Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
617 Keith J. Devlin
The Axiom of Constructibility: A Gui...
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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
617 Keith J. Devlin
The Axiom of Constructibility: A Guide for the Mathematician I
II
Springer-Verlag Berlin Heidelberg NewYork 1977
Author Keith J. Devlin Department of Mathematics University of Lancaster Lancaster/England
Library of Congress Cataloging in Publication Data Devli% Keith J The axiom of constructibility. (Lecture notes in mathematics ; 617) Bibliography: p. Includes index. i. Axiom of eonstruetibility. I. Title. II- Series: Lecture notes in mathematics (Berlin) ~ 617. QA3oL°8 no, 617 [QA248] 510'.8s [511'.3] 77-17119
AMS Subject Classifications (1970): 02K15, 02K25, 02K05, 04-01, 04A30, 20A10, 20K35, 54D15 ISBN 3-540-08520-3 ISBN 0-38?-08520-3
Springer-Verlag Berlin Heidelberg NewYork Springer-Verlag NewYork Heidelberg Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1977 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2140/3140-543210
PREFACE
Consider the following four theorems of pure mathematics.
The ~ h - - B a n a c h
Theorem of /~nalysis:
If F is a bounded linear functional
defined on a subspaee M of a Banach space B, there is an extension of F to a linear functional G on B such that
The Nielsen-Schreier
llG~l= llF~l .
Theorem of Group Theory:
If G is a free group and
is a subgroup of G, then H is a free group.
The Tychonoff Product Theorem of General Topology~
The product of any
family of compact t o p o l o g i c a l spaces i s compact. The Zermelo l~ell-0rdering Theorem of Set Theory:
~Very set can be well-
ordered.
The above theorems have
two
things in common. Firstly they are all
fundamental results in contemporary mathematics.
Secondly, none of them can be proved
without the aid of some powerful set theoretic~l ass~a~ption:
in this case the ~xiom
of Choice.
Now, there is nothing wrong about assuming the Axiom of Choice. But let us be sure about one thing: we are making an assumption here. ~e are saying, in effect, that when we speak of *'set theory", the Axiom of Choice is one of the basic properties of sets which we intend to use. This is a perfectly reasonable assumption to make, as most pure mathematicians would agree. Horeover (and here we are at a distinct advantage over those who first advocated the use of the Axiom of Choice), we know for sure that assuming the Axiom of Choice does not lead to a contradiction with our other (more fundamental)
assumptions about sets.
In Chapter I of this book we describe four classic open problems of mathematics, as above one from Analysis,
one fromAlgebra,
one from General Topology, and
IV
one f r o m S e t T h e o r y . S i n c e we c a l l i% s h o u l d be o b v i o u s t h a t Indeed,
it
"problems" rather
they are not quite
c a n be shown t h a t
of any of these
these
problems.
than "theorems",
however,
t h e same a s o u r f o u r s t a t e m e n t s
above.
a s s u m i n g t h e a x i o m o f c h o i c e d o e s n o t l e a d %o a s o l u t i o n
But by making a fuzf~her assumption about sets,
we a r e a b l e
t o s o l v e e a c h o f t h e s e p r o b l e m s ( a n d many more p r o b l e m s known %o be u n s o l v a b l e without
such an assumption).
This assumption
The Axiom o f C o n s t r u c t i b i l i t y axiom, closely It
is an axiom of set
b o u n d u p w i t h w h a t we mean b y " s e t " .
i s known n o t t o c o n t r a d i c t
already
i s t h e Axiom Of C o n s t r u c t i b i l i t y .
indicated,
its
t h e more b a s i c
It
theory.
implies
assumptions
assumption leads to the solution
It
is a natural
the axiom of choice.
about sets.
And a s we h a v e
o f many p r o b l e m s known t o be r
unsolvable
f r o m t h e Axiom o f C h o i c e a l o n e .
axiom is eventually situation
accepted
is net unlike
that
as a basic involving
Time a l o n e w i l l
tell
whether or not this
assumption in mathematics.
Currently,
t h e Axiom o f C h o i c e some s i x t y
years
the ago.
