NORTH-HOLLAND MATHEMATICS STUDIES
51
Notas de Matematica (78) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester
Axiomatic Set Theory Impredicative Theories of Classes
ROLAND0 BASIM CHUAQUI lnstituto de Maternatica
Universidad Catdlica de Chile Santiago, Chile
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of Coagrtrr C8tmIogiag in Ptlblieation Data
Chuaqui, R. Axiomatic s e t theory. (North-Holland mathematics studies ; 51) (Notas d e m a t d t i c a ; 78) Bibliography: p. Includes index. A. Axiomatic s e t theory. I. Title. 11. Series. 111. Series: Notas de m a t d t i c a (North-Holland pub, c0.) ; 78.
[email protected] no. 78 [QA2481 510s [5ll.3'221 81-4631 ISBN 0-444-86178-5 RAcR2
PRINTED IN THE NETHERLANDS
PREFACE
T h i s book c o n t a i n s a x i o m a t i c p r e s e n t a t i o n s i n f i r s t - o r d e r l o g i c o f vers i o n s o f Set Theory based on an i m p r e d i c a t i v e axiom o f c l a s s s p e c i f i c a t i o n . T h i s axiom a s s e r t s t h e e x i s t e n c e o f t h e c l a s s o f a l l s e t s which s a t i s f y a g i v e n a r b i t r a r y f i r s t o r d e r formula. T h i s axiom i s i m p r e d i c a t i v e because t h e d e f i n i n g f o r m u l a may c o n t a i n q u a n t i f i c a t i o n over a r b i t r a r y classes, i n c l u d i n g t h e c l a s s being d e f i n e d , ( c f . A. Fraenckel, Foundations o f Set Theo r y , N o r t h - H o l l a n d Pub. Co. pp 138-140).
A l l theorems i n t h e book can be deduced f r o m an e l e g a n t and v e r y s t r o n g a x i o m a t i c system B C o f Bernays, which uses a r e f l e c t i o n p r i n c i p l e . However, f o r most o f t h e book a weaker system (Morse-Kelley-Tarski o r MKT) i s s u f f i c i e n t . The t h e o r y based on t h i s l a t t e r system has a c o m p l i c a t e d h i s t o r y ; p r o b a b l y t h e f i r s t e x p o s i t i o n was by A.P. Morse. H i s axiom system, however, i s n o t standard. The f i r s t a x i o m a t i c v e r s i o n presented as a standard f i r s t o r d e r t h e o r y i s t h a t appearing i n t h e appendix t o K e l l e y ' s book General Topology. The axioms used i n t h e p r e s e n t book a r e b a s i c a l l y due t o A.Tarski. The p r e s e n t a t i o n owes much t o T a r s k i ' s courses on S e t Theory a t t h e U n i v e r s i t y o f C a l i f o r n i a , Berkeley. The i m p r e d j c a t i v e comprehension axiom f o r c l a s s e s i s s t r o n g e r than t h e corresponding p r i n c i p l e s i n t h e usual t h e o r i e s o f Zermelo-Fraenckel and van I have t r i e d t o use t h i s e x t r a s t r e n g t h as much as Neumann-Bernays-Gudel. p o s s i b l e i n o r d e r t o s i m p l i f y t h e development and t h u s show t h e t e c h n i c a l advantages o f i m p r e d i c a t i v e t h e o r i e s . I n o r d e r t o i s o l a t e t h i s f e a t u r e , I have d e v i c e d a subtheory o f MKT which I c a l l General Class Theory ( C ) . T h i s weak t h e o r y i s s l i g h t l y s t r o n g e r than one w i t h t h e same name which appears i n may papers Chuaqui 1978 and 1980. Other p r e s e n t a t i o n s o f i m p r e d i c a t i v e t h e o r i e s have appeared i n p r i n t ; f o r i n s t a n c e , t h a t o f Monk 1969. Monk's book, however covers l e s s m a t e r i a l and has a d i f f e r e n t approach t h a n t h i s one, T h i s book can be used as a t e x t b o o k f o r a graduate o r advanced undergraduate course. The m a t e r i a l c o u l d be covered i n two semesters o r t w o quarters. An e a r l i e r v e r s i o n has been used a s a t e x t b o o k a t t h e C a t h o l i c U n i v e r s i t y o f C h i l e by t h e a u t h o r and o t h e r t e a c h e r s f o r s e v e r a l years. The vii
PREFACE
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main requirements f o r t h e study o f t h i s book a r e mathematical m a t u r i t y a n d some knowledge o f Elementary Logic. T h i s knowledge can be provided, f o r i n stance, by Enderton 1972 Chapters 1-2. Acquaintance w i t h s e t t h e o r y a t l e a s t on t h e l e v e l o f Halmos 1965 o r Enderton 1977 i s a l s o necessary a l though n o t s t r i c t l y r e q u i r e d from a formal p o i n t o f view. The formal s t y l e o f t h i s book would make i t d i f f i c u l t t o understand w i t h o u t o r e v i o u s a c quaintance w i t h t h e i n t u i t i v e n o t i o n s . The scope o f Set Theory c l e a r l y cannot be covered i n one book o f r e a sonable l e n g t h . Thus, i t was necessary t o make a s e l e c t i o n . The p r i n c i p l e s F i r s t , metarnathematical g u i d i n g t h i s s e l e c t i o n have been t h e f o l l o w i n g . q u e s t i o n s and s u b j e c t s t h a t can be b e s t t r e a t e d metamathematically h a v e been avoided. Second, I have t r i e d t o i n c l u d e a l l s u b j e c t s which I b e l i e v e have a b e t t e r p r e s e n t a t i o n i n an i m p r e d i c a t i v e t h e o r y o f classes. T h i r d , I have avotded t h e d i s c u s s i o n o f a l l e n t i t i e s whose e x i s t e n c e cannot be deduced form B C
.
The main s u b j e c t s excluded a r e D e s c r i p t i v e Set Theory, t h e P a r t i t i o n Calculus, and those l a r g e c a r d i n a l s t h a t can be b e t t e r t r e a t e d rnetamathemat i c a l l y o r whose e x i s t e n c e cannot be proved i n B C . Good p r e s e n t a t i o n s o f these s u b j e c t s a r e t o be found i n Kuratowski-Mostowski 1978, Jech 1978 o r Levy 1979. Another p e c u l i a r i t y o f t h e book i s i t s separate p r e s e n t a t i o n o f t h e t h e o r i e s w i t h o u t t h e axiom o f choice. T h i s has p e r m i t t e d an e x t e n s i v e t r e a t m e n t o f what can be proved f o r c a r d i n a l s w i t h o u t choice.
I b e l i e v e t h a t one o f t h e main f e a t u r e s o f t h i s book i s t h e e x t e n s i v e use o f s t r o n g p r i n c i p l e s o f d e f i n i t i o n by r e c u r s i o n . Thus, f o r i n s t a n c e , t h e r a n k f u n c t i o n i s d e f i n e d r e c u r s i v e l y b e f o r e o r d i n a l s , which on t h e i r t u r n , a r e o b t a i n e d as t h e values o f t h i s f u n c t i o n . T h i s procedure i s due t o T a r s k i ; i t was adapted by me f o r t h e o r i e s w i t h o u t t h e Axiom o f Foundations. I n f a c t , my main indebtedness i s t o A l f r e d T a r s k i . The c o r e o f t h e book had i t s o r i g i n i n a course on Set Theory g i v e n by him i n B e r k e l e y i n t h e Academic Year 1936-64. H i s i n f l u e n c e can be e s p e c i a l l y seen i n t h e axioms and t h e d e f i n l t i o n s by r e c u r s i o n a l r e a d y mentioned, p l u s t h e t r e a t ment o f c a r d i n a l s w i t h o u t c h o i c e and t h e a r i t h m e t i c o f o r d i n a l s . A few words a r e i n o r d e r w i t h r e s p e c t t o t h e s t y l e o f p r e s e n t a t i o n o f t h e book. I have chosen t o s t a t e theorems and d e f i n i t i o n s q u i t e f o r m a l l y i n f i r s t - o r d e r language. The p r o o f s , however, a r e i n f o r m a l . I n o r d e r t o l i g h t e n t h e burden o f understanding t h e formulas, I have g e n e r a l l y added i n f o r m a l remarks e x p l a i n i n g theorems and d e f i n i t i o n s . One o f my f r i e n d s has s a i d t o me t h a t t h e book was w r i t t e n i n t h e s t y l e o f 1950 and n o t o f 1980. I agree w i t h t h i s remark, b u t I bel-ieve t h a t t h e 1950 s t y l e i s b e t t e r . I t h i n k t h a t s t u d e n t s should l e a r n t o read formulas, even complicated ones. There was a m a j o r advance i n Mathematics when mathematicians l e a r n e d t o w r i t e and read equations. The r e a d i n g and w r i t i n g o f l o g i c a l formulas i s a l s o an advance, a l t h o u g h perhaps n o t so c r u c i a l as t h a t of equations.
PREFACE
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Several people have helped me by reading t h e manuscript. I would l i k e t o e s p e c i a l l y thank Newton da Costa, U l r i c h Felgner, I r e n e Mikenberg, Manuel Corrada and Cesar Mizuno. The e x c e l l e n t t y p i n g j o b was done by M.Eliana Cabaiias, whom I warmly thank.
I would a l s o l i k e t o thank the Regional E d u c a t i o n a l , S c i e n t i f i c a n d Technological Development Program of t h e Organization of A m e r i c a n S t a t e s f o r i t s financial support.
R. Chuaqui S a n t i a g o , 1981.
GUIDE TO USE THIS BOOK
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3.7.3 3.8.1
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- 4.3
PART 1
Introduction CHAPTER 1 . 1
Informal background
There a r e two main points of view with respect t o axiomatic s y s t e m s . According t o the point of view t h a t may be c a l l e d a l g e b r a i c , the axioms a r e t r u e f o r a l a r g e number of concepts. In t h i s case, t h e axioms t h e m s e l v e s c h a r a c t e r i z e the corresponding mathematical theory completely. For instance in Group Theory we define a Group as a s e t and operations on t h i s s e t t h a t s a t i s f y t h e axioms of Group Theory.
e assume t h a t s e t s and The point of view of t h i s book i s d i f f e r e n t . W c l a s s e s a r e o b j e c t s e x i s t i n g independently of our minds. We c h o o s e some sentences which a r e true of these concepts as axioms. From these axioms,we t r y t o derive as many t r u e sentences (our theorems) a s possible. The ideal s i t u a t i o n would be t o derive a l l t r u e sentences about s e t s and c l a s s e s . We know t h a t t h i s i s impossible (by G'ddel I s Incompleteness Theorem). Therefore we have t o be content w i t h deriving a l l what we need f o r t h e purpose a t hand. In order t o proceed according t o t h i s second point of view, i t will be necessary t o present t h e basic concepts of the theory and e x p l a i n them enough so as t o be able t o show t h a t t h e chosen axioms a r e t r u e . Obviously, t h i s will be an informal explanation not of a s t r i c t l y mathematical character. The basic notions a r e those of s e t and c l a s s , and t h e fundamental r e l a t i o n between s e t s and c l a s s e s i s t h a t of elementhood. I t i s necessary t o d e l i m i t these notions because t h e informal notions of s e t and c l a s s a r e not c l e a r l y determined; a t l e a s t i n p r i n c i p l e , t h e r e a r e several possible no-t i o n s of s e t o r c l a s s .
A d u h . ~i s an a r b i t r a r y c o l l e c t i o n of o b j e c t s , which may be n u m b e r s , functions, physical o b j e c t s , s e t s , e t c . Since t h e r e a r e no r e s t r i c t i o n s with respect t o t h e nature of these o b j e c t s , we might t h i n k of a c l a s s a s s p e c i f i e d when, f o r each object , i t i s possible t o determine whether i t belongs t o t h e c l a s s o r not. In p a r t i c u l a r , this would mean t h a t t o each property, defined i n any way whatsoever corresponds the c o l l e c t i o n o f t h o s e o b j e c t s which have the given property. I t i s well known t h a t t h i s way o f considering c l a s s e s leads t o R u s s e l l ' s paradox: Let us specify the c l a s s A by indicating t h a t an o b j e c t x i s a member of A i f and only i f x i s a c l a s s and x i s not a member of x. Then, A i s a member of A i f and only i f A i s not a member of A. T h i s i s a contradiction. Therefore, we cannot consider c l a s s e s as extensions of a r b i t r a r y properties. The course we s h a l l follow here i s t o l i m i t c l a s s extensions t o a giv-
3
4
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en universe o r domain V. All the elements of t h i s universeare a l s o c l a s s e s , the s e t s . I t i s a l s o possible t o include elements t h a t a r e n o t c l a s s e s ( t h e Urelemente), b u t we s h a l l r e s t r i c t our at.tention t o the case of no Urele- mente. Since c l a s s e s a r e extensions, they a r e determined by t h e i r elements. Thus, c l a s s e s with t h e same elements a r e equal.
...
The main i n t u i t i v e concepts which we s h a l l formalize a r e those o f eX06 and ncZ 0 4 &erne& 06 ( i . e . s e t of ) where instead of we p u t a given a r b i t r a r y c l a s s C.
...
...
...,
There i s in V , our universe, an init.ia1 c o l l e c t i o n of elements u , the c o l l e c t i o n of Urelemente. In our case, s i n c e we a r e dealing with pure s e t s , u will be empty. V i s closed under the two notions we want t o study. Whenever a s e t u i s in V a l l elements of u , and a l l s e t s of elements of u ( i . e . subsets of a ) a r e a l s o in V. Since c l a s s e s a r e extensions of properties limited t o V , a l l c l a s s e s a r e subclasses o f V ; therefore every element of a c l a s s i s i n V . V i t s e l f i s , obviously, a l s o a c l a s s . Thus, an object i s in V i f and only i f i t belongs t o a c l a s s . That i s , X i s a s e t i f and only i f X belongs t o a c l a s s . Classes t h a t a r e n o t s e t s ( i . e . t h a t do n o t belong t o any c l a s s ) will be called proper c l a s s e s . A l l objects d e a l t with in our t h e o r i e s , in particul a r s e t s , a r e a l s o classes. The c o l l e c t i o n u of Urelemente i s completely a r b i t r a r y . We only need t h a t u be a well s p e c i f i e d c o l l e c t i o n . Thus, once we have a domain Vclosed under the notions s p e c i f i e d above, we can take t h i s V a s a s e t of Urelemente and c l o s e i t t o form another universe V' This process can be r e p e a t e d with V ' instead of V and continued i n d e f i n i t e l y .
.
In the next section I shall introduce a strong axiomatic system (Bernays c l a s s theory B C ) which s u f f i c e s f o r proving a l l theorems in the book. Besides t h i s system , t h r e e other systems will be developed. These systems have axioms which a r e theorems of B C and, hence, weaker than i t . The f i r s t one i s General Class Theory ( C ) a very weak system. I n G, however, i t i s possible t o develop most o f the theory of d e f i n i t i o n s by recursion. The second system i s e s s e n t i a l l y equivalent t o t h a t contained i n t h e Appendix t o Kelley's book General Topology (Kelley 1955) without the axiom of choice. Since t h i s theory was f i r s t developed by A.Morse ( s e e Morse 1965) and t h e axioms used here a r e due t o Tarski, I s h a l l c a l l i t MorseKelley-Tarski ( M K T ) . The t h i r d system considered i s M K T whith c h o i c e ( M K T C ) . M K T C i s s u f f i c i e n t f o r most of Mathematics, as i s shown i n Kelley's book. I t i s the impredicative c l a s s theory corresponding t o Z F C (Zermelo - Fraenckel with c h o i c e ) , although M K T C i s d e f i n i t e l y stronger.
CHAPTER
1.2
Axioms
The axiom who f o r m u l a t e d p l e s discussed can be deduced
1.2.1
system t o be i n t r o d u c e d i n t h i s s e c t i o n i s due t o P. Bernays i t i n 1961 (see Bernays 1976) i n s p i r e d by r e f l e c t i o n p r i n c i b y Montague and LPvy (LPvy 1960). A l l theorems i n t h i s book from i t .
GENERAL AXIOMS,
We f i r s t d i s c u s s a s e t o f axioms which can be j u s t i f i e d by t h e conside r a t i o n s s e t f o r t h i n t h e p r e v i o u s s e c t i o n . A l a r g e p a r t o f s e t t h e o r y can be developed from them. 1.2.1.1
I M P R E D I C A T I V E A X I O M OF C L A S S S P E C I F I C A T I O N ,
T h i s axiom has been proposed by s e v e r a l people. The f i r s t mentions of i t seems t o be by Q u i n e ( Q u i n e 19511, Morse (Morse 19651, K e l l e y ( K e l l e y 1955, Appendix) and Mostowski (Mostowski 1950). I t a s s e r t s t h e e x i s t e n c e o f a c l a s s t h a t c o n t a i n s a l l s e t s s a t i s f y i n g a g i v e n c o n d i t i o n we m i g h t f o r m u l a t e t h i s axiom by:
Fa& w a y p m p e n t y P thane A a d a b s A calzclidfincj
t h a t P (XI,
04
th.e
hcth
x nuch
However, t h i s f o r m u l a t i o n where P i s a v a r i a b l e r e f e r r i n g t o p r o p e r t i e s . i s i n second-order l o g i c , because i t c o n t a i n s q u a n t i f i c a t i o n over propert i e s . Thus, we have t o r e p l a c e t h i s axiom b y f i r s t - o r d e r axioms.
A n a t u r a l method f o r t h i s purpose, i s t o r e p l a c e p r o p e r t i e s by f i r s t o r d e r formulas. T h i s procedure was proposed by Skolem and Fraenckel
.
I n o r d e r t o do t h i s , we have t o d e f i n e c l e a r l y t h e l o g i c a l f i r s t - o r d e r language. A l l axioms w i l l be formulated i n t h e p r i m i t i v e language e which P w i l l be c o n s t i t u t e d as f o l l o w s :
A ) V a r i a b l e s . V a r i a b l e s a r e lower-case and c a p i t a l s c r i p t l e t t e r s w i t h o r w i t h o u t s u b s c r i p t . For i n s t a n c e A , x 0,
....
B) Constants.
-
( i ) L o g i c a l constants: 1 ( t h e n e g a t i o n symbol), + ( i m p l i c a t i o n sym(equivalence b o l ) , V ( d i s j u n c t i o n symbol), A ( c o n j u n c t i o n symbol), V (universal quantifier), 3 (existential quantifier), 3 ! (the symbol), q u a n t i f i e r ' t h e r e e x i s t s e x a c t l y one'), = ( t h e i d e n t i t y symbol), and (,) parentheses).
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(ii) N o n l o g i c a l constants. The b i n a r y pred c a t e An e x p / r e h i o n o f
eP
E
( t h e membership
r e l a t i o n symbo ).
i s a f i n i t e sequence o f symbols.
Among t h e ex-
p r e s s i ons, we d i s t i n g u i s h t h e formulas: (1) I f x , y a r e v a r i a b l e s , t h e n
x
=
q and x € y a r e formulas.
( 2 ) I f J, and 9 a r e formulas, and x i s a v a r i a b l e , then 1 9 , ( 9 + $ ) ( @ " G), f@-* $1, (9 s)@'+ Wx 9, 3 x 9 , and I ! x 9 a r e formulas. ( 3 ) A l l formulas o f L
o f (1) and ( 2 ) .
P
,
a r e o b t a i n e d by a f i n i t e number o f a p p l i c a t i o n
Greek l e t t e r s w i l l denote expressions.
9 , J,, 0 w i l l be formulas.
Bound and f r e e v a r i a b l e s a r e d e f i n e d as usual. Sentences a r e formul a s w i t h o u t f r e e v a r i a b l e s . We say t h a t t h e f o r m u l a J, i s a ( u n i v e r s a l ) W xn 9. c l o s u r e o f t h e formula 9 , i f J, i s a sentence o f t h e form W xo
...
A l l axioms, d e f i n i t i o n s , and theorems o f o u r t h e o r i e s a r e sentences. When a s s e r t i n g such sentences t h e i n i t i a l s t r i n g o f u n i v e r s a l q u a n t i f i e r s w i l l be g e n e r a l l y o m i t t e d . I n o t h e r words, when a s s e r t i n g a c l o s u r e o f 9 , we s h a l l u s u a l l y w r i t e 9 . I n w r i t i n g formulas, some parentheses w i l l be g e n e r a l l y o m i t t e d . Those o m i t t e d should be r e s t o r e d as f o l l o w s : F i r s t proceed from l e f t t o r i g h t and when a q u a n t i f i e r i s reached, we a s s i g n t o i t t h e s m a l l e s t p o s s i b l e scope. We r e p e a t t h i s process f i r s t f o r 1 , t h e n f o r A and V , t h e n f o r +, G X [ y ] is t h e formula o b t a i n e d from 9 by s u b s t i t u t and, f i n a l l y f o r f-r i n g a l l f r e e occurrences o f x by Y. L a t e r we s h a l l i n t r o d u c e e x t e n s i o n s o f L by adding new n o n l o g i c a l P symbol s.
.
Our f i r s t axiom, o r r a t h e r axiom-schema i s t h e f o l l o w i n g : AxClass (Schema), Fon m y 6ohmuf.u 9 0 6 I: which d o u n o t contain A P @ee, any d o ~ w r e06 ,the 6oUowing dotunLLea A an axiom: 3 A tlx(xEA-@~A 3 U x E U ) .
Note t h a t t h i s schema (as a l l schemata) r e p r e s e n t s an i n f i n i t e number @ may c o n t a i n any o t h e r f r e e vao f axioms, one f o r each f o r m u l a @ o f L P' r i a b l e s besides x ; t h e o n l y f r e e v a r i a b l e excluded i s A. '3 U x E U ' expresses ' x i s a s e t ' .
1.2.1.2
AXIOM OF EXTENSIONALITY
8
T h i s axiom s t a t e s t h e e s s e n t i a l p r o p e r t y o f c l a s s e s as extensions o f p r o p e r t i e s , namely, t h a t c l a s s e s a r e determined b y t h e i r elements.
AXIOMATIC S E T T H E O R Y
7
The converse implication i s a l s o t r u e by v i r t u e of the laws of logic.
T h i s axiom excludes Urelemente, s i n c e i t implies t h a t t h e r e i s j u s t one o b j e c t with no elements, namely, t h e empty c l a s s .
AXIOM OF SUBSETS,
1.2.1.3
This axiom expresses t h e f a c t t h a t our universe i s closed under ' s e t of . . . I where instead of we p u t any element of t h e universe, i.e. any s e t . T h u s , a subclass of a s e t should be a s e t .
...
3 U bell A Wx(xEa+ xEb) + 3 U aeU.
Ax Sub. 1.2.1.4
AXIOM OF REFLECTION,
This fourth axiom enbodies t h e p r i n c i p l e t h a t i f we have a universe Y already given, then we can take V as a s e t of Urelemente in order t o form another universe V . T h u s , V will be a s e t in t h i s new universe V ' and any c l a s s A (subclass of V ) will a l s o be a s e t i n V ' .
...',
We assumed V t o be closed under 'element of i.e. any element o f an element of V i s an element of V . Thus, I/ a s a s e t i n V ' should have t h i s same property. Before introducing t h e Axiom of Reflection, we need some notation. For each @ of I: t h a t does not contain U, will be the formula obtained by P r e l a t i v i z i n g each bound v a r i a b l e X t o the formula V q ( y E X --* y E U ) (i.e. t o t h e formula t h a t says t h a t X i s a subclass of U). This formula will be abbreviated by X 5 U. More p r e c i s e l y , @u i s defined .by recursion a s follows: ( i ) If @ i s x E y ( i i ) If @ i s 0 vel y.
or
-,dJ
x = y
or 10
( i i i ) I f @ i s WX 0
, then
(iv) If @ i s 3 X 0
, then
@'
, then
, then
@
U
@
U
is @ itself.
i s O U + dJu or 1 OU, U
is
W X(X c -U
is
3X(X c - U A 0').
+
respecti-
0 ).
All other logical connectives can be defined in terms of these, so we do not need t o include them i n the d e f i n i t i o n . Notice t h a t i f @ i s 3 ! X 0 , then @ U i s equivalent t o 3 ! X(X c - U A 0'). The axiom schema of r e f l e c t i o n i s , then, AxRef. (Schema). b and containn at W A(@ YEA
+
m0b.t
3 u( 3 U
A y E u ) +@:[
bl
Fox each am& @ 06 I: t h a t doen n o t c o n t a i n u o h P A Q~e-e,t h e 6o&eowing 0 an axiom: uEU
1.
A W
X W q ( x ~ q ~ ux e u ) -+
A W b( W q(yEb
++
8
R O L A N D 0 CHUAOUI
T h i s axiom says t h a t i f A i s a c l a s s t h a t s a t i s f i e s t h e p r o p e r t y dJ, then t h e r e i s a s e t u, which i s t r a n s i t i v e (i.e. X E U i m p l i e s x C - u), such t h a t t h e common p a r t o f A and u, i.e. A n u , s a t i s f i e s dJu. The j u s t i f i c a t i o n o f Ax Ref based on t h e i n f o r m a l n o t i o n s runs as f o l Suppose we have a c l a s s A p o s e s i i n g a p r o p e r t y dJ, i.e. such t h a t I ; t h i s c l a s s i s a subclass o f our u n i v e r s e Y . I t i s c l e a r t h a t @[A] V i s e q u i v a l e n t t o dJ [ A ] , because a l l c l a s s e s a r e subclasses o f V . We now t a k e V and form another u n i v e r s e V ' such t h a t V E V ' . Then, i n t h i s new universe, V i s a s e t u. Since V i s c l o s e d under 'element o f . . . I , u satisfies:
lows. 9 [A
x W y(x€q€u Also, A c - u. i s t h e same as A . V as dJ [ A ] .
B.
Thus, i f b s a t i s f i e s
v
-+
XEU)
y(qEb
Therefore, s i n c e V i s u , dJ:[bl
.
-
Y E A A ~ E u ) ,t h e n i t
expresses t h e same f a c t
The t h e o r y o b t a i n e d w i t h t h e axioms o f t h i s s e c t i o n w i l l be denotedby 1.2.2
L I M I T I N G AXIOMS,
The f o l l o w i n g axioms exciude c e r t a i n c l a s s e s and make t h e development o f t h e t h e o r y simpler. However, t h e y a r e n o t as w e l l j u s t i f i e d as t h e prev i o u s ones. These new axioms have been e x t e n s i v e l y by s t u d i e d metamathem a t i c a l l y , e s p e c i a l l y w i t h r e g a r d t o q u e s t i o n s about t h e i r c o n s i s t e n c y and independence. Thus, i t i s convenient t o deal s e p a r a t e l y w i t h them. 1.2.2.1
AXIOM OF GLOBAL CHOICE,
T h i s axiom excludes s e t s and c l a s s e s t h a t cannot be w e l l ordered (see 2.2.3.21 f o r a d e f i n i t i o n o f w e l l o r d e r i n g s ) . Ax GC.
v x tj y ( x , q E A A x # q 3 z Z E XA 1 3 z ( z E x A z E y ) ) 3 8 V x ( x E A -, 3! ~ ( q E A x y E B ) ) A 3 C( W U ( W x ( x E U x E C ) A 3 x XED 3 x(xgU A W y ( y ~ P 4 Vz(zEx+zEy))) A Wx ( 3 U x E U +3y x E y E C ) ) . -+
-+
--f
+
Ax GC i s composed o f two p a r t s . The f i r s t , which w i l l be c a l l e d AxC, says t h a t f o r any c l a s s A o f d i s j o i n t nonempty sets, t h e r e i s a c l a s s B t h a t c o n t a i n s e x a c t l y one element o f each s e t i n A. (Thus, AxC i s 3 z Z E XA 1 3 z ( z E x A z E y ) ) WXW g ( x , q E A A x # q 3 B W x(xEA 3! q ( y E 8 A EX)).) The second, a s s e r t s t h a t o u r u n i v e r s e V i s t h e u n i o n of a c l a s s C of s e t s w e l l - o r d e r e d by i n c l u s i o n . T h i s axiom w i l l n o t be used u n t i l P a r t 4, where a c l a r i f i c a t i o n o f i t s meaning w i l l be given. +
-+
-+
Many doubts about t h e t r u t h o f t h i s axiom have been expressed. T h i s i s due, on t h e one hand, t o t h e f a c t t h a t Ax C a s s e r t s t h e e x i s t e n c e o f a c l a s s , which i s n o t unique, w i t h o u t d e f i n i n g i t , and, on t h e o t h e r hand, t o some o f i t s consequences which a r e strange.
9
A X I O M A T I C SET THEORY
However, Ax C i s i n d i s p e n s a b l e f o r numerous p r o o f s i n many d i f f e r e n t mathematical d i s c i p l i n e s , and i t has been shown c o n s i s t e n t w i t h t h e o t h e r axioms ( i f these a r e themselves c o n s i s t e n t ) . The t h e o r y i n c l u d i n g Ax GC besides those i n t h e p r e v i o u s s e c t i o n w i l l be denoted B C .
1.2.2.2
A X I O M S OF
F O U N D A T I O N S OR R E G U L A R I T Y
8
T h i s axiom excludes classes A f o r which t h e r e i s an i n f i n i t e sequence such t h a t E X E X E X ~ EA . *.. 2 1
...
xo, x l , ..., x n
I n o u r i n f o r m a l model, we s t a r t from an i n i t i a l c o l l e c t i o n o f U r e l e ment ( i n o u r case empty). A l l c l a s s e s a r e formed from t h i s c o l l e c t i o n . If a c l a s s A e x i s t e d w i t h t h e above mentioned p r o p e r t y , then A would n o t be o b t a i n e d from Urelemente. I n o r d e r t o c l a r i f y t h i s m a t t e r , l e t us c a l l t h e k e r n e l o f a c l a s s B , t h e c o l l e c t i o n o f Urelemente which e i t h e r b e l o n g t o B y o r t h e elements o f B, o r t o elements o f elements o f B y e t c . The Axiom of R e g u l a r i t y c o u l d be k e t i m L f r . Combining t h i s p r i n c i p l e w i t h t h e paraphrased: "EvefLy &!cud buu non e x i s t e n c e o f Urelemente ( o b t a i n e d f r o m Ax E x t ) we a r r i v e t o : "EvefLq c&& A b u X t 6hum Rhe empty some.
The f o r m u l a t i o n o f t h i s s o r t o f axiom i n d: would be e x t r e m e l y cumberTherefore, we adopt: P Ax Reg.
3 x xEA
+
3 x(xEA A V Y ( q E x
+
q P A)).
I t i s easy t o show t h a t Ax Reg excludes i n f i n i t e €-descending sequences o f t h e form x E X E x O , by c o n s i d e r i n g t h e class A = {xo, x1 , 2 1 x2, 1.
...
...
We s h a l l n o t o f f i c i a l l y adopt t h i s axiom i n o u r t h e o r i e s . However, we s h a l l mention t h e o r i e s i n which t h i s axiom i s v a l i d . I f T i s any t h e o r y T R w i l l be T w i t h Ax Reg added.
PART 2 General Class Theory CHAPTER 2 . 1
Introduction t o General Class Theory
2.1.1
AXIOMS FOR G ,
General Class Theory ( C ) i s the theory with t h e following axioms: (i) (ii) (iii) (iv)
Ax Ax Ax Ax
Class. Ext. a + 3 U a€U. Em. W x x Num. W X ( X E U - x € b V x = c ) A 3 U b E U - . l U a E U .
Ax Class and Ax E x t a r e the same axioms already introduced f o r B . Ax Em a s s e r t s t h a t t h e empty c l a s s i s a set. Ax Num says t h a t i f b i s s e t and a is formed by adding c t o b , t h e n a i s a l s o a s e t . This axiom will allow
us t o construct a l l natural numbers. those i n 2.1.4 a r e theorems of G .
All theorem in this Chapter except
The primitive language 1 will be extended t o allow f o r t h e possibiP l i t y of introducing defined symbols. The extended language will be c a l l e d In o r d e r t o define t h e new symbols, we introduce v a r i a b l e b i n d i n g term 1:. operators ( o r b r i e f l y v b t o s ) , Variable binding term operator symbols a r e symbols t h a t bind v a r i a b l e s of terms o r formulas t o form terms. The des c r i p t i o n operator U i s an example. I f 4 i s a formula and x a v a r i a b l e , U { x : $ I i s a term. The f r e e v a r i a b l e s of t h i s term a r e the f r e e variables of $ except x , which i s bound. In t h e intended i n t e r p r e t a t i o n , U Ex : 41 denotes t h e unique o b j e c t t h a t s a t i s f i e s $ i f t h e r e i s one and i s unique, or an a r b i t r a r y b u t fixed o b j e c t otherwise. Thus, a s w i t h a l l terms U C x : $1 denotes an o b j e c t , i n our case, a c l a s s . There will be o t h e r vbtos. In p a r t i c u l a r t h e c l a s s i f i e r Ex : @ I , i.e. t h e c l a s s of a l l s e t s t h a t s a t i s f y 4. Although from the point of view of pure l o g i c i t i s more convenient t o take U a s p r i m i t i v e and define a l l other vbtos i n terms of i t , i n s e t theory t h e c l a s s i f i e r works b e t t e r a s primit i v e . T h u s , I s h a l l follow t h i s course a n d consider the c l a s s i f i e r a s t h e only primitive vbto (introduced v i a an axiom) and define a l l o t h e r s i n terms of i t . Among the defined vbtos t h e r e a r e those composed of terms and formuFor instance, U x I r : $1, i .e. t h e union o f t h e r ( X ) f o r X s a t i s f y i n g @ ( X ) . This term has a s f r e e v a r i a b l e s a l l those f r e e in r and 4 except X. las.
13
ROLAND0 CHUAQUI
14
The description operator and the c l a s s i f i e r , which a r e vbtos t h a t b i n d variables of formulas, will be c a l l e d the vbtos of t h e f i r s t c l a s s . All o t h e r , which bind v a r i a b l e s of a term and a formula, are c a l l e d the secondc l a s s vbtos. We now pass t o t h e formal d e f i n i t i o n of I. In Axiomatic Set Theory, i t i s important t o define c l e a r l y t h e logical language. This i s so, because i n formulating some of the axioms, theorems, and d e f i n i t i o n s i t i s necessary to use the notions o f term and formula of t h e language. T h u s , Logic appears n o t only i n the formalization of proofs, b u t i t i s a l s o i n t r i n s i c a l l y enmeshed i n Axiomatic Set Theory through t h e m e t a l i n g u i s t i c grammatical notions. In order t o form L , we add t o L
t h e following types of symbols: P Predicates. Predicates with a fixed a r i t y will be introduced a s needed. Predicates will be bold-face c a p i t a l Roman l e t t e r s (with o r w i t h out s u b s c r i p t s ) , combinations of bold-face l e t t e r s , o r e s p e c i a l l y designed symbols (For instance A , B O , C ) .
-
-
Operation Symbols. Operation symbols of a fixed a r i t y w i l l be introduced a s needed. 0-ary operation symbols a r e t h e individual constants, These symbols will be bold-face c a p i t a l i t a l i c l e t t e r s (with o r without subs c r i p t s ) combinations of bold-face i t a l i c l e t t e r s , o r e s p e c i a l l y designed , Y , 0). symbols. (For instance, F , G 1 , U ,
-
-
Variable b i n d i n g term operators (vbtos). These symbols will a l s o be introduced a s needed. They will be denoted by e s p e c i a l l y designed symbols. There will be operators of f i r s t a n d second c l a s s .
An e x p m ~ h i o nof 1: i s a f i n i t e sequence of symbols. Among t h e expres, recursively a s follows: sion, we d i s t i n g u i s h t e m h and ~ a m h defined (1) A v a r i a b l e i s a term, ( 2 ) I f F is an n-ary operation symbol and F(r0,..., r n is a term.
T
~
..., ,
7
n -1 a r e terms, then
( 3 ) I f $ i s a formula, x a v a r i a b l e , and 0 a f i r s t c l a s s operator, then O{x : $1 i s a term.
...,
( 4 ) I f T i s a term, J/ a formula, xo, x ~ variables - ~ and 0 a second c l a s s operator, then 0 IT : # I i s a term. x o , . . Y ‘n 1
. -
( 5 ) I f 7 and u a r e terms, T = u and T E U a r e formulas. (6) I f A i s an n-ary predicate and T ~ , T n - 1 a r e terms, then i s a formula. A ( T ~ , . . . , rn
-
(4 V
-
...,
( 7 ) I f $ and 4 a r e formulas, and x i s a v a r i a b l e , thenl4I , (41 (4 A $1, (4 $1, V x 4 , 3 x 4 , and 3 ! x4I a r e formulas.
$1,
+
$),
(8) All formulas and terms a r e obtained by a f i n i t e number of applica-
AXIOMATIC SET THEORY t i o n s of (1)
-
15
(7).
Greek l e t t e r s w i l l denote expressions @, , ) I 0 , and and r , u, terms.
K
w i l l be formulas,
Bound and f r e e v a r i a b l e s a r e d e f i n e d as usual, n o t i n g t h a t v b t o s and q u a n t i f i e r s b i n d v a r i a b l e s . Sentences a r e formulas w i t h o u t f r e e v a r i a b l e s . I f y i s a formula o r a term, YX
o...Xn-1[?0'..
r
n-1
]
...,
T ~ , 7 a r e terms, t h e n n- 1 w i l l be t h e f o r m u l a o r term o b t a i n e d from y by s i -
multaneous proper s u b s t i t u t i o n o f a l l f r e e occurrences o f xo, To
...
