)) 14.2.8
~ H.G II).
be a l a w l l k e
= HO,
.
sequence
H'n = H n - H n _ I
such that
for
A I ( H I < HI+l).
n 9 1.
14.2.11. Lemma. A n ( H n ~ Jn ) § lim H n ~ lim Jn" n n G . llm H n ~ llm G . H n n n
(II)
Proof. (II)
(I)
is i m m e d i a t e 9
O . llm H n : G(H 0 + (H i - No)
+ (H 2 - H1)
+...)
=
n
G . H 0 + G . ( H 1 - HO)
+...
~ GHO + (GH 1 -
GHo)
§ (GH 2 -
GH1)
+...
= llm n
In v l e w contains
of the n e x t exactly
one
definition or m o r e
we r e m a r k
t h a n one
that
element.
It is d e c i d a b l e
whether
an
F
GHn .
-
14.2.12.
Definltlon.
For any
8 2 -
F, G ~ W0
we define
F G = F exp G
inductively
as follows 9 For
F ~ {0}
we put
FG = F
for any
(a)
F {0}= F .
~b)
F exp (G O +...+ Gn) = F
(c)
F exp Z G I = llm F 0 i
GO FG1
m 9149
Lemma.
lawllke
sequence
Let
F
Gn 9
n
be a strictly
of natural numbers,
A n ( H n < Hn+1).
increasing
and let
n
(In the natural
ordering)
be a lawllke sequence
such
Then
llm H n ~ llm Hk(n) n
.
n
Proof. We remark that If since
we put
Gn Oe.
F
F , {0}
n
14.2.13. that
G. For
F 1 < F 2 < F3, then
F 5 - F I ~ (F 2 - F I) + (F 3 - F2) ,
F i +((F 2 - FI) + (F 3 - F2) ) ~ (F i + (F 2 - F1) ) + (F 3 - F2) ~ F 2 + (F 3 - F2) ~ F 3
Repeated Hk(n+l)
application - Hk(n)
of thls result yields:
~ (Hk(n)+l
- Hk(n))
9 ..+ (Hk(n+l) One verifies
~l(Gi
obtains
Z G i ~ Z Jl" Hence by (4) I i
14.2.14.
Lemma.
sequence
of natural numbers,
(I) If
n
- Hk(n)+1)
+...
(4)
- Hk(n+l)_l )) .
easily that from
Let
+ (Hk(n)+2
~ Jw(i)
+'''+ Jw(i+l)-I )' w(O)
we conclude
be a strictly and let
n
to
llm H ~ llm Hk(n) n n n
increasing
(In the natural
be a lawlike sequence.
~ n ( H n < Hn+1) , F 9 {0}, then llm F exp (H0+...+ Hn) ~ llm F exp (Ho+...+ Hk(n)). n
(II) Let
n
A I ( H I = Gk(i)
+...+ Gk(l+l)_l),
k(O)
= 0. Then
F exp Z G I ~ F exp Z H i . I I (III) Let
~ l < n ( H I = Gk(1)+...+JGk(l+l)_l) , k(O) H n = ~ Gk(n)+j
. Then
F exp
= O, one
= O, and
IZ G I ~ F exp(Ho+...+
Hn).
.
order)
-
83
-
Proof. F 9 (0}, F G > {0}. Hence
(I) For F exp
(H 0 +...+ Hi).
AI(F exp (H 0 +...+ Hi+ 1) >
Then apply 14.2.13.
(II) F exp Z H i = lim F exp (H 0 +...+ H i ) = limF exp(G 0 +...+ Gk(n+1)_l) I i i lim F exp (G O +...+ Gi) (by (I)) ~ F exp Z Gn. i n (III) F exp (H 0 +...+ Hn) F exp
= F exp H 0 . . . . .
(G O +...+ Gk(n)_1)
F exp H n =
F exp Z Gk(n)+j
=
J F exp
(G O +...+ Gk(n)_l)
lim F exp (G O + . . . +
lim P exp (Gk(n)
J
Gk(n)+j)
(14.2.11,
+...+ Gk(n)+j)
II)
~ l l m P exp (G O + . . . + J
J
= F exp
Gj)
Z Gj (by (I)).
J 14.2.15.
Definition,
H
is called a refinement
of
F
(notation
F = H) if
+...+ H n (H = Z Hi) ' H i = r i = 0r... , n i (i ~ N resp.), F = S O U . . . u S n (F = n ~ S n resp,)~
H
= H0
AIAJAx 6 SlAY E Sj (i < J § x ~y). 14.2.16.
Remark.
The relation
14.2.17.
Lemma. When
=
is evidently not symmetrical.
G = H, then
F G ~ F H.
Proof. We prove this lemma by induction
over
distinguish
step.
four cases in the induction
Case I. Let Let
Hi = r i
Si =
for
to
G. We
i ~ n, S O U . . . u S n = F. Ai AJAx 6 s i A y ~ sj (l<J ~ x~y).
TO,... , T k
such that
<xi>@Ta(i) v . . . a(O)
v < x i + y i > ~ Ta(i+l)_l , i ~ n,
= O, Yi = a(i+l)-
G I = Tb(i) u . . . U T b ( i + l ) _ l CTi = Ji
for
<1> 9 eTa(i)+1 Gi = Jb(i)
a(1)-l, A l ( a ( i ) < a ( l + l ) ) .
, I ~ m, b(O)
0 ~ i ~ k. Then
~(<xi>eTa(i)U<x
Also
with respect
G = G O +...+ Gn, G = H 0 +...+ H m .
Now we can find
Take
WO
+'''+ Jb(i+l)-l'
b(l+1)).
Hi = r i =
i + 1> 9 T a ( i ) + l U . . . ) ... = eTa(i)
= O, A l ( b ( i ) <
= 9 C T a ( i ) ~
+...+ eTa(i+1)_1
i < m, b(O)
= O.
= Ja(i)
+'''+ Ja(i+l)-I
'
-
Now, using our induction F G = F GO . . . . .
hypothesis
84
-
with respect
F Gn ~ F J0 . . . . .
to
F Jk ~ F HO
GO,... , Gn:
.....
F Hm = F H.
~s~_~. G = G O +...+ Gn, G - z H i . Dy a reasoning analogous the previous case, we can find iJi, i E N such that Gi = Ja(o) +'''+ Ja(l+l)-1'
i < n, a(O)
to the argument
= O, Al(a(1)
in
~ a(l+l)),
Gn = ~ Ja(n)+k Hi = Jb(i) Then,
+'''+ Jb(i+i)-l'
with the use of the induction F G = FGo ... F Gn F exp (Ja(o)
Case Ill.
9 b(i+l)).