The axiom is being applied more and more, and what is more it tends %o decide problems in the "correct" direction. And one can provide persuasive arguments which justify the adoption of the axiom. (Again as with the Axiom of Choice in the past, there are also argaments against its adoption.) However, since the axiom is being applied in different areas of pure mathematics, it is a proposition of interest to the mathematician at large regardless of the final outcome concerning its "validity".
Until
recently
the notion
only by the mathematical considerable particular,
acquaintance the notions
logician°
of formal languages,
theory.
in areas
of set theory,
who do n o t p o s s e s s t h a t we h a v e w r i t t e n whilst
all
logic,
satisfaction,
study requires logic
model theory,
t h e a x i o m h a s become o f i n t e r e s t
of these prerequisites
this
this
any kind of in-depth
extensively
--
a in
and a good
B u t w i t h t h e g r o w i n g u s e o f t h e Axiom o f C o n s t r u c t i b i l i t y
short
it would be very nice
mathematical
Indeed,
s e t was s t u d i e d
with the ideas and methods of mathematical
deal of pure set outside
of a constructible
account.
from logic.
Our b a s i c
is for this
premise in writing
if everyone had at least
is almost certainly
It
to mathematicians
a basic
not the case.
audience
has been that,
knowledge of elementary
We t h e r e f o r e
a s s u m e no
p r i o r knowledge of m a t h e m a t i c a l l o g i c . i s d e s i g n e d so t h a t t h i s else.)
(The one e x c e p t i o n i s C h a p t e r V, b u t t h e book
c h a p t e r c a n he t o t a l l y
ignored without affecting
anything
S i n c e i t would c l e a r l y he f a r t o o g r e a t a t a s k t o d e v e l o p t h i s m a t e r i a l t o a
l e v e l a d e q u a t e f o r a n y t h i n g a p p r o a c h i n g a c o m p r e h e n s i v e trea~nen% o f e o n s t r u c t i b i l i t y , we c h o o s e i n s t e a d %o c u t some c o r n e r s and a r r i v e a t t h e r e q u i r e d d e f i n i t i o n s
very
q u i c k l y . I n o t h e r w o r d s , we p r e s e n t h e r e a d e s c r i p t i o n of s e t t h e o r y and t h e Axiom of Constructibili%y, not the theory itself. Admittedly this approach may prove annoying to logicians -- but they do not need to read this account, being well equipped to consult a more mathematical account.
The book i s d i v i d e d up a s f o l l o w s . I n C h a p t e r I we d i s c u s s some w e l l known p r o b l e m s o f p u r e m a t h e m a t i c s . S i n c e e a c h o f t h e s e p r o b l e m s i s u n s o l v a h l e on t h e b a s i s o f t h e c u r r e n t s y s t e m of s e t t h e o r y , h u t can be s o l v e d i f one a s s u m e s t h e ,ixiom of Constructibility, as i l l u s t r a t i o n s
t h e y p r o v i d e b o t h a m o t i v a t i o n f o r c o n s i d e r i n g t h e axiom, a s w e l l of i t s a p p l i c a t i o n .
I n C h a p t e r I1 we g i v e a b r i e f
t h e o r y . T h i s forms t h e b a s i s o f our d e s c r i p t i o n of c o n s t r u c t i b i l i t y C h a p t e r IV a p p l i e s t h e Axiom o f C o n s t r u c t i b i l i t y considered in Chapter I.
Chapter V is different
some knowledge o f l o g i c i s a s s u m e d .(At l e a s t , i o n a p r i o r knowledge of l o g i c i s r e q u i r e d .
in Chapter III.
in order to solve the problems from t h e r e s t
for a full
of t h i s book i n t h a t
a p p r e c i a t i o n of o u r d i s c u s s -
The r e a d e r may be a b l e t o g a i n some i d e a
o f what i s g o i n g on w i t h o u t s u c h knowledge. We c e r t a i n l y as p o s s i b l e . )
account of s e t
t r y %o k e e p t h i n g s a s s i m p l e
I n C h a p t e r V we t r y t o e x p l a i n j u s t how i t i s t h a t t h e Axiom o f
Constructibility
e n a b l e s one %o answer q u e s t i o n s of m a t h e m a t i c s of t h e k i n d c o n s i d e r e d
in the previous chapters.