..., 'n-1
by Proper s u b s t i t u t i o n means t h a t no f r e e v a r i a b l e o f r i becomes
7 n-1' bound, when ri i s s u b s t i t u t e d f o r xi.
s i o n s we w r i t e y
J/
[
S i m i l a r l y , when J/ and @ a r e expres-
$ 1 f o r t h e e x p r e s s i o n o b t a i n e d from y by s u b s t i t u t i n g
a l l occurrences o f J/ by 4 . I f 7 i s a t e r m o r a formula, y ' i s a vahiant o f y i f y ' i s o b t a i n e d from y changing some bound v a r i a b l e s . As i s well-known, y and y ' a r e l o g i c a l l y e q u i v a l e n t , i f formulas, and equal, i f terms.
A l l o b j e c t s denoted by terms a r e classes. n - a r y p r e d i c a t e s a r e i n t e r p r e t e d as n-ary r e l a t i o n s between classes. These n - a r y r e l a t i o n s a r e n o t i n general, classes. I n t h e course o f development o f s e t theory, r e l a t i o n s which a r e classes, i.e. o b j e c t s o f t h e t h e o r y (i.e. p o s s i b l e values o f t h e v a r i a b l e s ) a r e i n t r o d u c e d . Thus, i n o r d e r t o d i s t i n g u i s h these two types o f r e l a t i o n s , t h e i n t e r p r e t a t i o n o f n - a r y p r e d i c a t e s w i l l be c a l l e d n-ary n o a o n n , r e s e r v i n g t h e name r e l a t i o n f o r those which a r e classes. Unary not i o n s w i l l be c a l l e d coUectionn. The n o t i o n denoted by ' E l i s t h e b i n a r y n o t i o n o f belonging. Thus, I x E y ' i s r e a d ' x belongs t o y ' , o r ' x i s an element o f y ' . n - a r y o p e r a t i o n symbols denote n - a r y o p e r a t i o n s over classes, i.e., Xn-l assign the class mappings F such t h a t t o t h e n c l a s s e s Xo,...,
O,...,
Operations F a r e n o t themselves, i n general, classes. A F(X Xnml). 0-ary o p e r a t i o n symbol, i.e., an i n d i v i d u a l c o n s t a n t , denotes a c l a s s . I n o r d e r t o c l a r i f y these m a t t e r s , l e t us c o n s i d e r a p o s s i b l e model 02 of o u r theory. a h a s t o be o f t h e f o r ( A , E, Ri, 0 . ) iEI A, the unijEJ
.
verse, c o n t a i n s a l l t h e classes. E i s a b i n a r y n o t i o n i n t e r p r e t i n g E. The Rils a r e t h e o t h e r n o t i o n s which i n t e r p r e t t h e o t h e r p r e d i c a t e s ; and t h e
U . ' s are t h e operations. J There w i l l be o n l y two v b t o s o f t h e f i r s t c l a s s : t h e c l a s s i f i e r and U.
E: 1
As was mentioned before, t h e term U I x : @ I denotes t h e unique c l a s s (i. e. element of t h e u n i v e r s e o f t h e model 0 2 ) which s a t i s f i e s @ i f t h i s c l a s s
ROLAND0 C H U A Q U I
16
e x i s t s , and an a r b i t r a r y b u t f i x e d c l a s s , otherwise. I n s e t theory, t h i s a r b i t r a r y c l a s s i s chosen as t h e empty c l a s s , i.e., t h e c l a s s w i t h no members.
@.
U or the The t e r m { x :@ I denotes t h e c l a s s o f s e t s x t h a t s a t i s f y c l a s s i f i e r c o u l d be used as p r i m i t i v e . For convenience we s h a l l use t h e c l a s s i f i e r as p r i m i t i v e . Thus, i n s t e a d of axioms f o r U t h e f o l l o w i n g axiom schema i s added.
Fah each 6ohmda 4 t h e d o ~ ~ 0~6 ht hee 6oUawLng AvtunLLea
Ax Def.
axiom,
an
x E t x : @ } - c p A 3 U x E U .
U i s t h e n d e f i n e d as f o l l o w s : 2.1.1.1
D E F I N I T I O N SCHEMA,
L e t @ be a formula, t h e n
U { A : @ } = { x : 3 !A @ A 3 A ( @ A x E A ) l , where x does n o t appear i n 4 . As w i t h a l l schemata t h i s d e f i n i t i o n r e p r e s e n t s an i n f i n i t e c o l l e c t i o n o f d e f i n i t i o n s , one f o r each formula $. As an immediate consequence o f Ax Class, Ax Ext, Ax Def and Def. 2.1.1.1 we g e t : 2.1.1.2
THEOREM SCHEMA,
(i)3 1 A @ + W y ( g = U { A : 4 1
.in
@.
-
LeR @ be a ~ohmLLea,t h e n
@A 1 g
I ) whetle
y doer, not OCCUR
(ii)1 ( 3 ! A @ ) + U { A : @ =l U I A : W z z $ ! A } . PROOF, Assume t h a t t h e r e i s a unique A such t h a t @ and l e t y = U C A : We have t h a t x E y i f and o n l y i f t h e r e i s an A such t h a t cp and x € A . A i s unique; thus we have x c y i f and o n l y i f x E A f o r t h i s A. By Ax Ext, A = Y and $ A t y 1
41.
.
IfQ A
y 1 t h e n y i s t h e unique A such t h a t
y = U{A:@I.
.
Assume, now, -I ( 3 ! A9 ). empty. Thus ( i i ) f o l l o w s .
2.1.1.2
Then,
I x :1
@.Hence,
!A
@ A
we a l s o o b t a i n
3 A(@ A x
E A)}
2.1.1.3 REMARKS, We c o u l d a l s o t a k e as p r i m i t i v e U w i t h axioms ( i ) and ( i i ) and d e f i n e t h e c l a s s i f i e r by:
{ x :@ } = U { A : W x ( x € A
-
@ A 3 U xEU)}.
is
AXIOMATIC SET T H E O R Y
17
Then we would have Ax Def a s theorem. 2.1.1.2 ( i ) and ( i i ) a r e i n e s s e n t i a l v a r i a t i o n of t h e axioms introduced f o r t i n da Costa 1980 p. 138. T h u s , theorems shown t h e r e a r e s t i l l v a l i d f o r our U.
The t h e o r i e s of the present book w i l l be t h e o r i e s in t h e usual logical sense, i.e., s e t s of sentences of I: closed under l o g i c a l consequence (or der i v a b i l i t y ) . They will be axiomatizable t h e o r i e s , i.e. w i t h a recursive s e t of axioms. There will be two types of axioms: plropeh axioms and deiiniLLtion6. Proper axioms will always be given in C Proper axioms a r e r e f e r red t o simply as axioms. Ax Def i s considered pas a d e f i n i t i o n .
.
Definitions, other than Ax Def, a r e of t h e following types (see Gadel 1940) : ( A ) Definition of special c l a s s e s . s t a n t s and a r e of t h e form
These introduce individual con-
A = T ,
where A i s a new individual constant and T i s a term with primitive o r previously defined symbols and without f r e e v a r i a b l e s (i.e. a constant term). A denotes a c l a s s . ( B ) Definitions of n-ary notions. predicates and a r e o f the form
These d e f i n i t i o n s introduce n-ary
...,
+
where B i s a new n-ary predicate, Xo, Xn-l a r e d i s t i n c t v a r i a b l e s and i s a formula i n primitive o r previously defined symbols, containing a t most Xo, X n - 1 f r e e . In p a r t i c u l a r c o l l e c t i o n s a r e defined by formulas w i t h a t most one f r e e variable.
...,
+
( C ) Definitions of n-ary operations. These d e f i n i t i o n s introduce na r y operation symbols and a r e o f t h e form
...,
where F i s a new n-ary operation symbol, X,, ,,X, are distinct variables and T i s a term i n primitive o r previously-defined symbols w i t h a t most Xo Xn-l f r e e .
,...,
Occasionally, a d e f i n i t i o n is given by c a s e s , i . e . we define,
ROLAND0 C H U A Q U I
18
where r and o a r e terms and 4 i s a formula, a l l w i t h a t most Xo,..., Xn-l f r e e . This type of d e f i n i t i o n can be e a s i l y reduced t o t h e regular form by, F(XO,
...,X n-1 ) = U { z :
(z =
7
A $) V (z = u A 1
@)I.
(D) Definitions of vbtos. These d e f i n i t i o n s a r e schematic ( i . e . , defi n i t i o n schemas) i n the sense t h a t the schema r e p r e s e n t s a d e f i n i t i o n for each formula, o r formula and term o f C.
Schematic d e f i n i t i o n s f o r vbtos o f t h e second c l a s s have one of t h e f o l 1 owing two forms : ( i ) Let 0 be a vbto of the f i r s t c l a s s , $ , 8 formulas i n primitive o r previously defined notation. Let 0' be a new vbto. Then f o r every s e t o f d i s t i n c t variables Xo,..., Xnml, every term T and every formula $,
where x i s a v a r i a b l e occurring n e i t h e r i n 8 ' nor i n and d i f f e r e n t from 8 ' i s a v a r i a n t o f 8 t h a t i s f r e e f o r r and $J.
XO'.
..
$J
( i i ) The second form i s t h e same a s t h e f i r s t , w i t h W Xo,..., s u b s t i t u t i n g 3X0, 1 Xn-1 *
W Xn-l
...,
( E ) S t i p u l a t i o n of variables.. These a r e not properly d e f i n i t i o n , b u t can be considered a s abbreviations. In this case, we introduce a p a r t i c u l a r as variables s t i p u l a t i n g t h a t formulas of t h e type of l e t t e r s a ,P ,7 , form W a O x [ a I stand f o r W x(C(x) O) ,
...
-+
3 a 9, [ aI
stand for
3 x( C ( x ) A 0)
,
where C denotes a previously defined c o l l e c t i o n . For instance, l a t e r we s h a l l introduce Greek lower - c a s e l e t t e r s for ordinal s. Equality o f notions and operations i s defined extensionally. That is, we say t h a t F and G a r e t h e bame opmmXon and w r i t e F = G i f they a r e both n-ary and F = G can be considered an abbreviation of this l a s t formula.
Similarly two notions A and B a r e t h e they a r e both n-ary and,
bame
n o a o n (written A = B ) if
AXIOMATIC SET T H E O R Y
19
For each theory T, t h e primitive theory T i s t h a t i n E w i t h t h e propP P e r axioms a s only axioms. Using extensions of t h e usual theorems of l o g i c ( s e e Shoenfield 1967 p. 57-65 o r Enderton, pp. 154-163, together with da Costa 1980) i t can be proved t h a t T is a conservative extension of T and P t h a t a l l defined symbols a r e eliminable. T h i s means, i n p a r t i c u l a r , t h a t f o r any formula $ i n 1: t h e r e i s a formula +* i n I: w i t h t h e same f r e e varP i a b l e s , such t h a t any c l o s u r e of $ +* can be derived from T.
-
In order t o eliminate defined symbols from formulas we proceed as f o l lows: First replaced defined symbols by t h e i r d e f i n i t i o n , using v a r i a n t s i f necessary. T h e n , when only t h e c l a s s i f i e r i s l e f t replace recursively the formulas of t h e form $JJ I x : $1 I by 3 v(w = I x :$1 A $J). F i n a l l y , replace v = Ex : + I by V x ( x E v $ A 3 U u E U ) . The formula obtained a f t e r a f i n i t e number of such s t e p s w i l l be i n I: and equivalent t o t h e o r i g i n a l i n E. P
-
When we have a schema (axiom, theorem o r d e f i n i t i o n ) and a new symbol i s defined, unless otherwise noted, an i n f i n i t e set of axioms, theorems o r d e f i n i t i o n s i s added, one f o r each formula in the new symbol. T h u s , we immediately extend Ax Class to: Foh any 6omLLea $ 06 I: which does not contain A dhee, any d!obwLe t h e 6oUocuing ,johmLLea 0 a theohem. 3 A V x(xEA
-
$ A
3U x E U )
06
.
Ax Ref, however, i s an exception. I t s extension t o formulas o f L i s not so simple s i n c e r e l a t i v i z e d formulas occur. I t s extension t o I: will be discussed i n Section 3.1.4. Finally, i n order t o simplify the notation, t h e following conventions wi 11 be adopted. ( a ) On writing formulas o r terms some parentheses will be g e n e r a l l y omitted. Those omitted should be restored according t o t h e following procedure: First we proceed from r i g h t t o l e f t and when an operation symbol i s reached, we assign t o i t t h e smallest p o s s i b l e scope. Then, we again proceed from right o t l e f t and when a predicate i s reached we assign t o i t t h e smallest possible scope. We repeat this process f i r s t f o r t h e q u a n t i f i e r s , then f o r 1 , then f o r A , then f o r V , then f o r , and, f i n a l l y f o r
-.
+
(b) When T and o a r e terms, and A i s a binary predicate, we w r i t e T A o instead of A ( T , a ) arid T A U instead of 1A ( T , u ) ( f o r instance x E q , x q q , rn,l, o a r e terms, we w r i t e T~ T x = q , x # y ) . Also, i f r o n Ao instead of r O A o A r l A o A . . . A T ~ - ~A o ( f o r instance x , y, Z E B f o r
,...,
xEB A yEB A ~ € 8 ) .
,...,
20
ROLAND0 C H U A Q U I
( c ) A l l axioms, d e f i n i t i o n s and theorems o f o u r t h e o r i e s a r e sentences. When a s s e r t i n g such sentences t h e i n i t i a l s t r i n g o f u n i v e r s a l q u a n t i f i e r s w i l l be g e n e r a l l y o m i t t e d , I n o t h e r words, when a s s e r t i n g a c l o s u r e o f @, we s h a l l u s u a l l y w r i t e 9.
,...,
...,An,l a r e b i n a r y p r e d i c a t e s , ... nl, Aln, , instead o f
( d ) When r o T a r e terms a n d A o , n we s h a l l sometimes w r i t e r o A o rlA1 r 2
...
T,
T
A ( T ~ An,l - ~ T ~ ) . For instance, X E y E z i n 0 A 0 T 1 ) A (rlA1 r 2 )A stead o f X E y A y E z ; o r x E y C z, i n s t e a d of x E y A y C z. (T
2.1.1.4
DEFINITION (SPECIAL CLASS)
I
v=
Ex:
X = XI
.
V i s the u n i w m d U n . An obvious consequence o f 2.1.1.2
w
X(XE
V-
and t h i s d e f i n i t i o n i s :
3 U(X€U)).
We can say i t by We now have a s h o r t e r e x p r e s s i o n f o r ' x i s a s e t ' . I i n s t e a d o f I 3 U xEU'. 'x V ' s a y s t h a t x i s a p h o p ~Ceanb (i.
'x E V
e. a c l a s s t h a t i s n o t a s e t ) .
2.1.1.5 STIPULATION OF VARIABLESl From now on, we s h a l l use t h e f o l l o w i n g convention. A lower-case s c r i p t l e t t e r appearing i n a q u a n t i f i e r w i l l be understood as r e f e r r i n g t o sets. Thus, f o r m a l l y , i f x i s a lowercase i t a l i c l e t t e r and @ a formula, W x @ stands f o r W x(x E V - t # ) ; and 3 x # stands f o r 3 x(x E V A @). Since when a f o r m u l a @ w i t h x f r e e i s a s s e r t e d i t means t h a t tl asserted, 4 w i t h lower-case x stands f o r (when a s s e r t e d ) x E V-+ #
.
x@ i s
C a p i t a l s c r i p t l e t t e r s denote a r b i t r a r y c l a s s e s ( i . e s e t s o r p r o p e r classes). 2.1.1.6
DEFINITION (SPECIAL CLASS),
0 i s t h e m p t y Ceann.
0
=
Ex : x
f
XI.
AXIOMATIC SET THEORY
2.1.1.7
21
THEOREM i
(i)w x x q 0 (ii)0 E Y
PROOF,
.
( i ) i s o b t a i n e d by Def. 2.1.1.6
and Ax Em. ( i i ) by (i)
and 2.1.1.2.
DEFINITION (NOTIONS)
2.1.1.8 (i) A
CB
-
(ii)A C B -
I
W x(xEA
+
xEB).
(A&BAA+B).
I f A C B we say t h a t A LAa nubceabo 06 8. A C B can be read by A JA a pmpetr AYbcKLmn 06 8. We s h a l l a l s o w r i t e 8 > A and B 3 A, i n s t e a d O f A c - B and A C 8 , r e s p e c t i v e l y . The f o l l o w i n g theorem can be e a s i l y proved. 2.1.1.9
THEOREM,
2.1.1.10
--
DEFINITION (OPERATIONS)
I
(i) A u 8 =
(x:XEA
V XEB)
(ii) A n 8 = {x: x€A A
XEB)
(iii) A 8 = { x : xGA A x$B} (iv) 8 = V% 8
. . .
U (union) i s a b i n a r y o p e r a t i o n which assigns t o each p a i r o f classes A, 8, t h e c l a s s A U B. S i m i l a r l y w i t h n ( i n t m e c t i a n ) and (diddehence). The compLement - 8 i s a unary o p e r a t i o n .
The p r o o f
o f t h e f o l l o w i n g theorem i s easy and i s l e f t t o t h e reader.
22
ROLAND0 C H U A Q U I
2.1.1.11
THEOREM
( i ) A u 8 = 8 u A .
(ii) A n B = 8nA. ( i i i ) A U ( 8 U C ) = ( A U 8) U C . ( i v ) A n ( 8 n C ) = ( A n 8) n C . ( v ) A n (8
U
C) = (A n 8 )
. .
u (A n C)
( v i ) A u ( 8 n C ) = ( A u 8) n ( A u C ) (vii) A U A = A = A n A . (viii) A
U
(8
(ix) A
%
(8 n C ) = (A?.B)
(x) A
%
(8 u C)
(xi) (A (xiii) A
%
A) = A
8.
U
U
(A 2,
(8
%
(8
-V
C) = ( A
(B
U
C) = (A
-P ) ( x i v ) (AcC A 8 c
U
. . C) . C)
( A % 8 ) n (A
B ) % C = (A?.C)
U
(xii) A
%
C)
.
8 ) u ( A n C) 8)
%
C.
(A u 8 c -C
+
u
2) A
A n 8c C n0).
(xv) A n 8 C - A C- A u 8 .
(xvi) O n A = O A O u A = A .
Vn A
(xvii)
= A A
V u A = Y.
(xviii) A
%
0 = A A A
(xix) A
%
8
(xx) A n 8
%
V = 0.
A n (% 8) A
(xxi) A n 8 = A
% ++
(A
%
.
8)
A u 8
(xxii) A n g = 0 - A 5
.
8
2.8.
-
A c -8
The c o l l e c t i o n o f a17 c l a s s e s w i t h t h e o p e r a t i o n s U, n,%, and t h e cos t a n t s 0, V , form what i s c a l l e d a Boolean Algebra. These o p e r a t i o n s a r e c a l l e d BooLeufl o p e r a t i o n s . 2.1.1.12
D E F I N I T I O N (OPERATIONS)
I
(i) { A ) = {x : x = A). ( i i ) { A , 83 = { x : x = A V x = 8 ) . ( i i i ) { A , 8, C), { A , 8, C, P I , a r e d e f i n e d s i m i l a r l y as (ii).
...
2.1.1.13 (i) A
(ii) 8
THEOREM,
4 V E
PROOF,
V
+
+
{A1 = 0 8 u {CI
. E
V .
(i) i s e a s i l y o b t a i n e d f r o m Def. 2.1.1.12
(i).
23
AXIOMATIC SET THEORY ( i i ) Suppose t h a t 8 E V and l e t A = 8 U { C } . I f C 4 V , t h e n A = 8 and, hence, A E V . I f , on t h e o t h e r hand, C E V , t h e n we have W x ( r E A t f x E 8
Hence, by Ax Num, A 2.1.1.14
U
0= 8,
V x = C ) .
V .
E
THEOREM
(i) {Al E V .
( i i ) { A , B) E V .
( i i i ) { A , B, C ) ,
{A, 8 , C, D}
E
V
PROOF, (i)We have t h a t { A } = 0 (ii)and 2.1.1.13 (ii).
Rl =
( i i ) We have t h a t { A , 2.1.1.13.
{A}
The p r o o f o f ( i i i ) i s s i m i l a r .
...
U
{A}.
Hence { A )
E
V , by 2.1.1.7
u 181. Hence { A , B } E V by ( i ) a n d
.
I n o r d e r t o s h o r t e n some expressions, t h e following d e f i n i t i o n i n t r o duces t h e f i r s t few number. 2.1.1.15'
DEFINITION
2.1.1.16
THEOREM,
.
3 = {0,1,21
( S P E C I A L CLASSES).
1,2,3
1=
IO),
2
=
IO, 1) ,
E V .
ORDERED P A I R S AND CARTESIAN PRODUCT,
2.1.2
We need an o b j e c t ( i . e . a c l a s s ) r e p r e s e n t i n g an o r d e r e d p a i r (a,b ) o f a r b i t r a r y s e t s a, b. That i s , a c l a s s ( a , b ) w i t h t h e f o l l o w i n g properties. (1) ( 4 6
)
(2) (a,b
) =
(3)
-
i s a class, whenenver
(c,d)
a, 6 E Y + ( a , b )
a, 6
a = c A b = d.
E V .
E
V.
ROLAND0 C H U A Q U I
24
The f i r s t d e f i n i t i o n chronologically was given by Hausdorff in 1912 : {{O,al, {1,6}1. Instead of 0,1, i t i s possible t o use any p a i r o f d i f f e r e n t sets.
(a,b) =
The d e f i n i t i o n adopted in this book i s due t o Kuratowski and Wiener. 2.1.2.1
DEFINITION ( O P E R A T I O N ) ( ~ ~ 6= {)( a } , C a , b I I
2.1.2.2
THEOREM,
I
( i )( a , b ) E V
(ii)(u,b)=(c,d)- + a = c A b = d .
PROOF, ( i ) i s e a s i l y obtained from 2.1.1.14 PROOF OF ( i i ) . Suppose t h a t ( a , b ) = ( c , d ) . follows t h a t {cl, Ec,dl E ( a , b ) . Hence, and
(1)
{cl = { a }
(3)
Cc,dl
=
{Ul
(ii).
From Def. 2.1.2.1
,
or
(2)
Ccl
or
(4)
{ c y d l = Ea,bl
= {a,bl
it
.
CASE I , Suppose ( 2 ) Then c = a = b. This implies t h a t ( 3 ) and ( 4 ) a r e equivalent and c = d = a. Hence a = b = c = d and the conclusion of ( i i) hol ds
.
.
Then c = d = a. This implies t h a t (1) and ( 2 ) CASE I I , Suppose (3 a r e equivalent and a = b = c. T h u s , again a = b = c = d.
.
C A S E 1 1 1 , Suppose (1) and ( 4 ) . Then c = a and ( c = b o r d = b ) . I f c = b then ( 2 ) i s t r u e and we a r e in Case I. I f d = b y then a = c and b = d which proves the theorem. 2.1.2.3 2.1.2.4
COROLLARYl (a,b)
= (b,u)
DEFINITION (OPERATION A
x
:
c--f
a =b.
Cantadian prroduct).
B = l z : 3 x 3 q ( z = ( x , q )A x E A A q E B ) } .
A x B i s t h e c l a s s of the ordered p a i r s whose f i r s t member belongs t o A and second member belongs t o 8 . The Cartesian product i s an operation of a d i f f e r e n t character than union, i n t e r s e c t i o n and difference. Whi 1 e i f A , B C C, we have t h a t A U 8 , A n B , and A 2, B C - C , i t i s not t r u e in genera1 t h a t A x B C -C
.
2.1.2.5
THEOREM,
(i) A x B = 0 - A = O V B = O . (ii) A x B = C x D A A # O A B + O - + A
= C A B = D .
AXIOMATIC SET THEORY
PROOF,
25
(i)i s c l e a r from Def. 2.1.2.4.
PROOF OF ( i i ) . Suppose t h a t A x B = C x D A A f 0 A 8 f 0. Then t h e r e a r e a, b w i t h u E A and b E B ; hence ( a , b ) E A x B = C X U and ( f r o m Def.2.1.2.4) aEC and b E D . T h i s shows t h a t
(l)c+o A D#O Suppose now, t h a t x E A ; t h e n ( x , b ) -C. therefore, A C
E
Thus x e C
AxB = CxD.
and,
-
S i m i l a r l y we can show t h a t B C - 0 , and u s i n g ( l ) , t h a t C c - A and l7cB.S We need, besides t h e ordered p a i r o f two s e t s u , b y t h e concept o f an that ordered p a i r o f two a r b i t r a r y c l a s s e s A, B. Namely an o b j e c t [ A , B ] satisfies:
(1') [A, 81 i s a c l a s s , f o r any A, 8 , ( 2 ' ) [ A , 81 = [ C ,
D]-A
= C A
B = D
. .
shows t h a t A x B serves t h i s purpose whenever A # O and 8 +; 0 2.1.2.5 I n o r d e r t o l i f t these l a s t l i m i t a t i o n s , t h e f o l l o w i n g d e f i n i t i o n i s i n t r o duced. 2.1.2.6
DEFINITION,
IA,Bl = A x
{O)
U
Bxll}.
We have, 2.1.2.7
THEOREM, [ A , B l = [ C , D I
-,A
= C
A 8
=
D.
The p r o o f i s l e f t t o t h e reader. F i n a l l y we have t h e f o l l o w i n g theorem whose p r o o f i s a l s o l e f t t o t h e reader. I n w r i t i n g formulas we assume t h a t x has a s m a l l e r scope t h a n u, n ,% Thus, f o r i n s t a n c e , A x B u C stands f o r ( A x B ) u C .
.
2.1.2.8
THEOREM ,
( i ) A x (BUC) =A x B u A
xC
.
( i i ) (BUC) x A = B x A u C x A . (iii) Ax(BnC) = AxBnAxC. (iv) (BnC) xA = B x A n C x A . (v) Ax(B%C) = A x B % A x C . (vi) (BsC) xA = B x A % C x A . PROBLEMS 1. Show t h a t H a u s d o r f f ' s d e f i n i t i o n o f an o r d e r e d p a i r o f s e t s , s a t i s f i e s (11, ( 2 ) and ( 3 ) . 2. Prove 2.1.2.7
ROLAND0 CHUAQUI
26 3. Prove 2.1.2.8
4. The concept of an ordered t r i p l e of s e t s i s similar t o t h a t of ordered pair of sets. I t should s a t i s f y :
a ) ( a , b , c ) i s a c l a s s , i f a, b, c € V b) a, b, c E Y + ( a , b , c )
Y
E
c ) ( a , b , c ) = ( d , e , 6 ) t--) a
=
dA b = e A c =
6.
Show t h a t ,
( i ) EIO,a), {1,61, I2,cI) and { E d , Ea,bl, {a,b,c)l do n o t s a t i s f y ( a ) - ( c ) , and ( i i ) EC O,a), ( l,b), ( 2 , ~ 1) and ( ( a , b ) , c), s a t i s f y ( a ) - ( c ) . 5. Show,
A x B = BxA
2.1.3
f-*
A
B V A = 0 V B = 0 ,
GENERALIZED OPERATIONS,
The f i r s t definition of t h i s section, introduces the union and inta&e.Otion 0 6 Rhe &emem2 0 6 a g i v e n A, and t h e powen Ceans 0 6 a CAUA. 2.1.3.1
D E F I N I T I O N (OPERATIONS).
(i) u A = {x : 3 y ( x E y € A ) l . ( i i ) n A = Ex : W y ( y € A -+ x ~ y ) ) . -A). ( i i i ) P A = Ex : x c
S i n c e A = { x : x E A I , we have t h a t U A = u ( { x : x € A ) ) a n d n A = n ( { x : x E A 1 ) . Other notations f o r these operations t h a t we shall n o t use, a r e u x for u A and n x f o r n A .
*A
xEA
The following theorem gives elementary properties of these operations. 2.1.3.2
THEOREM,
( i ) A -c B + ( u A-c u B A n B c n A A P A c PB) (ii) u 0 = 0 = n V . (iii) n 0 = V = u V . (iv) P V = V . (v) u P A = A . (vi) A E P u A . ( v i i ) W x ( x € A -* n A c - x). (viii) u E d = a = n { a ) . ( i x ) U {a,b) = a u b .
27
AXIOMATIC SET T H E O R Y
PROOF, I s h a l l give a proof of ( v ) and ( v i ) , leaving t h e r e s t of t h e theorem t o t h e reader.
( v ) : I f x E U P A , then x E q c A f o r some q , t h e r e f o r e , x E A ; and hence U P A 2 A . On the other hand, l e t x E A ; then { x ) E P A by Def. 2.1.3.1 and 2.1.1.14 (i).Hence X E { x } € P A and X E u P A ; i . e . A L u P A . ( v i ) : Let x E A ; then x C - U A.
Since x E V , x
E P U A
We s h a l l c a l l a c l a s s A LttrransiLLve i f U A Z A . equivalent conditions f o r t h i s notion.
.B
The next theorem g i v e s
PROOF, We have, tf y t f x ( y E x € A Y E A ) i s equivalent t o V y ( 3 x ( y € x € A ) -, q E A ) . T h u s , these expressions a r e e q u i v a l e n t t o V q(q E u A Y E A ) and, hence t o u A 5 A. +
-+
I t i s also clear that We have t o show,
-
A S P A i s equivalent t o W x ( x E A
u A C_ A
-,
x C- A ) .
A C_P A .
Suppose u A L A ; then, by 2.1.3.2
( i ) and ( v i ) , A C -P u A S P A .
Finally, suppose A C P A ; then, by 2.1.3.2
( i ) and ( v ) ,
U
AcUPA=A..
In the r e s t o f t h i s s e c t i o n , new operators will be discussed. T h i s will permit t o g e n e r a l i z e unions and i n t e r s e c t i o n s t o a r b i t r a r y c o l l e c t i o n s o f classes. Xo,
2.1.3.4 ..., Xn-l
(iii) n
DEFINITION SCHEMA, Let d i s t i n c t variables.
xo
... 'n-
1
IT
Then
: $1 = I x : tf
T
xo
be a term, 0 a formula, a n d x ,
... w xn-l
(4
-+
x
E 7))
.
In general, i t i s necessary t o i n d i c a t e the v a r i a b l e s which a r e bound by t h e operator. Thus, t h e term I { x , q l : x + q ) without p u t t i n g which varia b l e s a r e bound could be any o f t h e following terms:
(1) , I { x , y )
: x
+ q)
= I z : 3 x ( z = Cx,ql A x +
q)}
.
28
ROLAND0 CHUAQUI
These four terms a r e d i f f e r e n t . (1) defines a unary operation t h a t assigns t o each given y the c l a s s of p a i r s { x , y l w i t h x # y . S i m i l a r l y ( 2 ) i s a l s o a unary operation assigning t o each x the c l a s s o f p a i r s { x , q l w i t h + y. ( 3 ) i s the special c l a s s formed by a l l p a i r s Cx,yl with x f q. ( 4 ) i s a binary operation t h a t f o r x , q with x f y gives { { x , ~ } } a n d f o r x,y with x = q gives 0.
x
,...,
I s h a l l omit t h e variables Xo Xn-,, when t h e s e a r e a l l t h e f r e e variable i n 7 . Thus, { { x , q l : x # y l stands f o r {{x,ql : x f y l . XY
With t h i s newly introduced notation we can reformulate t h e d e f i n i t i o n of t h e Cartesian product: A X B = { z :3 x 3 q ( z = ( x , y ) A x E A A x E 8 ) ) = {( x , y ) : x E A A y E B } .
(Def. 2.1.2.4)
Also, the Cartesian product i s r e l a t e d t o t h e generalized union by: AxB = =
I f we take have ,
7
U
X
,
{ A x C x } : xEB}
uy{{q}
x
.
B : yEA3
= X in ( i i ) and ( i i i ) we get U {X :
u {X : $1 n
{x
:
$1 and
n {X : 61.
We
:3 X(qEX A @ ) I { q : W x(6 + y E X I > .
= {y
@ l=
Suppose, now, t h a t 4 has only X f r e e .
Then $J defines the c o l l e c t i o n A
In t h i s case, U { X : @ l and n { X : @ } a r e t h e union and i n t e r s e c t i o n o f A , and a r e defined even when A does not c o n s t i t u t e a c l a s s .
On the other hand, ( x : $1 i s always a c l a s s . u
({x:@))
= u
Thus,
{ x : @ } and n ( { x : @ } )
= n
Also, f o r an a r b i t r a r y c l a s s A , we have U A = u ( ( x : xEA}) = u { x : xEA}
and
{x:@l
.
29
AXIOMATIC SET THEORY n A = n ( { x : xEAI) = n
{x:xEA1.
Hence, we a r e j u s t i f i e d i n u s i n g t h e same symbol f o r t h e union o r i n t e r s e c t i o n o f t h e elements o f a c l a s s and t h e corresponding o p e r a t o r f o r t h e elements o f a c o l l e c t i o n . The b i n a r y u n i o n A u B and i n t e r s e c t i o n A n B can be d e f i n e d u s i n g t h e g e n e r a l i z e d n o t i o n s , thus: 2.1.3.5
THEOREM I
( i ) A u B = U { X : X = A V X = B) (ii) Ant3 = n { X : X = A V X = B)
.
.
Therefore, theorems about g e n e r a l i z e d unions o r i n t e r s e c t i o n i n c l u d e as p a r t i c u l a r cases t h e corresponding ones about t h e b i n a r y o p e r a t i o n .
PROOF, A l l these statements a r e easy t o prove. show ( i v ) l e a v i n g t h e r e s t t o t h e reader.
As an example I s h a l l
30
ROLAND0 CHUAQUI
PROBLEMS 1. Complete t h e p r o o f o f 2.1.3.2. 2. Show t h a t ,
xCCx,yI : x
ql
f
=
ttx,ql : x Y
f
ql
-
x
=
Y *
3. Describe t h e f o l l o w i n g c l a s s e s and determine which can be proved t o be sets. (i)
,Cx u Iql : q XY
E
tx u Cy) : q E
XI , XI
(ii) Same as (i) with x
(iii) ,Cx: X E yl XY
Cx:xE ql
Y
E
y
,
“CX : x E
, x
-A n B . ( i ) n A nnBC
B)
= U A U
U B ,
5 . Complete t h e p r o o f o f 2.1.3.6.
.(ql : y
Cx u {q3 : q
Ix:xE
4. Show:
U (AU
u
E
xl
E XI
instead o f qEx.
Y
( i v ) Same as (iii) with
(ii)
V
{x
q3 , ql
= q instead o f
.
x
E
q.
>
.
31
AXIOMATIC SET THEORY
(+) 2.1.4.
IS A SUBTHEORY OF B .
G
I n t h i s s e c t i o n , we assume B as o u r t h e o r y and deduce t h e axioms o f G from t h o s e o f B. Since Ax Class and Ax E x t a r e common t o b o t h t h e o r i e s , i t i s enough t o show Ax Em and Ax Num as theorems o f B. A l l theorems i n t h i s s e c t i o n a r e theorems o f B.
A l l d e f i n i t i o n s , s t i p u l a t i o n o f v a r i a b l e s , and many o f t h e theorems o f 2.1.3 depend o n l y on Ax Class and Ax Ext. Thus, t h e y c a n S e c t i o n s 2.1.1 a l s o be used f o r B.
-
We d e f i n e d i n 1.2.1.4 @' f o r formulas $ w r i t t e n i n t h e p r i m i t i v e n o t a t i o n . I s h a l l l a t e r extend t h i s r e l a t i v i z a t i o n t o d e f i n e d concepts b u t we s h a l l n o t need i t i n t h i s s e c t i o n . Using t h e d e f i n i t i o n s o f t h e p r e v i o u s s e c t i o n s (2.1.1.5, 2.1.3.1 and 2.1.3.3), Ax Ref can be w r i t t e n ,
2.1.1.10,
W A ( @ + 3 u ( U u-C u A @ ~ [ A n u ] ,)
Ax Ref
where $ i s a f o r m u l a i n L which does n o t c o n t a i n u a n d c o n t a i n s a t most A free. P lows:
Also, Ax Sub can be w r i t t e n ( b y 2.1.1.4, Ax Sub
W A W b ( Ac b - A €
2.1.4.2
THEOREM ( B b Ax Em). 0 E V
PROOF,
From Ax Ref, we o b t a i n ,
A = A
+
3
U(U E
2.1.1.5
and 2.1.1.8)
V ) .
V A uu c -u A A n u = A n
Hence, t h e r e i s a s e t u.
as f o l -
But 0
a, b E V
Cu
.
U)
.
Therefore, by Ax Sub 0 E Y..
( { a , b ) E V A 3 U(U u c -u A
a
u)).
2.1.4.3
LEMMA ( B )
PROOF,
Suppose t h a t a, b E V and a + 6 ( t h e p r o o f f o r t h e c a s e L e t A = { a , b } . We have,
-*
E
a = b i s similar).
3x 3 y ( x # y A x , y E A ) . Using Ax Ref we o b t a i n ,
3u(
Uu
c -u
A
3x 3y
(x,y
5u A
x
#
y A x,y E A n u ) ) .
Hence, t h e r e i s a t r a n s i t i v e s e t u, such t h a t ,
(1) 3 x 3 y ( x
#
y A x,y
E
A n u).
Therefore, s i n c e A n u
-A C
=
( a , b } , A n u = 0,
A n u = { a } , An u = (6)
32
ROLAND0 C H U A Q U I
o r Anu = { a , b } . I n t h e f i r s t t h r e e cases, (1) would be false. ( 2 ) Anu = { a , b l C u
.