F exp Ja(n)-I
+'''+ Ja(n)-I ) limk F exp
(Ja(n)
Ja(n)+k ) ~ lim F exp
F exp Z Ja(n)+k k +'''+ Ja(n)+k) ~
(JO+...+
G = Z Gi, G = H 0 +...+ H n = H. Now we can find i
Jk )
Jl, i ~ N, such that
Gi = Ja(i)
+'''+ Ja(i+l)-l'
i ~ N, a(O)
= O, A l ( a ( i )
< a(i+l))
HI = Jb(1)
+'''+ Jb(i+1)-l'
i < n, b(0)
= O, A i ( b ( i )
< b(i+l))
"
Then F G = F H Is proved the previous case.
from the induction
hypothesis
G = z Gi, G = Z H i = H. Now we can find i I GI = Ja(i) +'''+ Ja(i+l)-l'
a(O)
in the same manner as in
Ji' i ~ N, such that
= O, i ~ N, Ai(a(i)
HI = Jb(1) +'''§ Jb(l+l)-l' b(O) = O, i 6
N, Al(b(~)
< a(i+l))
< b(•
ete.etc.
14.2.18. Theorem. Proof.
for
G ~ H §
By induction
Then to every
~
14.2.14).
Hn = ~ Jb(n)+k
Case IV.
= O, Ai(b(1)
hypothesis:
F exp Ja(o)''"
lim F exp (Ja(o)+...+ k
(14.2.11 ( I I ) ,
i ~ N, b(O)
Gi
over
FG ~
WO
F H. wlth respect
we can find a
i < n, Jn : CH[Vn)'
Jl ~ Gi
Ji'JO for
to
G. Let
= cH(Vo)' i ~ n.
G = GO+...+
Ji = CH[vi-1'
vi)
Gn .
- 85
FGo
... F
Gn
~
FJo
So
FG =
If
G = Z Gi, we can find I
Jo = ~H(Vo)'
... F
Jn
FH
~
-
(by the previous
Ji' i ~ N, with
Ji ~ Gi
lemma,
for every
since
H = Jo+...+
i,
Vl) and H a z Jl" Therefore FG = I lim F GO +'''+ Gn = l~m F Jo +. ..+ Jn = F exp z Ji ~ F exp H n i
~y
Jl = ~H[Vl-1'
the previous
14.2.19.
lemma).
Theorem.
G ~ H § G F ~ H F.
Proof. by induction i4.2.20.
over
WO
with respect
to
F.
Th.eorem. F exp GH ~ (F exp G) exp H.
Proof. We apply induction
over
WO
with respect
to
H.
F exp G. {0} = F exp G = (F exp G) exp {0}. Now let F exp G(Ho+...+ Hn)
= F exp
H = HO+...+ H n. ~nen
(GH 0 +...+ GH n) H0
F exp GH 0 . . . . . Let
H = z H i . Then i
F exp GH n ~ (F exp G)
Hn ... (F exp G)
(F exp G) exp H.
F exp G(z H i ) = F exp r GH i = i i
l~m F exp (GH 0 +...+ GH n) ~ l~m(F exp G) exp H0
(H O +...+ H n)
Hn
l~m (F exp G)
... (F exp G)
= (F exp G) exp H.
14.5. 14.3.i. Definit%on. Ax 6 G (F x ~ W0)
We define
by induction
~ Z{F x
an ordered over
{F x : x g G}
with respect
G:
(a)
G = {0}
(b)
G = G O +...+ G n § Z{F x : x 6 G} = Z(F,o, t y
(C)
G = Z GI§
Z{F x
: x G G}
WO
sum of a species
= F0 +
: y ~ GO} +'''+ Z { F , n , g y : y 6 Gn}. : x 6
G}
= Z
I 14.3.2. Suppose
Lemma.
(Z(F,z,m
z Let
to
with
F -- G, and let
r
Y
Gz})
be an o r d e r - i s o m o r p h i s m
Ax ~ F (H x = H'r ), then E{H x : x g F} ~ 2{H'r
: y E
: x ~ G~,
9
from
F onto
H.
Jn ).
-
Proof.
As in 14.2.17,
86
we apply induction
-
over
WO
with respect
have to distinguish four cases. Let us treat as a typical case Then we can find Ji such that
F = Z Fi, G = Z G i. i i
+'''+ Ja(i+i)-1'
a(O)
= O, i ~ N, ~ i ( a ( i )
< a(i+l)),
Gi = Jb(i)
+'''+
b(O)
= O, i ~ N, Ai(b(i)
9 b(i+i)).
Jb(i+l)-i'
= Zz Z{H~ y : y 6 F z} ~ Z{Z{H z z
where
is such that if
Hz
,Y
Ja(z)
+'''+ Ja(z+l)-I
The coneluslon Hence
F, and we
Fi = Ja(i)
Z{H x : x ~ ~
and
to
~y. r
for every
then follows
AzAvAy (o ~ v ~ a(z+i)-a(z)
AzAuAy 14.3.3.
-I
: y ~ G z} = z{H'y
(O~u~b(z+i)-b(z)-1 Theorem.
Let
^
Completely
14.3.4.
Theorem.
Proof.
by induction
~z,y
: y ~ Ja(z)}+...+
G m H, and let
between
~
Fz
y) = H < z > e y ) -
such that
r
§
H' < z , u > ~ y
Z{~a(z+i).i ' y : y 6 Ja(z+l)-i H'y
to be such that
be an o r d e r - l s o m o r p h l s m
to the derivation
between
G
and H;
: x E H}.
of 14.2.18
and {H x : x 6 G}
from 14.2.17.
be given such that
Z{F x : x ~ G} = Z{H x : x 6_G}.
over
W0
with respect to
G.
14.4. 14.4.1.
Definition.
species
of
Ordinals
WO. The ordinal
therefore
defined
Ord ({0})
= 1, Ord ({
The theorems
as
are the equivalence m = Ord
classes
(F) associated
of order-isomorphic
with a species
F ~ WO
is
{G : G * F}. : i ~ N})
= ~ .
of 14.2 permit us to define
}}=
~b(z)+u~y)~
:
Z{F x : x G G} = Z{Jcx
{F x : x ~ G}
Ax 6 G (F x ~ Hx) , then
Az AY(Hz,r
))
y ~ Ja(z)+v § ~z0
^ y E Jb(z)+u
analogous Let
z, then
: y E G}, if we take
suppose ~ x ~ G (F x = Jcx); then Proof.
is an order-lsomorphlsm
: y 6 Jz}}, with
: Y 6 Jz }} ~ ~{Z{~a(z),y
z{H~z > @ y
+'''+ Ja(z+l)-i
from the induction hypothesis.