I n o r d e r %o i l l u s t r a t e
o u r d e s c r i p t i o n we p r e s e n t a f u r t h e r
a p p l i c a t i o n of t h e axiom, t h i s t i m e i n Measure T h e o r y . (We t h e r e b y p r o v i d e some c o n s o l a t i o n f o r measure t h e o r i s t s in Chapter I . )
who may h a v e f e l t
The hook i s s t r u c t u r e d
left
out by our choice of problems
on t h e a s s u m p t i o n t h a t many r e a d e r s w i l l n o t
w i s h t o go i n t o t h e s u b j e c t m a t t e r of C h a p t e r V v e r y t h o r o u g h l y , i f a t a l l .
It is to be hoped that mathematicians may wish to use the Axiom of Constructihility. For this reason the proofs in Chapter lV are given in some detail, except
VI
that
i n e a c h c a s e we s t a t e
without proof a very general combinatorial
i s a c o n s e q u e n c e o f t h e Axiom o f C o n s t r u c t i b i l i t y , order to prove the desired result.
The a d v a n t a g e o f t h i s
may u s e t h e p r o o f a s a model f o r o t h e r p r o o f s , of time investigating
Finally this
and t h e n u s e t h i s
principle principle
approach is that
in
the reader
without having to spend a great
t h e Axiom o f C o n s t r u c t i b i l i t y
which
deal
itself.
a word a b o u t o u r u s e o f t h e p h r a s e " p u r e m a t h e m a t i c s " . I n ~ r i t i n g
book i t has been c o n v e n i e n t to r e s t r i c t
mathematics other than set theory". "mathematician" in our title.
A similar
the meaning of t h i s remark applies
phrase to "pure
t o o u r u s e o f t h e word
CONTENTS
Preface
Four Famous Problems
Chapter I.
1
1. A Problem i n Real A n a l y s i s
1
2. A Problem in A l g e b r a
2
3. A Problem i n G e n e r a l Topology
5
4. A Problem in Set Theory
6
~at
Chapter If.
i s S e t Theory,?
11
1. S e t Theory a s a Framework for ~athematics
11
2. Set Theory Under t h e Microscope
13
3. A Language f o r S e t Theory
14
4. The Set-Theoretic Hierarchy
18
5. The Axiom of Choice
25
The Axiom of C o n s t r u c t i b i l i t y
27
1. The C o n s t r u c t i b l e L i e r a r c h y
28
2. The Axiom of Constructibility
30
Chapter III.
3. The Generalised Continuum l~pothesis 4. H i s t o r i c a l
33 ~emarks
34
A p p l i c a t i o n s of V= L i n M a t h e m a t i c s
Chapter IV.
1. C o m b i n a t o r i a l P r i n c i p l e s
from V= L
36 36
2. The S o u s l i n IToblem
39
3. The \~itehead Problem
~
4. Collectionwise I ~ u s d o r f f Spaces
58
5.
6~
F u r t h e r Remarks
VIII Chapter V
A Problem in Measure The ery
65
I. Lktensions of Lebesgue Measure
65
2. The Measure Problem
66
3. A Theorem in Model Theory
72
4. The Condensation Lemma
76
5. Solution to the Measure Problem
78
6. ~[istorical ]lemark
80
Appendix I
~ i e m s for Set Theory,
81
Appendix II
Independence Froofs in Set, Theo,ry
86
Glossary of key Terms
92
Special Symbols
94
Suggested Further Reading
96
Chapter I
FOUR FAMOUS PROBLENS
In order
both to motivate
and to illustrate
its
of pure mathematics, and one from set solved
we g i v e h e r e a b r i e f
one f r o m a n a l y s i s ,
theory.
on t h e b a s i s
solvable
use,
the consideration
set
account
o f f o u r w e l l known p r o b l e m s
one from algebra,
These problems all
of the usual
o f t h e ~ixiom o f C o n s t r u c t i b i l i t y ,
one from ~eneral
have one thing
theoretical
topology,
i n con~non: t h e y c a n n o t b e
assumptions
(axioms),
but they are
i f we a s s u m e t h e i~xiom o f C o n s t r u c t i b i l i t y .