.
Applying, now, Ax Sub we o b t a i n t h a t {a,bl tive
E
V.
From (2) t h e second c o n c l u s i o n o f t h e Lemma (i.e. i s obtained. THEOREM ( B I- Ax Num).
2.1.4.4
Hence,
B E V-. B
f o r u transi-
LIEU
u {C)
E
V
PROOF , L e t B E V . I f C V then B U { C } = B E V and t h e theorem i s proved. So suppose C E V . By 2.1.4.3, {B,C} E V and, hence, t h e r e i s a t r a n s i t i v e s e t u such t h a t { B , C } E P . We have, B , C E u , so { C } C u. S i n c e - u . Using, now, Ax Sub. we o b t a i n u i s t r a n s i t i v e , B E u. Hence, B u { C ) c BU{C} E Y .
.
F i n a l l y , we have 2.1.4.5
METATHEOREM,
G LA a oubtheaay
by Ax Class, Ax Ext, 2.1.4.2,
PROOF,:
06
B
.
and 2.1.4.4.
.
PROBLEMS
1. Prove i n B (a) A
U
B
,
E
V -
A E V A €3
E
V .
( b ) A E V V B E V + A n B E V . (c)A+O 2.
A
B+O-+(AxBEV++AEVA
-
B E V ) .
Prove t h a t i f we e l i m i n a t e t h e r e s t r i c t i o n which does n o t c o n t a i n A f r e e i n Ax Class, G and, hence, B , become i n c o n s i s t e n t . H i n t : Consider t h e formula:3AW
x(x
E
A
x
$Z A A
3u x
E
u).
CHAPTER 2.2 Re1a t i ons
OPERATIONS WITH RELATIONS,
2.2.1
A b i n m g heRdtian R i s any subclass o f V x V . Since o n l y b i n a r y r e l a t i o n s w i l l be considered, i f R 5 V x V we say, simply, t h a t R i s a hela-tion. R i s a r e l a t i o n t h a t i s a s e t i f and o n l y i f R E P ( V x V ) . Sometimes t h e f o l l o w i n g n o t a t i o n s w i l be used i n case R i s a r e l a t i o n :
(g,x) E R , x R g stands f o r ( g , x ) $ R
xR
g stands f o r
.
Some o f t h e b i n a r v n o t ons used so f a r do n o t denote r e l a t i o n s . i . e . classes. T h i s i s t r u e , f o r instance, f o r =, E , 2 . However, i t i s p o s s i b l e t o i n t r o d u c e r e l a t i o n s which r e p r e s e n t t h e s e n o t i o n s r e s t r i c t e d t o sets.
2.2.1.1
D E F I N I T I O N( S P E C I A L CLASSES). I D = {Cg,x):
{C g,x
x = 9) , q}
,
D Y = {(g,x): x f y)
.
EL
=
I N = {C g,x
)
: x
)
: x
E
5 gl ,
It i s clear that,
2.2.1.2 REMARKS, The Boolean o p e r a t i o n s d e f i n e d i n general f o r c l a s s e s a r e a l s o used f o r r e l a t i o n s . I n p a r t i c u l a r , unions and i n t e r s e c t i o n s f o r r e l a t i o n s a r e again r e l a t i o n . The complement S R o f a r e l a t i o n R , however, i s never a r e l a t i o n , s i n c e i t always c o n t a i n s elements which a r e n o t ordered p a i r s . A new r e l a t i o n a l complement w i l l be i n t r o d u c e d below. Besides t h i s , t h e n e x t d e f i n i t i o n g i v e s o t h e r o p e r a t i o n s which have i n t e r e s t o n l y f o r r e l a t i o n s .
2.2.1.3 (j)
-
D E F I N I T I O N( O P E R A T I O N S ) I
R= V x VSR, 33
34
ROLAND0 CHUAQUI
.
( i i ) R-' = { ( x , y ) : ( y , x ) E RI ( i i i ) R o S = {( x , z ) : 3 y (( x , g ) E R A ( y , z ( i v ) RO = r D n ( ( R o ( v x v ) ) u ( ( v X v
R ~ = R ~ v R~2 = v R , ~
E
)
0
S)}.
R)) ,
R3 = R o R o R , . . ,
R ,
- R i s t h e treeative campLement of R ; R - l i s t h e canweh6e r e l a t i o n of R ; R o S i s t h e camponLtian of R and S (we have, z R o S x 3 q (y&A z S y ) ) 0 R i s t h e i d e n t i t y r e l a t i o n r e s t r i c t e d t o the f i e l d of R.
.
The operations defined in 2.2.1.3 can be applied ( a s a l l operations) t o a l l c l a s s e s . However, they a r e of i n t e r e s t only f o r r e l a t i o n s . The values of these operations a r e always r e l a t i o n s and these values a r e d e t e r mined by t h e r e l a t i o n a l p a r t ( i . e . t h e c l a s s of ordered p a i r s ) o f thearguments.
T h u s , we have: - R , R-l, R o S , R o , R1, - R = -
R2,
,... C-
Vx V
;
R-l= (RnVxV)-',
( R n V x V ) ,
ROS=(R
R3
n v x V ) o . ( s n V x V ) .
In order t o simplify the expression of theorems, i t i s assumed in t h e r e s t of t h i s chapter t h a t R , S, T a r e r e l a t i o n s , i.e. these v a r i a b l e s a r e assumed r e l a t i v i z e d t o subclasses o f Y x V .
2.2.1.4 EXAMPLES, The following a r e examples of a p p l i c a t i o n s of these operations: D V
IN
= - I D , I D = I N ~ I N - ~I ,N ~ E L= E L , = ( - ( E L ~ ( - E L ) - ~ ) ) D- V~ O, D
(AxB)'~ = BxA
,
V
(AxB) o (CxU) = 0
( A x B ) o ( C x U ) = Ax17
if
Cn B + O
= v x v ,
if
.
C n 8 = 0,
-
and
The c o l l e c t i o n o f a l l r e l a t i o n s with t h e operations u , n , , 0 , -1 , and the constants 0, V x V , I D c o n s t i t u t e s what i s c a l l e d a Relation A l gebra. The following theorem l i s t s some of the i d e n t i t i e s (and inequalit i e s ) s a t i s f i e d by Relation Algebras, in p a r t i c u l a r , by the algebra of a l l re1 a t i o n s . 2.2.1.5
THEOREM,
AXIOMATIC SET THEORY
(iii) (R n
s1-l
( i v ) ( - R)-' ( v ) R o (S
= R - ~n
35
s-'.
-(R-').
=
o T)
= (R
o S ) o T.
( v i ) R o I D = R = I D o R. ( v i i ) R o (S U T ) = ( R o S ) ( v i i i ) (S
U
T)
o R
= (S
U
(R o T).
o R ) u ( T o R).
- R o T A S o R c- T (ix) S E T + R o S c ( x ) R o (S n T )
2
(R o S) n (R o T).
( x i ) (S n T ) o R
C
( S o R ) n (T o R ) .
o R
(xii) R o 0 = 0
= 0.
-
o R.
( x i i i ) ( V X V ) O R O( V X V ) = V X V - R + O . (xiv) (V
x
V )o R o (V x V ) = 0
( x v ) ( R o S)-'
= S-l
o R-'.
R = 0.
( x v i ) R n ( - R ) - ~c -D Y ( x v i i ) R o ( - R)-'
C -
D
~
PROOF, The proof o f these statements i s q u i t e easy. t h e l a s t two w i l l be g i v e n as an example. PROOF and
(x,y)
OF ( x v i ) .
E ( - R)-',
PROOF OF
Suppose
i.e.
(xvii).
(
y,x
(
x,y
) $ R.
)
E R n (- R ) - l .
Therefore
y
#
x.
Suppose ( x , y ) E R o ( - R ) - ' . ( - R -1) , i . e . ( y , z ) $ R .
suchthat(x,z)ERand(z,y)€
The p r o o f
Then (x,y
)E
of
R
Then t h e r e i s a z Thus, x + y .
2.2.1.6 REMARKS, I t i s i n t e r e s t i n g t o n o t e t h a t u s i n g some o f t h e s e i d e n t i t i e s , i t i s p o s s i b l e t o r e p l a c e e v e r y Boolean ( p r o p o s i t i o n a l c a l c u l u s ) combination o f e q u a t i o n s and i n e q u a l i t i e s by one equation. For i n s t a n c e , take
(1)
R = S V R ' = S'
s t a r t with, l( R = S V R' = S ' ) . T h i s i s e q u i v a l e n t t o f? f S A R' f S ' . Taking A €3 = ( A €3) U ( 8 A ) , R f S, R' # S' i s c l e a r l y e q u i v a l e n t t o R I S f 0 A R ' L S' # 0 . By 2.2.2.5 ( x i i i ) , t h i s i s transformed i n t o ( V x V ) o ( R I S ) o ( V x V ) = V x V A ( V x V ) o ( R ' L S ' ) o ( V x V ) = V x V; and t h i s c o n j u n c t i o n i s e q u i v a l e n t t o
-
( 2 ) ( ( V x V ) o ( R IS)
-
o (V
x V ) ) n ( ( V x V ) O ( R ' I S ' ) o ( V x V)) =
36
ROLAND0 CHUAQUI
=
vx
V .
Since t h i s l a s t e q u a t i o n i s e q u i v a l e n t t o t h e n e g a t i o n o f ( l ) ,we have t h a t ( 1 ) i s equivalent t o t h e negation o f ( 2 ) , i.e. t o
(3)
-
( ( ( V x V ) o (RIS)
0
(Vx V)))
Let
T
-
=
o ( V X V ) ) n ( ( V XV ) O (R’ 2 S ’ ) o
0.
f
( ( V X V ) o ( R A S ) o ( V x V ) ) n ( ( V x V ) O( R ’ L S ’ ) o
Then, b y 2.2.2.5
( x i i i ) , (3) i s f i n a l l y equivalent t o
V X V ) oT o ( V x V ) = Y x V .
( 1 ) has been transformed t o t h e e q u i v a l e n t e q u a t i o n (4).
.7
DEFINITION
(OPERATION).
t x : 3 g x R ql
D R =
D R s h a l l be c a l l e d t h e darnuifl D R u D R”, 2.2.1.8
the 6 i d d
06
06 R ;
D R-’,
t h e mange
04
R ; and
R.
D E F I N I T I O N S (OPERATIONS).
(i)
AIR
= R nA x
V.
(ii)
RIB
= R
B .
(iii)
R*A
=
n
v x
Cy . : 3 x ( x € A A x R y ) }
AIR i s R r e s t r i c t e d i n i t s range t o A ; R I B i s R r e s t r i c t e d i n i t s domain t o B; R*A i s t h e image o f A by R ; and R-l*B i s t h e counterimage o f 8 by R * . 2.2.1.9
THEOREM,
AXIOMATIC SET THEORY
(ix) D R-'
37
= R* D R .
The proof i s easy. 2.2.1.10
THEOREM,
(i) (R* A ) n 8
=
o-
A
(ii) A n (~-l* 2. R* A ) (iii) A C -8
+
R* A
=
- R*
n
(R-'* B)
o
A R- '*%
R* ( A U B ) = (R* A )
(vi) R*
(A1.8)
(vii) R* ( A
%
2
(R* A )
R-'* 8)
(viii) R* ( A n R-
8)
0.
R* A c -A .
8.
C
(iv) R* ( A n 8 ) c (R* A ) n (R* 8 )
(v)
=
U
(R*
%
(R* 8 ) .
5 (R* 2
A)
.
8).
8 .
(R* A ) n 8 .
(ix) B n D R - c ~ R* R - ~ 8*. (x) R* A (xi) R*
%
=
R* ( A n R-'* R* A ) c - R* R-'*
R-'*
2.
R* A
=
R* A .
R* A .
PROOF, (i) will be proved from the definitions and the rest will be proved from (i) and 2.2.2.7.
Since (R-')-' = R, it is enough to show, AnR* 8 # O + -1* +. 8 n R A .f 0. So suppose q E A n R* 8 . Then , Y EA A 3x(xE 8 A x R g ) ; hence, there is an x such that x E B A 3 g ( g € A A g R-'x), i.e. xEBnR-l*A. Thus, 8 n R-'*A f 0 , PROOF OF (i).
PROOF OF (ii).
A n (~-l* % R*A) =
(1.R*A)
n (R*A) =
OF (iii). By (ii), we have that 8 n(R-'* R*8) is assumed, A n (R-l*2. R*B) = 0. By (i) (?.R*B) n R*A
PROOF
R*A
o
o
But the right side is obviously true. Hence the left side, which i s
(ii).
if A
-
Applying (i) we obtain,
58
- R*B
C
.
PROOF
OF (iv):
by (iii).
= =
0. Hence, 0, i.e.
ROLAND0 C H U A Q U I
38
P R O O F OF (v). (R*A) U (R*B) c R * ( A U 8 ) i s e a s i l y deduced from (iii).So (*) R*(A U 8 ) C - (R*A) U (R*B) w i l l be shown. L e t us suppose t h a t ((R*A) u ( R * B ) ) n C = 0.
Then, R * A n C = O = R * B n C .
By (i), we o b t a i n A n R - l * C = O = BnR-'*C. Hence, ( A u 8) n R-'*C = 0. Using a g a i n (i), (R*(A U €4)) n C = 0. Now, i f we t a k e C = % ( ( R * A ) U ( R * B ) ) , (*) i s obtained.
P R O O F OF ( v i ) .
By ( v ) R*A
2 R*B
U
(R*(A?.B)).
Hence ( v i ) .
P R O O F O F ( v i i ) . By ( i i i ) , R*(A%R-'*B) 5 R*A. Also, s i n c e R - l * B n = 0 we g e t from ( i ) , ( R * ( A % R - l * B ) ) n B = 0. Hence ( v i i ) .
n (A%R-'*B)
By ( v i i ) , we have, ( R * A ) n B = R*A%(R*A
PROOF O F ( v i i i ) .
c R*A % (R*(A%R"*B)). c - R*(A 2, ( A % R - l * B ) )
%
B)
A p p l y i n g ( v i ) , we o b t a i n , R*A?.(R*(A%R-'*B)) = R*(AnR-l*L?).
= (R*V)
.
( x ) and ( x i ) a r e l e f t t o t h e - r e a d e r .
.
n B.
By ( v i i i ) ( R * Y ) n
The f o l l o w i n g theorem g i v e s t h e behaviour o f images w i t h r e s p e c t r e l a t i o n a l operators.
THEOREM (R
5
Hence ( v i i i ) .
P R O O F O F ( i x ) . We have 8 n D Rn gc - R * ( v n R - ~ * B ) = R* R - ~ * B
2.2.1.11
E
to
I
S ) * A = (R*A) LJ ( S * A ) .
U
S)*A C - (R*A) n (S*A).
(R
( R n S)* {a) = (R* {a)) n (S* {a}). (R A
%
#
S)*A
0
+
2
(R*A)
(S*A).
(S*A) C_ (?.S)*A
.
( R o S)*A = R* S*A
-
.
-
D (R o S) = S'l* D R A D ( R o S ) - l = R* D S. R
CS
The proof is easy.
V X(R* X
- S*
C
X)
W x (R*(x)
-
C S*{x).).
F i n a l l y , some o f t h e p r e v i o u s r e s u l t s a r e extended t o g e n e r a l i z e d Boolean o p e r a t i o n s . The p r o o f is l e f t t o t h e reader.
A X I O M A T I C S E T THEORY 2.2.1.12
(iii)
THEOREM SCHEMA,
(nx 0"'
Xn-1
Let
7
39
be a term and 4 a formula.
CT : @ } ) * A C- r-X0".
Xn-1
Then
IT* A : $1.
PROBLEMS
( x ) and ( x i ) , from ( i ) .
1.
Prove 2.2.1.10
2.
Prove 2.2.1.12.
3.
Show t h a t : ( i ) R o ( S n T ) = ( R o S ) n ( R o T ) i s not t r u e i n general, ( i i ) (R = 0 V S = 0 )
-
R o ( V x V ) o S = 0,
( i i i ) R n S n T C- R o S - ' o T . ( i v ) R* (AnB) = (R*A) n (R*B) i s not true i n general. 4.
Find d e f i n i t i o n s using E L and r e l a t i o n a l operations o f t h e following re1 a t i ons : (i)
CC q , x )
: q =
{XI),
( i i ) {( g , x ) : x n q = 01 ( i i i ) {( g , x ) : 3 z(q = I x , z l ) l (iv) C(q,x) : q = P x l
( v ) {C q,x 5.-
)
,
: 3 z(q = ( x , z ) ) l
,
.
Prove t h a t : E L * A = % P % A , E L - ~ * A= u A , Dv*A =
6.-
A.
Prove P e i r c e ' s law, i.e.
(RoS) n T = 0
-
(R'l o T ) n S = 0
ROLAND0 CHUAQUI
40
2.2.2
RELATIONS A S SUPERCLASSES,
A c o l l e c t i o n , o r any n o t i o n , i s defined by a formula. In general, c o l l e c t i o n s a r e not c l a s s e s . However, t h e r e a r e some c o l l e c t i o n s which can be represented by r e l a t i o n s which a r e themselves c l a s s e s . This p o s s i b i l i t y i s based on the following theorem. 2.2.2.1
THEOREM SCHEMA, L e L r he a t m ; then: 3R(R C -V
PROOF, Let
Hence, 2.2.2.2
T
V A i x
x
=
T
R*{x}).
be a term and d e f i n e R by,
R*{xI = { q : ( q , x )
E
R} = {y :q
E T } = T
.
rn
DEFINITION SCHEMA, Let 4 be a formula and r a term. Then,
In order t o explain t h e use of t h i s d e f i n i t i o n , I s h a l l discuss a p a r t i c u l a r case f i r s t . Let F be a unary operation and A an a r b i t r a r y c l a s s . The formula 3x(X = F ( x ) A x E A ) defines a c o l l e c t i o n A by, A (X)
-
3x(X
=
F ( x ) A x E A).
There may be no c l a s s t h a t contains a l l elements of A, since f o r some
x, F ( x ) might be a proper c l a s s . Let R = [ F ( x ) : x c A ] = u { F ( x ) x{x} : : xEA1. The p a i r [ R , A l (defined in 2.1.2.6) i s a c l a s s and, in a sense, i t represents A : Given any X with A ( X ) , t h e r e i s an x € A s u c h t h a t X = R*CxI; on t h e o t h e r hand, i f X = R*{xI f o r some x f A , then A(X). Moreover, each p a i r [ R , A ] where R defined by 3x(X = R*{xI A x E A).
5VxV
represents t h e c o l l e c t i o n
-
I f A i s a c o l l e c t i o n t h a t can be represented by a p a i r [ R , A l ( i . e . A(X) 3x(X = R*{x1 A x E A ) ) then A w i l l be c a l l e d a superclass. A i s a b c t ad codecl dotl A in [ R , A I . R i s not enough t o determine A ; because A might be d i f f e r e n t from D R. W i t h j u s t R we could not have superclasse A with A ( O ) , because f o r a l l x E D R, R*{x} f 0. In p a r t i c u l a r , we have, 2.2.2.3
THEOREM SCHEMA, L e t F be a u m q aperraLLiun. Then, Wx(xEB + F ( x ) = 0 )
+
[ F (x) : x E A ] = [F(x) : x e A U B ]
AXIOMATIC S E T T H E O R Y
PROOF, Suppose t h a t
F ( x ) = 0, f o r
X E
8.
41
We have,
[F(x) : x g A u B ] = u (F(x) x ( x } :x E A u B ) =
B u t , by hypothesis, Therefore,
: x E B ) = 0.
u { F ( x ) x { x ] :x E A } U u { F ( x ) x { X I :x E B I
F(x)
{ x } = 0, for x € B .
x
Thus,
U
{F(x) x(x} :
[F(x) : x € A U B ] = u {F(x) x { x l :xEA} = [F(x) : x E A ]
.
Now, in general, when 7 i s a term and 4 a formula, the c o l l e c t i o n deI x ( 9 A X = 7 ) i s c a l l e d the nupetlCeael~ dettenmined by 7 and {x : $1.
fined by
An easy theorem i s t h e following. 2.2.2.4 THEOREM SCHEMA, L e t A be a A U p e t l C h b b and F a unatry apehaZLan. Then Rhe caUeeection F * A dedined by 3 Y( A(Y) A X = F ( Y ) ) ih dno a h u p e t l c h A b .
.
P R O O F , Let A be determined by determined by F ( 7 ) and t h e same A .
7
and A .
Then F * A i s a superclass
PROBLEMS
1. Show t h a t e x a c t l y f o r the empty s u p e r c l a s s , a n y superclass A can be r e presented by just a r e l a t i o n R. 1.e. we have A(X)
2.
-
3x
X = R*Cx}
.
Prove,
R C- V x V + R = [ R * { x } : x E D R ] / \ R ~ A = [ R * ( x ) : x ~ A l .
42
ROLAND0 CHUAQUI
T Y P E S OF R E L A T I O N S ,
2.2.3
There a r e s e v e r a l types o f r e l a t i o n s whose d e f i n i t i o n s w i l l be g i v e n The d e t a i l e d study o f some of them w i l l occupy l a t e r s e c t i o n s o f the The f i r s t t y p e i s t h a t o f r e f l e x i v e r e l a t i o n s . We say t h a t R i s nLkongLy hedLexiue ifI D C R. G e n e r a l l y , t h i s requirement i s t o o s t r o n g t o be u s e f u l . The usual tTpe o f r e f l e x i v e r e l a t i o n i s t h a t o f r e f l e x i v e i n A , where A i s a c l a s s . We say t h a t R i s he&fkxiue i n A i f D R C A a n d I DI'A C R. These two c o n d i t i o n s i m p l y t h a t D R = A. A weaker concept of C R (i.e. if R i s r e r e f l e x i v e i s used i n t h i s book: R i s hedlexiue i f Ro here. book.
f l e x i v e i n D R U DR-', s i n c e Ro = I D I ( D R U D R - l ) . Since we have a s h o r t expression f o r R i s r e f l e x i v e , i.e. Ru C_ R , no f o r m a l d e f i n i t i o n w i l l be i n troduced. A s i m i l a r procedure i s used f o r many n o t i o n s i n t h i s book: i n t h e i n f o r m a l d i s c u s s i o n s a name f o r t h e n o t i o n i s used, b u t i t i s n o t introduced formally.
We say t h a t R i s h e , $ L e ~ L w ei f R C D v (i.e. W x ( ( x , x ) R ) ) . R i s nymnettLic if R-1 5 R (i.e.W xW x ( x R q -, q R x ) ) . I t i s easy t o see t h a t R-l 5 R i s e q u i v a l e n t t o R 5 R-', and, hence t o R = 8 - I . R i s asymne&ic I t i s c l e a r t h a t i f R i s asymmetric t h e n R i s i f R n R-' = 0 (or-R 5 R-I). i r r e f l e x i v e . Also, R o i s n o t n e c e s s a r i l y symmetric, i f R and S are. But we have t h e f o l l o w i n g theorem whose p r o o f i s l e f t t o t h e reader.
2.2.3.1
THEOREM,
(i)RO
5R
A
soc -s
R = R - ~A S =
(ii)
-,
c_ R-'
(~-1)'
s-'
A R
~ = SS
A ( R ~ S ) O 5R
~
--f
RR
~
.S
~ =S( R ~ s ) - ' .
2
R i s t u n n X u e i f R 5 R (i.e., i f W x W q Wz ( x R y A q R z -, x R z ) ) . We have t h a t V x V , 0 , I D , and I N a r e t r a n s i t i v e , b u t EL and D v a r e n o t . However, E L l A i s t r a n s i t i v e i f and o n l y i f A i s t r a n s i t i v e (i.e. U A C -A ) . V x V , 0, I D , and D v a r e symmetric, w h i l e t h e o t h e r s a r e not.
The i n t e r s e c t i o n o f an a r b i t r a r y c o l l e c t i o n o f t r a n s i t i v e r e l a t i o n s i s transitive:
2.2.3.2
PROOF, theorem.
q r z.
THEOREM SCHEMA,
Let
Suppose
Hence x r z .
R
n
'0
0 . .
xRq A yRz. Therefore,
le,t r be a tehm and 4 a jjotvnuh.
Then,
{ r : @ I and assume t h e hypotheses o f t h e Xn-1 Then f o r a l l Xo,... Xnml, such t h a t 4 , x r q A
x R z.
.
43
AXIOMATIC SET THEORY
The c o m p o s i t i o n o f two t r a n s i t i v e r e l a t i o n s i s n o t , i n genera1,trans i t i v e . However, we have:
2.2.3.3. PROOF,
(RoS)
THEOREM, Assume
R
R2
-R C
A S2
C S A R -
2 C _ R A S2 C -S A R o S
oS = S OR
= SoR.
2
'JxWy(xRy
(i.e.
+
(R oS)2 C - R oS.
Then,
2 = R o ( S o R ) o S = ( R o R ) o ( S o S ) = R 2 O S2
R i s den6e, i f R C R
-+
5
R o S .
. R i s an
3 z(xRz A z R y ) ) .
eqLwaLence trelation, i f R i s symmetric and t r a n s i t i v e , i.e. i f R = R - l and 2 R C R. There a r e s e v e r a l o t h e r ways o f s a y i n g t h a t R i s an equivalence relation:
(i)says t h a t i f R i s symmetric and t r a n s i t i v e , then i t i s r e f l e x i v e . N o t i c e t h a t t h i s i s n o t t r u e f o r r e f l e x i v e i n A. The usual n o t i o n o f e q u i v a l e n c e r e l a t i o n i s eqLLiuaLence treeation i n A ( i . e . symmetric, t r a n s i t i v e , and r e f l e x i v e i n A ) . I t i s n o t t r u e t h a t symmetric and t r a n s i t i v e i m p l y e q u i v a l e n c e r e l a t i o n i n A. However, we s h a l l o n l y use t h e n o t i o n o f equiva l e n c e r e l a t i o n i n i t s f i e l d and, thus, i t i s enough t o r e q u i r e symmetry and transitivity.
.
2 Then PROOF OF ( i ) . L e t R C R = R - l and suppose x E D R u D R - ' t h e r e i s a y such t h a t x R q . Then y R x , by symmetry, and X R X , by t r a n s i t iv y t y
-
.
R
2
5 R.
PROOF
2 OF (ii). (1) Suppose R
5R
= R-'.
By ( 7 )
it i s clear
that
Hence R = R - l = R2.
R = R - ~= R*
R = R - ~A R 2 c -R
clearly implies
Thus, we have proved t h e f i r s t equivalence.
( 2 ) Assume, now, R = R - l = R2. R = R-'o
Then R = R O R = R-'
O R . i.e.
R = R-'
= R2
+
R , i s proved.
( 3 ) Assume, R = R - l o R. Then R - l = ( R - l o R ) ' l a l s o R2 = R o R = R - ' o R = R. Thus, R = R - ' o R
= R-' -+
o (R-l)-l
R = R-'
= R2.
= R-l O R =
R;
ROLAND0 C H U A Q U I
44
.
The l a s t equivalence i s proved s i m i l a r l y .
We pass now t o g i v e t h e main p r o p e r t i e s o f equivalence r e l a t i o n s . F i r s t , t h e d e f i n i t i o n o f c l a s s o f d i s j o i n t nonempty s e t s .
2.2.3.5
DEFINITION (COLLECTION)
-
A A
I
0 @ A A W xW y(x,yEA
+
x
x ng
= g V
=
0).
2.2.3.6 A B B R E V I A T I O N , We sometimes w i l l need t o say t h a t A i s a c o l l e c t i o n of nonernpty and d i s j o i n t classes. In general, we would say t h a t t h e c o l l e c t i o n of r such t h a t @ (7 a t e r m and # a f o r m u l a ) i s such a c o l 1e c t i on by,
...
WXo
@xo
where Yo,
WXn-l(@+r
... xn-l
..., Yn-l
0) A WXo
f
I
[ Y 0”. Yn-l
-b
7
=
7
... Xn-l WYo...WYn-l xo ... ‘n-1 [ yo.. . yn-ll
a r e new v a r i a b l e s n o t appearing i n
7
(4 A
v
n o r @.
T h i s formula w i l l be a b b r e v i a t e d by,
2.2.3.7 X = R*Cx}).
D E F I N I T I O N(NOTION), E(R,X)
-
~ ~ ( X E D R D U R-’/\
We have t h a t i f R i s an e q u i v a l e n c e r e l a t i o n , t h e n E(R,X) means t h a t X i s an equivalence c l a s s o f R.
2.2.3.8
A B B R E V I A T I O N , F o r any f o r m u l a #, t h e t e r m 4’
be a b b r e v i a t e d by R 2.2.3.9
THEOREM ( P A R T I T I O NTHEOREM).
A { X : E (R,X)}
A R = R
E (R,x))-
I t i s easy t o see t h a t ,
and t h a t ,
R E (R,X)
=
u {XXX : E(R,X)}
A { X : E (R,X)} =
u {R*{x} X
P R O O F , (1) Assume, f i r s t , t h a t R
x
-
R =
R-’OR
A$R*{x}
U
{X X X
-
:x E D R U DR-l},
R*{x) : X E D R U D R-’}
= R-l o R.
:(PI w i l l
A X I O M A T I C S E T THEORY
-
45
L e t x,y E DR and z E R * { x } n R * { y } ; t h e n x R z and y R Z . Therefore, by symmetry z R y , and, by t r a n s i t i v i t y , x R y . Now, we have, by symmetry and q R u . Thus, R*{x} = T(*{q}. Since x E D R , R*{x) # 0. transitivity, X R U So we get,
Ax{R*{x}
Now i f
: xEDR UDR-l},
i.e.
A
( x , ~ ) E 2, t h e n x,gEJ?*{c_i); t h e n Thus, (*) R C_u { X : E(R,X)).
u { X : E(R,X)).
{x
:E(R,x)~.
( x , y ) E T?*{g} x T ? * { g } C -
On t h e o t h e r hand i f ( x , z ) E R*Cg} x l?*{y}, t h e n x R y A y R z. Thus, ( x , y ) E R. Hence, u {X : E(T?,X)} C - R. From (*) we deduce i? = u {X: E(R,X)}. Thus, we have proved t h e i m p l i c a t i o n from l e f t t o r i g h t . ( 2 ) Assume now
A {X :E(R,X)} A R = R
E (R,X)'
We have,
R -1
OR
(u { x x x : E ( R , x ) } )
= (U { x x x : E ( R , x ) } ) - ~0
(u { X x X : E(R,X)}) o
=
= U{(XxX)
But, s i n c e
o ( Y x Y ) : E(R,X) A E(R,Y)}
A {X : E(R,X)}
, we 0
0 (YXY)
(XXX)
R-l o R =
2.2.3.10
have t h a t
,
if
X
f
Y
, if
X
=
Y
(since
x
n Y = 0)
=
XxX Therefore,
(u { Y x Y : E(R,Y)})
U
{ X x X : E(R,X)} = R.
COROLLARY,
R = R-' O R A x , y E DR+ (R*{x} = R*{y}
-
xRy).
From t h i s c o r o l l a r y , we see t h a t we c o u l d d e f i n e t h e .type 0 6 x by R as t h e e q u i v a l e n c e c l a s s o f x, i f x E DR and R i s an equivalence r e l a t i o n , T h i s c l a s s i f i e s a l l elements o f D R i n t o d i s i n symbols, t R ( x ) = R*Ix). j o i n t classes. However, t h i s procedure m i g h t n o t be v e r y convenient s i n c e R*{x} may be a p r o p e r c l a s s and we would n o t be a b l e o f having c l a s s e s o f types. 2.2.3.11
THEOREM SCHEMA,
L e A 4 be a
~0munlLea.
Then,
The u n i o n o f equivalence r e l a t i o n s i s n o t , i n general, an equivalence r e l a t i o n . Since, however, i n t e r s e c t i o n s p r e s e r v e symmetry and t r a n s i t i v i t y , t h e y a l s o preserve equivalence r e l a t i o n s . Thus, 2.2.3.12
THEOREM SCHEMA,
LeR
7
be a R m and 4 a 6umuRa.
Then,
ROLAND0 CHUAQUI
46
The composition of two equivalence relation R , S, in general, is not an equivalence relation. However, from 2.2.3.1 and 2.2.3.3, we obtain THEOREM,
2.2.3.13 =
R = R-loR A S = S-'oS
(R~s)-~~(R~s).
A R'oS = S o R
--f
RoS =
We now pass to ordering relations. We say that R i s antioymm&c
RnR-' C I D (i.e. if tf x v y x R y / I yRx
DEFINITION, PO(R)
2.2.3.14
-
+
RnR-'
x
if
Y). = Ro A R
2
=
R.
If PO(R), we say that R is a p a h t i a e ohdetLing. R is a partial ordering if and only if R is reflexive (Ro 5 R ) , antisymmetric (RnR-1 L I D ), and transitive ( R 2 2 R ) . DEFINITION,
2.2.3.15
R
V A = U { z : W x ( x € A + x R z ) A Wu ( W x ( x € A - + x R u ) zRu)). R A A = U ( z : W x ( x € A + z R x ) A Wu ( W x ( x € A + x R u ) + u R z ) l .
(i)
(ii) (iii)
+
LubR(z, A)
(iv) G l b R ( z , A )
(v) a
R
R
-
-
W x(xEA
--t
x R z ) AW u( W x ( x € A
+
xRu)
+
zRu)).
Wx(xEA+ zRx) AWu(Wx(x€A+uRx) + u R z ) ) .
V b = V {a,bI. R R a A b = A (a,bI
(vi)
R
If R is a partial ordering, V A , denotes the L e a s t uppm bound (accordR ing to R ) 0 4 A (lub) if it exists. Similarly, A A is the greatest lower bound (glb). L u b X ( z , A ) means that z is the least upper bound of A. Thus P
we can express that this least upper bound exists by LubR(J)A,A). Similarly for greatest lower bounds. If P O ( R ) , then the existence of these bounds implies their uniqueness. 2.2.3.16
DEF IN IT ION
(i)U L O ( R )
(if)
LLO(R)
+ +
+ +
I
PO(R) AWxWy(x,g
E
DR+ 3 z ( x , y R z AVu(x,gRu+zRu))).
PO(R) AWxWq(x,qEDR + 3 z ( z R x , y A b u ( u R x , q + u R z ) ) .
AXIOMATIC S E T T H E O R Y
-
( i i i ) LO(R)
( i v ) CULO(R) CLLO(R)
(v) (vi )
C L O (R)
-
47
ULO(R) A LLO(R).
-
-
PO(R)A W A ( A c - DR Wu( Wx(x€A
PO(R)A WA(A C- DR
Vu( Wx(xEA
+.
3 z ( Wx(xEA
+.
xRu)
-+
3 z ( Wx(xEA
-+
xRu)
+.
xRz) A
+.
zRx) A
zRu)).
+.
uRz)).
+
C U L O ( R ) A C L L O (R).
U L O ( R ) i s read R i s an upper s e m i l a t t i c e ordering i . e . R i s a p a r t i a l ordering and every p a i r of elements has a 1.u.b. LLO(R) i s read R i s a lower s e m i l a t t i c e ordering, i . e . R i s a p a r t i a l ordering and every p a i r of elements has a g.1.b. L O ( R ) i s R i s a l a t t i c e ordering. When a C i s added we get complete l a t t i c e (upper s e m i l a t t i c e , lower s e m i l a t t i c e ) orderings i.e. p a r t i a l orderings i n which every subclass of t h e domain h a s lub and g l b ( l u b , g l b ) .
DEFINITION,
2.2.3.17
-
( i ) CO(R) * V x V q ( x , y E D R UDR ( i i ) SO(R)
C O ( R ) i s read, R i s connected.
2.2.3.18 THEOREM I RO
-R c
-
+.
( i i ) SO(R)
x R y V qRx).
+
PO(R) A C O ( R ) .
nem o t d e h i n g ) .
(i)
-1
(CO(R)
-
SO(R), R i s a b h p L e ohdehing ( o r & -
R o (vxv)oR c -RUR-~).
R ~ R -=I R O A R~ = R A R o
( v x v ) oR c RUR-~.
( i i i ) SO(R) +.LO(R) The proof i s l e f t t o the reader. F i n a l l y , we study well-founded r e l a t i o n s . 2.2.3.19
(i)
DEF IN IT ION,
-
oR(x)=
( i i ) WF(R)
(R-'*{X>)
%
1x1
.
W A(A c -DRu D R - l A A
f
0
-+
3 x ( x e A A A n OR(x)=O)).
OR(x) i s the c l a s s of s e t s d i f f e r e n t from x t h a t a r e r e l a t e d t o x by R , i.e.
OR(x) = ( q : y
+
x A yRx1.
T h u s , we may write
48
WF (R)
-
2.2.3.20
ROLAND0 CHUAQUI VA(A C - D R U DU-'
A A # 0
3 x(xEA A Wy(yEA A qRx+y=x))).