Z{H x : x 6 F} ~ Z{Z{~z,y z
~{Z(~z,y
y)
'Y : y 6 Ja(z)
a few arithmetical
operations
on ordinals.
-
14.4.2. Definition.
87
-
Ord(F + ~ } )
= Ord(F)
Ord(F + G)
= 0rd(F) + Ord(S).
Ord(F G)
= Ord(F) 0rd(G)
Ord(FO)
= 0rd(F).
Some simple identities
+ I.
of ordinal arlthmetlc
proved for w e l l - o r d e r e d
= Ord(F) exp Ord(G).
Ord(O). follow readily
from the theorems
species:
Ca + B) + v = ~ + (s + y).
8+y (8)y
14.4.3.
If we define
~ < 8
8 y9 = ~8y =
by
VY(~
+ Y 9 s) (recall that we have excluded
as an ordinal for reasons of technical convenience)
then it may be concluded from
14.3 that we can define well ordered sums of ordinals unambiguously: 14.4.4.
It is a consequence
ordered for any given species
of 14.1.5,
14.2.7
0
(II), that
z{aS:
{~ : s < 8}
8 < y}.
is totally
B . But on the other hand we have no hope of p r o v i n g the
of all ordinals to be totally ordered,
as is illustrated by the following
(weak) counterexample. Let
a
be an arbltrary lawllke function of
Define the sequence
(N)N.
n : an = 0 § F n = {0}, an ~ 0 § F n = { : 1 6 N}. Then
we have no general m e t h o d to decide
Ord
(~ F n) = ~ v m <
Ord (~ F n)
since this would imply Aa(An(an
= O) v Vn (an ~ 0)).
For a concrete illustration we only have to m a k e a depend on the decimals
of
in the usual way. We shall refrain here from a further development reader is referred to
of ordinal arlthmetlc~
bhe
[Br 1926].
14.5. 14.5.1. and
It will be clear that there is much similarity in b e h a v l o u r between
WO. By a quite general t h e o r e m one obtains immediately that
definable in terms of
K. (For a statement of the theorem,
WO
K
is explicitly
see section 15.)
-
Here we shall demonstrate are explicitly and induction
definable
B8
-
by a more straightforward in terms of each other,
scheme for
K, WO
respectively
argument that
K
and that the closure
are provable
and
WO
conditions
from these explicit
definitions. For an easy comparison, elements
of
WO
we study
instead of
W, the species
WO
of characteristic
functions
of
itself.
If we put A W (P,a) ~D
[a = ~x.(iZx) I v [aO = 0 ^ Ay
~x.a(y,
x)@
P]v
v [aO = O A VzAy((y ~ z ^ ~x.a(9 9 x) e P) v (y 9 z * A ~ a(y m x) = O))] then the introduction by
of
Aw(W,a)
W
K
inductive definition
is expressed
+ Wa, +
I%.5.2. Theorem.
by a generalized
Qa]
+
is explicitly
definable
in terms of
W
such that
AK(K,a) + Ka, Aa[AK(Q,a) become provable Proof.
+ Qa] § [K __.Q~
(by induction
If we replace
Vz[a = Ax.(z+i)],
W).
a = Ax.(1~x)
we o b t a i n
If we define a species
over
in the definition
of
W* by
a E We+-+ Vb @ W Vc(a = kn.bn, sg h(b,n),(ch(b,n) where m'(m'
AW(Q,a)
~(Q,a).
sg x = i A (i z x), h(b,n)
= O
= m + i
otherwise.
Then one proves easily
Aa(A w, CW',a) + w'a) + + Qa] + The proof is left to the reader.
if
Q].
bm ~ O, m o
+ i)) n,
by
-
Now
K
-
is explicitly definable by a g K,-* a g W * A AK(K,a)
Is readily proved.
AnVy
a(n @ y) ~ o.
§ Ka
Suppose now
Aa[AK(Q,a) Take
89
(4)
§ Qa].
QOa ~D AnVy a(n t y) ~ O § Qa.
If a : ~x.z+l, then Now let
AK(Q,a) , hence
Qa.
aO = o ^ Ay(AnVz a(y e n 9 z) ~ O § ~m.a(y @ m) ~ Q), then aO : O ^ AnVz(a(n @ z) ~ o) § ~y ~m.a(y m m) ~ Q, hence ~nVz(a(n • z) ~ O) § (aO = O ^ therefore by (4) AnVz a(n m z) ~ O § a ~ Q, so
Ay
~m.a(y 9 m) ~ Q),
a g QO.
If aO : O ^ V z Ay[(y < z ^ ~x.a(y ~ x) ~ QO) v (y > z ^ Ax a(y ~ x) = o)] we have trivially
QOa, since
AnVy a(n e y) ~ o
clearly does not hold.
Hence we have shown that (4) § Aa[&./~(QO,a) and therefore
We~_QO
-~ QOa] i.e. Aa(a ~ W~§ a 6 QO).
Hence Aa(a E We^ AnVy a(n 9 y) ~ o § a 6 Q O ^ so
AnVy a(n , y) ~ o)
K ~___Q.
14.5.3. Theorem.
W
is explicitly definable in terms of
K, such that
Aa [Aw(W,a): § Wa] Aa[Aw(Q,a) becomes provable Proof. Let
V
§ Qa S §
(by induction
~__Q] over
K).
denote a class of spread directions
satisfying
-
Va
=-D aO ~ 0 A An(an
90
-
~ o § lax a(n ~ x) ~ o v
v V z { A x ~ z(a(n ~ x) ~ o) ^ Ax 9 z(a(n ~ x) : o) 1~). we define
W~:
b ~ W ~ -D V a 6 V
Ve An(bn
~ 0 ,-~V~(m < o n ^ sg(am).em
: bn ~ 0)).
We prove easily
AW*( w" ,a) ~ Wa. Now suppose Aa[~-(Q,a)
§
Qa].
(5)
We apply induction with respect to e to prove Wa~__Q. Let us introduce y(epa) (e g K, a E ~ v a -- ~x.O) for the unique b
function
such that An(bn
~ O .-~Vm(m
~o n ^ bn = sg(am).em
Now we want to prove by induction
over
~ O)).
K
AeAa g V (~(e,a) ~ Q). If e = Ax.z+1, Let now
then
w(e,a)
= Ax.z+l,
hence
u
g Q.
eO : 0 ^ AxAc 6 v (~(~n.e(x ~ n),c) 6 Q) and let
(6)
a g V. We remark that a ~ V § Ax(~n.a(x ~ n) 6 v) v Vz(Ax
< z(knoa(X ~ n ) g V ) ^
lm
Ax 9 z(~na(x e n )
= o)).
Furthermore ax ~ 0 A e O Suppose
first
= 0 § An.w(eja)(x
Ax(kn.a(x
Ax(ln.v(e,a)(x u and thus
@ n) 6 V ) .