1. A P r o b l e m i n R e a l A n a l y s i s l 1
Let X be an infinite set, < a linear ordering of X. We may define a topology on X by taking as an open basis all intervals (a,b) = { x e X
la < x < b
} for a , b e X
with a < b. A classic theorem of Cantor says that if X has no largest member and no smallest member, and if the above topology on X is both connected and separable, then X is (considered as an ordered topological space) homeomorphic to the real line, ~, (considered as an ordered topological space). The basic idea behind the proof is to take a countable dense subset of X (by separability), prove that this set is isomorphic to the rationals, @, and then show that X must be isomorphic to the Dedekind completion of the dense subset, and hence isomorphic to ~, the Dedekind completion of Q. Use is made of the fact that the connectedness of X is equivalent to the two facts (a) that for each pair a,b of elements of X with a < b
there is a third element,c, of X with
a < c f ( 0 ) . can always find
C
)~ < K •
. We wish to prove that A n B is closed and unbounded in ~
ving. Then supf[~] e
C.
every set of ordinals has a suprenmm of course. )
The closed unbounded subsets of K generate a
Let ~ K
is closed if, whenever
is a limit ordinal and f:6---* C is order preserving, then sup~ will be a W-group. Hence Z/nZ is a W-group.
~ )Z/nZ
be the canonical projection. Clearly, Ker(~) ~ ~. Thus there is an
exact sequence
,Z
0
,Z
~ "~In~
,0.
Since Ext(Z/nZ,Z) = O, this sequence must split. Hence there is an embeddi~
~/~
, ~.
But Z/nZ has torsion and Z is torsion free, so this is impossible.
3.4 Lemma Let H ~ G . there is a
Proof:
H
Suppose that G is a W-group but that G/H is not a W-group. Then
~Z
which does not extend to a
G
~)Z.
• G/H
)0
Let 0
,H
~, G
be exact. By 3.I there is an exact sequence
~o~(G,Z) ~ >~om(H,Z) e >~t(G/~,Z) ~ ,~t(G,Z), where 1~(~) = ~ O l H =
~ restricted
t o H, f o r a l l
~ e Hom(G,Z).
Now, b y h y p o t h e s i s on G/H, E x t ( G / H , Z ) ¢ 0. 2Lnd by h y p o t h e s i s on G,
~t(G,z) = 0.
I~(I~) ¢ ~o~(~,Z).
~ence any
~euce, ~.(~)
G
~ .
= Ker(T) = ~t(G/~,Z) ¢
T h i s means t h a t
In other words,
~
there is a
0. Hence ~ - ~
Ker(e) ¢ Hom(H,Z). ~ ¢ 1~(r)
with
is not the restriction to H of any
G
for
~ ~Z. Thus
is as required.
Suppose now t h a t we a r e g i v e n a W-group, G, a n d we w i s h t o show t h a t G i s free.
How m i g h t we p r o c e e d ? One way i s t o t r y t o c o n s t r u c t ,
f o r G. A u s e f u l t o o l t h e n would be t h e f o l l o w i n g r e s u l t ,
explicitly,
a direct
a basis
consequence of
48
the proof of 1.2.1.
3.3 Lemm Let H ~ G. If G/H and H are both free, then G is free and any basis of H can be extended to a basis of G.
We might try to use 3.5 as follows. Suppose we can represent G as the union
o f an i n c r e a s i n g ,
continuous chain
( I n c r e a s i n ~ h e r e means t h a t limit ordinal ~ ~ C( /
, so
we ~ u s t
have
~(~) ~ ~(~')
#
In
o t h e r w o r d s , n o t h i n g can go wrong.
Set
= 13 Clearly,
H and K a r e d i s j o i n t .
~
,
K=
B e i n g s u b s e t s of
U~X~. ~1'
they are discrete,
and hence
63
closed. By normalitywe
P i c k some f u n c t i o n
can find disjoint open sets U, V such that H K U and K K V.
f: ~1
,~
so that
E Z
Since H u K =
ml
' this
~
~f(~)(~)
definition
is
~
sound.
V .
Let
E= { ~ e ~ l I ft~ = f ~ } .
By assumption, E is a stationary subset of
Let
C = {o(~ coI
Af .
I oc is a limit ordinal & (V~~
take a little
whenever B a K has positive
, w h e r e A ~ ~ . We s a y t h a t
and g(~) < f ( ~ )
only important
~
on B ' .
each ~