+
THEOREM,
( i ) and ( i i ) s i m p l i f y t h e d e f i n i t i o n o f well-foundedness f o r r e f l e x i v e ( i i i ) asserts t h a t i f R i s welland i r r e f l e x i v e r e l a t i o n s , r e s p e c t i v e l y . founded t h e n by making i t r e f l e x i v e (adding Ro) i t remains well-founded. The p r o o f i s easy. Ax Reg which says: W A (A f 0
-+
3 x ( x € A A x n A = 0))
a s s e r t s ( i n G ) i n f a c t , t h a t EL i s well-founded. We have, f i r s t , t h a t Ax Reg + G i m p l i e s t h a t EL i s i r r e f l e x i v e : Suppose X E X ; l e t A = 1 x 1 . Then i f Y E A , y = x ; since, x E x n A we have t h a t t h e r e i s no y i n A d i s j o i n t from A , c o n t r a d i c t i n g Ax Reg. Also, EL-'*{x} = { y : y EL
x}
=
{y: q E x }
hence f o r a l l c l a s s e s A , A L D R u D R - l . ( i i ) we get, i v e , u s i n g 2.2.3.20 WF(EL)-VA
=
X ; and DEL u D E L "
= Y ;
Therefore, s i n c e E L i s i r r e f l e x -
(A#O+
3x(x€AAxnA=O)).
Therefore, we have proved i n G : 2.2.3.21
THEOREM,
WF (EL) A EL c -D v -
Ax Reg.
The f o l l o w i n g i n d u c t i o n p r i n c i p l e f o r well-founded r e l a t i o n s i s v e r y important. 2.2.3.22 D R U DR-'
THEOREM,
WF (R)
+
(v
x(x
E DRU
A).
PROOF , Suppose R i s well-founded and f o r
OR(x) c -A
uDR-l)
D R -o R ~( x ) c - A+XEA)-+
x
E DRU
DR-'
i m p l i e s x E A . Suppose, a l s o t h a t D R U D R ' l - A. A f 0. Since W F (R), t h e r e i s a y E (DRu D R - ' )
we have t h a t Then ( D R %
U
A such t h a t
A X I O M A T I C SET THEORY
O R ( y ) n ( ( D R u DR-l ) Therefore
A)
Ax Reg
THEOREM,
PROOF,
B u t OR(y) c - R-'*{yI
0.
f
t h i s c o n t r a d i c t s y E ( D R U DR-' )
YEA;
2.2.3.23
1 ,
.
T h e r e f o r e OEL ( x ) = EL-'*{x} = x
w
D EL u D E L - 1 ,
x(xc A
-+
(P A C A
-+
X E
A)
+
-+
A
1,
A =
By 2.2.3.22,
v=
5DR.
Hence O R ( y ) cA.
A.
v ).
we have W F ( E L ) A EL C D Y.
By 2.2.3.21,
Assume Ax Reg.
49
we g e t , s i n c e
.
V =
But V > A . I t i s i n t e r e s t i n g t o note, t h a t t h e i n d u c t i o n p r i n c i p l e f o r R s t a t e d i n 2.2.3.22 i s a c t u a l l y e q u i v a l e n t t o W F ( R ) . Thus we c o u l d r e p l a c e i n 2.2.3.22 by The same i s t r u e i n 2.2.3.23.
-.
+
F i n a l l y , the important well-ordering r e l a t i o n s a r e introduced.
PROBLEMS
R = R-lo R
1.
Prove:
2.
Show t h a t
3.
Prove: A {X =
4.
:$I u
I!'
-(R o
A A {X:
(- R ) 2
- R-l) $1
.+
=
((- R)')-l
o ( - R)';
characterize ( - R ) 2
i s always t r a n s i t i v e .
R C R -u 4- $
{X : 4 } =
u { x : $1 AWX(4 - + x =
: ICx [ Y I A Y c - XI)).
Define:
R i s extensional Prove:
R i s extensional 5.
-+
Define:
R i s complete
-
-
-
w x ~ y ( x , yE D PO ( - ( R
vA(A
o
R
-+
- R-')).
- DR+ 3 z V x ( x R z C
-
R-'*C~I = R - ~ * c ~ I x
-
xEA)).
=
Y)).
.
ROLAND0 C H U A Q U I
50
6.
Prove: R i s extensional and complete Prove: (a) ( w x ( x
E
Prove: WO(R)
-
CLLO(-(R o
D RD ~ R-~A oR(x)cA
(b) WA(PA c -A
7.
+
WF
-,A (R)
= V )
-+
A CO(R)
Ax Reg.
.
+
XEA)
--f
- R-I)). DRU
DR-'
c A)
+
WF ( R )
CHAPTER
2.3
F u n c t i o n s and O p e r a t i o n s
2.3.1
FUNCT I O N A L R E L A T I O N S
The r e l a t i o n R i s a Bunotiotr i f
(y,x) E R
and ( z , x ) E R , i m p l y
Thus, f u n c t i o n s R can be c h a r a c t e r i z e d by R 0 R - l
q = z .
ZID.
The f o l l o w i n g theorem, whose p r o o f i s l e f t t o t h e r e a d e r , g i v e s severI n t h i s book, R o R - l c I D a l equivalent possible d e f i n i t i o n s o f functions. i s used t o say t h a t R i s a f u n c t i o n . We a l s o assume i n t h i s Chapter t h a t R, S, T a r e r e l a t i o n s .
THEOREM,
2.3.1.1
(i) R0R-l cID(ii)
R O R - ~ 510-
R O R - ~
(v) R
OR-^
LIDc -D I
(vi) RoR-'cID-
( ~ v Ro) n R
=
0.
WSWT(SnT)oR = (SoR) n ( T O R ) .
( i i i ) R o R - lcID(iv)
( R o R - 1) n D v = 0 .
-
W S W T ( S ~ T ) =~ R ( s ~ R 2 ), (TOR).
wx
WY
R - ~ * ( xn Y ) = ( R - ~ * x )n ( R - ~ * Y ) .
V X W Y (R* X)
f-
Y = R*(XnR-'*Y).
I t i s c l e a r t h a t o f o u r c o n s t a n t r e l a t i o n s , I D and 0 a r e f u n c t i o n s , w h i l e V x V , E L , I N , and Dv a r e n o t .
The i n t e r s e c t i o n o f f u n c t i o n s i s a f u n c t i q n , b u t t h e u n i o n i s n o t , i n general, a function. Under some r e s t r i c t i o n s , i t i s : 2.3.1.2
THEOREM SCHEMA,
LeL
T
51
be a ,tm and @ a l;otvnu&;
then
ROLAND0 C H U A Q U I
52
The r e l a t i v e complement o f a f u n c t i o n i s never a f u n c t i o n . A subclass o f a f u n c t i o n i s a l s o a f u n c t i o n and t h e composition o f f u n c t i o n s i s a f u n c t i o n . The o t h e r r e l a t i o n a l o p e r a t i o n , t h e converse o r i n v e r s e , w i l l be discussed l a t e r . The f o l l o w i n g theorem, easy t o prove, summarizes these f a c t s .
2.3.1.4
DEFINITION,
R'x =
R*{x}
*
R ' x i s Lhc value o d x by R.
R*{x}
PROOF OF ( i ) . Suppose t h a t R i s a f u n c t i o n and = { q } f o r some y. Thus n R {XI = y.
x
E
D R . Then
L e t F be a unary o p e r a t i o n . The r e s t r i c t i o n o f F t o t h e c l a s s o f w i t h F(x) E V c a n be represented by t h e f u n c t i o n F
x
I t i s c l e a r t h a t F i s a f u n c t i o n , w i t h domain D F = {x : F ( x ) E V 1 , and F ' x = F(x) f o r x E D F . Thus F and F have t h e same v a l u e s f o r x w i t h
F(x) E
v.
However, t h e r e p r e s e n t a t i o n o f o p e r a t i o n s by r e l a t i o n s g i v e n i n 2 . 2 . 2 i s more general, because i t does n o t r e s t r i c t t h e domain t o those X w i t h
53
AXIOMATIC S E T T H E O R Y
But, when p o s s i b l e , t h e p r e s e n t r e p r e s e n t a t i o n by f u n c t i o n s i s E V. more convenient.
F(x)
The f o l l o w i n g d e f i n i t i o n f o r m a l i z e s t h e i n t r o d u c t i o n o f such f u n c t i o n s .
2.3.1.6
D E F I N I T I O N SCHEMA, ( 7
X
=
Let
T
:q5) = ( ( 7 , x ) :
be a t e r m and q5 a formula. Then
GI.
For i n s t a n c e , we have when F i s a unary o p e r a t i o n , { ( F ( x ) , x ) : F(x) E V )
.
(F(x): F(x)E V ) =
X
We can i n t r o d u c e f u n c t i o n s f o r a l l t h e o p e r a t i o n s a l r e a d y d e f i n e d . For example, f o r t h e image o p e r a t i o n ,
R* = ( R * x : x E
v)
.
I n t h i s case and o t h e r s , t h e same symbol w i l l be used f o r t h e operat i o n and t h e corresponding f u n c t i o n . With the value n o t a t i o n defined i n 2.3.1.4 can be g i v e n by, F*A = { F ' x : x E A } , and F-l*A have e a s i l y ,
tl x ( x
t h e image o f a f u n c t i o n F, = { x : F'xEA}. Also, we
2.3.1.7 THEOREM, R0R-l c CZD-I D A S0S-l E DR+ R ' x = S I X ) ) .
I n t h e r e s t o f t h i s s e c t i o n , t h e l e t t e r s F, G, H, s t r i c t e d t o functions.
2.3.1.8 (i) (ii)
= B n D F -1
F* F-'*B
.
F* F - ~ * B c -B . A n B =
o
-+
=
(F*A) n B .
(F-~*A) n (F-~*B) = 0 .
(v)
F-l*(AnB)
= (F-'*A)
n (F-l*B).
(vi)
F-l*(A%B)
= (F-'*A)
%
(vii) (viii) (ix)
6, g,
t--f
DR = D S A
and h. a r e r e -
THEOREM (PROPERTIES OF IMAGES OF FUNCTIONS)
(iii) F*(AnF-l*B) (iv)
(R = S
8
5 F*A
v B 3A
-+
3 C'(C'
B n D F-'
B n D F - ~ =F*
( X I F-~*A = F-~*B
%
-A
C
=
(F-'*&).
A B = F*C).
F*A.
~-l* % B .
+
A nDF = B nDF
.
I
ROLAND0 CHUAQUI
54
These p r o p e r t i e s o f images o f f u n c t i o n a r e n o t c h a r a c t e r i s t i c o f f u n c t i o n i n t h e sense t h a t t h e r e a r e o t h e r r e l a t i o n s w h i c h s a t i s f y them. B u t ; we have t h a t ( v i i i ) , ( i x ) , and ( x ) a r e e q u i v a l e n t and so a r e ( x i ) and ( x i i ) . PROOF,
2.2.1.10.
and ( i ) c a n be p r o v e d d i r e c t l y and t h e r e s t o b t a i n e d f r o m (i) We have, F 0 F - l
PROOF OF ( i ) .
I D I D ( F o F-') (ID IDF-')*B
=
I D I D F -I.
-ID.
By 2 . 3 . 1 . 3
C
T h e r e f o r e F*
F-'*E
( i i ) F o F-l =
o F-l)*
= (F
=
= B n DF-'.
PROOFOF ( i i ) :
By (i).
PROOF OF (iii). By 2 . 2 . 1 . 1 0
( v i i i ) , we h a v e (F*A) n B C - F*(AnF-'*B).
On t h e o t h e r hand, b y 2 . 2 . 1 . 1 0 ( i v ) , F*(AnF-'*E) U s i n g , now, ( i i ) we o b t a i n , F * ( A n F - l * B ) C - (F*A)
c ( F * A ) n ( F * F"*B).
17-8.
n B = 0. By ( i i i ) PROOF O F ( i v ) . Assume A n t 3 = 0. By ( i i ) (F* F-'*A) F*(f-l*AnF-'*B) = (F* F-'*A) n 8 = 0. T h e r e f o r e , F-l*A n F-l*B n D F = 0 . F-l*B C - D F . Thus, F-l*A n F-l*B = 0. B u t F-'*A
PROOFOF ( v ) .
By ( i v ) , we have ( F - 1 * ( A n 8 ) ) n ( F - 1 * ( 8 n J A ) )
= ( F - l * ( A n B ) n (F-'*(A%E))
( v ) , we o b t a i n , F-l*A
= (F-l*(A%B))
=
0 =
From 2 . 2 . 1 . 1 0
n (F-l*(B\A)).
(F-'*(AnB)) U (F-l*(A-E)) a n d F-'*B = T h e r e f o r e (Fql*A) n (F-'*E)=((( F - l * ( A n E ) ) U = (F-l*(AnB)) u (F-l*(B%A)). (F-'*(A
=
* 8 ) ) )) n (( (F-l*(A
PROOF OF ( v i ) .
n8 ) )
U
(F-'*( E % A ) ) ) = F-l*(A n B )
By 2 . 2 . 1 . 1 0
( v ) , F-l*A
t h e n , b y ( v ) , F-l*A = ( F - l * ( A % B ) ) U ( ( F - l * A )
(F-'*(A*B)) PROOF
n (F-l*B)
OF ( v i i ) .
= (F*A) n 8 = B.
= 0.
Therefore,
Suppose
Therefore, t a k e
E
= (F-l*(A%E))U(F-'*(ArlB));
(F-l*S)).
(F-l*A) \(F-'*E)
5 F*A. c'
Also, b y ( i v ) =
.
-DF-' c
= F*
,
F-'*(A%B).
Then, b y ( i i i ) , F * ( A n F - l * B )
= A n F-l*B
PROOF OF ( v i i i ) . We have, B n D F - ' i n g ( v i i ) , we o b t a i n ( v i i i ) .
.
=
V. Hence, a p p l y -
55
AXIOMATIC SET THEORY
PROOF OF ( i x ) .
Using 2.2.1.10
By ( v i i i ) , we have B n DF-'
( B n D F - ' ) = B n DF-'. But, by 2.2.1.10 ( F - ' * % 8 ) u (F-'* x D 0 F - l ) = F - l * % B . (xi,
=
F*A
f o r some A .
F* % F-'* % ( v ) , F - l * % ( 8 n OF-') = = F*A,
( x i ) , we o b t a i n , F*%F-'*%F*A
i.e.
( x i ) and f x i i ) a r e ' l e f t t o t h e r e a d e r .
We a l s o have d i s t r i b u t i v i t y o f i n v e r s e image o f f u n c t i o n s w i t h generalized intersection.
2.3.1.9
THEOREM SCHEMA, L& r be a t m and 6 a 6o/un&.
On t h e o t h e r hand, i f x $ implies that
f o r a l l such r . IT
x
6
F-'*r
E
nx
, i.e. y
Therefore
FIX
E
DEFINITION
BA i s t h e & a n
06
F'x
=
:$I,
t h e n f o r a l l Xo...
f o r some y
nx
0.. .x n-1
:$I. 2.3.1.10
r
0"' Xn-1
{r
:$I,
Then,
E 7,
i.e.
Xn-l,
and hence F'xE r
x c ~ - l *n
'0'
*
'Xn-1
I
6uncLionn w L t h domain 8 and m n g e included i n A.
Since i n G we almost never can prove t h a t f u n c t i o n which i s a s e t w i t h domain 8, 'A 8
BA
f
0, i.e.
t h a t t h e r e is a
w i l l n o t be much used i n P a r t 2.
On t h e o t h e r hand A ( ? ) i s t h e no;tion t h a t says t h a t f 12 a 6unOtiun w a h domain 8 m d tange i n c h d e d in A. W i t h t h i s n o t i o n , we can express t h a t ? i s a f u n c t i o n by D F V ( F ) . T h i s w i l l o f t e n be used.
2.3.1.11 DEFINITION SCHEMA (GENERALIZED CARTESIAN PRODUCT). Let be a term and $ a formula. Then n ( 7 :q,) = Ed : 6 & x V A
~ ; O ~ - ' C J D A DI x~: = $1 A
Vx($
+
6kE
r)).
X
ROLAND0 CHUAQUI
56
2.3.1.12
For instance, we have
F, by F ( 0 ) = A xEA A
"6 = nX (6'~:x
DEFINITION,
gEBl.
nX
( A : xEB) =
E
06).
G
A.
I f we d e f i n e t h e o p e r a t i o n
XIx (F(x) : X E 2 ) = (I( x,O) , ( y , 1 ) I : I ( x,O) , ( g , l ) l i s a p o s s i b l e d e f i n i t i o n o f t h e o r -
and F ( 1 ) = 8, then The s e t
2
dered p a i r o f x and g. I t i s t h e f u n c t i o n 1; E {x,gl, w i t h 6'0= x a n d 4'1 = y . I f we ' i d e n t i f y ' t h i s 6 w i t h t h e ordered p a i r ( X , L J ) , t h e n t h e Ilx (F(x): x E 2 ) i s ' i d e n t i f i e d ' w i t h A x B .
I n some c o n t e x t , ( x , g ) i s b e t t e r as ordered p a i r than {C x,O), ( y , l ) } ; i n o t h e r s t h e o p p o s i t e i s t h e case. The same i s t r u e f o r t h e t w o operat i o n s of C a r t e s i a n Droduct. T h i s m u l t i p l i c i t y o f d i f f e r e n t p o s s i b l e d e f i n i t i o n s f o r t h e same conc e p t i s an i n e l e g a n t c h a r a c t e r i s t i c o f a l l s e t t h e o r i e s and i t seems unavoidable. We say t h a t F i s a biunLquc
2.3.1.13
(ii)
one-one i j u n c t i o n i f F and F - l a r e func-
i f D F ~ ~ - Al D ( F~- l) D F (F-').
t i o n s , i.e.
(i)
ofi
-
DEFINITIONa
!A(F)
! A = (1;:
.
A ~ ( A~ A) ~ ( ~ - l ) A
A(ij)l,
! A ( F ) says t h a t F i s a pe,trnLLtCLtion o f A , i.e. F i s a b i u n i q u e funct i o n from A o n t o A. !A i s t h e c l a s s o f a l l permutations o f A.
PROBLEMS
Prove 2.3.1.1
1. 2. 3.
Characterize a l l r e l a t i o n s R t h a t s a t i s f y R = R o R - l o R
4.
Show t h a t F i s a b i u n i q u e f u n c t i o n i f and o n l y i f D F * = V A DF-'"
Prove 2.3.1.3
= PDF-'
A (F*
I
.
P D F i s a biunique function).
-
5.
Prove 2.3.1.8
6.
F i n d a r e l a t i o n R which i s n o t a f u n c t i o n b u t s a t i s f i e s
7.
Characterize t h e r e l a t i o n s
(x)
(xii).
W B 3 A ( B n D R - ~= R*A).
R
that satisfy
R* R'l*
R*A
=
R*A
.
=
A X I O M A T I C SET THEORY 2.3.2
57
MONOTONE O P E R A T I O N S ,
z
A unary o p e r a t i o n F i s Y,Z-monutone i f i t s a t i s f i e s : Y LA c B C Y c F [ A ) C_ F ( B ) 5 2 . We say t h a t F i s rnoncdone i f i t i s 0, V-KonoTone. S i m i l a r l y , f o r f u n c t i o n s we have: 2.3.2.1
F ' x C- F ' y ) .
DEFINITION,
Mo ( F )
- DbF
(F) Avxt'q(x,y
E
DFA
x
--f
5q
-+
The f o l l o w i n g theorem g i v e s c o n d i t i o n s e q u i v a l e n t t o monotony. - 1. 2.3.2.2 THEOREM SCHEMA, L e R F be a unmy opehCLtion and Y c Then t h e doflowing CvnditioMn CVLQ e.qLvaRen2:
( i ) W A WB(Y c - A c- 8 c- Z
( i v ) W A WB(Y ( v ) W A WB(Y
c - A, B c- Z
- A, C
B C -Z
+
--t
+
Y c - F(A) c - F ( B ) c- Z ) .
Y c - F(A) u F ( B ) C_F(AUB)C - Z).
-
Y C - F ( A n B ) C F(A) n F ( B )
5 Z).
Suppose t h a t V AW B ( Y C A C B C Z Y C F(A) C F ( B ) 2 Z ) . PROOF, Therefore, f o r e v e r y X t h a t s a t i s f i e s - @ A-Y C-X c Z , we have-Y z F ( X ) 5 F( u(X : 4 A Y 5 X c Z}) c Y c - X C- Z } ) C_F(X)& Z . - Z and Y & F ( n{X : @ +
n
Thus,
( i ) i m p l i e s ( i i ) and ( i i i ) .
I t i s c l e a r t h a t ( i i ) i m p l i e s ( i v ) and i t i s enough t o show t h a t ( i v ) o r ( v ) i m p l y i m p l i e s ( i ) . Suppose Y S A 5 8 C Z. Then, Thus, Z ? F ( B ) 2 F(A) 2 2 F(A) u F ( B ) 3 Y .
-
( i i i ) i m p l i e s ( v ) . Therefore, ( i ) .I s h a l l prove t h a t ( i v ) by ( i v ) , Z > F ( B ) = F(AUB) Y.
The i m p l i c a t i o n o f ( v ) t o ( i ) i s proved s i m i l a r l y . I n a s i m i l a r way, t h e f o l l o w i n g can be proved.
.
58
ROLAND0 CHUAQUI
(iv)
t!xtlq F(x) u F ( q ) gF(xuq),
(v)
W x w q F ( x n q ) 5F(x) " F ( Y ) .
We now begin t h e s t u d y o f f i x e d p o i n t s o f monotone o p e r a t i o n s . T h i s study i s based on T a r s k i 1955. The theorems proved here w i l l be useful i n several chapters o f t h e book. We say t h a t a c l a s s X i s a hixed p a i d of a unary o p e r a t i o n F, i f We have t h a t Y, Z-monotone o p e r a t i o n s always have f i x e d p o i n t s . F(X) = X. F i r s t , t h e theorem f o r 0, 2-monotone o p e r a t i o n s . THEOREM SCHEMA, L e t F be a unmq opehation.
2.3.2.4
W A W B ( A-c B c Z +
F ( A )CF(B)LZ)+
n {X : X c - Z A F(X) = X ] A U =
LJ
Then
~C~U(F(C)=CAF(U)=DAC=
{ X :X C - Z A F(X) = A } ) .
F can be considered as a monotone o p e r a t i o n from subclasses o f Z t o subclasses o f Z .C i s t h e l e a s t f i x e d p o i n t and Q i s t h e g r e a t e s t f i x e d p o i n t . Thus, t h e c o n c l u s i o n can a l s o be w r i t t e n , WA W B ( A - c B-c Z + F ( A ) & F ( B ) 5 Z ) + F ( n { X : X-C Z A F ( X )= X } ) =
= n
{x : x
A F(X) =
c -
z
A F ( x ) = X I A F( u { x : x c -
z
A F(x)=
XI.
PROOF, Assume F(A) c F(B) C Z). L e t that C Z. Suppose, By t h e assumption, we we g e t t h a t F(C) 5 X,
c
X I )= u { x : x c -z
A
t h a t F i s a unary o p e r a t i o n such t h a t , V A W B ( A c-B C-Z + C = n { X : F ( X ) 2 X C Z l . Since F ( Z ) 5 Z, we have now t h a t X i s such t h a t F ( X ) 5 X 5 Z. Then C C X have F(C) L F ( X ) . S i n c e we assumed t h a t F(X)-z X , f o r a l l X w i t h F(X) 5 X E Z. T h e r e f o r e F(C) 5 C.
.
On t h e o t h e r hand, from F(C) c C
5 Z,
by t h e monotony o f F , we deduce
F (F(C))E F ( C )5 Z. Thus F ( C ) i s o n e o f t h e c l a s s e s whose i n t e r s e c t i o n i s - F(C),and, hence, F(C) = C. C. Therefore, C c
c
=
Also, C = n { X : F ( X ) c c Z} c -X - n { X :F(X) = X n { x : F ( X )= x c z).
5 Z } C- C.
Therefore
.
I n o r d e r t o p r o v e t h e r e s t o f t h e theorem, t a k e U = LJ { X :X c Z A X c F ( X ) } . The p r o o f t h a t U = F ( U ) and D = u { X : X = F ( X ) 5 2 1 i s s i m i l a r t o t h e above.
2.3.2.5
THEOREM SCHEMA,
L e t F be a unmq opetration. Then
Y C - F ( A ) C F ( B ) 5 Z) F ( n{x :Y c x = F ( x ) c z } ) = n { x : Y c- X = F(X) 5 Z } A -
Y c - Z A vAWB(Y c- A c- 8 c- Z
F ( u { X :Y
-X C
=
F(X) C Z})
+
= U {X :Y
+
5X
= F(X)
5 Z}.
AXIOMATIC S E T T H E O R Y
59
The proof i s s i m i l a r t o t h a t of 2.3.2.4.
PROOF,
By 2.3.2.2,
we have
F ( n {X : @ A Y c X = F(X) c C X = F(X) - Z } ) -c n { F ( X ) : @ A Y -
.
5Zl
=
n {X:@ A Y c - X = F(X) 5 Z ) . S i m i l a r l y f o r unions.
n
{x : u
PROOF,
{ V :Q A Y c - V = F(V)
By 2.3.2.6
and
5 Z}
C_X = F ( X
CZ)
.
2.3.2.5.
* 2 . 3 . 2 . 8 E X A M P L E , In a topological space X, l e t F ( A ) be t h e c l a s s o f accumulation points of A f o r A C X . Let Z be a closed subset of X. T h u s , we have A C 8 C Z + F ( A ) CF(i3) Z. Therefore, by 2.3.2.4, t h e r e i s a l a r g e s t D such t h a t D = F ( D ) . Then D i s p e r f e c t and Z % D i s s c a t t e r e d (Theorem o f Cantor-Bendixon. ) From t h e theorems proved, we now deduce theorems f o r two unary operations. 2.3.2.9
THEOREM SCHEMA,
W XW Y ( ( X c -
Let F and G be u w y ope&onb.
Yc A -,F ( X ) C F(Y))
A (X
5Y 5 8
+
G(X)
G(Y)))
3 A 1 3 B1(A1 Z G ( 8 ) A B1 c F ( A ) A F(d%A1)= B1 A G(B%B1) =
Then, *
All.
60
ROLAND0
CHUAQUI
PROOF, Assume t h a t F and C a r e a u n a r y o p e r a t i o n s such t h a t W X W Y F(X) C F(Y))A X 5 Y 2 B G(X) c G(Y))). D e f i n e t h e operat i o n H,-by
((X
5Y c A
-
+
+
Then 8 Suppose X 2 Y c F ( A ) . and, hence A % G ( B % X ) 5 A%LG(B%Y)
A p p l y i n g 2.3.2.4
1 B%X
5 A.
2 &%Y, t h u s , G(B%X) > G ( B Therefore,
Y)
t o H a n d A, we o b t a i n a B1 such t h a t H(B1) = B1.
We
have, B1 = H(B1) = F ( A % G ( B % B 1 ) ) c F ( A ) . Let
.
A1 = G ( B % B 1 ) .
8 1 = H(B1) = F ( A % A 1 ) . 2.3.2.10
S i n c e 8 % B , C 8 , we have A1 z G ( 8 ) .
Finally,
1 -
THEOREM SCHEMA,
Le.L F a n d G be unarry 0perrcc;tiunn.
Then
W X W Y ( ( X-c Y c-A - + F ( X ) cF(Y))A(XcY5B+G(X) cG(Y)))A F(A) c 8 A G(B) L A A nA = 0 = 1 2
B 1 nB 2
+
3 A1 3 A 2
3B1
3B2 (A = A U A A 1 2
A F ( A 2 ) = B1 A G ( B 2 ) = A1)
.
B
=
B1UB2 A
L e t F and G be monotone o p e r a t i o n s f o r s u b c l a s s e s o f A and PROOF, r e s p e c t i v e l y and suppose F ( A ) 2 B and G ( B ) 5 A. By 2.3.2.5, there a r e A1, B1 such t h a t A1 E G ( B ) , B1 c F ( A ) , F ( A % A 1 ) = B1, and G ( B % B l ) =A1.
B,
L e t A2 = A % A 1 and
B2
=
8%Bl.
We have, A1 L G ( B )
5A
t h u s A1 C A and B1 5 B. T h e r e f o r e A = A1 u A2 and B = rn c l e a r t h a t A1 n A2 = 0 = 8 n 8 1 2'
and B1 s F ( A )
B 1 LJ
B2.
5
It i s also
PROBLEMS
1.
Prove:
DFDF(F)A CLO(R)ADF=DR A
WaWb(aRb+F'aRF'b) R 3 c 3 d ( F ' c = c A F ' d = d A c = A E x : F ' x = x} A d = V { x : F ' x = x } ) .
R
8;
+
61
A X I O M A T I C S E T THEORY
2.
Prove:
CLO(R)AW6Wg(6,gES-+6ED6D6 A D 6 = D g = D R A 6 o g = g o d ) - + n
*3.
A p p l y 2.3.2.9
2.3.3
t o r e a l numbers, r e p l a c i n g
-
by s u b s t r a c t i o n .
ADDITIVE AND MULTIPLICATIVE OPERATIONS,
An o p e r a t i o n F i s compLdAQeii a d d t t i v e , i f f o r a l l t e r m s i and f o r m u l a s t h e g e n e r a l i z e d u n i o n i s c o m p l e t e l y a d d i t i v e . Also, b y 2.2.1.12 ( i i ) R* i s c o m p l e t e l y a d d i t i v e and, by 2.2.1.12 ( i v ) , i f f o r a c l a s s A we d e f i n e t h e o p e r a t i o n A b y A ( X ) = X*A , so i s t h i s A . An a p p a r e n t l y weaker n o t i o n i s t h a t o f c l a s s a d d i t i v e . F i s C&b add i t i v e , i f f o r a l l c l a s s e s A, F U A = u C F ( y ) : Y E A } . I t terms o u t t h a t t h e s e two n o t i o n s a r e e q u i v a l e n t :
( i i i ) W A F(A) = u I F ( I y I ) :
YEA}
.
(iv) IRWAF(A) =R*A. I ( i ) C l e a r l y i m p l i e s ( i i ) , and ( i i ) i m p l i e s ( i i i ) , because CCyl : y E A l
PROOF
A =
U
.
The i m p l i c a t i o n o f ( i i i ) t o ( i v ) i s proved, as f o l l o w s : F(A) = u {F(Iy})
: Y E A ) f o r a l l A.
R
=
Suppose
D e f i n e R, by
[F({x})
: xEV].
By 2.2.2.1 and Def. 2.2.2.2, R*CxI = F ( I x 1 ) f o r a l l x E V . Therefore, by 2.2.1.12 ( i v ) , R*A = u {R*{xI : X E A I = u { F ( C x I ) : x E A 1 = F ( A ) .
ROLAND0 CHUAQUI
62
F(UCr
The i m p l i c a t i o n from ( i v ) t o ( i ) i s o b t a i n e d from 2.2.1.12 : $ I ) = R*(U{P : $1) = UER* 7 : $j = UCF(r) : $1.
( i v ) , i.e.
A n o t i o n weaker than complete a d d i t i v i t y i s s e t a d d i t i v i t y . An ope r a t i o n F i s neA a d d i t i v e i f f o r a l l s e t s x we have F(u x) = uCF(y) : EX}. T h i s i s d e f i n i t e l y weaker as i s shown b y t h e o p e r a t i o n F d e f i n e d by,
[uA,
if
A E V ,
This F i s set a d d i t i v e but n o t completely a d d i t i v e . i t i s p o s s i b l e t o prove. 2.3.3.1, 2.3.3.2 THEOREM SCHEMA, i n c o n d i t i a a a.te e q u i v d e n i .
LeA F be a n opemuXon.
(i)
w
x F ( u x ) = u I F ( y ) : y ~ x 1,
(ii)
w
xF(x)
(iii)
=
Similarly
as
Then ,the d0Uvw-
u W((g1):y~xl,
3 R W x F ( x ) = R*x
.
A s l i g h t g e n e r a l i z a t i o n of these theorems i s t h e f o l l o w i n g : 2.3.3.3
THEOREM SCHEMA,
ing c o n d i t i o n b me equivaLent.
L e A F be an opmaaXon.
( i ) F a t e v m y t m r and ~vhmvnuRa9
PROOF,
Take
G ( A ) = F(A)
%
Then t h e dvUow-
,
F ( 0 ) and a p p l y t h e p r e v i o u s theorems..
An o p e r a t i o n F i s c a l l e d compLetdy &pLica.LLue r a n d e v e r y f o r m u l a 4 we have
i f f o r every t e r m
AXIOMATIC S E T T H E O R Y
f o r every s e t x , F ( n . x ) = n { F ( y ) : y E x ) . m u l t i p l i c a t i v e operations. 2.3.3.4 THEOREM SCHEMA, ing condLtiorzcs me e q u i w d e n t
63
We have s i m i l a r theorems for
L e t F be a n a p e h a t i o n .
Then t h e 6 a U a w -
( i ) Fvk e v e h y tm r and 6vzmvnLLea @,
( i i ) WA(A
f
0
+
F(nA)
=
n IF(y) : YEA})
,
( i i i ) 3R3CWAF(A) = C % R * $ A . &ned
.tax
The t h e e c v n d i t i a a &emmain equivalent id t h e ned-OukZion AA eRiminated dhvm (i)and (L], and C = V p u t in &a c o n d i t i o a m e equivaLent kepeacing A by x.
t a nonempty
(AX). Tha
The proof i s l e f t t o the reader. The theorems we have s t a t e d show t h a t completely a d d i t i v e operations can be represented by c l a s s e s ; i.e. by r e l a t i o n s . An a r b i t r a r y operation, on t h e other hand, can be represented by a r e l a t i o n , only r e s t r i c t e d t o
sets.
An operation F i s AiniteLy a d d i t i v e , i f F ( A u 8 ) = F(A) U F ( B ) , f o r a l l A , B ; ~ i n i t d yn e t a d d i t i v e i f t h i s condition i s t r u e with x, y rei f F(An8) = F(A) n F ( B ) placing A , 8. F i s &kLteLy ( n e t ) muLt.Lp&cat.ive, ( F ( x n y ) = F(x) n F ( y ) ) f o r a l l A , B ( f o r a l l x, y ) . I t i s c l e a r by 2.1.3.5 ( i ) ( o r ( i i ) ) t h a t completely a d d i t i v e ( o r m u l t i p l i c a t i v e ) operations a r e f i n i t e l y a d d i t i v e ( o r m u l t i p l i c a t i v e ) . S i m i l a r l y , by 2.1.1.15, s e t a d d i t i v i t y ( o r m u l t i p l i c a t i v i t y ) implies f i n i t e s e t a d d i t i v i t y ( o r multiplicativity).
PROOF, The implication from ( i i ) t o ( i ) i s obtained from 2.3.1.8 ( v ) and 2.2.1.12 ( i i ) .
In order t o prove t h e converse implication, assume t h a t F i s complete-
By 2 . 3 . 3 . 1 , t h e r e i s an R n ( V x V ), and suppose t h a t y G x and
l y a d d i t i v e and f i n i t e l y s e t m u l t i p l i c a t i v e .
Take
such t h a t R*A = F ( A ) .
G-l =
R
.
z G x . Then x E ( G - ' * C y 1 ) n (G-'*{z>) = F ( C y 1 ) n F ( C z ) ) = F ( C q I n ( Z 1 ) + 0. But; s i n c e F i s c o m p l e t e l y a d d i t i v e , F ( 0 ) = 0. T h e r e f o r e , C g ) n C z 1 f 0 , and hence y = z. Thus, we have p r o v e d t h a t G i s a f u n c t i o n .
A v a r i a n t o f 2 . 2 . 3 . 5 w h i c h i s easy t o p r o o f i s : L e R F be a n opehation.
THEOREM SCHEMA,
2.3.3.6
ing atre equivalent.
(i)W A ( A f 0 + F ( u A ) = u{F(q) : q f A } ) g v z (F(qnz) = F(q) n F(z))
T h u e btatemevdh atre & o
A
,
3 G ~ c ( ~ ~ v A( GA )F ( A ) = ( G - ' * A )
(ii)
Then Lhe 6oMow-
equivaLevLt w d h
u
c)
x hephcing
A thhoughoLLt.
The n e x t theorem g i v e s a c h a r a c t e r i z a t i o n o f o p e r a t i o n s t h a t a r e f i n i t e l y a d d i t i v e and m u l t i p l i c a t i v e . 2.3.3.7
L e t F be a n openation.
THEOREM SCHEMA,
Then
W A V B (F(AuB) = F ( A ) uF(B) AF(AnB) = F ( A ) n F ( B ) ) A
F(0) = 0-
T k i n hXaXement
WAWBF(A%B) = F ( A ) %F(8). heYnain6
D ~ u ewhen x,g atre buhbaXuZed t h t a u g k t doh A,B.
The p r o o f i s l e f t t o t h e r e a d e r . O p e r a t i o n s w h i c h a r e c o m p l e t e l y a d d i t i v e and f i n i t e l y s e t m u l t i p l i c a t i v e , are a l s o completely m u l t i p l i c a t i v e . S i m i l a r l y , complete m u l t i p l i c a t i v i t y and f i n i t e a d d i t i v i t y i m p l y c o m p l e t e a d d i t i v i t y . 2.3.3.8
L e t F be a n opehation.
THEOREM SCHEMA,
ing c o n d i t i o n 6 atre e q u i v d e n t : (i) (ii)
W A ( A f 0 -+ F ( u A ) = u { F ( y ) : g E A 1 A F ( n A ) = n { F ( y ) : Y E A ) ) W A(A # 0
+
F(uA)=
u
F(Y) "F(z)
(iii)
Then t h e doUow-
W A(A # 0
+
{ F ( y ) : y € A } ) A W y Wz F ( g n z ) = 3
F ( n A ) = n { F ( g ) : Y E A ) ) A W A WB
F(AUB) = F(A)
U
F(B)
.
T h u e thhee c a n d i t i o n h hemain equivaLerCt when x and z m e h u b h U u Z e d
doh A and 8.