= w(An.e(x ~ n), a(x e n)).
Then from
46):
9 n) ~ Q )
47)
5
= 0
from (5) and
, n)
(7) (which implies
Now let Vz(Ax ( z(kn.a(x @ n ) ~ Then from (6): VzCAx < z(kn~165
V)
~(Qbu
A AX > z(~n a(x 9 n) = 0)).
9 n) E Q) a A x
>z ~n(u
and (e,a)
we conclude
: O,
therefore Aw~(Q,~(e,a)) , hence with (5) Q~(e,a). Thus we have proved AeAa ~ V(w(e,a) s Q), i.e. Wm_~Q.
9 n) = 0))
to
Q~ (eja).
-
w 15. Species revisited|the
91
-
role qf the comprehension
prlnciple
15.1. In the IntultionIstic the fundamental
theory of species,
question of classical
we are confronted with the analogue
axiomatic
set theory:
said to exist? Or in terms of a theory of constructions: structions? Let
X
The problem is illustrated
be a certain given species.
there is nothing problematic Let
~
be a language
prehension principle
(Y
elements
(Yx~
is accepted Xn
for any
x, y
as a well-deflned
of
may be expressed by a schema: y
we accept the universal
(1)
for subspecies
principle ~
of
relative
x.)
to
X
clearly depends
is a first order language with variables
X, the resulting predicative
seems to us to be quite unobjectionable.
version of the comprehension
The strength 15.1.1.
of full comprehension
Theorem
arithmetic
(see e.g.
[Kr
~
AX,VX,
for
Let us call this weak comprehension.
contains
quantifiers
is illustrated
1968 AS). Suppose
with the language extended by variables
and quantifiers and N~
on the
principle
Evidently we can build a ramified hierarchy by repeated weak comprehension, in classical ramified analysis. or ful ! comprehension, where the other extreme possibility.
object,
X. The com-
F(x))
of a comprehension
principle.
n ~ N.
for elements
not containing
a variable
power of .C . When
of
, X
F(x) of #~
VYAx
expressive
X
containing variables
For every formula closure of
The strength
Once
relative to ~
which species may be
which notions are con-
by the role of the comprehension
in accepting
of
and the comprehension
as
AY, VY, represents
by ~
to consist X, Y, ...
of Intultionistic for subspecies
of N,
axiom relative to thls language
If ~ + denotes the corresponding classical system, then the G6del translation of w 3 extends to ~ (taking (AX F(X))- = AX F'(X), (%/X F(X))- = n A X ~ F - ( X ) , and preserves validity. Proof. We only have to verify that the translation for F Is a consequence this Is straightforward. Therefore
~+
of the comprehension
is consistent
if ~
of the comprehension
scheme in ~
is consistent.
applied to
scheme F-
and
(I)
-
92
-
15.2. Inductive definitions
like those of
represent examples of intermediate If we accept full comprehension,
is Justified
K
in w 9, and the definition of
WO
in
PA
satisfying
w
14
forms of comprehension.
then the introduction of a predicate
Ax(A(PA,
x) § PA x )
Ax(A(Q,
x) § Qx) § PAC__ Q
(2)
)
(classically as well as Intuitionistically)
whenever
A
satisfies
the condition of monotonlcity: A(P, x) ^ P ~ PA
(3)
P' § A(P', x).
Is~ said to be introduced by a generalized inductive definition
Justification is given by remarking that
PAy Since
PA
--AX[Ax (A(X. x) + Xx)
PA
(4)
over species, we might,
if we think
Just as well accept full comprehension outright.
But if we impose more stringent requirements for the introduction of
may be defined by
Xy ] .
(4) requires universal quantification
this Justification satisfactory,
(g.i.d). The
on
A, sometimes better Justifications
can be given. For example, the introduction of
K
(introduced and discussed in section 9) is essentially Justified by observing that and
AK(P , a) § Pa K
expresses closure of
under certain simple operations;
is then viewed as the species such that
using these simple closure properties e g K
P
e ~ K
only. Moreover,
iff this can be proved it is to be remarked that any
may be said (in a sense) to codify itself a standard proof of
Likewise we may Justify the introduction of to be explicitly definable in terms of
e g K.
WO. In fact, we have even shown
WO
K.
Once we have accepted
K, a quite general class of g.i.d.'s also becomes acceptable,
since the species
required to exist by the g.i.d, may also be defined
PA
plicitly in terms of
K. We have the following result:
15.1.2. Theorem.
IDK ~
w I0, and let such that talning
A P
Let
A(P, a)
be an extension of the system
be any formula of
is constructed by means of
IDK ~
IDK
as described in
with a single predicate
^,v, Ax~x,Va
~1,b2,...,
xl,x2,..,
yS), t
letter
from formulae not con-
and formulae of the form P(~y.t
ex-
a term of
IDK ~.
P,
-
Then we can explicitly
define
a)
Aa(A(PA, Aa[A(Q, for any
Q
§
(in
IDK r
-
a predicate
PA
such that
PAa)P
a) § Qa] §
in the language
93
of
AaKPAa § Qa]
IDK e.
Proof. We shall not present a full proof here;
for more details
see
[Kr, T]. The
essential idea is that for A(P, a) of the form described above, a ~ PA must have a standard "cut-free" proof, which may be codified by a well-founded tree, hence by a function Let for example
of
K.
A(P, a)
be of the form
~JbAx(RCa, b, x) v PC[a, b, I ] ) (r
b, x] : ky.t
Then we take'as PA a
[a, b, x, y]
our explicit
~D VeVcVd
§ d TM * = r
for a suitable term
t).
definition:
{d ~ : a ^ eO = 0 ^ AmAy TM, c m, y])
((e(m e y) = O §
^ (e(m m y) ~ 0 ^ em # 0 ~ R(d m, cTM, Y))}
where d TM : ~x.d {m, x}, c TM : ~x.c {m, x}.
15.3. A typical application definition
union of all connected not find
of the full comprehension
of a component
of a point Y~__X
YI' Y2' closed in
p
such that
Y, such that
We shall present here a typical
principle
in a topological p ~ Y. ( Y ~ YI ~ Y 2
(and essential)
X
is given by the space
<X,~>
is connected if we can-
= Y' Yi ~ Y2 = ~)"
application
of the predieatlve
prehension principle, which clearly shows the role of the comprehension as a creator of new objects. 15.3.1. We write X~IY if there are (lawlike) mappings r r such that for some Z C_r r162 (r162 = {r : y~ r Let X ~ 2 Y mean: there Is a (lawlike) mapping may be properly included in X.) 15.3.2.
Theorem.
X~2Y
§ X~IY
r
as the
such that
(IT 1967 A], lemma 2.4).
[x]
r
principle
bi-uniqu%
: Y.
com-
(dom
-
Proof.
Let
r
We p u t
for
any
,
,
,
,
= Y, dora r C
X, where
x 6 Dom r
We define
-
dora r = {x
:
Vy(r
= y)}.
x 6 X:
Yx = {y : y g Dom r ^ x E D o m If
94
then
r
Yx
on {u
r A r
is inhabited,
: V z ( z ~ Yy)}
#'(u
= # x.
~'[(u
: Vz(z E vy)}]
= Cy},
and conversely.
by p u t t i n g
Clearly
There remains
to be proved that
$'(u
= $'(u
§ u (We use 15.3.3.
u
= r162
Remark.
is bi-unique. $(x)
= $(y)
: u
= {y : y g Dora r A r X ~2 Y
: X ~ X ^ y ~
The proof uses the comprehension application
r §
The hypothesis
{~:xjy:.
= Y.
Y ^ y
requires
for
the existence
x ~ Dora r of
X, Ym and
= r
principle
of weak comprehension,
= r
see e.g.
relative to these species. IT 1967 A], theorem 4.3.
For another
-
95
-
16. Brouwer's theory of the creative subject i6.1. In a number of papers published after 1945 (e.g. [Br 1949~,
[Br 1948~, p. 1246,
[Br 1948 A],
[Br 1949 A]), Brouwer introduced the Idea of the creative subject
(or the idealized mathematician).
Thls concept gave rlse to much discussion and
It is likely to do so for some period of tlme to come. A systematic and coherent theory has not yet been developed,
so I am restricted to presenting a few fragments.
The central idea Is that of an idealized mathematician Jectivlstic viewpoint of Intuitlonlsm, to obtain the required idealization,
~conslstent wlth the sub-
we may think of ourselves;
or even better,
we may think of ourselves as we should like
to be), who performs hls mathematical activities in a certain order (you may think of the order given by time). The process of bls mathematical activity proceeds in discrete stages. Therefore we introduce a basic notion:
~m
A
to be read as: "the creative subject has a proof of
A
"the creative subject has evidence for
m II .
We suppose
~m A
~ --mA At stage for
m
A
at stage
to be a decidable relation,
v ~m
at stage
m"
or better
l.e.
A.
we know If we have evidence for
A
or If we do not have evidence
A.
Clearly (Vm ~ m A) § A. "If we have evidence for
A
(2) at stage
m, then we can find a proof of
A".
In order to simplify the interpretation we also suppose
(3)
( l---mA) A (n > m) § ( l--nA) . "The evidence at stage
m
Is also contained In all following stages".
If we boldly identify the provable assertions with the assertions
for which we
can obtain evidence at a certain stage, we also have A *-~ Vm( ~ m A)
(4)
- 96
-
or in combination with (2)
Vm( ~ - m A) ~-* A
(5)
.
If we want to be more cautious, we may satisfy ourselves with the following assertion instead of (4): (6)
A § -1-1Vm( ~--mA) . (6) may be read as follows: there is a proof of
"I am completely free in making deductions. Hence if
A, it is absurd that I would be able to prove that I will
never find a proof of If we want to assert
A
(at no stage will have evidence for
A)".
(6), without asserting (4), this means that we do not want
to identify all possible constructive proofs with the collection of proofs whose existence becomes evident to me at a certain time (stage). Let us call in the sequel the theory based on (6) instead of (4) the "weak theory", and the theory based on (4) the "strong theory". 16.2. In the existing literature, most of the deductions are based on the weak theory. In developing consequences from the weak or the strong theory, I shall try to be cautious and hence proceed more or less axiomatically, in order to show what is actually used in certain deductions. In the weak system, we can derive the following scheme (called Kripke's scheme in the literature) :
Vx[(Ax(xx
- o) + ~
In the strong system, Vx(Vx(xx
~A)
^ (Vx(•
~ O)
(7)
+ A)].
(7) can be strengthened to
(8)
# o) +~ A).
This is seen by defining
A)
x
relative to a given assertion
A
by:
:I ]
(n ~--nA) § xn = O. If we have a definite prescription involving the actions of the creative subject (by means of a relation like
~ n A) for determining the values of a sequence, we
speak of an empirical sequence (as is done e.g. in [M 1968]).
-
97
-
Our idea of lawlike sequence does not exclude empirical
sequences,
at least not
as lon E as we are willing to consider reference to our own course of activity by means of
l"-n a s " d e f i n i t e " .
It is clear however, stricter,
that e.g. primitive
more objective
recursive
functions
sense~ their values are independent
are lawlike in a of future decisions
about the order in which we want to make deductions. Let us call a sequence which is given by a complete description a mathematical
([M 1968])
If a sequence
~
or absqlutely
is defined by a complete description
We shall return to the distinction
between empirical
is really evident the use of
~-n
§
since it is conceAvable
(even if
~n'
in 16.8. A
VaAx A(x, ax)
A
that
sequences
from a proof of
AxVy A(x, y)
(A
we ought to restrict
(This can be done, I believe,
from the species
AxVy A(x, y)
itself does not refer to
Va~x A(x, ax) with a mathematiaal, not Involving
...,
or
in section(~ i.e. for an
only if we do not exclude empirical
In order to conclude
Xl, X2,
mathematical
for lawllke objects onl~
A(x, y)
of lawlike sequences,
~
and mathematical
It seems to me that the axiom of choice as discussed
AxYy
from sequences
we shall call
X2 , . . . .
Xl ,
containing free variables
I n
lawlike sequence.
without reference to the creative subject, lawlike in
not involving
is proved with
~-n ) . not involving ourselves
consistently
~ n ) to
to arguments
throughout
the
preceding sections.) For lawllke sequences,
we obtain from (7)
AaVb~(Ax(ax If we use
= 0)*-* Vy(by
= 0))
(9)
(8) instead, we obtain the stronger form: AaVb(Ax(ax
This is seen by applying
= 0)*-~ Vy(by
= 0))
"weak" counterexamples
turn out to be essentially
collapse,
equivalent.
(1o)
.
(7) and (8) respectively
In virtue of (9), various types of unsolvable tlonlstlc
.
to the formula
problems
A
-D Ax(ax = 0 ) .
that are used in intui-
i.e. some of these classes of problems
-
A few of these types by the following
of problems,
formulae,
98
-
some of which we have met before,
which express
assertions
are represented
which we have no hope of p r o v i n g
intuitionistically: Ab(Vx(bx
: O)v-IVx(bx
Ab(-7-1Vx(bx
: O) § V x ( b x
AD(-)Vx(Dx
: 0))
= O)v~IVx(bx
AaAb[-7(Vx(ax
(11)
: 0))
= 0))
= 0) ^ V x ( b x
(11) and (13) are restricted
forms
the principle of testability Markov's principle.
respectively.