65
AXIOMATIC SET THEORY
I t i s c l e a r t h a t ( i ) i m p l i e s ( i i ) and ( i i i ) . and 2.3.1.10.
PROOF,
( i )by 2.3.3.6
( i i ) implies
I n o r d e r t o prove t h e r e m a i n i n g i m p l i c a t i o n , assume ( i i i ) and suppose F ( A % 8 ) = F ( A ) % F ( B ) f o r a l l A , 8. A l s o , f i r s t t h a t F ( 0 ) = 0. By 2.3.3.7, f i n i t e s e t a d d i t i v i t y i m p l i e s , by 2.3.2.3, monotony, and t h i s , a g a i n by 2.3.2.3, implies W F ( y ) : Y E A } C F ( U A ) . Now ,
F(uA)
u W y )
-.
YEA} = n {F(uA) %F(y) : YEA} Y =
n {F(UA Y
=
Fin { U A % y : y E A } }
.
But, i f u E U A , t h e n u E y E A f o r a c e r t a i n y . Thus, n { u A % y : Y E A } = 0 . Y Therefore, s i n c e we assumed F ( 0 ) = 0
F ( u A ) ~u Thus,
,
y) : YEA}
Y
Y
Hence, u
4
u A%y.
,
{ F ( y ) : Y E A } = F ( 0 ) = 0.
F ( u A ) L u {F(y): Y E A ) , and complete a d d i t i v i t y i s proved.
Now we prove (iii)i m p l i e s ( i ) w i t h o u t t h e assumption F(0) =. 0. L e t t h e o p e r a t i o n G be d e f i n e d f o r a l l A , by G ( A ) = F ( A ) % F ( O ) . We have, G ( 0 ) = 0. Also, i f A Z 0,
G ( n A ) = F ( n A ) %F(O) = n{F(y) : Y E A } %F(O) = n
{F(y)%F(O): Y E A }
,
= n {C(y) : YEA]
S i m i l a r l y we can show, G ( A u 8 ) = G ( A ) U G(8). Applying, t h e case proved above, we have f o r A f 0, G ( u A ) = U CG(y) : Y E A } ; i.e. F ( u A ) F ( 0 ) = u { F ( y ) % F ( O ) : Y E A } . Since f i n i t e a d d i t i v i t y i m p l i e s , monotony, we have f o r a l l 8, F ( 0 ) 5 F ( 8 ) . Hence,
F(uA) = (F(uA)%F(O)) u F(0) = = u
U
{F(y) % F ( O ) : Y E A }
U
.
F(O)
{ ( F ( y ) s F ( 0 ) ) u F ( 0 ) : Y E A ] = u EF(y) : Y E A } .
The requirement t h a t F s a t i s f y t h e c o n d i t i o n s f o r a l l c l a s s e s or a l l s e t s i s e s s e n t i a l i n these theorems. For i n s t a n c e , t h e r e a r e o p e r a t i o n s which a r e c o m p l e t e l y m u l t i p l i c a t i v e and f i n i t e l y a d d i t i v e ( t h e t o p o l o g i c a l c l o s u r e ) i n some c o l l e c t i o n , b u t a r e n o t c o m p l e t e l y a d d i t i v e i n t h e same collection. W i t h t h e axiom o f c h o i c e i t i s p o s s i b l e t o f i n d a f i n i t e l y a d d i t i v e and m u l t i p l i c a t i v e o p e r a t i o n t h a t i s n o t c o m p l e t e l y a d d i t i v e . Without A x C , t h i s problem seems t o be open.
66
ROLAND0 CHUAQUI
PROBLEMS
1.
P r o v e 2.3.3.4
2.
P r o v e 2.3.3.7
CHAPTER 2.4 N a t u r a l Numbers
I n t h i s c h a p t e r t h e c l a s s o f ncLtuhd numbm w i l l be s t u d i e d . T h e s t u d y o P t h i s c l a s s c o u l d be postponed and many o f t h e theorems here would become p a r t i c u l a r cases o f more general theorems which w i l l be proved l a t e r . However, s i n c e many o f t h e techniques used i n t h i s p a r t i c u l a r case a r e simpler, t h e u n d e r s t a n d i n g o f t h i s techniques w i l l be made e a s i e r by u s i n g t h e m f i r s t i n t h e n a t u r a l numbers.
2.4.1
FUNDAMENTAL
2.4.1.1
DEF I N IT1 ON a
PROPERTIES,
( i ) w = n {X : O E X A
(ii) S x
= x u (x}
.
x ( x ~ X-+ x u
{XI
E
X)}
.
w i s the class o f n a t W m m b m . The empty s e t 0 i s , as a n a t u r a l number, zmu. For each s e t x ( i n p a r t i c u l a r , i f x i s a n a t u r a l number),Sx i s Ahe buccaak od x. We have t h a t 1 = 0 U { O } i s t h e succesor o f 0, 2 = 1 U ( 1 3 , t h e succesor o f 1, e t c . S i n c e by 2.1.1.13, S x E V, f o r every x E V , t h e f u n c t i o n S = ( S x : x E Y ) has v a l u e S'x = S x f o r a l l x. The same symbol S w i l l be used f o r t h i s f u n c t i o n and f o r t h e o p e r a t i o n .
2.4.1.2
THEOREM,
(i)O E A A W x ( x € A n w - + S x ~ A ) - + -w c A . (ii) 0
E 0 ,
(iii) X E w + S x E w. (iv) W x ( x E w - x = O V
Wy(y~wAx=Sy).
( i ) i s t h e usual inducLLofl phincipte f o r n a t u r a l numbers. ( i i ) says t h a t w c o n t a i n s z e r o and i s c l o s e d under successor,and t h a t any n a t u r a l number i s e i t h e r z e r o o r successor o f a n o t h e r number. PROOF, ( i ) i s immediate from Def. 2.4.1.1. ( i i ) ) and x t h i s d e f i n i t i o n t h a t 0 E w (by 2.1.1.7 67
I t i s a l s o c l e a r from x u {x} E (by
E w
--f
ROLAND0 C H U A Q U I
68
2.1.1.13 left.
( i i i ) ; t h u s we have ( i i ) ,
( i i i ) , and hence, ( i v ) from r i g h t t o
I n o r d e r t o prove ( i v ) from l e f t t o r i g h t , assume t h a t x f 0 and x f S q f o r e v e r y q E w. Consider A = w 2. Ex}. We have, 0 E A, because Suppose t h a t Y E A . Then S q E w. Since x f Sq, Sy € A . 0 E w and 0 # x Therefore, by ( i ) , w 5 A and, hence x $2 w. = From 2.4.1.2
( i i ) , the following i s evident.
2.4.1.3
THEOREM,
2.4.1.4
THEOREM,
(i) (ii) (iii)
0,1,2,3,
... E
w.
V x ( x ~ w + O + S x ) . W x ( x ~w
+ U
x c- x ) .
uw = w.
( i ) says t h a t 0 i s n o t a successor; ( i i ) , t h a t e v e r y n a t u r a l number i s t r a n s i t i v e (see 2.1.3.3 f o r characterizations o f t r a n s i t i v e classes); ( i i i ) i m p l i e s t h a t w i s a l s o t r a n s i t i v e , i.e. t h a t EL 1 w i s t r a n s i t i v e .
P R O O F , (i)i s obvious. PROOF OF ( i i ) . L e t A = { x : x E w A U x C XI. We have, 0 E A. Suppose, now t h a t x E A ; i.e. X E W and u x c x. Since t h e u n i o n i s a d d i t i v e , Thus, by i n d u c t i o n U(X u { x } ) = u x u x = x x u {XI. Therefore,Sx€A. 2.4.1.2 ( i ) , w ZA.
c
PROOF OF ( i i i ) . F i r s t we show t h a t w i s t r a n s i t i v e , i . e . U w C _ w . Let 8 = w n P W . I t i s c l e a r t h a t 0 E 8. NOW, if x E B we have t h a t X E A~ x c W. Thus, x u 1 x 1 E w A X u t x } c E B. BY i n - w, i.e. x u {XI d u c t i o n 2 7 3 . 1 . 2 ( i ) , w-c B = w n P w C- P w , i.e., b y 2 . 1 . 3 . 3 , u w c w . On t h e o t h e r hand, x E w i m p l i e s t h a t x E x U { x } Therefore, w C u w and, hence, w = U w. 2.4.1.5 ur v r ments o f w. K, A ,
2.4.1.6
TI
w;
thus x E
U
w
.
STIPULATION OF VARIABLES, The lower-case Greek l e t t e r s w i t h o r w i t h o u t s u b s c r i p t s o r primes w i l l be used f o r e l e THEOREMA,
( i )~ $ K2 (ii) 0 f x
.
5K
+
3 y ( y E x A y n x = 0)
( i i) says t h a t EL I'K
i s we1 1 -founded.
.
A X I O M A T I C S E T THEORY
69
I t i s c l e a r t h a t 0 E A. Suppose PROOF OF (i).L e t A = { K : K 4 K } . Then K u { K } E K o r K u { K } E K, and by t h e t r a n t h a t K U IK}E K LJ { K } . s i t i v i t y o f K, K E K . A l s o , i f K ~ { K }= K, K E K . Thus, i f K U { K } E K U { K ) we o b t a i n i n any case t h a t K E K , i . e . SK '$ A i m p l i e s K '$ A. By i n d u c t i o n 2.4.1.2 (i), w = A .
3 q ( q E x A x n q = 0)). PROOF OF ( i i ) . L e t B = { K : W x ( 0 # x c K I t i s obvious t h a t 0 E B. Suppose t h a t K E 8 and t h a t 0 f x C_SK = K U { K } , We s h a l l show t h a t 3 q ( y E x A x n q = 0 ) . I f K $! x, t h i s r e s u l t i s c l e a r , because t h e n x C K E 8. I f x = { K } , we t a k e q = K and a p p l y ( i ) . The l a s t case remaining Ts t h a t K E x and K n x # 0. Since K E B y t h e r e i s a q e l c n x such t h a t q n K 0 x = 0. But, x = ( x n K ) u { K } and, s i n c e y E K, by ( i ) and t h e t r a n s i t i v i t y o f K, K 4 Y. T h e r e f o r e x n q = 0. Thus, we have shown t h a t SK E B and, by i n d u c t i o n 2.4.1.2 (i), w c - B. +
.
( i ) 0 it A c -w
+
3 q(qEA A q n A
= 0).
(ii) W F (ELlw). ( i i i ) W K ( K -C A + K E A )
+ w & A ,
( i i i ) i s the s o - c a l l e d "second i n d u c t i o n p r i n c i p l e " . PROOF OF ( i ) . L e t 0 # A C w; t h e n K E A f o r some K. If K n A = 0 , t h e n take Y = K , and ( i ) i s proved, So suppose t h a t K n A # 0. By 2.4.1.6 ( i i ) , t h e r e i s A E K - n A such t h a t X n K n A = 0; b u t X ~ K , by t h e t r a n s i t i v i t y o f K. T h e r e f o r e AnA = 0.
(ii)i s j u s t a r e f o r m u l a t i o n o f (i). PROOF OF ( i i i ) . Assume t h a t W K ( K C A K E A ) and w P A ; then By ( i ) , t h e r e i s a K E U 2. A witF K n (w 2. A ) = 0; 6 u t , by t h e t r a n s i t i v i t y o f w, K & U and, hence, K c A. T h e r e f o r e K E A, c o n t r a d i c t i n g K E W2 . A .
w % A # 0.
.
2.4.1.8
+
THEOREM.
KEX V X ~ K .
PROOF, Let A = C K : W X ( K E X V X ~ K ) } and, i n o r d e r t o use 2.4.1.7 If K = 0, t h e n i t i s c l e a r t h a t K E A . We s h a l l ( i i i ) , assume t h a t K c A. now show K E A under fhe h y p o t h e s i s K f 0. T h i s w i l l be done by showing t h a t X E K under t h e assumption A ~ K . By t h i s l a s t assumption, we haveX'%= 0; by 2.4.1.6 ( i i ) , these i s a ~ ? X % K such t h a t p n ( X % K ) = 0 (by t h e t r a n s i t i v i t y o f w, P E W ) . From t h e t r a n s i t i v i t y o f A , we deduce p C X and, hence, U C K . By assuming K ~ we U w i l l reach a c o n t r a d i c t i o n . U n d k the h y p o t h e s i s K P _ ~, we o b t a i n ~ 2 . uf 0; a g a i n by 2.4.1.6 ( i i ) , there i s v E K 1-1 w i t h v n (ic2.p) = 0; b u t as above, V C _ K and, hence, vcp. Since V E K and K c - A, we have v E A; t h e r e f o r e v E p V p c v ; b u t v q p ( s i n c e V E K % ~ ) Thus, p s u and, hence, v = p ; b u t t h i s c o n t r a d T c t s V E K A U ~ K . So we have
.
ROLAND0 CHUAQUI
70
reached a c o n t r a d i c t i o n and, t h e r e f o r e , t h e h y p o t h e s i s K ~ is U f a l s e . Hence But, we a l s o had P E A , i.e. K E A Since we had ~ C _ K , we g e t V = K . Thus, we have shown K E X V X C-K , i.e. K E A . By t h e i n d u c t i o n p r i n c i p l e 2.4.1.7 (iii), w C - A.
.
K ~ U .
2.4.1.9
THEOREM,
(i) X E K - X C K A X E W . (ii)
K
= (PK 2,
( i i i ) IN]^
=
{K))
nw.
(EL u Z D ) ( w
PROOF, Since ( i i ) i s j u s t a r e f o r m u l a t i o n o f ( i ) and ( i i i ) an immeI f x E K then, by t r a n s i t i d i a t e consequence, we o n l y need t o prove (i). vity, x C - K ; by 2.4.1.6 (i), x + K, and, by 2.4.1.5 ( i i i ) , x E w .
Xc
On t h e o t h e r hand, if
K..
2.4.1.10.
A
x
then
E w,
K
Q - x , and, by 2.4.1.8,
THEOREM i
(i) K E X V
i.e.
xcw
K
=
A
V AEK
.
PROOF OF (i).Suppose t h a t K
K=A.
~ AX ) \ ? K ; by 2.4.1.8,
KC_A A
)\&K
,
PROOF OF ( i i ) . By ( i ) and 2.4.1.9. 2.4.1.11
THEOREM a
PROOF, ( i ) i s a consequence o f (ii).By 2.4.1.10 (ii), w/lNIw i s a simple o r d e r i n g . Since w / E L I w i s well-founded and w / I N I w = w / ( E L U Z D ) l w , by 2.2.3.18 ( i i i ) , WF(w/ZNIw).
w / l N I w i s t h e n a t u r a l o r d e r i n g o f w. Thus, KCX means K is l e s s than o r equal t o K. K i s l e s s than A can be expressed by K E A o r by K C A . I f 0f A c - w, t h e n t h e l e a s t element of A i s n A . 2.4.1.12
THEOREM,
sK
=
SA
+
K
=
A
71
AXIOMATIC SET THEORY
K
A .
PROOF, Assume t h a t K U { K } = A U {A}; t h e n K E A u { A ) , i.e. K E A V From 2.4.1.9, KCX. S i m i l a r l y , we can show A C-K , and hence, K =
= A .
2.4.1.2 (i), (ii) and ( i i i ) , 2.4.1.4 (i)and 2.4.1.12 a r e c a l l e d Peano's axioms f o r n a t u r a l numbers. Namely,
I. 11.
111. IV. V.
c o n s t i t u t e what
O E W .
X E
W+SXE w.
v x 0 +sx sx = s y x = Y . 0 E A A v x(xEA -+
-+
SxCA)
+
w
5A.
These axioms a r e t h e standard system f o r t h e n a t u r a l numbers. They were f i r s t f o r m u l a t e d by Dedeking i n "Was s i n d und was s a l l e n d i e Z'a'hlen?" Dedekind, however, d i d n o t have a c l e a r c o n c e p t i o n o f an a x i o m a t i c system and i t was Peano who f o r m u l a t e them e x p l i c i t l y as axioms. Since t h e y a r e t r u e i n G , number t h e o r y can be developed i n G. Dedekind a l s o j u s t i f i e d i n h i s essay, r e c u r s i v e d e f i n i t i o n s . Namely, d e f i n i t i o n s o f t h e f o l l o w i n g type. F o r any f u n c t i o n ffand s e t c, we want t o d e f i n e a f u n c t i o n F s a t i s f y i n g t h e f o r m u l a $ g i v e n by
I$ [ F ] - D F = w A F'O
= c A
F ' S K = H'F'K
.
i.e. a f u n c t i o n F w i t h domain w and such t h a t F'O = c and F'SK = H ' F ' K Such d e f i n i t i o n s a r e n o t d e f i n i t i o n s i n t h e usual sense s i n c e i n t h e l a s t c l a u s e F occurs i n b o t h sides, 1.e. F i s d e f i n e d i n terms o f F.
.
However, i t would be p o s s i b l e t o j u s t i f y such d e f i n i t i o n s by p r o v i n g I f t h i s i s proved, we c o u l d d e f i n e F by F = U {G : I$ [ G I 1 Now i n t h e r i g h t side, F does n o t occur. R. Dedekind was t h e f i h s t t o prove 3 ! F I$ [ F]
3 ! F $ [ F].
.
.
The n e x t theorem 2.4.1.13 j u s t i f i e s , i n t h i s sense, r e c u r s i v e d e f i n i t i o n s o f o p e r a t i o n s . The p r o o f t h a t i s g i v e n here i s e s s e n t i a l l y Dedekind's L a t e r a s t r o n g e r theorem w i l l be proved by a n o t h e r method. S i n c e we a r e d e a l i n g w i t h o p e r a t i o n s and n o t f u n c t i o n s i t i s n o t poss i b l e t o p r o v e a s i m p l e statement o f t h e form 3: F @(F). The statement has t o be m o d i f i e d somewhat. L e t $ (F)
+ +
F(O) =
C A W
K
F (SK) = H ( F ( K ) )
A W X ( X ~ W -,F ( x ) =
V)
.
We s h a l l say i n t h e theorem t h a t t h e r e i s a unique o p e r a t i o n F such t h a t I/J [ Q . T h i s means t h a t i t i s p o s s i b l e t o d e f i n e such an o p e r a t i o n , and t h a t any f o r G such t h a t QF [ G I we have F = G (i.e. F c o i n c i d e s w i t h G for a l l classes.
72
ROLAND0 C H U A Q U I
The c l a u s e ' F ( X ) = V , f o r a l l X 9 w ' i s necessary s i n c e o p e r a t i o n s must be d e f i n e d f o r a l l classes. Any o t h e r v a l u e c o u l d have been chosen instead o f V .
2.4.1.13 CURSION),
THEOREM SCHEMA,
(PRINCIPLE
L e L H be a unmy a p W a n .
OF DEFINITIONS
BY RE-
Then
( i ) 3 ! S (S C V x V A D S = w A S*{O} = C A V
K
S*{S
K)
= H(S*{K)).
( i i) Thehe LA a unique opehaZion F nuch t h a t F ( 0 ) = C , F (SK ) = H (FK) d o t & K, and F ( X ) = V doti UU X $ a . PROOF, By 2.2.2.1, ( i ) implies ( i i ) ( i t i s a c t u a l l y equivalent t o I f we have a r e l a t i o n S s a t i s f y i n g ( i ) we d e f i n e F ( K )= S*{K) f o r K E W and F ( X ) = I/, o t h e r w i s e . T h i s F s a t i s f i e s ( i i ) . On t h e o t h e r hand ifF and G s a t i s f y ( i i ) d e f i n e S = [ F ( K ): K E w ] , T = [ G ( K ) : K E w ] ; t h e n S and T s a t i s f y ( i ) and hence S = T . T h e r e f o r e F ( K ) = S*{K) = T*{K) = G(K) f o r K E W , and F(X) = V = G(X) otherwise. Thus F i s unique. (ii)).
So we j u s t prove ( i ) :
F i r s t , we prove t h a t f o r e v e r y t h e formula 4 [ K I g i v e n by:
K
t h e r e i s a unique R t h a t s a t i s f i e s
L e t A = { K : 3 !R I$[ K ] } ; we s h a l l prove by i n d u c t i o n 2.4.1.2 t h a t w c A . We have t h a t C x {O} i s t h e unique R which s a t i s f i e s hence 0-€ A .
(i),
I$ [ 01 ;
Suppose, now, t h a t K E A and l e t T be t h e unique R which s a t i s f i e s D e f i n e T ' = T U ( H ( T * { K ) ) x {SK}). We s h a l l show t h a t T I i s t h e unique R which s a t i s f i e s 4 [ S K I .
@[K].
We have, T ' * { O } = T*{O) = C. I f X E K , t h e n T ' * { S A 1 = T*{S A] = H(T*{A)) = H ( T'*{X)). F i n a l l y , TI*{SK} = H ( T * { K ~ ) . Thus, T I i s an R which s a t i s f y 4 [ S K I . I n o r d e r t o show t h a t T I i s t h e o n l y such R , assume t h a t T" i s another We s h a l l prove by i n d u c t i o n t h a t f o r e v e r y A E S S K T ' * { h } = Tl'*{A}. Since D T ' , D T " C S S K , t h i s shows t h a t T ' = TI'. We have, T ' * { o ) = c = T ~ ~ * { o a) l; s o , T'*{s T) = H ( T ' * { X } ) = H ( T ~ I * { X I ) = T " * { s A I , f o r a l l X E SSX.
R s a t i s f y i n g I$[ S K I
.
Now, d e f i n e R ( K ) = U {R : 0 [ K ] 1. I t i s easy t o see t h a t i f XCK , t h e n R ( K ) I S X s a t i s f i e s 4 [ X ] ; t h u s R ( K )IS h = R ( X ) , t h e r e f o r e R(K)*{u)= R(X)*{uI if 1.151. Let S = u {R(K):
KEwI.
We have,
AXIOMATIC SET THEORY
73
PROOF, L e t S be t h e r e l a t i o n o b t a i n e d u s i n g 2.4.1.13, such t h a t S * { O } = c and S*{SK} = H ( S* { K } ) , where H i s t h e o p e r a t i o n d e f i n e d by Since V V ( H ) , we can prove by i n d u c t i o n 2.4.1.2 (i), that H ( X ) = H'X. S*{K} E V f o r a l l K. Hence F can be d e f i n e d by F = ( S * { K } : K E w ) .
U n i c i t y i s e a s i l y shown by i n d u c t i o n . When w r i t i n g r e c u r s i v e d e f i n i t i o n s , t h e y w i l l be w r i t t e n i n t h e usual f a s h i o n , i n d i c a t i n g t h a t t h e y a r e o f t h i s type. Thus, if we want t o d e f i n e a f u n c t i o n F , we s h a l l w r i t e
w V ( F ) A F'O = c A F 'SK = instead o f
HI
F'K
,
-
F = U { G . w V ( G ) A G'O = c A G'SK = f f ' G ' ~ } 2.4.2
.
A R I T H M E T I C OF N A T U R A L NUMBERS,
Recursive d e f i n i t i o n s w i l l be used f o r d e f i n ng a r i t h m e t c operations. F i r s t , addition. 2.4.2.1
D E F I N I T I O N BY R E C U R S I O N ,
( i ) W V ( +) A P t 0 = 11 A We w r i t e
p
+ u
for
( i i ) +" = ( p + v : u E u )
.
+I
)1
P
Su
+'v
.
= S
Let P ,
(
p
IJEU.
Then,
ROLAND0 C H U A Q U I
74
u,
I n t h i s d e f i n i t i o n , 2.4.1.14
p+ i s a f u n c t i o n . p e w , was defined.
was used w i t h c = p and H = S .
For each
Thus, an i n f i n i t e f a m i l y o f f u n c t i o n s , one f o r
each
Although, as i t w i l l be seen l a t e r , a d d i t i o n o f n a t u r a l numbers i s commutative, and, thus, +v = v+, I have i n t r o d u c e d a d d i t i o n on t h e r i g h t ,
+
i n o r d e r t o d e f i n e m u l t i p l i c a t i o n i n t h e same way as i t i s done v ' o r d i n a l s , where a d d i t i o n i s n o t commutative. 2.4.2.2
for
THEOREM,
(i) p + O = p . (ii)
+ 1-1
0
u +
(iii)
1-1
(v)
"w
.
(,+I
(p + v) +
(vi)
(viii)
1 = Sp.
+ S V = S(LI+ v).
(iv)
(vii)
= p.
p
+v
= v
u +
TI
=
+
p.
(V
+
TI).
p c u + v A ( v > O + u c p + v ) .
PROOF, ( i ) , ( i i i ) and ( i v ) a r e o b t a i n e d from Def. 2.4.2.1 and 2.4.1.14. ( i i ) , ( v ) , and ( v i ) a r e e a s i l y shown by i n d u c t i o n 2.4.1.2 (i). PROOF OF ( v i i ) . F i r s t , show by i n d u c t i o n on p, t h a t p + 1 = l + p We have, 0 + 1 = 1 = 1 + 0. On t h e o t h e r hand, i f p + 1 = 1 + p, t h e n Sp + 1 = (p + 1) -t 1 = (1 + p) + 1 = 1 + (p f 1 ) = 1 f Sp
.
S i m i l a r l y , by i n d u c t i o n on v, ( v i i ) i s shown. PROOF OF ( v i i i ) . 2.4.2.3 (i) v
(ii) V
THEOREM ,
& v'
-
C V ~-
(iii) v = v' PROOF, (1) v c
-
By i n d u c t i o n .
p
+ v
5p
+ v'
= p
+
+ v c p +v'.
p
1.1
+ v
v'
F i r s t , we show, VI
+
p
.
+
v c p +
VI
,
.
I
.
A X I O M A T I C SET T H E O R Y
75
by i n d u c t on on v: I f v = 0, now t h a t then v' 2 pothesis,
(2) v 5 v '
(1) i s o b t a i n e d f r o m 2.4.2.2 ( i ) and ( v i i i ) . Suppose i ) i s t r u e f o r v and a l l v ' , and assume v + 1 C v ' ; 0 and, so v ' = IT + 1 w i t h v C IT. By t h e i n d u c t i o n hyp + v c LJ + IT; t h e r e f o r e , p + (v + 1) C p + v ' .
5 p +
+v
! . l
--t
v'
,
i s o b t a i n e d from ( i ) and t h e l o g i c a l t r u t h v = v ' p + v = pfv'. I f v g v ' , then v ' C v and, hence by ( l ) ,v + v ' C p + v. T h e r e f o r e p t v ' ~ L + J v. T h i s shows, +
(3) p + v
5 LI
+ v'
-+
v
5v ' .
S i m i l a r l y , we can show,
.
(4) p + v c p + v' + v c v ' .
From ( l ) , ( 2 ) , ( 3 ) and ( 4 ) , ( i ) and (ii) a r e obtained. deduced from ( i ) .
2.4.2.4
THEOREM,
PROOF,
p C_ v
+
IT LJ
I!
+
IT =
v
(iii) is
.
U n i c i t y i s immediately o b t a i n e d from 2.4.2.3
(iii).
The p r o o f o f t h e e x i s t e n c e o f IT i s by i n d u c t i o n on p , assuming LJ C v . IfLI = 0, we t a k e v = IT. Suppose, now, t h a t f o r e v e r y v 2. p , t h e r e i s a IT w i t h p + 7~ = v and assume p + 1 c v. Then 1 ~ .c v and, hence, t h e r e i s a IT' 3 0 such t h a t p + IT' = v. T h i s , IT' = TI + 1. Therefore, v = p + ( r + 1 ) = = p + ( 1 + IT) = (p + 1) + IT.
.
2.4.2.5
DEFINITION,
From 2.4.2.4,
= U { T : V + I T = p].
we e a s i l y d e r i v e :
-
v = 0 ) A (V c p
2.4.2.6
THEOREM,
2.4.2.7
D E F I N I T I O N BY RECURSION,
(p
C
v
+
p
= R
R O = I D ~ ( D R U D R -A~ )R~~
(i)
R-'
(ii)
Rv.
U - V
=
-+
v
+
(p
- v ) = p)
.
( I T E R A T I O N OF R E L A T I O N S ) ,
~ R ~ .
(R-~)'.
We have d e f i n e d i n ( i ) , an o p e r a t i o n F such t h a t f o r each V E W , F ( v ) = Thus, we have a p p l i e d 2.4.1.13 w i t h C = R 0 and H(Y) = R O Y . The f o l l o w i n g theorem i s an immediate consequence of t h e d e f i n i t i o n .
2.4.2.8 (if
THEOREM,
F 0 F - ~c ID
+
FV 0
(~'1-l c -I D
.
ROLAND0 CHUAQUI
76
PROOF, As an example, ( v ) w i l l be shown by i n d u c t i o n (RO1-l = I D
'
( D R U D R - ~ )= ( R - ' ) ' ; also, ( R 1)-1 R-l = ( R - l ) v i l = R - ( v t l )
=
(R-1y
(R oRV)-l
=
(Rv)-l 0 R-l
I
=
As an example o f i t e r a t i o n , we have,
2.4.2.9
THEOREM,
p+V =
V
s
)-I
.
PROOF, By i n d u c t i o n on v ; p t o = s
S(S"
u)
=
sv+1 p.
0 p = p .
ptSv=S(u+v)=
We pass, now, t o m u l t i p l c a t i o n d e f i n e d by i t e r a t i o n o f a d d i t i o n .
2.4.2.10 (i)
x =
(tV
IJ We w r i t e p v P
Let
DEFINITIONs
.
o
p, Y E W . Then,
: v~w).
for
PX
I,
-
( i i ) xv = ( p -.v : u ~ w ) . As f o r a d d i t i o n , Xy i s i n t r o d u c e d i n o r d e r t o d e f i n e e x p o n e n t i a t i o n as w i l l be done f o r o r d i n a l s .
We have t h e f o l l o w i n g p r o p e r t i e s o f m u l t i p l i c a t i o n . We use t h e usual conventions r e g a r d i n g parentheses, i.e. m u l t i p l i c a t i o n takes precedence over a d d i t i o n .
2.4.2.11 (i) p
(ii) 0
(iii) p (iv) p
THEOREM,
-
-
v = 0. ( v t l ) = 1-1 1 = p.
.
($)
(v) (vi) p
(vii) p
0 = 0.
(v 6
(v
T)
+
( v i i i ) ( v t 1)
TI)
. 1-1
= (p = 1-1 =
v + 1-1
*
v
-
v) v + p
. 71
-
-P+P.
. ,
AXIOMATIC
77
SET T H E O R Y
PROOF, ( i ) , ( i i i ) , and ( i v ) a r e immediate from Def. 2.4.2.10. and ( v ) , and ( v i ) a r e e a s i l y shown by i n d u c t i o n .
(ii),
( v i i ) w i l l be shown by i n d u c t i o n on IT :
-
p
(v+O) = p * v (u t (71+1)) =
p
p
= (p
. .
= l l *
v +
p
.
0 ((v+Tr) + 1) = p (VtTI) + p v + p ' IT) + p = p ' w + ( p . v t p ' (TI+ 1).
= p '
*
-
71 t
p)
( v i i i ) i s e a s i l y proved by i n d u c t i o n on p . w i l l be shown by i n d u c t i o n on v:
(ix)
o =o =
p
p
*
0 .p. (v+1) = p - v + p = w - p + p = (v+1) ' p .
.
( x ) and ( x i ) a r e e a s i l y shown by i n d u c t i o n .
THEOREM ( E U C L I D ' S ' A L G O R I T H M ) .
2.4.2.12
PROOF, we have u * K
p
-
=
K +
v
Let
Ev
.
K
= n {TI : p
1-1 But, 1-1 C
K
+
6
K
u
+
.
X
C
+ 1)
(71
p '
K +
-
Then, s i n c e p (v+l)>v, t h e r e is a unique X such t h a t p ; t h e r e f o r e , by 2.4.2.3, A C p.
3
By 2.4.2.4,
v}.
-
I n o r d e r t o show t h e u n i c i t y o f K, l e t us suppose, a l s o , t h a t p K ' C -(K' +I). We have, p K' C v c p ( K t 1); thus, K ' c K + 1, and, hence K ' 5 K Also, p ( K ' + 1) and, hence, K 5 K ' . There K C v C p fore, K = K '
V
cp
-
. .
. -
The proof of t h e n e x t theorem, s i m i l a r t o t h a t o f 2.4.2.3, t h e reader.
2.4.2.13
i s l e f t to
THEOREM,
(i p + O + ( w ~ w ' - p . v ~ p - w ' ) . (ii
p#O-'(v~v'-p.vcp-v').
(iii p , v, v'
f
0
+
(v = v l - p
-v
= p *
v
1 -
NOW, e x p o n e n t i a t i o n is d e f i n e d as t h e i t e r a t on o f m u l t i p l i c a t i o n .
78
ROLAND0 C H U A Q U I
2.4.2.14
DEFINITION,
We w r i t e 1-I’ 2.4.2.15
for
exp
u
expP =
-v .
(xv 1-I
’
1 :v€w)
THEOREM,
(i) vo = 1 ( i i ) v1-1”
.
.
= v’av
The p r o o f i s l e f t t o t h e reader. 2.4.2.16
THEOREM,
A O C X C V A
TI
2
5v
A 1
L e t 2 C v and 1 C 1-1. PROOFl K ’ 71 hence, { K ’ : p C v 1 f 0. L e t
1.1 C v K ”
= vK
C 1-1
+
CVK).
- .
L e t X and
v
TI
3!
K
Then, 1-1 C v’” K
= n {K’
:
I! X 3!
IT (p = v K . h
(2.4.2.15
cv
K’
}.
+TI
( v i i ) ) and, We have, vK c
be t h e unique numbers t h a t s a t i s f y 2.4.212
w i t h vK r e p l a c i n g v, i.e., (1) 1-1 = v K . X t
I f we had w wK
*
A
5 AK
*
A
TI
5 A,
t TI = !-’
(2) A c v .
and
T I C V ~ .
then, s i n c e p
, that
vK
’
.
we would have, 1-1 c vK v i s a c o n t r a d i c t i o n . Therefore, C
c
.
( 1 ) and ( 2 ) prove t h e e x i s t e n c e o f K, X, TI. The uniqueness o f K i s proved as i n 2.4.2.12. Then, by t h i s same 2.4.2.12, X and IT a r e unique.
A X I O M A T I C S E T THEORY
79
PROBLEMS ( i ) - (iv).
1.
Prove 2.4.2.8
2.
Prove 2.4.2.13. Prove 2.4.2.15.
3.
4.
0 = 0, Let (2)
(V + 1
+ v.
) = ();
Define j with w x o w ( j )
by j ( v ,1-1
) =
.
( a ) Show t h a t " w ~ w ( j - ~ ) ( b ) Show t h a t there a r e k and R with ww(k), " w ( a ) , ""(k-'),
and "w(L-')
and such t h a t
( b l ) j ' ( k ' v ,L'v) (b2) k ' v
for all 5.
A L'v
.
Define for V E O , by i t e r a t i o n of exp
Y
hw =
( a ) Compute
hV' 0, hv' 1, hvt 2, hv' 3 .
( b ) Compute
hv'
( c ) Show, 6.
-v c
vEw
, &v , = v
v
3
(
ex(
'
1 :
( u + ~ ) ,h V ' ( u .IT),hv(u').
1A
u
2
0
+
3 ! II ( h d
With hv of Problem 2 , d e f i n e j ,
'P
=
71
c
p
c hv'(v+l))
PEW).
.
h 'J ' 1 V
Solve ( a ) , ( b ) , and ( c ) of Problem 2 with j v i n s t e a d o f h v .
T Generalized Recursive D e f i n i t i o n s
2.5.1
T H E ANCESTRY R E L A T I O N ,
The a n c e s t r y o f a r e l a t i o n R , denoted by Rw, i s t h e l e a s t t r a n s i t i v e I t was i n t r o d u c e d by Whitehead and and r e f l e x i v e r e l a t i o n i n c l u d i n g Russel i n t h e i r P r i n c i p i a Mathematica and b e f o r e them by Frege. I t i s c a l l e d t h e a n c e s t r y o f R because i f we t a k e R a s t h e r e l a t i o n o f being a par-
.
ent,
RW corresponds t o b e i n g an a n c e s t o r . 2.5.1.1
DEFINITION,
RW = n { X :
R n VxV
C
X A Xo
U
X
2
C -
X}
The f o l l o w i n g p r o p e r t i e s o f t h e a n c e s t r y of R a r e easy t o show.
2.5.1.3
THEOREM,
RW = u {Rv : v
E w)
.
T h i s theorem g i v e s another p o s s i b l e d e f i n i t i o n o f RW, t h i s t i m e from below. 2.5.1.1 d e f i n e s RW as t h e l e a s t upper bound ( a c c o r d i n g t o 2) o f on t h e o t h e r hand, some r e l a t i o n s i.e. d e f i n e s i t from above. 2.5.1.3, g i v e s RW as t h e g r e a t e s t l o w e r bound o f o t h e r r e l a t i o n s , i.e. f rom be1ow. PROOF,
Let
S = u ( R V : v E w}.
80
defines i t
We have t h a t R n V x V = R
1C S
.
81
AXIOMATIC SET THEORY
Also, by 2.4.2.8 ( i i ) , DR" u (DR")-' LDRU D R - l f o r a l l V E W ; and DS u DS-' = u (DR' U D(R")-' : v E w ) . Therefore, So 5 Ro 5 S. NOW, S O S = u { Rv o R' : v ,u E to} = U {R" I.r : v ,p E w } 5 U CR" : W E W } = S. Thus, from +
2.5.1.2
( i i i ) we deduce t h a t RW C -S.