~a~b(Ay~(Vx Vc~y[(Vx
b{x,y}
A
a{x,y} = O ^ V x
a{x,y}=
of the excluded third and
(12) is an intuitionistic
analogue
of
to : O)
b{x,y} = O) §
o § cy = o) ~ (~x b { x , y } =
0 + cy ~ 0)]).
the analgon of the assertion:
set is recursive,
the second one expresses
not containing non-lawlike V b ( A 4-~ Vx(bx
Since the members
(14)
: O) ^ (cy ~ 0 §
of disJolng r.e. sets can be separated For an
.
to
The first formula expresses enumerable
: 0))]
of the principle
(11) Is equivalent
AbVc~y~cy : o § and (14) is equivalent
(13)
= o)) §
("lVx(ax = O)v"lVx(bx
Intuitionistically,
(12)
the analogon
by a recursive variables,
every recurslvely of: every pair
set.
(8) simplifies
to
= 0)).
(15)
of the pairs ~IA
§ A, ~ A
v A
and (A^B) possess
equal strength
§ (~A v ~ B ) , ~ A v ~ A
as axiom schemes when added to intuitionistic
see easily that as a consequence lawllke parameters (11) *--'* ( 1 2 ) , (13) It is worthwhile using
of (15),
only; likewise ~ (14).
knowing however,
(9) only, without
further
(11) implies
(13) implies
that
~ Av~IA
(11) *-~ (12),
reference
Av
~A
logic, we
for all
for such
A
with
A. Therefore
(15) 4-+ (14) can be proved
to the creative
subject.
-
Prqof,
(i) (11) ~ (12) is immediate,
Suppose
(12), and take any
99
-
even without
(9).
b. According to (9) we can find a
-*1(-IVy(by
: 0)~9
V'y(cy = 0))
b
such that
.
So -rVy(by = O) ~-~ -~-lVy(cy = 0). Since
-11(Vy(by = O) v-IVy(by : 0)), it follows that -7~(Vy(by = o)v,-iVz(cz
P v-~'IQ §
-I"1(P v Q)
: 0))
.
is a t h e o r e m of intuitionistic p r o p o s i t i o n a l
nn(VyCby
: 0))v
VzCCcz
logic, hence
= o)),
so ~Vx(bx.cx Applying
: 0).
(12), we conclude to
Vx(bx.cx
: 0), hence
Vx(bx
= O) v V x ( c x
: o),
which is equivalent to Vx(bx (ii). Suppose
= O)v-/Wx(bx
(13). For
~Vx(ax
= o)~
a, b
= 0). sueM that
conversely, Take any
~Vx(ax suppose
b, and let "~(Ax(bx
Then and
~
Ax(bx ~Ax(bx
= o)^Vx(bx
= 0)) we conclude to
Vx(bx = O)
-iVx(ax = O)~-~-iVx(bx Then by 413)
1(Vx(ax
: 0).
= o)v~Vx(ax
= o).
(14). a
be such that
~ O) ~
Vy(ay
= 0)).
# O)*-~-~Vy(ay = o)) ~ O)*-*nVy(ay
= 0).
From (14) and the fact that "~(Vx(bx = O ) a V y ( a y = 0 ) ) i t follows that -~Vx(ax = O) v - I V x ( b x = O) therefore we may conclude to I V x ( b x = O ) v ~ w V x ( b x = O).
Now we shall proceed with somewhat more i n t e r e s t i n g theorems.
-
iO0
-
16.3. Theorem.
We can prove in the weak system (in fact, using
~VbAaVxAz[Vy
a(z,y}
= 0 *-~Vh b{X,{Z,U}}:
This result has been called a refutation sequences
(16)
O] .
of "Church's
thesis",
but since empiric~
are rather far removed from the idea of "mechanically
functions,
it is perhaps better to describe
Myhill proposed Proof.
(9) only):
computable"
it as a non-enumerability
result,
as
([M 1967]).
Suppose AaVxAz{Vy
a(z,y}
= o ~Vu
b{x,
(17)
(z,u}}: O} .
We remark that ~ V u b{x,{x,u}}:
0 ~Au(1
Now we can find (by (9)) a
: b{x,(x,u}}: c
~ n ( A u bKx,{x,u}}# In virtue of our hypothesis
O) *-~u(b{x,{x,u}}~
O) .
such that
O ~Vv
(18)
c{x,v} : o) .
(17), there exists an
Ax(Vv c{x,v} = o ~ V w
b{Xo,{X,W}}:
X0
such that
(19)
O .
So we obtain
InVw
b{Xo,{Xo,W}}:
O~Vv
+-~Au
b{Xo,{Xo,V}}~
0
~-~Au b{Xo,{Xo,V}}~
0
~Vw contradiction.
C{Xo,V} = O
b{Xo,{Xo,W}}=
This disproves
(from (19))
(from (18))
O :
(17).
16.4. Theorem
([M 1967]).
AxVx' B(X,X'),
There exists au extensional
but for no continuous
functional
AxB(x,rx). Proof.
Apply Kripke's
scheme
(7) to
AX -DVx~y 9 x(xy = 0).
predicate r
of type
B
such that ((N)N)CN)N
-
101
-
Then we obtain
AxVx'[('~VxAy
= o))^~
> x(xy = o).,-,-Ax(x'x
(20)
^(Vxx'x / o-. Vx4y 9 O(x~' : o))] Let us denote the part of (20) within the square brackets by suppose that for some continuous
B(X, X ') , and
r
AxB(x,rx). In case Since
(tX)X # O, it follows that r
is a continuous
VxAy 9 x ( x y
funct&onal,
= o).
there exists an initial segment
~y
of
•
such that
Ax"C~y
= ~"y § (rx)x = (rx")x)
and therefore
Ax"(~y
= ~"y ~ VxAz
9 x(•
= o)).
This is obviously false; we only have to take for ~"y = ~y, X"(y+z) Hence
(r•
= 0
the function such that
for all z.
leads to a contradiction;
the zero functional. tradiction,
= I
•
therefore A • 2 1 5
As a consequence A x n V x A y
= O, i.e.
r
is
9 x(xy = o), which is a plain con-
since the zero function provides a counterexample.
16.5. 16.5.1. Theorem.
-TAX(n~x(•
Proof. We apply (7) to Vx[(nnVx(xx Take for Since
A•
= O) + ~x(• -DVx(•
~nA6
we find a
holds x
(AS
Vx(xx
= O)
:
= O) § AX')].