We s h a l l show by i n d u c t i o n on v , t h a t RV
S&RW:
- RW
C
f o r a l l W E N and thus,
Since R C - RW, Ro C- R W o C- RW. Also, assuming RV c- RW, we have R v + ' RoRv C RoRWC - RWoRWC - Rw (2.5.1.2 ( i ) and ( i i ) ) .
=
PROOF, As an example, I s h a l l show RW = Ro U ( R W o R W ) , l e a v i n g t h e two o t h e r e q u a l i t i e s , whose p r o o f i s s i m i l a r , t o t h e reader. We have,
R o U ( R o R W ) = R o u ( R o u { R w : v ~ w }=)R o u U { R v t 1 :
v€w}=U{Rv:vE~)=
=RW. The f o l l o w i n g theorem w i l l be needed i n t h e n e x t s e c t i o n . WF(R) means t h a t R i s w e l l founded.
- DRWU D R w - ' and C DRW-' C DRU DR-'.
PROOF, L e t 0 f X
q
f
z A q RW z ) } .
X'
C
t h e r e i s a z € X ' such t h a t OR(z) n X I = 0.
Recall t h a t
d e f i n e X ' by X I = { z : ] q ( q E X A Since R i s w e l l founded, Since z E X ' , t h e r e i s a q
X
We s h a l l see t h a t 0 ( q ) n X = 0. Suppose n o t ; RW t h a t t h e r e i s a uEX w i t h u f g A u RW q. I n t h i s case, q e X ' . From
such t h a t q RW z A g f z . i.e.
z V 3 t ( g RW t A t R z ) . But q + z , SO t h e r e e x i s t s a t w i t h q RW t and t R z. Since gEX and OR(z) n X ' = 0, t h e n t # q because t E OR(z). Thus, t E X ' , b u t t h i s i s a l s o i m p o s s i b l e be-
2.5.1.4,
s i n c e g RW z we deduce,
cause o f t h e same reason. 0 ( q ) n x = o . RW When
=o..
=
We have a r r i v e d a t a c o n t r a d i c t i o n .
Therefore
X ' = 0 , we t a k e any gEX, and we prove e a s i l y t h a t 0 ( y ) n X = RW
As an a p p l i c a t i o n o f t h e a n c e s t r y r e l a t i o n we s h a l l s t u d y t h e IxunhiLLue cLonwie 06 a d u n . Remember t h a t a c l a s s A i s c a l l e d t r a n s i t i v e i f Thus, t h e t r a n s i t i v e c l o s u r e of A i s t h e l e a s t U A C - A ( c f . 2.1.3.3). t r a n s i t i v e class containing A .
82
ROLAND0 CHUAQUI
2.5.1.6
DEFINITION, T A
= n {X : A c c X) -X A u X -
I n o r d e r t o d e f i n e T A from below, t h e f o l l o w i n g d e f i n i t i o n i s i n t r o duced.
2.5.1.7
v -1* A .
DEFINITION, UvA
= (EL )
The f o l l o w i n g theorem i s proved by i n d u c t i o n .
2.5.1.8
.
THEOREM,
( i ) u0 A = A
(ii)
U " + ~ A=
,
u(u"A).
NOW, we have t h e promised c h a r a c t e r i z a t i o n from below o f T A .
2.5.1.9
THEOREM,
(i) T A = u W"A
:V E ~ I .
(ii) T A = EL~-'*A. PROOF OF (i). Let 8 =
{u"A :
U
~ E W } .
.
We have t h a t A = uoA C -8
A l s o , U B = U U {U'A : ~ E w =} U { U (U'A) * v € u } = T h e r e f o r e A U U €3 c - 8 and, hence, T A C_ B . On t h e o t h e r hand, l e t A u'A
C -X
u(u"A)
.
5u
We have, X c -X
.
Uo
U
{U"'"'A:
vEu}
C
8.
u X C- X.
A = A C -X
.
We s h a l l show by i n d u c t i o n t h a t Suppose UvA 5 X. Then U"' A =
Thus, we have shown t h a t 8 = W " A : v E u }
.
PROOF OF ( i i ) . We have, EL^)-' = (U = u {(EL")-' : v ~ u = } u {(EL-')' : v 6 u ) = = u {(EL")-'*A:
U
vEu} = u {u" A :vEu}
.
5TA. : v ~ u 1 1 - l=
Thus,
(ELu)-l*A=
The f o l l o w i n g theorems g i v e a l i s t o f p r o p e r t i e s o f T t h a t w i l l be useful l a t e r
2.5.1.10
THEOREM
(i)A C -T A .
I
.
83
AXIOMATIC SET THEORY
by Def. 2.5.1.6 PROOF OF (i):
TA.
We have PROOF OF (ii): 2.5.1.11
U
TA= U u{uvA
:VEW
} = u{uvt ' A : V E U } ~
THEOREM
(i)AC - B - + T ACTB. ( i i ) T T A = TA
(iii) u A
-A f
++
.
TA = A
-
3X(TX = A )
,
PROOF, ( i ) i s c l e a r from Def. 2.5.16. Since T A C T A and T A i s - T A . But, by 2.5.1.7 ( i ) , TAC- T T A t r a n s i t i v e (2.5.1.10 ( i i ) ) , T TA c Hence we have (ii).
.
PROOF OF ( i i i ) . Suppose U A C A . We a l s o have A C A . Hence T A A, and, t h u s T A = A . T h i s l a s t statement i m p l i e s by l o g i c 3 X T X = A . -A . F i n a l l y , from 2.5.1.10 ( i i ) , 3 X T X = A implies U A C 2.5.1.12 u CTX : $ 1 ,
THEOREM SCHEMA,
L e A $ be a aomLLea.
.
-
C
Then T ( u EX : $ 1 ) =
T h i s theorem expresses t h e complete a d d i t i v i t y o f T .
.
PROOF, I t i s o b t a i n e d from 2.5.1.9 i s completely additive. 2.5.1.13
= T n { T X :$ 1 .
THEOREM SCHEMA,
( i i ) , s i n c e t h e image o p e r a t i o n
L e A $I be a 6omuRa.
Then n tT X : $1
=
That i s , t h e i n t e r s e c t i o n o f t r a n s i t i v e c l a s s e s i s t r a n s i t i v e .
.
PROOF, I t i s c l e a r from 2.5.1.10 ( i ) t h a t n { X : $ I c _ T n { T X : $1. A l s o , i f $ , t h e n n { T X : + } z T X , i.e. Tn{TX $ )C T T X = T X . Hence T n { T X : $ I c_n { T X : $ I . From t h e complete a d d i t i v i t y o f T , 2.5.1.11 o b t a inc.
( i i i ) , and 2.5.1.13
we
ROLAND0 C H U A Q U I
84
2.5.1.16
T{a}=
THEOREM,
*
a } = U'a. PROOF, Since u { a } = a , we have U"" Hence, by 2.5.1.9, T { a } = u {u" { a } : V E W } = { a } U u {U" {+ 1{ a } : U E W } = U { U u u : aEw3 u { a } = T a U I a I .
'
T a = a --t T ( a
2.5.1.17
COROLLARY,
2.5.1.18
THEOREM, T A = u { T x : x E A }
PROOF,
TA = T
U
= U {Txu
{{XI
: xEA} =
{x} : x E A }
U
U
{a))= a U
COROLLARY,
{a} ,
A = (TUA)
U
A.
{T{x} : x € A ]
= U { T x :xEA} U A .
Also, by complete a d d i t i v i t y o f T , T u A =
2.5.1.20
U
Vx(xEA
+
Tx
=
U
{Tx : X E A }
x)+ ( U A ) u A
.'
= T(u A U A ) ) .
That i s , i f every element o f A i s t r a n s i t i v e , then ( U A ) u A transi tive.
i s also
The following theorem i s l e f t t o the reader. 2.5.1.21
THEOREM a
( i ) T u A = U T A .
(Ti) A f -u A
+
T A = UTA.
PROBLEMS
1.
Let Rm = n { X : R u X 2 c - X I ; i.e. Rm i s the t r a n s i t i v e closure of R Prove the following properties o f R m . (1) R C - Rm. (2)
- Rm
Rm2C
.
(3) R c S A S 2 c-S - R m c-S .
( 4 ) R2 g R
+
R = R".
.
AXIOMATIC SET (5) R
5S
Rm C - SOo.
+
(6) Rmm
85
THEORY
= R”.
(7) R-lm= Rm-l. ( 8 ) Rm = R (9) (10)
R
O
R
(RoRm)
U
~= R
~
(RoS C -S
(11) R
-S
C
( 1 2 ) R o (S
( 1 3 ) R~
(14) R~~
= R
OR)^ U
= R 2m
5
++
R=
R
U
(Rm0R) = R u R w 2 .
m 2
- S)/\ ( S o R C- S
C
- S \I S o R c- S ) - R m c-
C
~ (
~
RU O R ~ =~ )R
+
R A S2
S o R m
5s).
S .
(- R ) m = (-R)
-S C
U
-,
R U ~( R ~* ~ O R ) .
U
u ( R ~ R ~ ” A” ) R
O R
-
(R O S ) ~ O R .
=
R
R
RmoS
A (RoS
( 1 5 ) R = R-’
(16) R2
O
=
= R 2. U
(RUS)”
(S o (R o S)”)
(-R) 2
U
O R .
(-R) 3
.
= R U S U ( R O S ) ~U ( S O R ) ~ U
(R o (S o R)m).
( 1 7 ) R o S Z S o T * R m o S C- S o T m . (18) R o S = S o R * (RUS)m = RmUS””U(RmoSm). 2 - R A S 2 C S A R o S = S o R + (RUS)m = R U S U ( R o S ) . (19) R C (20)
-
-
R O R - ~C Z D / \ S = ~ D R / R ~ - , S O S - ~ C Z D .
-
2.
Modify ( 9 ) ( 2 0 ) o f t h e p r e v i o u s problem so t h a t t h e m o d i f i e d propert i e s a r e t r u e o f RW and p r o v e them.
3.
Prove
WF ( R )
-f
WF (Rm)
.
4.
Prove 2.5.1.21.
5.
S u p p o s e t h a t X E V + P X C V . D e f i n e u P A = u ( P y : y ~ A ) . We say t h a t A i s s u p e r t r a n s i t i v e i f U A U U P * ( A ) C A . We d e f i n e t h e s t r o n g t r a n s i t i v e c l o s u r e by T A = n { X : A u U X U G P * ( A ) C A}. Prove t h e theorems shown f o r T ( m o d i f y i n g them i f necessary) f o r T .
*
ROLAND0 CHUAQUI
86
2.5.2
R E C U R S I O N OVER WELL-FOUNDED
RELATIONS,
The main purpose o f t h i s s e c t i o n i s t o j u s t i f y d e f i n i t i o n s o f operat h a t a r e g i v e n by (*) F ( x ) = H ( x , [ F ( q ) : q € O R ( x ) I ) , f o r a l l x E DRu D R - ' , where H i s a g i v e n o p e r a t i o n and R i s a well-founded r e l a t i o n . Since o p e r a t i o n s must be d e f i n e d f o r a l l classes, (*) i s extended tions
arbitrarily for X
9
D R U DR-';
f o r instance, here t h e e x t e n s i o n w i l l
be
g i v e n by (**) F ( X ) = V f o r X $ D R U DR-' Thus, F ( x ) i s d e f i n e d i n terms o f H a n d t h e values o b t a i n e d f o r a l l k(q) where q i s an R-predecessor o f x . I n o r d e r t o j u s t i f y t h i s t y p e o f d e f i n i t i o n s i t i s necessary t o prove t h a t under t h e c o n d i t i o n s assumed, t h e r e i s a unique F s a t i s f y i n g (*) and (**). Remember t h a t t o say t h a t t h e r e i s a unique o p e r a t i o n F s a t i s f y i n g some c o n d i t i o n $, means t h a t t h e r e i s such an F and t h a t any operat i o n G s a t i s f y i n g $ i s t h e same a s F , i.e. F = G ( s e e S e c t i o n 2.1.1. f o r meaning o f =). Theorem 2.5.2.1, below, i s a g e n e r a l i z a t i o n o f 2.4.1.13 marks made b e f o r e t h i s l a s t theorem a r e a l s o h e l p f u l here.
and t h e r e -
I f we were d e f i n i n g a f u n c t i o n F we would have t o prove
3 ! F ( D R uDR-l Y ( F ) A
v x(x€DRU
DR-'
+
F'x = ff'FIOR(x))). Thus, F ' x would be d e f i n e d i n terms o f s t r i c t e d t o t h e R-predecessors o f x . I n (*), [ F ( q ) : q
E
H and t h e f u n c t i o n F i t s e l f re-
O R ( x ) ] represents the operation F r e s t r i c t e d
to
O R ( x ) . We need t o have x e x p l i c i t y i n H (i.e. H has t o be b i n a r y ) because D ( [ F ( q ) : y E OR(x)] i s not, i n general, O R ( x ) and, t h e r e f o r e x cannot be o b t a i n e d from [ F ( q ) : q E O R ( x ) ] and R. x i s needed i n t h e p r o o f o f
2.5.2.1.
On t h e o t h e r hand, D ( F I O R ( x ) )= O R ( x ) and so p l i c i t y i n t h e f u n c t i o n H.
x i s n o t neededex-
However, t h e f a c t t h a t H h a s t o be b i n a r y i s n o t a r e a l r e s t r i c t i o n , because i f we had a u n a r y J t o s t a r t w i t h , we c o u l d d e f i n e a b i n a r y H by H(X,Y)
=
J(Y),
f o r a l l X, Y .
There a r e e s s e n t i a l l y two t y p e s o f p r o o f s f o r t h e e x i s t e n c e o f F ' s s a t i s f y i n g t h e r e c u r s i v e c o n d i t i o n s . T h a t used f o r 2.4.1.13, which i s b a s i c a l l y Dedekind's b u i l d s F from below c o n s t r u c t i n g p a r t i a l o p e r a t i o n s ( o r r e l a t i o n s ) f o r t h e i n i t i a l segments o f w. The p r o o f g i v e n f o r 2.5.2.1 i s o f a d i f f e r e n t type. F i s o b t a i n e d as t h e l e a s t o p e r a t i o n t h a t s a t i s f y t h e f i x e d p o i n t theorem 2.3.2.4 some c o n d i t i o n s . I n t h e p r o o f o f 2.5.2.1, i s a c t u a l l y used. But an a n a l y s i s o f t h e p r o o f o f 2.3.2.4 shows t h a t f i x ed p o i n t s a r e o b t a i n e d as t h e l e a s t c l a s s e s t h a t s a t i s f y c e r t a i n c o n d i tions.
A p r o o f from below c o u l d a l s o be g i v e n f o r 2.5.2.1
(as a l s o a p r o o f
87
AXIOMATIC SET THEORY
from above f o r 2.4.1.13).
I t i s l e f t as an e x c e r s i c e t o t h e reader.
The f o l l o w i n g theorem i s t h e j u s t i f i c a t i o n f o r r e c u r s i v e d e f i n i t i o n s
2.5.2.1
L d f f be a b i n m y upehatian.
THEOREM SCHEMA,
(i)V R ( W F ( R )'31
( i i) LeR WF ( R )
.
Then
S(S = [ H ( x, S I O R ( x ) ) : x E D R U D R - ' I ) ) .
a u n i q u e o p a a t i u n F huch t h a t , H(X, [ F ( y ) : q E O R ( X ) ]), if X E D R U DR- 1 ,
Then t h a t
F(X) =
42,
PROOF, F i r s t , we s h a l l see t h a t ( i ) i m p l i e s (ii)( ( i ) i s a c t u a l l y e q u i v a l e n t t o ( i i ) ) . L e t S s a t i s f y t h e c o n c l u s i o n o f ( i ) . Then d e f i n e F ( X ) = S * { X ] , f o r X E DRU DR-' and F ( X ) = V , o t h e r w i s e . I t i s easy t o see t h a t F s a t i s f i e s ( i i ) . NOW, suppose F and G s a t i s f y ( i i ) and l e t 3 = [F(x): x E D R U O R m 1 ] Therefore, F(x) = S*{x} = G ( x ) f o r and T = [ G ( x ) : x E D R U D R - ' 1 . But F(X) = V = G(X), i f X $! D R u D R - l . Thus, V X F ( X ) = x E D R u DR-'. = G ( X ) , i.e. F = G. Now, we proceed t o prove (i). The u n i c i t y o f S i s proved by i n d u c t i o n (2.2.3.20): L e t S = [ H ( x , S I O R ( x ) ) : x E D R U O R -'] and T = [ f f ( x , T I O R ( x ) ):
x
E
We s h a l l prove by i n d u c t i o n t h a t S*{x} = T * { x } f o r e v e r y x E
DRU D R - ' 1 . - DRU D R - l , we w i l l have S = T . D R u D R - ' ; thus, s i n c e D S , D T C
L e t A = { x : x E DRU D R - l A S*{x} = T * { x } ) , and suppose x E DRUDR-' w i t h O R ( x ) LA. Then S I O R ( x ) = T ( O R ( x ) . Thus, S*{xl = H ( x , S I o R ( x ) )=
H ( x, T I O R ( x ) ) = T * { x } , i.e.
xEA.
Therefore, DRU DR-'
5 A.
Now, we prove t h e e x i s t e n c e o f an S w i t h S = [ H ( x, S I O R ( x ) ) : x E D R U
DR-'1 DR-'.
, i.e.
a r e l a t i o n S such t h a t S * { x } = H(x,SIOR(x))f o r every xEDRU
F i r s t we assume t h a t R i s t r a n s i t i v e . F o r each X, d e f i n e t h e o p e r a t i o n Gx by, (0)
G X ( T ) = [If(x, T ] o R ( x ) ) : x E X ]
.
We see t h a t what we need i s a f i x e d p o i n t o f GDRu D R - l ,i.e. such t h a t S = GDRu D R -(lS ) .
I n o r d e r t o use 2.3.2.4
p o i n t , Gx would have t o be monotone.
an S
t o obtain a fixed
U n f o r t u n a t e l y , Gx i s n o t monotone
f o r many H I S ; and so we need a more c o m p l i c a t e d o p e r a t i o n .
Define,
ROLAND0 C H U A Q U I
88
J(T) =
[U
: Y C_TIOu(x) A Y = G
{ H ( x,Y)
(Y)} :
OR(X)
x
E
DRu OR-'].
J i s monotone; because i f T C - T ' , t h e n Y c- TIOR(x)i m p l i e s YcT'lOR(x); thus
U
: Y STlOR(x) A Y = G
{ H ( x,Y)
Y = GOR(x)(Y)I.
s = ~ ( s= )[ u {XI =
{ H ( x , Y ) : Y c T-I I O R ( x )
A
we o b t a i n an S such t h a t , { H ( ~ , Y ) : Y C_
sIoR(x)A
Y = G
H ( x,
sl~,(x))f o r
From (1) we o b t a i n ,
S*Ix}
x = u
E
1 (Y)} :XEDRU DR- 1.
OR(X)
We have t o prove t h a t ,
(2) S*
gu
Therefore J ( T ) c J ( S l ) .
Using 2.3.2.4,
(1)
(Y)}
ORW
D R U DR-'.
Iff(
: Y c - SIOR(x)A Y =GOu(,)(Y)j
x,Y)
.
Thus, i n o r d e r t o prove (2), we should show, - S]OR(x)A Y = GO ( x ) ( Y ) } . ( 3 ) ff( X, SIOR(x))= u IH (x,Y) : Y c
R
Consider t h e statement, ( 4 ) Y C _ S I O R ( x )A Y = G
OR
(Y)
-+
.
Y = SIOR(x)
I f we have ( 4 ) , then, U
{ff (x,Y)
y = G
OR(4
: Y C SIOR(x)
-
( Y ) > = ff
Y = G
ORb)( Y ) >
=
u iff ( x , SlO,(x)): Y c - SIOR(x)A
(x, SIOR(X)).
Thus ( 4 ) i m p l i e s ( 3 ) and, hence, ( 2 ) . The p r o o f w i l l be completed f o r R t r a n s i t i v e when we show ( 4 ) by i n d u c t i o n ( 2 . 2 . 3 . 2 0 ) : Let 8 = O R ( x ) C_ B
(5)
{x: Y
, and
C
-
SIOR(x)A Y
=
G (Y) ORb)
-+
Y = SIOR(x)l , suppose t h a t
ss(oR(x) A = GOu(x)(y).
L e t q E OR(x);s i n c e R i s t r a n s i t i v e , OR(q)z O R ( x ) ( i . e . z R y + z R x ) . Thus, from (5), we o b t a i n
YP, Also, from ( 5 ) and ( 0 ) ,
89
AXIOMATIC S E T THEORY
( 7 ) y[OR(y) = SIoR(q) for
y
But ( 5 ) , ( 7 ) and ( 0 ) imply,
'
= =
(2).
OR(4( y )
=
[H(qYyIOR(q)):
[H(q,SIoR(q)) : q
OR(')]
q *
Since y E 8 , ( 4 ) i s t r u e w i t h q s u b s t i t u t e d f o r Hence H(qYs[oR(q)l = S*Iql.
x.
But ( 4 ) i m p l i e s
OR(x)]= SIOR(x). Thus ( 4 ) i s proved f o r x, and, hence, x E B. By 2.2.3.20, DRU DR-' 5 8, i.e. ( 4 ) i s t r u e f o r ev e r y x E DRU DR". Since (4) i m p l i e s ( 2 ) , ( 2 ) i s a l s o proved f o r e v e r y x EDRUDR-']. So we have obsuch x. Therefore, S = [H(x,SIOR(x)): t a i n e d t h e d e s i r e d S when R i s t r a n s i t i v e . Therefore, Y = [S*Iyl : q
E
Now l e t R be an a r b i t r a r y w e l l - f o u n d e d r e l a t i o n . Then, by 2.5.1.5, and 2.5.1.2, RW i s w e l l founded and t r a n s i t i v e . L e t H be g i v e n as i n t h e h y p o t h e s i s o f t h e theorem and d e f i n e H by,
From t h e p r e v i o u s p a r t of t h e p r o o f , we o b t a i n a r e l a t i o n S such t h a t
S*(x) = H(x,SlO (x)) f o r a l l x E DRW U DRW-' = DRu DR-'. We have, RW OR(x)C- 0 (x); hence, SIOR(x)= (SlO (x))lOR(x). RW RW Therefore,
S*(xl = Z(x,SlO (x)) = H(x,(SIO (X))lOR(X)) = H(x,sIOR(x)).
RW
RW
90
ROLAND0 CHUAQUI
PROBLEM
Prove 2.5.2.1
wx(x 0
RW
E DRU
from below by using t h e following lemma schema:
DR-'
3 !T ( T
--t
[ H ( ~ , T I O ~ ( Y ) :) y
=
(x) 1
E 0
RW
T h u s , S.will be obtained a s the union of p a r t i a l r e l a t i o n s defined on
(4.
2.5.3
WELL-FOUNDED CLASSES
8
We say t h a t a cea6b A 0 w d - d o u n d e d i f ELlA i s well-founded. The purpose of t h i s s e c t i o n s i s t o study c l a s s e s t h a t a r e h e r e d i t a r i l y well founded, i.e. c l a s s e s A such t h a t TA i s well-founded ( t h i s means t h a t A , the elements of A , the elements of elements of A , e t c . a r e well-founded.)
First, W i s the c l a s s o f a l l h e r e d i t a r i l y well founded sets. 2.5.3.1
DEFINITIONl
W = {x : i Y ( Y
5 Tx A
Y # 0
Y = 0))).
2.5.3.2
--t
3 z(z€Y A z n
THEOREM,
(i) u WC W. (ii) TW=W.
PROOF Suppose t h a t Y E x E W . Then g T h u s , Tg 5 Tx. I t i s c l e a r from Def. 2.5.3.1 u WC -W. I
u W E W i s equivalent t o TW
= W ,
E
Tx, and hence, g C_ T x . t h a t y E W. Therefore,
by 2.5.1.11
(iii).
We have, t h a t t h e subclasses o f W a r e t h e h e r e d i t a r i l y well founded classes: 2.5.3.3
THEOREM 1
(i) A c - W c * W Y(0
.
( i i ) WF (ELIW)
#
Y c - TA
+
3 z ( z € Y A z n Y = 0)).
AXIOMATIC S E T THEORY
91
PROOF OF ( i ) . Suppose t h a t A c W . Then, by 2.5.3.2 and 2.5.1.11 - TA; then Y 7 W . There i s a y such t h a t y E Y . ( i i ) TA c W. Let 0 f Y C 0, take z = y . Suppose, then t h a t y n Y If 0. Then, s i n c e y C If y n Y - Ty and y EW. Therefore t h e r e i s a z E T y n-Y TY 9 Ty n Y # 0, b u t Ty n Y c such t h a t z n Ty n Y = 0. Since z E Ty, z c Ty and, hence z n Ty = z . Therefore z n Y = 0. T h u s , we have proved t h e implication from l e f t t o right. C Suppose, now, t h e r i g h t hand s i d e , a n d l e t X E A . I f Y c Tx, then Y Tx C- TA. From this and Def. 2.5.3.1 we e a s i l y o b t a i n , t h a t W.
PROOF OF ( i i ) :
by ( i ) noting t h a t x = O,(x).
Similarly a s i n 2.2.3.19,
from 2.5.3.3
.
( i i ) we e a s i l y obtain
2.5.3.4
THEOREM ( I N D U C T I O N P R I N C I P L E FOR W).
2.5.3.5
THEOREM ( I N D U C T I O N P R I N C I P L E FOR FUNCTIONS).
PROOFo Let A = { x : F'x = G'x). Suppose x c A. This means t h a t i f y E x , then Fly = G'y ; i.e. F*x = G*x. From t h e hypothesis we get FIX = c A and s i n c e D F = D G = W , F = G Glx , thus x E A . By 2.5.3.4, W -
..
THEOREM SCHEMA ( I N D U C T I O N SCHEMA FOR W).
2.5.3.6
LeA
Q be a 60muRa whme y and X do
n0.t OCCWL.
Then,
PROOF, Assume t h a t $ s a t i s f i e s t h e hypothesis of t h e theorem a n d CA. Let, n o w X-c W ; l e t A = { ~ : $ ~ [ y ] ] T. h e n P A c A . By2.5.3'.4, W B y the hypothesis of t h e theorem, then X C A ; i.e. Wy(y€X 0, F Y I ) . we have-$y [ X I
.
2.5.3.7
.
+
THEOREM,
(i) X E W + x $ x (ii) x
(iii)
X
W EW
E
+
-+
x $ Tx T x = O,,(x)
.
92
ROLAND0 CHUAQUI
PROOF OF ( i ) . Suppose x E W and X E X . Consider g = { X I . Then W . B u t the only element of g i s x , and g n x = x # 0 , contradicting 2.5.3.3 ( i ) . Thus, x 9 x .
q
C
PROOF OF ( i i ) . Let A = { g : g 9 T g } and suppose x C A. Assume x E T x . By 2.5.1.18, T x = x U U ( T g : g E x } . Thus, X E X or-there i s a q E x such t h a t x E Tq. By ( i ) , x @ x. Also, g E x E T g implies q E T g . B u t since x C A , Y E A . This i s a contradiction. Therefore x 9 T x and x C A . B y 2.5.3T4, W C - A and ( i i ) i s proved.
.
P R O O F OF ( i i i ) . By 2.5.1.16, TC X I = T x u { X I . Suppose x E W . Then T x = ( T C x l ) % Ex) , by ( i i ) . Using 2.5.1.9 ( i i ) we obtain, T ( x 3 = ELw '*{x}. Hence, T x = (ELw - 1* { x } ) % { X I = OELw(x).
-
2.5.3.8
THEOREM, 0
#
Ac -W
-+
W x ( x E AA T x n A = 0 ) .
P R O O F , By 2.5.3.3 ( i i ) , EL IW i s well founded. Then ( E L [ W ) w i s I t i s easy t o show t h a t ( E L I W ) w = E L w I W . a l s o well founded, by 2.5.1.5. Now, i f 0 f A C W , then A c D (ELwI W ) . Then by t h e d e f i n i t i o n of well foundedness, tFere i s an x F A , such t h a t 0 ( x ) n A = 0 . By 2.5.3.7 ( i i i )
EL^
we obtain T x n A = 0 .
We now deduce a new induction p r i n c i p l e f o r W . THEOREM (INDUCTION PRINCIPLE FOR W ) . W x(Tx C A W C -A .
2.5.3.9 x€A)
+
+
.
P R O O F , Suppose t h a t V x ( T x C A + x G A ) and W q A Then W - V A # 0. Since W i s By 2.5.3.8, t h e r e i s an x E W % A such t h a t T x n ( W % = 0. - A. Hence x E A , b u t t h i s t r a n s i t i v e and x E W , T x 5 W . Therefore T x c contradicts x E W % A .
.
2.5.3.10 +TB C A .
A)
THEOREM (INDUCTION PRINCIPLE).
B c - W A T B n P A c- A
The proof i s l e f t t o t h e reader. 2.5.3.11
THEOREM,
PROOF,
Let x
5W
x C_ W
.
-+
x
E
W
Then by 2.5.3.3
. (i), since x
E;
V, x E W .
The next theorem shows t h e f i r s t c l a s s t h a t can be proved proper 2.5.3.12
THEOREM,
W4 V.
AXIOMATIC SET T H E O R Y
.
PROOF, Suppose t h a t W E Y .
(ii) WP W .
by 2.5.3.7
2.5.3.13
93
Then, by 2.5.3.11,
W E
W.
B u t then
THEOREM a
( i ) 0 E W. (ii) X E W A ~ E W + X U { ~ ~ E W .
.
PROOF, ( i ) i s obvious. v i t y of Wand 2.5.3.11.
( i i ) i s e a s i l y o b t a i n e d from the t r a n s i t i -
From 2.5.3.13 we o b t a i n t h a t the c o l l e c t i o n o f s u b c l a s s e s of W s a t i s f i e s a l l axioms of G . 2.5.3.3 ( i ) , shows t h a t i t a l s o s a t i s f i e s Ax Reg.
T h u s we have o b t a i n e d an " i n n e r model" of G + Ax Reg, showing the r e l a t i v e c o n s i s t e n c y of Ax Reg with G , i . e . i f G i s c o n s i s t e n t , then G + Ax Reg i s a l s o c o n s i s t e n t . 2.5.3.14
-
THEOREM,
0 E W and i f
By 2.5.3.11,
PROOF,
W C W .
u s W.
1
2.5.3.15
x
E
W ,x
U
{XI
E
W.
Hence
THEOREM,
(i) u W =W. ( i i ) W = V + - + Ax Reg. (iii) 0
#
A C - W + 0 E TA.
The proof i s l e f t t o the r e a d e r . F i n a l l y , s i n c e EL I W i s well founded and O,,(x) = x we deduce from the general theorem on d e f i n i t i o n s by r e c u r s i o n 2.5.2.1: 2.5.3.16
THEOREM SCHEMA,
LeR H be a b i m q opanation.
Then
(i)3! S ( S = [ H ( x, SIX) : x E WI).
( i i ) T h u ~ eh a ~VLiqueu n m y opetration F duch t h a t , F ( X ) = H ( X , [ F ( y ) : Y E X ) ) 6o/r
aee x
c_W
and F(X) = V PROOF,
,
othmhe.
( i ) i s immediate from 2.5.2.1
From 2.5.2.1
( i i ) we o b t a i n an o p e r a t i o n
(i).
F, such
that F(X)
=
94
ROLAND0 C H U A Q U I
= H (X,
[F ( q )
: qEX I ) for a l l X F(X) =H(x, F(X) = V
(ii).
D e f i n e F by,
W.
E
[F(q) : Y E X I ) otherwise
.
or 2.5.3.6)
I t i s e a s i l y shown by i n d u c t i o n (2.5.3.4
2.5.3.17
for all
x CW that F satisfies
EXAMPLES,
(1) L e t H ( X ) = A, f o r a l l X and F t h e o p e r a t i o n d e f i n e d by, F(X) =H([F(y): YEX 1 ) . F(X) = A f o r a l l X C -W .
Then
( 2 ) L e t H ( X , Y ) = { Y * { x ) : x E X } , f o r a l l X , Y and F ( X ) = H ( X , [F(y): y E X ] ) f o r a l l X C W. We have, I D ( X ) = X . T h e r e f o r e [ I D ( x ) : xEX ] * { X I Hence, H ( X , I F D ( x ) : x E X ] ) X = I D ( X ) . Since F i s unique, F ( X ) = x. I D ( X ) = X , f o r a l l X c W.
-
( 3 ) L e t H ( X , Y ) = X u u { Y * { x } : x E X 1 , f o r a l l X , Y , and F ( X ) = H ( X , U u {Tx: xEX} = H ( X , -W . [ T x : x € X ] ) . Hence F ( X ) = T X f o r a l l X C
[ F ( x ) : x E X ] ) f o r a l l X C W . We have, TX = X
PROBLEMS
1.
Prove 2.5.3.13.
2.
Prove t h e f o l l o w i n g i n d u c t i o n p r i n c i p l e s , (1) B C W A ( B n P ( A u ~ B )-) c A + B -c A . ( 2 ) 2.5.3.10.
3.
F i n d t h e o p e r a t i o n F r e c u r s i v e l y generated by H ( a c c o r d i n g t o 2.5.3.14) when H i s :
(1) H ( X , Y ) = { Y * { x } : x E X n A } . (2)
H(X,Y)
= { Y * { x } :x E X
u 8)
( 3 ) H(X,Y) = { Y * { x } : X E (X n A )
. U
B)
.
CHAPTER 2 . 6 We1 1 -0rderings
ISOMORPHISMS AND SIMPLE ORDERINGS,
2.6.1
Some general properties of isomorphism of relations will be discussed in t h i s section. F i r s t , the definition. 2.6.1.1
9 is
R
t o S;
DEFINITION,
R
read F A a n Aomohphi.hm o d R a n t o S; R
2
S, R A A o m o h p k i c
? S , F A an Avmvtrpkinm ( o h embedding) oh R i n t o
R i~ embeddable i n
S; and R
2
S
,
S.
Some obvious properties of these notions are contained in the following theorem. 2.6.1.2
THEOREM,
(i) R=R. (ii) R = S - + S z R . (iii) R = S = T - * R = T . (iv) R
2 R. 95
96
ROLAND0 CHUAQUI
R i s a simple o r l i n e a r o r d e r i n g ( S O ( R ) , Def. 2.2.3.15 ( i i ) ) i f R i s r e f l e x i v e , t r a n s i t i v e , a n t i s y m m e t r i c , and connected. F o r simple orderings, t h e d e f i n i t i o n o f isomorphism i s s i m p l e r .
2.6.1.3 D
S ( ~~- 1 A ) ~w x PROOF,
SO ( R ) A SO ( S )
THEOREM,
w~
( X q R
+
F ~ X SF
.
--*
(R z S
-
3 F(DRDS ( F )
A
I ~ )
Assume t h a t R and S a r e s i m p l e o r d e r i n g s .
I t i s c l e a r t h a t i f R = F S , s i n c e D R = DRU DR'l F s a t i s f i e s t h e r i g h t hand s i d e o f t h e equivalence.
and D S = DSUDS-',
I n o r d e r t o prove t h e i m p l i c a t i o n from r i g h t t o l e f t , suppose t h a t t h e r e i s an F such t h a t D R D S ( F ) A DSDR (F-') F i r s t , D R u DR-'
= DR and DSU D S - '
= DS.
A W xtl q(xR y F'xS Fly). Hence D R u DR-'= D F and -+
F*(DRu DR-') = D S U D S - l . L e t x,q E DR and F ' x S F ' q ; suppose 1 x R y . Since, R i s connected (i.e. C O ( R ) ) , we have y R x A y # x. Then, f r o m t h e assumption on F, F ' y S F ' x A F l y # F ' x ; hence l ( F ' x S F ' q ) c o n t r a d i c t i n g o u r assumption t h a t F ' x S F ' y Therefore, we have proved t h a t
.
F'xSF'q Hence, R
2
+
xRy
.
Fs.
The f o l l o w i n g theorem i s easy t o prove.
2.6.1.4
THEOREM,
( i ) PO(R) A R
( i t ) SO(R) A R
S E
S
( i i i ) WF(R) A R z S ( i v ) WO(R) A R z S
. SO(S) .
PO(S)
-+
-+
WF(S).
-+
+
WO(S)
.
I f R i s a p a r t i a l o r d e r i n g we s h a l l w r i t e
forxRyAx+y. That i s , we have t h a t
x
G Rg
x
Q
Ry
f o r x R y , and
i f and o n l y i f ~ € R - ~ * { q land ,
x < Rq
x<
y
i f and o n l y if x E OR(y). Also, i t i s easy t o see t h a t R i s a s i m p l e orderi n g i f and o n l y i f i s t r a n s i t i v e , asymmetric, and s a t i s f i e s x < R y V
%
x
= y V
y<
x f o r a l l x,q
E DR.
AXIOMATIC S E T T H E O R Y
97
F i n a l l y , we r e c a l l Def. 2.3.1.3 and introduce a new notion, I f R i s R a r e l a t i o n and A c-D R , A A was defined t o be t h e g r e a t e s t lower bound of A , R i f i t existed. In p a r t i c u l a r , ADR i s the f i r s t element of R , i f t h i s Similarly, A i s the l e a s t upper bound of A , i f i t e x i s t s ; and DR i s t h e l a s t element of R , i f i t e x i s t s .
6
fists:
Recall t h a t R i s a well ordering (WO(R)) i f R i s a well founded simple ordering. For well orderings g r e a t e s t lower bounds always e x i s t . We have:
R
A A
THEOREM, W O ( R ) A 0
2.6-1.5
< Rx)
A W g( Wx(xEA
+
g
G
Rx)
#
+
A C- DR
R
R
+
A A
E
A A
x(xEA
+
y G R A A).
Let WO(R) and 0 # A c D R . Then t h e r e i s a zEA such t h a t R A n o ~ ( Z ) =o. W e s h a l l show t h a t z = A A. Let xEA; then x $ O R ( z ) , 1.e. X s I R Z . Since R i s connected, z < R ~ . Suppose, now t h a t g < R~ f o r a l l x E A . PROOF,
Since
zEA,
2.6.1.6
g < R ~ Thus, . we have z
R
= A A.
.
DEF I N IT1 ON a
( i ) Rx = RIOR(x). ( i i ) IS(A,R)
-
A C - DR A W x tl y(xEA A g R x
-+
YEA)
IS(A,R) i s read, A i s an irtitid begrnertt 06 R. I t i s c l e a r t h a t i f R i s a simple ordering then OR(x), f o r x E D R , i s an i n i t i a l segment o f R.
Since R i s t r a n s i t i v e , i f y k x , then OR(y) & O R ( x ) and, hence ( R ) =
RY
X Y
*
We have the following easy theorem. 2.6.1.7
THEOREM,
( i ) SO(R) ( i i ) WO( R )
+so(RIA). -+
WO( R I A )
.
98
ROLAND0 CHUAQUI
PROBLEMS
.
1.
Prove 2.6.1.2
2.
Prove 2.6.1.7.
3.
Assume t h a t i f 6 i s a f u n c t i o n w i t h D d = w, then the f o l l o w i n g r e l a t i o n s are simple orderings.
( 3 ) Show t h a t S i s n o t isomorphic w i t h I N
Iw
6 E Y.
Show t h a t
.
WELL-ORDER I NG R E L A T I O N S 8
2.6.2
Since w e l l o r d e r i n g r e l a t i o n s a r e w e l l founded, t h e i n d u c t i o n p r i n c i p l e 2.2.3.20 a l s o a p p l i e s here:
2.6.2.1
OR(x)5 B
THEOREM ( I N D U C T I O N P R I N C I P L E ) . XE
+
B)
-+
D RC B.
w O ( R ) A tf x ( x E D R
A
The f o l l o w i n g theo.rems general i z e 2.6.2.1.
2.6.2.2
THEOREM SCHEMA,
LeX Q be u 6~mLLeat h a t A a t i 6 6 i e 6 a
PROOF, L e t Q be a f o r m u l a which s a t i s f i e s t h e h y p o t h e s i s and suppose t h e r e i s a w e l l o r d e r i n g R such t h a t 1$I~[R ] . From t h e hypothesis, R we deduce t h a t t h e r e i s an x E D R such t h a t 1 $,[Rx]. Let y = A { x :
x
Qx [ R x I 1 .
E D R A 1
.
Qx [ R x l ) W x(x E D R tradicting
Y
By 2.6.1.5,
But R x = ( R ) Y X 9, [ ( R y ) x ] ) +
1$,I
R I. Y
.
.
and x
E
we have 1Qx [ R ] A W x( x < y Y D R y, f o r x < Ry , hence, we g e t
From t h e h y p o t h e s i s we o b t a i n
Qx [ R I , Y
+
con-
99
A X I O M A T I C S E T THEORY
2.6.2.3
la +
COROLLARY SCHEMA'
-,((R = S
W(R)
(Vx(xEDR
-+
+.
RI
(+,[
+.
be a
~0mvnLLeabuch
+x [ S I 1)
that,
A
3T(T=RxA@X[Tl) ++,[Rl)
Then, we have,
WR(WO(R)+. + , [ R l ) . A f u n c t i o n F i s s t r i c t l y R-increasing, i f D R D R ( F ) and V ~ V y ( x < ~ y + F I X < F ' y ) . We have t h e f o l l o w i n g p r o p e r t y o f s t r i c t l y R - i n c r e a s i n g functions
.
2.6.2.4 Vx(x
E
THEOREM, WO(R) A D
PROOF, an
w ~x v q ( x < R y
R ( F ) ~A
+.
DR+ x G R F ' x ) .
FIX<^
F I ~ ) +
Suppose t h a t t h e h y p o t h e s i s i s s a t i s f i e d and t h a t t h e r e i s
x such t h a t x p , F'x.
R
L e t y = A {x :
S i n c e R i s connected, F ' x C R x .
F @ X < ~ X ] .We have, by 2.6.1.5, F ' Y < ~ Y . From t h e hy o t h e s i s we g e t F ' F ' Y < ~F l y , i.e. F ' y E ( x : F ' x < ~ x } . and F ' Y < ~ Y= A { x : F ' x < ~ x ) , a
fi
contradiction.
2.6.2.5
THEOREM,
WO(R)
+
(R
-
=S
3! F R a F S )
.
PROOFl We only have t o show t h a t R = S i m p l i e s t h a t t h e r e i s a unique F such t h a t R z F S , s i n c e t h e converse i m p l i c a t i o n i s obvious. So G - " F ' x < ~ G-l'F'y f o r a l l suppose R z F S and R z G S . We have, -+
x,y E DR.
x G R G-"F'x, i.e. G ' x G S F ' x . can be shown, and, hence F ' x = G ' x f o r a l l x E D R . 2.6.2.6
By 2.6.2.4,
F' xGsGk
THEOREM,
( i ) WO(R) A IS(A,R) A A # DR ( i i ) WO(R)
+.
V x(x E DR
( i i i ) WO(R)
+.
Wx Wy(x,y
PROOF OF ( i ) .
R
Similarly,
x = A (DR-A)
+
E
+
E
DR A A = O R ( x ) ) .
R $ Rx).
DR A x # y
L e t W O ( R ) , A # DR
EDR-A.
3x(x
--*
Rx $ Rq )
, IS(A,R).
.
Since DR- A
+0,
I t i s easy t o show t h a t A = OR(x).
PROOF OF ( i i ) . Suppose W O ( R ) , x E D R , and R z F R x'
We h a v e ,
ROLAND0 C H U A Q U I
100
D R D R ( F ) and W u W y ( ~ < ~-, y all u E DR.
Fly).
=
PROOF OF ( i i i ) .
US
R D R x = O R ( x ) . Thus, F 1 x C R x
On t h e o t h e r hand, F ' x E
tradictory xGRF'x.
Hence, by 2.6.2.4,
Flu for
, con-
We have t h a t O R ( x ) C O R ( y ) o r O R ( y ) s O R ( x ) . Sup-
pose OR(g) C - O R ( x ) ; then R
= (Rx)y.
Y
Under t h e o t h e r
We then a p p l y ( i i ) .
hypothesis t h e p r o o f i s s i m i l a r . 2.6.2.7 THEOREM, 3y(yEDSA R z S ) . Y PROOF,
-
R
2
S V ~ X (EXD R A R s S ) V X
{ x : x E D R A 3 y ( y E D S A R x 2 S )}. Define the Y DS ( F ) , by F I X = U Cy : R x s S j ; by 2.6.2.6 ( i i i ) , we
- S F l xf o r
segment o f R.
+
Let C =
function F with have t h a t R
w O ( R ) A WO(S)
every x E C .
L e t x < u~E C .
see t h a t , s i n c e
x
E
Then
Y
We s h a l l prove t h a t C i s an i n i t i a l
Ru 2 GS F l u
D R U , R Z S ~ , ~hence ; xEC. X
I t i s easy t o
f o r some G.
S i m i l a r l y i t can be shown
t h a t F*C i s an i n i t i a l segment o f S , because F*C = { y : y E D S A 3 x ( x E D R A
Rx
2
Sy)l
.
From 2.6.2.6
( i ) , we g e t t h e f o l l o w i n g p o s s i b i l i t i e s
(1) C = D R A F*C = DS
.
( 2 ) C = D R A F*C = OS(y), f o r some y. ( 3 ) C = O R ( x ) A F*C = D S , f o r some
(4) C
= OR(x) A F*C = Os(y),
f o r some
We a l s o have R I C s F SIF*C, since: and R
Y
Thus R i.e.
2
y
2
S
f o r some G,
F'Y
G-IoH
H.
x. x
and y.
I f x < ~ Y x, , y E C , t h e n R (iii).
Now, ( 4 ) i s impossible, because we would have R
x
E
OR(x).
Then S F l y C_ SFlx.
Suppose t h a t F 1 y G S F I X .
R x c o n t r a d i c t i n g 2.6.2.6
x =G S FIX
X
E
S
Y
, and
hence x E C ,
( I ) , (2) and ( 3 ) g i v e t h e t h r e e p o s s i b i l i t i e s i n t h e c o n c l u s i o n o f t h e
theorem.
2.6.2.8
COROLLARY i
AXIOMATIC SET THEORY
101
PROOF, S i n c e (ii)i s e a s i l y o b t a i n e d from ( i ) by 2.6.2.7, need t o prove (i).
we o n l y
The i m p l i c a t i o n j r o m r i g h t t o l e f t i s obvious. So suppose R a n d S a r e w e l l o r d e r i n g s and R < F S . By 2.6.2.7, R S V 3 x ( x E DRA Rx = S ) V
3 y ( q E DS A R s S ).
Y
.
R x, c o n t r a d i c t i n g 2.6.2.6
Assume t h a t S g G R x ; then R n G
Thus, t h e o t h e r two p o s s i b i l i t i e s remain. 2.6.2.9
DEFINITION, { ( ( O y l ) ,
R + 1=
{((
y,O
(q,O)): qEDR}
U
,(
)
(ii).
~ ~ 0 :) ( )q , ~ E) R}
{((O,l)
U
,(0,l))).
IfR i s a w e l l o r d e r i n g t h e n R + 1 i s a successor o r d e r i n g f o r R , and we have:
(1) D ( R + l ) = [ D R , { O ) ]
=
DRx { O }
(2) ( x , O ) R + l (y,O) * xRq
(3)
(
.
x,O) R +1 ( 0 , l ) f o r every x
( 4 ) ( 0 , l ) R + 1 (0,D
U
.
E
DR
{CO,l)}
,
.
The f o l l o w i n g p r o p e r t i e s a r e easy t o prove and l e f t t o t h e reader. 2.6.2.10
WO(R)
THEOREM, +
R +1 (WO(R+l) A V D ( R t 1 ) = ( 0 , l ) A
R cz R ( o , l ,
A
R ? R + l A R $ Rtl). Thus, R + 1 i s o b t a i n e d by adding a l a s t element t o t h e o r d e r i n g of R . sion.
T h i s s e c t i o n i s completed w i t h some p r i n c i p l e s o f d e f i n i t i o n b y r e c u r -
2.6.2.11
THEOREM SCHEMA,
( i ) L e t H be an a p W o n and R a w& unique ope/ration F nuch t h a t F ( x ) = H ( [ F ( y ) : q E O,(X)l)
otdehing.
, if X
F(X) = V , o t h m h e . (ii)L e t H , G be unahy o p e h a t i o ~ nnuch t h a t
w x(x E
B
-+
wo (c(x))).
Then thehe A a
EDR
,
102
ROLAND0 CHUAQUI
Then t h e h e 0 a unique v p W o n F duch than
F(X)= H ( [F(y): G(q)2
C(X) A
+
G(y) G(X)A y
B 1)
E
F(X) = V , o t h W e . PROOF, ( i ) i s a s p e c i a l case o f 2.5.2.1. we d e f i n e R C_ B x B by
xRq then
R
t--*
4X E B ,
I n order t o obtain ( i i ) ,
C(x)2 G(q) ;
i s well-founded and we can a p p l y 2.5.2.1.
The f o l l o w i n g theorem i s an e x t e n s i o n o f 2.6.2.9. 2.6.2.12
THEOREM SCHEMA,
6oMveawing condition:
WO(R)A R T'*{q))
-+
S A V x v y(x
I
E
L e t H be a n op&on
DR
A y E
DS A Rx
t h a t ~ a . t d & i ethe ~ 2
Sy
-+
T*{x} =
H ( T) = H(T').
Then t h a t 0 a unique apehhation F duch t h a t
F(R) = H([F(Rx) : x EDRI F(R)= V , o t h m 0 e
.
9
4 wo (R),
PROOF, We s h a l l prove f i r s t t h e u n i c i t r by i n d u c t i o n (2.6.2.2). L e t F and G s a t i s f y t h e c o n c l u s i o n o f t..e t h e o r e m a n d suppose'R i s a we1 o r d e r i n g such t h a t f o r a l l x E D R , F(Rx)= G(Rx). Then i t i s c l e a r t h a t
F(R)= G(R). Hence, a p p l y i n g 2.6.2.2 we1 1- o r d e r i n g R.
we see t h a t
F(R)=G(R)f o r e v e r y
f o r each w e l l o r d e r i n g R Now, we s h a l l c o n s t r u c t F . By 2.6.2.9, t h e r e i s a unique o p e r a t i o n FR such t h a t , FR(x)= H ( [FR(y): y E OR(x)l) Define S 2 R w i t h S $ R, l e t Rx e S (2.6.2.8). For f o r every x E DR.
FR(S)= FR(x). F i n a l l y , we d e f i n e
We have t o prove t h a t we show:
(*)
If
R 2 S,
S',
by i n d u c t i o n (2.6.2.3). We have,
R
F s a t i s f i e s t h e c o n c l u s i o n o f t h e theorem. F i r s t
+ S, R $ S ' ,
then
FS(R)= FSl(R),
A X I O M A T I C SET T H E O R Y FS(R) = F S ( x ) ,
with
SX
=
H ( [FS(Y) :Y
=
H ( [ F (S 1 :q S
Y
E
103
=R
OS(X)l 1
E
OS(X)l )
.
2 R and S $ R f o r q E OR(x), by t h e i n d u c t i o n h y p o t h e s i s Y Y FS ( SY ) =- FT (SY ) f o r a l l y E OR(x). From t h e d e f i n i t i o n o f FT, i f S: S Y' we have F ( S ) = FS(SL). Suppose now t h a t R = S I X , , where x'E D S ' Then But s i n c e S
S'x,
T
Sx
oS,( x ' ) ] .
.
Y
T = [ F (S ) : y OS(x)J and T ' [FS,(Sz) :z E . I tConsider S Y i s easy t o see t h a t T and T ' s a t i s f y t h e h y p o t h e s i s f o r E
w i t h R = S x and S = S'
x'
*
=
H
Hence,
FS(R) = H ( T ) = H ( T ' )
-
=
H W S ' SZ) : z
= H([FS'
z) : z
=
FS'(S'x,)
=
FS,(R).
From (*), we o b t a i n t h a t F(R) = Therefore, s i n c e R = ( R + 1 ) (
But, i f q < R
+
(
0,l)
Os.t(x')1
E OS, (X'N
FS(R)f o r
1
any S w i t h R
2S
A R
+S.
0,l)
, there
i s a z E DR such t h a t ( R
.
+ 1) = RZ
Y
.
Using a g a i n t h e h y p o t h e s i s on H by a s i m i l a r argument as above, we f i n a l l y obtain,
F ( R ) = H ( [ F (RZ) : z E D R ] )
.
ROLAND0 CHUAQUI
104
PROBLEMS
1.
2.
Prove 2.6.2.10.
Prove that if R i s a well ordering, then one and only one of the following alternatives i s satisfied (a) R (b)
S + 1 for a certain well-ordering S.
Wx(x
f
DR
+
3 y(y
f
DR A y
f
x A Rx
2
R )). Y
If R satisfies (b), we say that R is a LimLt w&-md&ng.
3.
Prove the following induction principle
WO(R) ( b ' x V q ( xE D R n A A R z R x + 1 Y i s a limit well-ordering A OR(x)C -A + x€A) +
-+
YEA) -+
A V
x(x
DRU DR-'
E
D R A Rx
- A ). C
CHAPTER
2.7
Cardinality
2.7.1
EQUIPOLLENCY,
Since Cantor, two c l a s s e s have been defined t o have the same ( c a r d i n a l ) number of elements i f t h e r e i s a biunique function between them. More precisely.
DEF IN I T I O N
2.7.1.1
(i) A=
F
B - A ~ D FA P A = B
(ii) A = 8
-
<, 8
A D F ~- ~ ( F~A) ~ ~ D F ( F - ~ ) .
3 F AzF 8.
++
(iii) A < ~ B (iv) A
1
A c -D F A
PA c- B
A D F ~ ~ - lA (D F ~ - ')D F (F-1).
F AsFB.
-3
I f A = B we say t h a t A and 8 a r e equipo.Ueeent.
The following theorem i s an immediate consequence of the d e f i n i t i o n . 2.7.1.2
THEOREM,
( i ) A = B - A x A e B x B . (ii) A
5
8
(iii) A = B
-
++
-
(iv) A
5
B
(v) A
2
0 - A
(vi) A = 1 (vii) A
-
5 1-
A
A
X A
2
g x
B
.
3 F ( ~ B ( F )A ' A ( F - ~ ) ) .
3 F ( ~ B ( F )A D + A ( + ) ) .
5
0 - A
3b A =
= Cb)
-
= 0.
3 c A = {c}.
0 V 3 b A = (61.
(viii) A = A .
105
106
ROLAND0 C H U A Q I
(ix) A = 8 * B = A . ( x ) A = €3 = C + A = C . ( x i ) A 1 n A 2 = 0 = 8 nB A 1 2 (xii) A < A A ( A E B + A < (xiii) A
5
2.7.1.3
THEOREM,
PROOF,
8
5
C
-+
5
A
.
C
A = B 1 A A = B2 1
A 1 U A 2 = B1
+
B2 '
U
8).
(CANTOR-BERNSTEIN).
I t i s c l e a r t h a t i f A = B , then A
A
5
8
5
5
8
5
A.
A
-
A = B.
Suppose, then, t h a t A 5 €3 5 A. Thus, t h e r e a r e F,G such t h a t A S F B and 8 s G A . F* and G* a r e monotone o p e r a t i o n s (2.3.2) and we have, F*A c - 8 and G*B
A2,
B1,
- A.
A p p l y i n g t h e F i x e d P o i n t Theorem 2.3.2.10,
C
B2 such t h a t A1 n A2
F*A1 = El, and G*B2 = A2. ( i x ) we o b t a i n t h a t A = 8.
,
B1 n B2 = 0, A = A 1 U A 2' Then, A1 = B1 and A2 = B2. Now, u s i n g 2.7.1.2
.
2.7.1.4
THEOREM ( I N T E R P O L A T I O N T H E O R E M ) . C' + 3 & ' ( A ' C A A = A' A C - B' C- C' A B 8 ' ) . PROOF, Suppose t h a t A
- 8 C- C,
A ' 'FA,
C
A
and C z G C ' .
- 8 C- C C
A A' C - C'
BY t h e F i x e d
t h e r e a r e A1 and B1 such t h a t A1 & G * 8 , BICF*A' -8 We have, F * ( A ' - A ) c F * A ' = A c F*(A-A1) = B1, and G*(B-gl) = A1. 1 Put, A ' -A1 = A2 and 8-B1 = B2. We have, 8 = B1 u B2 and B1 n B2 = 0
P o i n t Theorem 2.3.2.9,
- A1u A2. -
Define 8 8
Since A 1 n A 2 = 0, A1 = B1,
= 8' (by 1.2 ( i x ) .
Also, B' = A1
U
)
and A2 = A'-
A
= G*(B-8
C
1 -
2.7.1.5
G*B
- C'
C
THEOREM,
,
we can f i n d A, 8 = B1 u B2
A
5
8
++
C
A'
.
and A2 = B 2 , we o b t a i n
A2 = A I U ( A ' - A
1-
.
,
C C'. -
3 B ' A = 8' c -8
1) 2 - A ' , and A1 =
Thus, 8' = A1
c--t
U
A2
5CI.m
3 A' A c - A' = 8 .
.
PROOF, The f i r s t e q u i v a l e n c e i s obvious from Def. 2.7.1.1. Suppose t h a t A = 8' c 8. T h e n 8' C_ 8 5 V , A C V , B' = A, and V = Y Applying 2.7.1.4, we o b t a i n A ' such t h a t A C - A' - V and A ' = B , i.e. A 5 A ' = B.
Suppose, now t h a t A C - A ' = B. Then 0 5 A 5 A ' , 0 5 8 , 0 = 0, and A ' = B. A p p l y i n g again 1.4, we o b t a i n a 8' such t h a t 0 5 8' 5 B and A = B ' , i.e. A = 8' c -8. m
107
AXIOMATIC SET THEORY
The f o l l o w i n g theorem i s immediate from Def. 2.7.1.1.
+
(A
2.7.1.6 (A = B
2 A A1 n A2 = o B2 A B1 n B2 = 0 A A1 = B1 A A2 = B 2 ) ) .
THEOREM (REFINEMENT).
3 8 3 8 (€3 1 2
++
2.7.1.7
5 B C- C A
=
B1
U
A c C A A' c - C' A A THEOREM, A' C 8' C C' A i? = 8 ' ) . - -
- C'
Suppose A C - C A A'
PROOF,
C
u A
A =
A A
5 C'
5 C'
5
A A'
A A'
5
C
C.
-+
3 8 3 8'
From 2.7.1.5,
we
o b t a i n A = A c C ' and A' = 2 C C . Also, A n A' C_ A = A, so again by 2.7.1.5, A n A' = P C_A. Besizes, A' = A'= ( A n p ) U (8'- A ) . Therefore, by 2.7.1.6, A' = E U E ' , E n E 1 = 0 , E = A n k ' , and E' = 2-A, Also, P C_ A C C', E C - C ' , and P = E . By 2.7.1.4, t h e r e i s an F , such t h a t E C F C- C ' F C - E' -= A ' - A ; by 2.7.1.5, t h e r e i s a G with a n d r = F. But, E'
-
E ' - F = G CA'-A.
Take B = A U G and B' = E l U F. We have, A C B , A C C , and G C A t - A c C . Therefore, A C B C C. Also, A' = EuE'-C E' U F = B' and F C' A ETC A' C C ' . T h e r e i o r e r A' c - 8' c - C'. On t h e o t h e r hand, 8' = ( E ' - F ) C F , E T - F = G , and F = K = A . Therefore, B = B'.
-
.
c
The p r o o f o f t h e f o l l o w i n g theorem i s l e f t t o t h e reader.
2.7.1.8
THEOREM,
C'
A, A'< C ,
-+
3 B(A,A'
5B5
C, C ' ) .
PROBLEMS
-
1.
Prove 2.7.1.8
2.
L e t A be a c l a s s and K a c l a s s o f b i u n i q u e f u n c t i o n s . For a , b E A , define: u = b * d ( d € K A a = b ) , and a s K , ! b , j ( d ~ KA u s b ) . 6 K,A 6 Assume t h a t A and K s a t i s f y t h e f o l l o w i n g c o n d i t i o n s : (1) u A E A ( l e t u = u A )
(2) x E A
-+
( 3 ) wA(x)
(4) I D
u % x E A,
+
u { x ' v :v E u 1
IuEK ,
(5) d E K
-+
,
6-l
E
K,
E A
,
ROLAND0 CHUAQUI
108
6Og'K
(6)
694EK
(7)
xEA A 6 E K
+
+
,
6lx'K
( 8 ) 6 , y E K A D d n Dg (9) x E A A 6 E K +
9
= 0 = D6-l n D g - l
+
bUg€K
,
~*xEA.
Prove t h a t under t h i s conditions a l l theorems o f t h i s section except possibly 2.7.1.2 ( i ) , ( i i ) , ( i i i ) , ( i v ) , a r e s t i l l valid with = K,A and 5 replacing = a n d K,A In Problem 2 , take A = PE where E i s a Euclidean space. Let $ I , the c l a s s of isometries in E and K = { 6 : 3 x 3 g 3 v ( v e w A "A(x) A K ( 9 ) A WLIWIT ~ C I T E V ( g 1 p * x l p ) n ( g 8 r * x 1 . i r ) = o = x 8 p n X'IT) A a = ~ { g ~ ~ l ~ '
<.
*3.
+
P E V l ) 1.
(1) Show t h a t A, K s a t i s f y ( 1 ) - ( 9 ) of Problem 2. ( 2 ) I n t e r p r e t geometrically t h e theorems of t h i s section.
2.7.2
C A R D I N A L ADDITION AND MULTIPLICATION,
In order t o have t h e cardinal sum of two c l a s s e s A , B , we need a class C such t h a t C i s t h e d i s j o i n t union of two c l a s s e s A ' , B' w h i c h a r e equiSimipollent t o A and B respectively. We have such a c l a s s C = [ A , B l l a r l y , f o r the cardinal sum of t h e superclass determined by F and 8 , ( i . e . t h e c o l l e c t i o n 3 x(X = F ( x ) A x E B), we already have [F(x): x E B ] We introduce some new notation t o emphasize this new aspect.
.
2.7.2.1
DEFINITION,
Recall t h a t
Then
2.7.2.2
[A,B] = A
Afc8 = [ A , B ] x
I01 u B
DEFINITION SCHEMA, C zx [ 7
Recall t h a t
x [ : ~$ ]
Let
{x}
.
Cl}.
x
:$1 =
= u {T x
.
7
be a term a n d $ a formula.
x[7
:$
1
.
:$}.
The cardinal product and i n f i n i t e cardinal product of c l a s s e s will be just t h e ordinary Cartesian product and generalized Cartesian product (Def. 2.3.1.11). No new symbols will be introduced f o r products. I t i s c l e a r t h a t A x B = Cc [ A : x E B ] ; t h i s j u s t i f i e s the i d e n t i f i c a t i o n of the prodX uct with t h e Cartesian product. The following theorem shows t h a t t h e d e f i n i t i o n of cardinal sum is ad-
109
A X I O M A T I C SET THEORY
equate. The corresponding theorems f o r t h e i n f i n i t e o p e r a t i o n i s n o t prova b l e w i t h o u t t h e axiom o f choice. A
2.7.2.2 THEOREM, A = A' A B = 8 ' ) .
8 = C * 3A' 3B'(C =
fC
PROOF, Suppose t h a t A
t CB
A'UB'
Take A ' = F * ( A
cFC.
F* B x C11). Then i t i s c l e a r t h a t C = A ' B = 8'.
U
B ' , A'
A A'nB'
=
0A
{ O } ) and B' = 8' = 0, A = A ' , and x
The i m p l i c a t i o n f r o m r i g h t t o l e f t i s easy t o o b t a i n .
2.7.2.3
THEOREM,
(j) A = A '
A 828' + A
(ii) A,Bc_W+A
tc
B=A' t
C
B' A A x B = A ' x B '
.
+c B, A x B c W .
The p r o o f i s l e f t t o t h e reader. The n e x t theorem g i v e s t h e main a r i t h m e t i c a l p r o p e r t i e s o f a d d i t i o n and m u l t i p l i c a t i o n . The p r o o f i s easy and l e f t t o t h e reader. R e c a l l t h a t !I i s t h e c o l l e c t i o n o f permutations o f Z ( D e f . 2.3.1.13).
2.7.2.4
THEOREM,
( i ) A +cB=B+c A A A x B = B x A ( i i ) 'v(F) A !I(G)
A n
. [ F ' i : i E I ]
.L
Z ; [ F ' ~ : ~ E I =I = IIi[F'G'i:iEI].
( i i i ) A +c ( 8 (iv)
'
tc C )
(v) A
= ( A +c B )
(A
x
B=0
++
C
A A x (BxC) = (AxB) x C.
t C(
O = O +
C
.
iEI]=Ilh[F'k:hEK].
Axe).
'V(G)+ C ; [ F ' i : i t l ] x
i,;
C [ F I G ' ~ : ~ E I ]
L
F ' ( j , i ) : j E J ' i ] :
J
[F'ixG'j:(i,j)EZxJ]
(vii)A+
tCC
c
L E I ] A %(F)-+C g [ C C [ F ' ( j , i ) : j E J ' i 1 : J
ni[ff.
(B+cC) = ( A x B )
x
(vi) ' V ( F ) A
= Zc
+
V ( J )A K = C : [ J ' i
Zi[F'h:kEK]A
i E I ] =
.
ZC[G'j:jEJ] J
A = A A ( A + B = O - A = B = O ) A A X O = O = O X A A
A = 0 V B = 0).
(viii) ( A + ~ = ~ - A = O A B = ~ V A = ~ A B = ~ ) A A ~ ~ = ~ X A = A .
110
ROLAND0 CHUAQUI
(ix) A
5 A'
A B
5
B'
-+
A+,B
5
A' +C B' A A
x
B
5
A' x B ' .
( X ) A ~ B + - + ~ C A + ~ C = € ~ . ( x i ) A t c B L . C + c D + + 3 E 3 E t 3 E 1 p 3 E " ' (A = E + c E ' A B ~E l ' +C E f " A C
^I
E + El' A C
D
L.
E'
+C E ' " ) .
PROBLEMS
1.
Prove 2.7.2.4
2.
Prove:
(1) ' V ( F ) A
V(G)
Z?[G'i:iEZ] L
+
[ F ' i + G ' i :i
Z :
C
.
E I ]
p
Zc[ F ' i : L E I ] +C 1
( 2 ) ' V ( F ) A J C- I - + C C [ F ' i : i E I ] = ZLC [ F ' i : i E J ] + 4.
(3) J
5I
=
2.7.3
A 'V(F) A W i ( i E I %J
+
F ' i = 0)
+
Z[!
L
C
Z L? [ F ' i : i E I % J ] .
F ' i :i E 7 ]
=
Z:[F'i:iEJ].
F I N I T E AND DENUMERABLE CLASSES,
A c l a s s i s dinite i f i t i s e q u i p o l l e n t t o a n a t u r a l number. c l a s s e s a r e sets: 2.7.3.1
THEOREM,
] V ( V E WA
v
A)
-+
A E V
Finite
.
PROOF, By i n d u c t i o n on v.
(i) A = 0 implies A
= 0 E V .
L e t B=A-F'vI
(ii) Suppose t h a t i f B = v , t h e n B E V and l e t S v
.
I t i s c l e a r t h a t v y F B and A = B u {F'v}.
Num, A E V . 2.7.3.2
DEFINITION,
If x = v for
VEU
, we
FN=
Then €3
E
V , and hence b y Ax
{ x : 3 V ( V E WA x = v ) ) .
say t h a t x has v elem&.
I f A = w , we s a y t h a t
111
AXIOMATIC SET THEORY
A
denwnaabke. 2.7.3.3
1.1
THEOREM,
+
v
~.I+~v.
2
PROOF, We f i r s t prove t h a t 1.1 i f vE1.1
, and
Fl1.1
=
(0,l)
.
+
I t i s c l e a r t h a t 1-1 + l = F ~ + 1. c
Now we show by i n d u c t i o n on v, 1.1 + v
I01
.
+ 0=
(2)
+ ( v + l ) = ( L l t v ) + 1 = (p +cv)
=
Ll +c 0
tC
!.l
2.7.3.4
THEOREM,
A +
C
PROOF I We f i r s t prove: If F't0,l) =
(
0,l)
, then
B + v
v
A
C
fC
pSCv.
2
(1) 1-1
1.I = 1.I x
D e f i n e F by F'v = ( v , O )
1 = 1.1 +c 1:
1 = B
+
A
+C 1
1 2 1.I 2
8.
+
Otherwise, i f F '
A = FB.
F-'( 0,1)+( 0,l). D e f i n e for x E A , G'x F'( O,l), otherwise. Then A z G B .
=
+JV+l).
A = €3. (
L e t A +cl z F B +cl.
0 , l ) H 0,l)
F'x, i f x
#
F-'( 0,l); and G ' x
The p r o o f f o r any v i s done by i n d u c t i o n , u s i n g 2.7.3.3. 2.7.3.5
THEOREM,
(i) a = 1.1 A b = v + ( a = b
-
, then
.
=
1-1'~).
(ii) 1.1=v-p=v. PROOF, (ii).
I t i s enough t o prove ( i i ) , because (i) i s a consequence o f
(ii) from r i g h t t o l e f t i s obvious. i n d u c t i o n on v.
(1)
LIP
We s h a l l show 1.1 = v
+
1-1 = v, by
o + p = 0.
( 2 ) Suppose t h a t w v i m p l i e s T = v , and t h a t 1.1 = v + 1. Then p f 0; hence p = ' T I + 1 f o r some T; i.e. IT + 1 = w + 1. By 2.7.3.3, 71 +cl = w +cl.
.
The i n d u c t i o n h y p o t h e s i s i m p l i e s t h a t n = v. Hence by 2.7.3.4, 'TI = v. Therefores u = TI + 1 = u + 1. 2.7.3.6 COROLLARY ( D I R I C H L E T ' S P R I N C I P L E ) . / \ ~ c ~ / \ ~ ~ V ( F ) A ~ ~ D b F+ Al F *-lV( ~ F-'). = The p r o o f i s o b t a i n e d immediately from 2.7.3.5.
a = 1-1 A b
2
v
112
ROLAND0 CHUAQUI
T h i s p r i n c i p l e can be paraphrased by s a y i n g t h a t i f i n w boxes, p o b j e c t s a r e p u t and wcu, then one o f t h e boxes, a t l e a s t , must c o n t a i n more t h a n one o b j e c t . 2.7.3.7 (i) w
THEOREM,
4
FN
.
(ii) w
.
PROOF OF (i).Assume t h a t w = v w i t h v E w . We have v Thus, w = v 5 v + 1 5 w Hence by Cantor B e r n s t e i n (2.7.1.3), c o n t r a d i c t i n g 2.7.3.5.
.
-
+ 1c
w
v =v
. 1,
+
PROOF OF ( i i ) : from ( i ) , by 2.7.1.3. PROOF OF (iii): D e f i n e F by F ' O
={
and F ' ( v + l ) = (v,O)
0,l)
,
Then w = F w +c 1. 2.7.3.8
THEOREM,
(A # 0
+-+
15 A) A (A
5
B
t--t
A = B V A t C l5 8 ) .
The p r o o f i s l e f t t o t h e reader. The n e x t theorem c h a r a c t e r i z e s i n f i n i t e c l a s s e s . THEOREM,
2.7.3.9
(1) I f A
PROOF, 0
A $ FN
E FN
,
tf
W v(vEw
+
v
5
A). A # v+l.
w i t h A = v, t h e n by 2.7.3.5,
( 2 ) I f A Ff F N we show by i n d u c t i o n t h a t v < A. I t i s e v i d e n t t h a t Suppose t h a t v < A ; then by 2.7.3.8, v f C 1 5 A. But v + 1 = v t C l
< A.
and, s i n c e A
Ff
FN, A
+ v+l.
T h e r e f o r e v + 1 < A.
We have shown t h a t i f w 5 A , t h e n A $ F N . W i t h o u t t h e axiom o f c h o i c e i t i s i m p o s s i b l e t o show t h e converse i m p l i c a t i o n , i.e. A $ F N , i m p l i e s w <, A. Thus, we have t h e f o l l o w i n g t h r e e t y p e s o f classes: ACFN, w 5 A , and w $ A A A 9 F N . The f o l l o w i n g theorem c h a r a c t e r i z e s c l a s s e s A w i t h w
2.7.3.10
THEOREM i
5
A.
AXIOMATIC SET THEORY
5
(ii) w
A
Vv(v€w
++
Assume t h a t w w +
c
5
A.
Then w
jGv:O,l)
((
i.e.
, because
(
x {O}.
H i s biunique. (2) A
t c l=
A
-
.
Thus,
4
0,l)
= A. therefore A
t C=l
1 tC
x
V E W ) (see
Def. 2.4.2.7
G'
) =
-"I(
0,l ) ; b u t
-
o f iteration). (
0,l )
f
Vv(vEu
-f
-
A +
Hence ~ 2 ~ 8 .
v = A ) i s proved by i n d u c t i o n .
C
3 B ( 8 c A A A = 8). C, 8 n C = 0 8 = A , and C 1. Hence 8
On t h e o t h e r hand, i f A = €3 Therefore, A = 8 + C = A C
C
+
I f A +cl = A , C
A and A = B.
A , t h e n A = B U C w i t h 8 n C = 0 and C 2 A + 1. But A 5 A tCl. Hence, by C
Cantor-Bernstein, A + 1 = A.
-
0,l) } ;
v f 0, we have
I t i s a l s o c l e a r t h a t H*w = 8.
1s C.
U {(
Define 8 =
Suppose t h a t H'v = H ' p w i t h v c y ;
0,l
(
{O}.
From t h i s c o n t r a d i c t i o n we o b t a i n t h a t H'v Z HIP ;
C
U
A
A x ( 0 1 and s i n c e p
( 3 ) We now prove: A + 1 = A then A = B
t C l=G
We s h a l l prove w z H B .
0,l) E A
Gy"'(
Then A
: V E U I and ti = ( G v ' I ( O y l j :
0,l)
t C1
We have, A +cl = (A x {O})
tcl).
Gv'( 0 , l ) = Gp'( 0 , l )
i.e.,
A
C = A , f o r some C;
t
tCl
F ' x , 0) : x E A
We have % ( H ) .
Gy - " I (
-
A
C Z ~ + ~ C = A .
Assume, now, t h a t A let G =
5
(1) We f i r s t prove w
PROOF,
c
A tC v = A ) .
~ B A B= C A .
(iii) USA-
A = l +
+
113
C
Dedekind i n t r o d u c e d t h e d e f i n i t i o n :
A i s finite
13B(B
C
A A A = 8).