B.
is an application
such that ~ n V x ( x x
would hold, then
= O)v ~Vx(x'x
= O) * ~ I A x ' ) ^ V x ( x x
x' any choice sequence
= O))([Kr 1967], p. 160).
of the principle
= 0), hence if
= 0), hence
AS. A8A8
Ax(~Vx(xx
of the excluded third) = o) § Vx(•
is contradictory
= o))
(compare 9.10 (I));
so the assertion of the theorem follows. 16.5.2.
Remark.
At first sight,
in section 9; but actually,
A x ~ 06 ( ~ V x ( • where
~
16.5.1
is Just weaker than the result 9.10
(V)
we have proved more, we have shown:
= o) ~ V x ( •
is the class of sequences
(but not necessarily extensional)
= o))
obtained from choice sequences by a lawllke
operation.
-
I02
-
16.6. Brouwer
gave
number
x
takes
([Br 1948 ~ ) a s i m p l e
such that
an a s s e r t i o n
know whether
x # O, w h i l e A
wA
or
w h i c h has not b e e n ~A),
and d e f i n e s
-I ~ n ( ~ A v - ~ n A )
§
rn
~m(IAvnnA)^
Then would
one v e r i f i e s imply
use
subject,
further
reference
xb
denote
a lawlike
equivalence
b
~
n(bm
(12) is e q u i v a l e n t
implies
bn
generator
:
^ n 9 m § r n = 2 -m.
or s t r o n g
a proof
another
system
of
x ~ O; but
x ~ 0
x ~ O.
(12)
relative
formulation
a real number
to the t h e o r y
of this
generator
with
§
rn
result
of the
without
defined w.r.t.
= 0
< m(bn'
= 0)) § r n = 2 - m
A b ( x b ~ 0 § x b ~ 0); but on the
x ~
Ax(x
Then
0 §247
o) ~ (11)
)
other hand
(12)
.
= 0 v x 4 0). F o r
let for a n y
/~n(r n - 2 -n < x < r n 9 2-n),
Therefore
other
~ o.
examples
(II)
xr
o) ~ A x ( x
of t h e o r e m s
A x(x ~ o §
implies
Ax(x ~
0 v x = 0).
x
= o vx~
to w h i c h
we can
o)
.
find w e a k
same k i n d :
Ay~z(y
be s u c h t h a t
n - 2 - n 9 0 v r n + 2 -n < O) ~ and i f we put
for l a w l l k e
Ax(x
x n
< r n - 2 " n > n 6 x m < r n + 2 - n > n 6 x.
= 0 4-~ r n . 2 -n 9 0 v r n + 2 -n < O, we see t h a t
Some
(that is, we d o not
by a real number
(ll) and
~ 0 ^ An'
~ o ~ x~
in t u r n
Anlx-rnl<2-n-i. Hence
up t i l l n o w
he
(II), t h e r e f o r e
Ax(x (II)
real
subject.
with
~ O)
(m & n A bm then
of a l a w l i k e
x @ O. To do this,
by
~Vm
implies
of
to o b t a i n
a real number
of
= 0
in the w e a k
to the c r e a t i v e
function
tested
"~m_l(-1Avn-lA)
that
in o r d e r
counterexample
x
so we d o n o t h a v e
of the
creative
Let
easily
~Av~A;
We can m a k e
(weak)
we h a v e n o p r o o f
x } o vx
# o) j
~ o a z ~ 0 4 Vx(yx
+ z : o))
.
counterexamples
of the
-
103
-
If we sufficiently enlarge our notion of real number, we can give m a t h e m a t i c a l disproofs by means of theorems
Ax(x
like 16.5.1
, e.g.
o)
o § x
is provable if the class of reals is sufficiently all reals obtained from real number generators enumeration of the rationals,
• 6~
large (roughly speaking:
n,
n
take
being a standard
as indicated in remark 16.5.2).
~6.7. 16.7.1.
Lemma.
mappings
r
Let
a, b
be two lawlike sequences
satisfying
~CRange
(a)~
~
Range
(b), r
Then there exists a bl-unique ~ERange (Or if
~
(a)]
= Range
~
(b)] ~ R a n g e
defined on Range
generated by
a'
such that
is defined likewise
Now we define
~
from
as follows.
such that
Lemma.
Consider
Now we put
We then use the existence Let
a lawlike sequence Range
X a
of
(y+l) th "new" number
= ax ^ ax ~ (aO,..., az} ^
y
different e l e m e n t s ) V ( a x
{r
ax, and let 0),...,
r
to demonstrate
AzVx(~ax
= bz).
such that
let us suppose
is further defined by: = aO
~ - m z 6 X § a{m,z}
= z.
Now one verifies easily that on account of Vm
-mX
y
~ax = bz.
~-m z 6 X § a(m,z}
x e x
= O)
@(ax)} must contain at least
(a) = X.
aO = O. a
= aO A y
a'x = y. Then we can find
be an i n h a b i t e d species of natural numbers.
Proof. For sake of simplicity, Then we put
(b).)
b.
b'z = y, because
different elements.
16.7.2.
is the
(6) = Range
a. Or explicitly:
{aO,...,a~} contains
bz
(a) such that
(a) only, Range
a'x = y ,-~ ax
a'x = y ,-~Vz < x(a(z+1)
a
(a).
(b) .
is taken to be defined on Range
Proof. We define
b'
such that there exist bl-unlque
x)
(an application of (6)) we have Range (a) = X.
0 6 X,~-oO ~ X.
Then there is
-
I04
-
Remark. The proof as described above proceeds directly from the axioms for the creative subject, but we could have obtained the result also from the specialization of (8) to species:
AxVa[Vy(ay hence
Ax[Vy(b{x,y}
= o) *-~ x 6 x I;
= 0 ~-* x ~ X]
for a certain
b~ and since
X
is inhabited,
we may construct in the usual way a constructive function which enumerates (x : V y b(x,y} : 0}. 16.7.3. Theorem. Let
X, Y
be hi-unique mappings, 6 (X)Y
be inhabited species of natural numbers, and let
r 6 (X)Y, ~ 6(Y)X.
Then there exists a hi-unique mapping
such that ~ [XI= Y (Intuitionistic
Prqqf. Immediate by 16.7.1, 16.7.4. Remark.
r
analogue of Cantor-Bernstein).
16.7.2.
The requirement that
X, Y
are inhabited can be omitted; the
proof is slightly more involved in that case. We would have had no hope of obtaining this result if we had required mathematical in
r
~
to be
This is demonstrated by the following counterexample:
X = N, Y = (2n+1 : n E N } v ( m
take
: ~VXHm(X)} , cn = 2n + I, ~n = n.
It is Instructive to compare 16.7.3 with the following theorem in the literature, which can be proved without reference to the creative subject: 16.7.5. Theorem. let
r
that
Let
X, Y
be inhabited subspecies of the natural numbers, and
be bi-unique mappings of ~X] is detachable in
mapping Proof.
~
from
X
onto
X
into
Y, ~ d e t a c h a b l e
Y, Y in
into
X
respectively,
such
X. Then there exists a bi-unique
Y.
(formulated with "recursive" instead of "lawlike")
[D, M 1960] theorem 13(b) or [Rog 1967] w T.4, theorem VI. As it stands,
16.?.5 le
Just weaker than 16.7.4. However, the existing proofs of
16.7.5 contain more information than is contained in 16.?.5, since they do not refer to the creative subject. Hence we may strengthen 16.7.5 to: 16.7.6. Theorem. Under the conditions from
X
onto
Y, ~ mathematical in
of 16.7.5, there exists a bi-unique mapping r
The formulation of many results in the preceding sections may be strengthened in the same manner. Our intuitive insight
that there is an essential difference between empirical
and mathematical sequences, the creative subject.
can be expressed to a certain extent in the theory of
-
lO5
-
Take the following property as an example. Let associated with a definite assertion variables),
aA
~-x~A Let
amath
A (i.e.
aA A
be an empirical sequence does not contain non-lawlike
defined by
§ aAx = 1,
~ x A § aAx = 2, ~ x ( A V ~ A )
§ aAx = 0.
be a variable for mathematical sequences.
Then (~x(AV~A)) expresses
§
: a A)
a difference between mathem&tlcal and empirical sequences.
16.8. Let me finish with the indication of some points in the theory of the creative subject in need of further clarification. creative subject has evidence to assert
Intentionally, A
downrightly "the creative subject proves
at stage A
at stage
I used the phrase "the
m" in preference to saying m".
The second formulation suggests the creative subject as making his conclusions by one, so we might ask ourselves if we could not assert that with a sufficient "refinement" in the distinction of the stages, the creative subject draws one conclusion at a time. This supposition can be expressed axiomatically by introducing the constant
A (m)
(denoting the new conclusion of stage
m) by the
statement: ~---mA(m) ^ An(n
9 m §
{(~-m B ^ An(n
9 m §
A(m))
^
§ (B *-* A (m))}.
In some respects this looks plausible; however, in case we accept this axiom, ~ m ~ X Ax would oblige us beforehand to make in the future all deductions AO, A1, A2, A3,... explicitly, since
~X AX ~
~xVm~---mAX ,
and this seems to be a rather unnatural restraint on the future activity of the creative subject. But even worse: our new axiom leads to a downright contradiction. For since it is natural to assume that we know when a conclusion has the form "a
is a lawlike sequence" we have: A (m)
is a conclusion of the form "a
A (m)
is a conclusion of another kind.
is a lawlike sequence" or
one
-
106
Then it is possible for us to enumerate the let
A (bx)'"
-
A (m)
of the form "a
is lawlike";
is a lawlike sequence".
be the x th conclusion of this form, stating "a x
Then AxVa(A (bx)
~ "a
is a lawllke sequence")
and so we conclude to the existence of b'{x, y} : ax(Y) Intuitively
c = ~x.b'{x,x} + 1
able to indicate a
z ~ N A (bz)
a
b'
such that
.
2s a lawlike sequence, but then we ought to be
such that
~ "c
is a lawlike sequence"
which implies: Ax(b'{x,x} which is contradictory,
+ I : b'{z,x})
Clearly, the axiom of "one-concluslon at a time" makes
the highly Impredlcative
character of the theory of the creative subject acutely
paradoxical. Therefore we must leave open the possibility of infinitely many assertions
A
for which ~ A ,
at
stage
and hence the cautious formulation "having evidence for
m". Example:
if we have proved
for any special number
AxAx
at stage
m, we also have
A ~mAX
x.
But since we also required ~-m A v 1 ~ m A , we are left with the quite genuine problem of finding a satisfactory interpretation of "evidence for
A
at stage
m". We
might try something like ['trivlally deducible from the available data at stage in the same manner as
An
is trivially deducible from
are left with the problem of a general interpretation
m",
AxAx, but once again we of "trivially deducible"
which ensures decidability at any stage. Another possibility which is suggested by the "paradox" is, that the primary cause of our trouble is not so much in the assumption of the "one-conclusion-at-atime" axiom, as well as in the unrestricted possibilities
for self-reference
implicit in our axioms for the creative subject. Specifically: llke ~--nA
and~-m(~nA)
reflection
= mathematical
activities)
do belong to different
although assertions
"levels of self-reflectlon"
(self-
consideration of the course of our own mathematical
we did not distinguish between them in this respect.
-
107
-
So the following possible alternative for a theory of the creative subject suggests Itself. To each mathematical assertion and construction we suppose a level (of selfreflection)
to be assigned. We restrict ourselves to assertions
and constructions
of finite level; the unions of all assertions and constructions of finite levels form the assertions and constructions of level m . Assertions which may be understood o r constructions which can be carried out without reference to ~--n are said to belong to level zero. Assertions which are described using ~--nA of level
p, are sald to belong to level
relative to ~ n A
for
A
of level
p
for
A
of level
p+l. Likewise,
p
and constructions
constructions
are sald to be of level
defined
p+l.
We make further the following basic general assumption: "When an assertion of level we use
a (p)
p
has a proof, It has a proof of level
to denote a constructive function of level
p". Now, If
p, we can assert
(instead
of ( l o ) ) Aa (p) Vb(P+l)(Ax(ax So If we use get at level
a, b
: O) ~ V y ( b y
= O)).
as variables for constructive
AaVb(Ax(ax
= O)+-~Vy(by
functions of level ~, we still
= O)).
On the other hand, our paradox cannot be derived anymore. I feel this approach deserves further investigation.
Another problem we dld not
touch upon, is the combination of the idea of incomplete
(choice-) objects In
connection wlth the theory of the creative subject. For example, how are the "stages" of the theory of lawless and choice sequences to be related with the "stages" In the theory of the creative subject? In conclusion we might say that the theory of the creative subject Is provocative, attractive,
and dangerous; It represents the extreme consequences
subJectlvlsm;
of intultlonlstlc
undoubtedly it deserves further study, precisely for this reason.
16.9. References.
An easily accessible exposition of Brouwer's counterexamples
the creative subject ls to be found In [H 19661, 8.8.1
involving
(pp. 115-I17). Krelsel
([Kr 1967]p. 160) introduced the r e l a t l o n ~ m (there appearing as Z ~ m ) . A further discussion In connection wlth analysis is to be found In [M 1967] and [M 1968].
-
lO8
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Offsetdruck: Julius Behz, Weinheim/Bergstr.