T h i s d e f i n i t i o n i s n o t e q u i v a l e n t t o o u r s w i t h o u t t h e axiom o f choice. We c a l l c l a s s e s A w i t h w 5 A , Dedekind i n f i n i t e . We show by i n d u c t i o n t h e P r i n c i p l e of Choice f o r f i n i t e f a m i l i e s . 2.7.3.11
Lct
THEOREM SCHEMA,
a # 0 A a E F N A Wx(xEa
-f
7
7
be a
+ 0)
P R O O F , L e t a = v f 0 and W x ( x t a by i n d u c t i o n on v. I f v = 0, i t i s c l e a r .
+
.tehn.
.+
Rx
T
#
( 7
Then
:x€a) f 0 ,
0). We show t h e conclusion
Let IIX
(7
: x € b ) f 0 f o r every
b c - a w i t h b =v, and assume v + 1 =F a . We have t h a t t h e r e i s a ~ E R (, 7 : We d e f i n e h = g u a {F'v}) Also, 7 J F ' v I f 0. L e t c E r X [F ' v ] E(c, F ' v ) 1 . Then h E n x ( 7 : x E a ) .
-
.
.
114
ROLAND0 CHUAQUI
Without Ax Choice i t i s i m p o s s i b l e t o prove 2.7.3.11
2.7 -3.12
av ( F )
( i i ) a,b E F N A a=b U E
E
FN ,
THEOREM I
(i) a E FN A
(iii)
omitting a
FN
-,
.+ F E
FN.
-,C'
b
C.
a + l c C ~ ~ C X C .
by i n d u c t i o n on t h e number o f elements o f a. PROOF OF (i): PROOF OF (ii). L e t a,b E F N and b Z F a .
D e f i n e H w i t h DH ,C'=
by
H'd = 6 o F . I t i s c l e a r t h a t D ( H ' 6 ) = b and D ( H ' d ) - ' C C. Hence by ( i ) , H ' 6 E V and, thus, H ' d E bC. I t i s easy t o prove t h a t H i s one-one and o n t o bC.
Define the function F w i t h D F = % x C { ( X , ( 0 , l ) ) }. Since a E F N , ax { b ) E a+c 1 C. The proof F N , and hence F ' ( 6, x ) E F N ( b y ( i ) ) ; thus, F t ( 6 , x )E t h a t F i s one-one and o n t o i s easy. PROOF OF ( i i i ) . L e t a E F N . by F t ( 4 , x ) = ( d t y : ( y,O) E a x {O})
U
F i n a l l y , we p r o v e t h a t o r d i n a l and c a r d i n a l m u l t i p l i c a t i o n (and expon e n t i a t i o n ) o f n a t u r a l numbers a r e e q u i p o l l e n t . 2.7.3.13
u
THEOREM,
PROOF OF ( i ) .
We d e f i n e an
F such t h a t u x v
c.
u - v by F ' ( K X)
=
.X+K. I n o r d e r t o show t h a t F i s b i u n i q u e we use E u c l i d ' s A l g o r i t h m (2.4.2.12), s i n c e f o r each TIE^ * V t h e r e i s e x a c t l y one K and one X such t h a t TI = u * ~ + hw i t h X E u . I t i s c l e a r t h a t a l s o K E V , s i n c e u*~g.rr C
U ' V .
PROOF OF ( i i ) .
(1) U0 = 1 (2)
=
0
By i n d u c t i o n on v.
.
v+l P Z v+c1 U
c.
vpxp
."
uV
xu
We have t h e f o l l o w i n g easy C o r o l l a r y .
."
uV
.p
c.
u
v + 1
..
AXIOMATIC SET THEORY
2.7.3.13
-
115
COROLLARY ( i ) a,b E F N
(ii) a,b E F N V
(iii) a,b E F N V
a +c b E F N .
a = 0 V b = 0 a = 0 V b = 0
-
+--*
a x b E FN
'b
E
+C
C).
FN
PROBLEMS
Prove:
-
1. A = A + c B - B x w S A . 2. A = A +
3. A $ w
4. C
E
C
+.
B = A W
+
C
V(V€W
-
+.
v
C
6. C = w - C = C +
+C
B+cC.
5 A).
W A ( A = A +cl + A = A
FN V C = w -
5. C E F N
A=A
# C +c 1 A V A ( A - A +cl A = A +C C ) . +
C
1 A W A ( A e A +c 1 +. A = A +c C).
7. 2.7.3.8.
2.7.4
F I N I T E AND DENUMERABLE O R D E R I N G S ,
I n t h i s s e c t i o n we deal w i t h f i n i t e and denumerable s i m p l e o r d e r i n g . F i n i t e s i m p l e o r d e r i n g s a r e c h a r a c t e r i z e d by t h e c a r d i n a l i t y o f t h e i r domain. I n o r d e r t o prove t h i s , we need t h e f o l l o w i n g lemma, which says t h a t f i n i t e s i m p l e o r d e r i n g s have f i r s t and l a s t elements.
2.7.4.1
THEOREM,
R R (i) SO(R)ADR EFN-l+ADR,VDR ( i i ) S O ( R )A D R
E
FN
+
w o (R)
EDR.
A WO(R-').
PROOF OF ( i ) . By i n d u c t i o n on v, where v = D R . For v = 0 t h e theorem i s obvious, s i n c e t h e h y p o t h e s i s i s f a l s e . Suppose, now, t h a t t h e
116
ROLAND0 CHUAQUI
theorem i s valid f o r v and l e t R be a simple ordering with D R = v + l . Let X = D R - { a ) , where a $ D R , and S = X / R I X . Then, SO ( S ) and DS = X = v S Let b = A D S , b2 = V D S . By t h e induction hypothesis, bl, b2 E DS.
.
1 Therefore, since S O ( R ) , a s R bl or b l S R a . Under t h e f i r s t a l t e r n a t i v e , a i s the f i r s t element o f Rand under the second, bl i s . Similarly, using b2 we obtain the l a s t element. PROOF OF ( i i ) : i s e a s i l y obtained from ( i ) .
2.7.4.2
THEOREM,
SO( R ) A SO( S ) A DR
PROOF, By induction on v, where v = D R . D S a n d R = S = 0.
FN A DR= DS
E
+
RES
.
If D R . 0 , then D R = 0 =
Suppose now, t h a t t h e theorem i s v a l i d f o r v a n d assume t h a t SO ( R ) , R SO(S),DR = D S = v + l . Leta=ADR, b - 3 D S . PutR'=R(DR-(al, and S ' = S I D S - Cb}. Then, by 2.7.4.1, R ' , S ' a r e simple orderings with DR' = D S ' 5 v . Thus, by t h e induction hypothesis, t h e r e i s a n 6 such t h a t R' z d S t . Let g = 6 U { ( b,a) 1. Then i t i s easy t o show t h a t R E S. 2.7.4.3
COROLLARY
,
SO( R ) A DR
E
FN
-
9
3 ! v Rev/ZNIv
.
PROOF, By 2.7.4.2,
main v.
noting t h a t v / I N l v i s a simple ordering withdoThe uniqueness follows by 2.7.3.5.
2.7.4.4 E X A M P L E , I t i s easy t o see t h a t 4.2 i s f a l s e f o r S O ( R ) and D R q F N . For instance, take S = w/ZN I w and R = I( v , p ) : (u,v a r e even A u c v ) V (p i s even A v i s odd) V ( p , v a r e odd A u C v ) } . I t i s c l e a r t h a t D R = D S = w, b u t R $ S. However, adding more conditions we can c h a r a c t e r i z e two types o f denumerable simple orderings. R w/ZN I w + + S O ( R ) A A D R E D R A Wx(x E D R + R R x < ~ { q : x < q~] ) A V A ( A C- D R A A D R E A A V x ( x € A + A { y : X < ~ Y ) E A), A = DR).
2.7.4.5
fi
THEOREM,
R
2.7.4.5 says t h a t t h e following conditions a r e necessary and s u f f i c i e n t f o r R t o be isomorphic t o t h e natural ordering of w: (1) R i s a simple ordering. ( 2 ) R has a f i r s t element. (3) Every element of D R has an immediate successor. ( 4 ) R s a t i s f i e s an induction p r i n c i p l e , i.e. i f A C D R contains t h e f i r s t element of R and i s closed under immediate successor, then A = D R .
A X I O M A T I C SET THEORY
117
PROOF, I t i s easy t o prove t h a t t h e c o n d i t i o n s on t h e r i g h t a r e i n v a r i a n t under isomorphisms and t h a t w/ZN I w s a t i s f i e s them. Thus, i f R = w / I N 1 w , t h e n ?I s a t i s f i e s these c o n d i t i o n s . Suppose, now, t h a t R s a t i s f i e s t h e c o n d i t i o n s ( 1 ) - ( 4 ) . D F = w by r e c u r s i o n on w, as f o l l o w s .
F'O
R
R
.
F'(v+ 1)= A { q : F ' v < ~ Y }
(F
By t h e hypotheses on R , % R can prove,
and F*w = D R .
F i s an
By i n d u c t i o n on v , we
.
p c v - + F p < R F'v. T h e r e f o r e by 2.6.1.3,
F, w i t h
,
A DR
=
Define
somorphism.
We need t h e f o l l o w i n g d e f i n i t i o n o f dense o r d e r i n g s . 2.7.4.6 DEFINITION, DO(R) SO( R ) A 3z(z+xAzfqAxRzAzRy)ADR~l.
x V y ( x R yA x + g
++
--L
I t i s easy t o see t h a t i f R i s a dense o r d e r i n g and D R has more than one element then D R i s i n f i n i t e .
We have t h e f o l l o w i n g theorem due t o Cantor. 2.7.4.7
THEOREM,
DO(R) A DO(S) A
W x ( x E DR+ 3 y 3 z Y < ~ x < ~ z )
A W x ( x E D S - 3 g 3 ~ g < ~ x
s
T h i s says t h a t i f R and S a r e denumerable dense o r d e r s w i t h n e i t h e r f i r s t n o r l a s t element t h e n t h e y a r e isomorphic. I n p a r t i c u l a r , t h e y a r e isomorphic t o t h e r a t i o n a l numbers w i t h t h e i r usual o r d e r i n g . PROOF, We prove f i r s t :
(*) I f a c -D R , a a y E D S such t h a t
RI
(a
E
"
Let a c -D R , a E F N
FN
,x
E
D R - a , and
%F u { ( g,x) } S I (F*a
,x
E
DR -a,
U
R I U = ~ S I F * U ,t h e n t h e r e i s
(~3).
and R ~ u = ~ S I F * U .
We have t o c o n s i d e r t h r e e cases. CASE I
,
There a r e zl,
z2
6
a
such t h a t
z
~
<
z2. ~
Then, s i n c e
S i s a dense o r d e r i n g , F ' z l # F ' z 2 , and F*a E F N , t h e r e i s a g E D S -F*a
118
ROLAND0 C H U A Q U I
such t h a t F'z1
T h i s i s t h e r e q u i r e d q.
CASE 1 1 1 F o r a l l z E a , z < x . Since S has no l a s t element and F*aE F N , t h e r e i s a y E D S -F*a w i t h y > R F'z f o r a l l z E a . T h i s i s t h e y. The p r o o f f o r t h e case x < z f o r a l l
CASE 111, (11).
Thus, we have proved (*). a1 so t r u e .
ZEU
i s similar
to
S i m i l a r l y (*) w i t h R and S interchanged i s
We now d e f i n e by r e c u r s i o n a f u n c t i o n F w i t h D F = w such t h a t F'v L e t w =G DR and w =,,DS. f i n i t e and R I D F ' v z F I v S I D ( F ' v ) - l .
P2 =
is
(a)
F'O = 0 .
(b)
L e t pl = n {K : G ' K DF' ( v - 1 ) l a n d v = ZIT + 1 f o r some 'IT. : R I ( D F ' ( v - 1 ) u {G'PI}) = F ~ ( ~ - ~ ) ~ K{ , G( 'H! J I~ l SID ( F ' ( v - 1))'l
(K.
+
u {ff'K})}.
- 1) U
D e f i n e F'v = F ' ( v ( c ) v = 2a 3 0. changed.
v
5
We r e p e a t t h e same process as ( b ) w i t h R and S i n t e r -
.
( b ) and ( c ) y i e l d t h e c o r r e c t r e s u l t by (*). F'v c - F'u +
L e t J = u {F'v : ~
G' v
{C H'u2, G a u l ) 1 ,
E w } .We
It i s clear, that i f
now show t h a t R ' J S .
I t i s enough t o p r o v e by i n d u c t i o n t h a t (1) D J = D R and D J - ' = DS. H'v = D ( F ' 2 ( v + l ) +1)'* f o r a l l V E W . We have t h e
E DF' 2 ( v + l ) and
d e t a i l s t o t h e reader.
( 2 ) J and J-' a r e f u n c t i o n s . T h i s i s because F'v, (F'v)-' F'p C - F'v f o r a l l p , v . t i o n s and p C_ v
a r e func-
+
( 3 ) x,y E DR A x S R q
-f
J'xSSJ'y. Then, s i n c e F'v i s an isomor-
I f x S R q , t h e n x,y E D F ' v f o r some v.
.
phism, ( F t v ) t x < S ( F ' v ) ' ~ , and, hence, J ' x S S J ' q . A p p l y i n g 2.6.13,
2.7.4.8
we o b t a i n t h e theorem.
THEOREM1 DO(R) A SO ( S ) A D S
The p r o o f , s i m i l a r t o t h a t o f 2.7.4.7
<w
+
3C(C C - DRA S I R I C ) .
i s l e f t t o t h e reader.
A X I O M A T I C S E T THEORY
119
PROBLEMS
1.
Prove: D O ( R ) A D O ( S ) A DR = D S = A R, S have f i r s t element but, have no
l a s t element
--t
R
S.
2.
Same a s 1, b u t R , S have f i r s t and l a s t element.
3.
Prove 2.7.4.8.
PART 3 Morse-Kel l e y - T a r s k i Class Theory CHAPTER 3.1 I n t r o d u c t i o n t o Morse-Kelley-Tarski
3.1.1
Class Theory
AXIOMATIC SYSTEM,
The system t h a t w i l l be s t u d i e d i n t h i s c h a p t e r i s e s s e n t i a l l y due t o A. Morse (Morse 1965). However, h i s p r e s e n t a t i o n i s n o t standard. The f i r s t standard p r e s e n t a t i o n as a f i r s t o r d e r t h e o r y i s t h a t appearing i n t h e Appendix t o K e l l e y ' s book General Topology ( K e l l e y 1955). The axioms u s e d here a r e due t o T a r s k i and I b e l i e v e a r e t h e f i r s t g i v e n i n t h e p r i m i t i v e language L o f f i r s t o r d e r l o g i c w i t h no d e s c r i p t i o n o p e r a t o r and E as o n l y P non l o g i c a l constant. The h i s t o r y o f t h e system i s , however, more complicated and I s h a l l n o t a t t e m p t a complete account. A few p o i n t s o f i n t e r e s t w i l l be noted. M K T (Morse-Kell ey-Tarski Theory) i s j u s t t h e t h e o r y o f von Neumann-Bernays-GGde1 (NBG)w i t h t h e axiom o f comprehension Ax Class, strengthened by a d m i t t i n g formulas w i t h a r b i t r a r y q u a n t i f i c a t i o n . I n N B G (see GGdel 1940), Ax Class i s v a l i d o n l y w i t h formulas which have t h e i r q u a n t i f i e r s r e s t r i c t e d t o set. As was mentioned i n 1.2.1.1 t h i s s t r e n g t h e n i n g o f Ax Class was suggested b e f o r e Morse, by Mostowski and Quine, among o t h e r s . So sometimes MKT i s c a l l e d Q M (Quine-Morse). I should a l s o mention t h a t MKT i s d e f i n i t e l y s t r o n g e r t h a n N B G and i s n o t a c o n v e r v a t i v e e x t e n s i o n w i t h r e s p e c t t o s e t s o f t h e usual Zermelo-Fraenckel ( Z F ) s e t theory. That i s t h e r e a r e theorems about s e t s t h a t a r e p r o v a b l e i n MKT b u t n o t i n Z F . I n MKT we do n o t i n c l u d e Ax Choice and Ax Reg. They w i l l be added l a t e r . The axioms o f M T K have t h e e f f e c t o f adding t o V a n i n f i n i t e s e t and c l o s i n g V under t h e o p e r a t i o n s o f ' b e t 06 . . . I , ' b e t 0 6 be& 06 I, I n place o f we p u t an a r b i t r a r y s e t i n V and ' A & 0 6 t h e - - Y 06 . . . I . and,for we p u t 'e,tments 06 7 ' where 7 i s a term which denotes a s e t i n V . For i n s t a n c e , i f t h e ' b e t 06 & cowb' i s i n V,we w i l l have a l s o i n V t h e ' b e t d 06 cowb' and t h e ' o e t d 06 b e t d 06 c o w b ' . Also, i f f o r each cow a, T o f a a r e i t s legs, which a r e i n V , t h e n we s h a l l a l s o have i n V , t h e s e t o f elements o f T o f a, i.e. t h e s e t of l e g s o f cows.
---
1:
...
...
The axioms w i l l be d i v i d e d i n t o t h r e e groups.
They w i l l be g i v e n i n
P' 3.1.1.1
GENERAL CLASS AXIOMS,
These a r e Ax Class and Ax Ext, which were discussed i n 1.2.1.1 1.2.1.2. 123
and
124
ROLAND0 CHUAQUI
Ax.Class.
Axiom Schema o f Class S p e c i f i c a t i o n .
Ax Class. L d 4 be a darn& i n which A doen nvL appem &e. t h e d o n m e ad t h e ~a&!vwing @rnvnLLea LA an axiom. 3AtlX(XEA
-
I$
A
Then
3 U XEU)).
Ax Ext. Axiom of E x t e n s i o n a l i t y . Ax Ext.
WAWB(WX(XEA - X € B ) -
A
= 8
.
Since d e f i n i t i o n s w i t h t h e o p e r a t o r { : I depend o n l y on these two axioms, we may use them i n MTK w i t h o u t r e s t r i c t i o n s . The o t h e r a x i o m s w i l l n o t i n g each t i m e how t h e y l o o k u s i n g d e f i n e d concepts. be g i v e n i n d: P' 3.1.1.2
CLOSURE AXIOMS,
The axioms o f t h i s group say, i n general, t h a t i f a c l a s s B i s a s e t t h a t i s i n r e l a t i o n t o A by some n o t i o n g i v e n by a p a r t i c u l a r formula J/, t h e n A i s a l s o a set. Thus, t h e general form o f these axioms i s : W A WB($[A,B]
A 3U(BfU)
--*
where $ [ A , B ] i s a f o r m u l a w i t h o n l y A,B f r e e . p a r t i c u l a r J/ and t h u s w i l l n o t be a schema. Ax Un. AX Un.
Axiom of unions. W A W B (WX(XEA
-
3 Y(XfYf
3U AEU))
Each axiom w i l l have
B ) ) A 3 ti(B€u)
-, 3 ~ ( A E U ) ) .
T h i s axiom can be w r i t t e n w i t h d e f i n e d n o t i o n s by, YA 'dB(A= U B A B E V - t A E V ) ,
or, even s h o r t e r ,
WB(B€ V - U B €
V )
or, f i n a l l y ,
tfb u b Ax Pow. Ax Pow.
f
Y .
-
Axiom o f power sets. V A WB(V X ( X E A
3 U(AEU)). I n defined notions,
W Y(YEX
-
a
YEB)) A 3U(BEU)
+
AXIOMATIC S E T THEORY
125
WB(BE V - P B E V ) ,
or finally,
VbP bE V . Ax Rep. Axiom o f replacement o r s u b s t i t u t i o n . T h i s axiom has a s i m i l a r e f f e c t t o F r a e n c k e l ' s axiom i n Z e r m e l o Fraenckel; i.e. i t s purpose i s t o achieve t h e f o l l o w i n g . " I f B i s a s e t and A has l e s s t h a n o r equal number o f elements than B, t h e n A i s a l s o s e t " . Since i t i s n o t easy t o express t h i s i n 2 we adopt t h e f o l l o w i n g p' which a l s o a l l o w s us t o dispense w i t h Ax Num. Ax Rep.
V A t/B VX(XEA
3
U(BEU)
+
-+
3 Y(YEB A VZ(ZEA
+
(YEZ * Z =X))) A
3 U(AEU).
Ax Rep. can a l s o be w r i t t e n by, WA t / B ( b X ( X E A - + l Y ( Y E B A X = U { Z : Z E A A Y E Z } ) ) A j U ( B E U ) jU(AEU)
.
-+
I f we d e f i n e t h e o p e r a t i o n FA by,
FA ( Y ) =
: Z E A A YEZ},
(I { Z
t h e n Ax Rep can be w r i t t e n by, WA WB(A = {FA(Y): Y E B ) A 3 U ( B E U )
i.e.
-+
3 U(AEU)).
i f A i s t h e image o f 8 by FA and B i s a set, t h e n A i s a s e t .
3.1.1.3
AXIOM OF E X I S T E N C E OF SETS,
We s h a l l j u s t need t h e e x i s t e n c e i n V o f an i n f i n i t e s e t . empty c l a s s i s a s e t w i l l be deduce from i t . Ax I n f .
That
the
Axiom o f i n f i n i t y . 3X3A3U(XEAEU
(WZWU(UEZ
-+
UEX)
-+
AWX(XEA+
3Y(XEYEA)
A
ZEA))).
I n defined notation, l A ( O # A E V
A
A C-U A A u { P X : X E A } Z A ) .
M K T i s t h e t h e o r y w i t h Ax Class, Ax Ext, Ax Un, Ax Pow, Ax Rep, and Ax I n f . The t h e o r y M K T ' i s o b t a i n e d by r e p l a c i n g Ax I n f by Ax Em i n M K T M K T w i t h Ax Reg, i s c a l l e d M K T R . S i m i l a r l y , M K T ' R i s M K T ' w i t h Ax Reg.
126
ROLAND0 CHUAQUI
3.1.2
ELEMENTARY CONSEQUENCES,
The main purpose o f t h i s s e c t i o n i s t o prove i n MKT, Ax Em and AxNum, showing thus t h a t G i s a subtheory o f MKT. I n t h i s way, we prove t h a t a l l theorems i n Chapter 2, except those o f 2.1.4 a r e a l s o theorems o f MKT. 3.1.2.1
0 E V.
THEOREM,
PROOF, By Ax I n f , t h e r e i s a s e t a # 0 such t h a t : x E a A z c x + But, s i n c e a # 0, t h e r e i s an x E a , and 0 5 x . Therefore, IT€ a and hence 0 E V. zEa.
3.1.2.2 PROOF,
-
{A) E V
THEOREM,
We have t h r e e cases:
.
(1) A 9 V . Then { A ) = 0 E V, by 3.1.2.1 and Ax Pow. (2) A = 0. Then { A } = {Ol = P O E V , by 3.1.2.1 ( 3 ) A E V A A f 0. We a p p l y Ax Rep. We have, s i n c e A E V ,
(*)
Vx(x
E {A)
-
x
Then A = U { Z : Y E Z A Z
Since A # 0, t h e r e i s a Y E A . Therefore, we have X
E {A}
-t
= A).
THEOREM,
A C -b
A
+
E
PROOF, L e t C be d e f i n e d by C = that C E Y :
now show t h a t
U
by (*).
.
V .
{{XI
:x € A ) .
We a p p l y Ax Rep t o show
L e t us suppose t h a t y E C ; t h e n q = { X I w i t h x E A C - b. q E C } . Hence, s i n c e b E V , by Ax Rep, C E V .
xEy A
We
{A}),
1 Y ( Y E A A X = U (Z : Y E Z A Z E A ) ) .
A p p l y i n g Ax Rep t o t h i s and A E V , we o b t a i n { A } E V. 3.1.2.3
E
Also, q = U ( z :
C=A:
L e t x E A , then, by 3.2.1.2, {XI E C, and t h u s x E U C i we have shown On t h e o t h e r hand, i f x E U C, then x E q E C , f o r some y; but, then q = { z } w i t h z E A ; t h e r e f o r e z = x E A ; t h u s we show t h a t U C C A , and t h e r e f o r e A = uC.
- U C.
A C
Since C E 3.1.2.4
V, by Ax Un, A
THEOREM,
(A,
E
V.
81
E
v.
127
AXIOMATIC SET THEORY
PROOF, We have two cases:
[A,B)
(1) A q V o r E
8
Y by 3.1.2.2.
( 2 ) A E V and B E V . (2.1) A = 0.
Then { A , B } C - { A } o r { A , B } C- { B } and we o b t a i n
q V.
We have t h r e e subcases:
Then { A , B } = { O , B } c _ P B E Y .
The r e s u l t i s o b t a i n e d by Ax Pow and 3.2.13. (2.2)
8
0.
=
S i m i l a r as (2.1).
(2.3) A # O a n d 8 + 0 . We h a v e t h a t A E P A and B E P B . .Thus, { A } E P P A , ( 8 ) E P P B . On t h e o t h e r hand, 0 E PB,and hence 1 = {O} E p p B . {O,{AI}, Therefore, { O , { A } l Z P P A , {1,{8}) c P P B . By Ax POW and 3.2.1.3, {1,{8)} E V . Also, {0,1l = P P O E V , by Ax Pow. Now we a r e ready t o a p p l y Ax Rep. L e t C = ( { O , { A I } , {l,{Blll; suppose z E C . Then z = i O , { A l l o r Since { A ) # 1 and ( 8 ) f 0, we have t h a t z = {l,{B}}.
0 and
E
z
1E z
-
-
z
=
CO,IAll
,
z = {l,{BH.
Therefore, we o b t a i n
wx(xEC Since P P O
E
+
1 q(y E P P O A x =
u (2:
V , we o b t a i n by Ax Rep, C E V .
Now, { A , B l C u u C, because A E C A I {l,{B}} E C. Hence b y Ax Un and 3.1.2.3, 3.1.2.5
ZEC A YEZl).
E
{O,{AIl
E C, and 8 E
{ A , B } E V.
181 E
THEOREM,
(i) AUB E V -
(ii)A E V +
A
A,B E V .
u {B} E
Y.
so we j u s t p r o v e . ( i ) . ( i i ) i s o b t a i n e d e a s i l y from (i), We have, A , B 4A u B; thus A u 8 E V i m p l i e s A , B E V. On t h e o t h e r hand, A , B E V i m p l i e s t h a t AuB = u { A , B ] . Therefore, A U B E V , by Ax Un and 3.1.2.4. PROOF,
3.1.2.6
PROOF,
METATHEOREM,
G A u Aubtheohq
By Ax Class, Ax Ext, 3.1.2.1,
06
MKT
and 3.1.2.5
. (ii).9
NOW, we can use a l l theorems i n Chapter 2, except those i n 2.1.4.
ROLAND0 CHUAQUI
128
3.1.2.7
THEOREM,
(i) A E Vt.
U
( i i ) A E V-PA
A E V. E
V.
A E V + u A E V i s Ax Un. On t h e o t h e r hand, s i n c e PROOF OF (i). Ac P u A (2.1.3.2 vi), b y A x P o w a n d 3 . 1 . 2 . 3 , u A E V + A ~ V . PROOF OF ( i i ) . A E V + P A E V A = U P A , by Ax Un,PAE V + A E V. 3.1.2.8
THEOREM,
V q V.
,
i s Ax Pow.
On t h e o t h e r hand, s i n c e
-
--
PROOF, Suppose t h a t V E V and l e t A = { x : x $ X I . We have: x E A B u t A c V ; thus, by X E V. Therefore, A E A A $ A A A E V. A E V . Hence, we g e t A E A A $ A , a contradiction. 3.1.2.3,
x$ xA
3.1.2.9
THEOREM,
( i ) A q V A B E V + A ~ B B V .
(ii) 2, 6
E
V.
-
PROOF OF (i).We have, A C ( A % 8 ) then A E V,
U
( B n A ) ; hence i f A
2,
8E V
PROOF OF ( i i ) . By ( i ) and 3.1.2.8. The g e n e r a l i z e d d i s t r i b u t i v e laws f o r g e n e r a l i z e d Boolean o p e r a t i o n s a r e e q u i v a l e n t t o an Axiom o f Choice, so t h e r e i s no hope of p r o v i n g them However we have t h i s s p e c i a l d i s t r i b u t i v e l a w and i t s dual. i n MTK
.
3.1.2.10 THEOREM, n { u X : X E a ) = u {n Y :
8 = { q :y Y E
C_U a
A W x(xEu
+
x n q
# 0))
+
81.
x n q + 0)). Suppose PROOF, Assume 8 = { q : y C U a A W x ( x E a TFen z E r- q' f o r some q' € 8 . NOW, i f x E a , f i r s t t h a t z E U {n q : q E 8). then x n y ' f 0 (because q ' E 8). Therefore, t h e r e i s a t E x A X E q ' . Then z E t ; i.e. z E t E x and z E U x . Thus, z E U x f o r a l l X E U , and, hence, z E n {u x : x E u } . We have proved, u { n y : q E B ) c_n {u x : x E a } . +
Suppose, now, t h a t z E n {u x : x E a ) . L e t q' = {t:tE u u A z E ,t}. We have, q' C_ U a E V ; t h u s y ' E V. Assume t h a t t h e r e i s an x' E a w i t h x'ny' 0. We would have t h a t : W t ( t E x ' + z 9 t). Hence, t h i s would i m p l y t h a t z 4 U x ' , c o n t r a d i c t i n g t h e h y p o t h e s i s on z. T h e r e f o r e x n y ' + O f o r every X E U , and q ' E 8. From t h e d e f i n i t i o n o f q ' , we g e t t h a t z E n q'. - u i n q : yE81. Hence z E u { n q : q E 8 1 , and n {u x x E a 1 C
AXIOMATIC SET THEORY
129
PROBLEMS
{x : a
+ x) 4 V .
1.
Show t h a t ,
2.
Show t h a t t h e f o l l o w i n g formula i s n o t t r u e i n general
-
Bqv -,%BEE. 3.
Show, P ( A U B ) = PA U P B
4.
L e t a = Ic,d} i n g t o 3.1.2.10.
5.
Formulate and prove t h e dual o f 3.1.2.10.
3.1.3
,c
=
A C - 8 V 8 C -A .
{ e , b ) , and d
=
Cg,hl.
F i n d ( e U h ) n ( g u h ) , accord-
R E L A T I O N S AND F U N C T I O N S i
The C a r t e s i a n p r o d u c t was d e f i n e d i n 2.1.2.4. THEOREM,
3.1.3.1
A,B
AxB E V -
PROOF, Suppose A x E E V Ax Un, A , B E V.
,A
f
For i t , we have,
VV A = 0
E
0 f 8.
Then A,B
v
8 = 0.
5u
u A
x
8.
Thus by
.
On t h e o t h e r hand, i f A = 0 o r 8 = 0, A x B = 0. Also, s i n c e A x B 5 P P ( A u B ) , A , B E V i m p l i e s by Ax Pow, A X B E V . Thus, we have proved t h e converse. We, now, t u r n t o t h e problem o f d e t e r m i n i n g when r e l a t i o n s , f u n c t i o n s , and superclasses a r e sets. T h i s w i l l g i v e us an e q u i v a l e n t f o r m u l a t i o n o f Ax Rep t h a t i s more usual t h a n T a r s k i ' s .
R
E
3.1.3.2 V.
THEOREM,
R c - V x V A DRE V A Wx(x E DR
+
R*{x)
E
v)-
PROOF, L e t R C V xV ,DRE V , and R*(x) E V , f o r x E D R . Then We have, R*{x} x 1x1 = RI{x}. Also, Rlx U R*{x) x {x} E V , by 3.1.3.1. {( x , x ) ) E V . L e t G be t h e f u n c t i o n d e f i n e d on DR , by G'x = R ( x U I ( x , x ) ) , and l e t 8 = { ( x , x ) : X E D R ] . Since B C P P D R , B E V . We now a p p l y Ax Rep w i t h A = G*(DR) t o show t h a t A E V.- Suppose t h a t z C A ; then z = G'x, x E D R . We have t h a t ( x , x ) E ( G ' x ) n B and t h a t i f q f x, then ( x , x ) e! G'y. Thus, G'x i s t h e o n l y element of A t o which ( x , x ) belongs. Using Ax Rep, s i n c e 8 E V , we o b t a i n A E V. T h e r e f o r e U A E V , by Ax Un. Now, R C u A , because i f ( q , x ) E R , then ( q , x ) E Rl{x} C - G'xEA. Thus, by 3.1.2.3, REV.
.
3.1.3.3
COROLLARY
m
D F V ( F ) -+ Flu
, F*uE
V.
ROLAND0 C H U A Q U I
130
Suppose D F V (F ) . Then Flu C V x V , D ( F l u ) E V ( s i n c e PROOF, D ( F l u ) c a E Y ), and i f x E D ( F l u ) , t h e n F * { x } = ( F ' x } E V (by3.1.2.2). - U U Flu; thus, b y 3.1.2.3, Therefore, by 3.1.3.2, F l u E V . Also, F*u C F*u E V. 8 COROLLARY SCHEMA,
3.1.3.4
v X(X€U
F(x)
E
{F(x) : x E u 1
E
-+
v)
-+
L e t F be a unmy a p W o n ; then
[ F ( x ) : xEa1 , u W x ) : x E u 1
,
V.
PROOF, L e t F ( x ) E V f o r XEU. I f we t a k e R = [ F ( x ) : x e u ] , i t i s Thus, R E V . easy t o see t h a t R s a t i s f i e s t h e h y p o t h e s i s o f 3.1.3.2. Now, U I F ( x ) : x E u 1 C u u [ F ( x ) : x E u ] and { F ( x ) : x E u 1 EF(x) : x E u 1 . T h u s , t h e o t h e r c o n c l u s i o n s f o l l o w by 3.1.2.3.
5
P
U
3.1.3.4 a s s e r t s t h a t i f A i s a s u p e r c l a s s o f s e t s and coded b y a set, then t h e c l a s s A d e f i n e d by A = ( x : A ( x ) } i s a set. Ax Rep can be r e p l a c e d by t h e c o n j u n c t i o n
1. D F V ( and
F) A
A
E
V + F*A E V ,
2. A , B E V - + C A , B ) E V . We have a l r e a d y seen t h a t 1 and 2 can be proved u s i n g Ax Rep (3.1.3.3 On t h e o t h e r hand, assuming 1, 2 and t h e h y p o t h e s i s o f Ax and 3.1.2.4). Then A = {FA(Y): Y E B } -and Rep, we d e f i n e t h e o p e r a t i o n FA as i n 3.1.2.3.
B E V . Thus, we o b t a i n A i m p l y Ax Rep.
E
V. Therefore, 1 and 2 ( p l u s t h e o t h e r axioms)
PROBLEM
(+) 3.1.4
MKT I S A SUBTHEORY OF B .
The purpose o f t h i s s e c t i o n , which i s a c o n t i n u a t i o n o f 2.1.4, i s to deduce t h e axioms o f MKT i n B . Since Ax Class and Ax E x t a r e common t o b o t h t h e o r i e s , i t i s enough t o show t h a t Ax Un, Ax Pow, Ax Rep, and Ax I n f a r e theorems o f B. A l l theorems o f t h i s s e c t i o n (as o f 2.1.4) a r e d e r i v a b l e from B.
AXIOMATIC
S E T THEORY
131
U We extend t h e d e f i n i t i o n of I$ t o a l l formulas @ o f 1, which
3.1.4.1.
may i n c l u d e t h e c l a s s i f i e r and d e f i n e d symbols. R e c a l l t h a t (W X 6)' is tl X(X C U * 0 U ) and ( 3 X 0 ) ' i s 3 X ( X C- U A 0 U). We use t h e c l a s s i f i e r as t h e onTy p r i m i t i v e vbto. (see 2.1.1. and add t h e f o l l o w i n g clauses t o ( i ) ( i v ) o f 1.2.1.4 i n o r d e r t o d e f i n e r h and f o r a l l terms T and formulas 0 o f L. ( v ) I f r i s a variable, 'r (vi) If
T
is
I x : $1, then
i s r itself. 7'
U i s { x : x G U A $ }.
( v i i ) I f u i s a term and A i s a s p e c i a l c l a s s d e f i n e d by A = u , then Au U = u
i s t h e s p e c i a l c l a s s d e f i n e d b y A'
.
( v i i i ) I f u i s a term and F i s an n-ary o p e r a t i o n d e f i n e d by F(XO,..,Xn.l) U U = u, t h e n F u i s t h e o p e r a t i o n d e f i n e d by F (Xo Xnm)l = u
-
.
,...,
-
( i x ) I f 0 i s a formula and B i s an n-ary n o t i o n d e f i n e d by B(X@..,X 0 , t h e n B u i s t h e n - a r y n o t i o n d e f i n e d b y B U (Xo, Xn-l) 0 u
...,
(x) I f r i s U {A
$1,
t h e n (17 { A : $1)'
is
Ix: 3 ! A$
A 3A(I$A x e A ) l U
( x i ) If 0 i s a vbto. of t h e second c l a s s d e f i n e d by OX
Cx : e l , t h e n (OX
0'
*
{T
Jn-1
: $3)'
09
i s d e f i n e d by ( { x :
( x i i ) I f I$ i s an atomic formula o f t h e f o r m B(ro,...;rn-l), U an n-ary n o t i o n and T ~ , rnmla r e terms, t h e n $ i s B ( r
...,
.
-)
nl
*
IT
,Xn- 1
:I$) =
where B i s
;,...,
rU n-l )
The main new p a r t o f t h i s d e f i n i t i o n i s ( v i ) . I t i s j u s t i f i e d because as was p o i n t e d o u t i n 1.2.1.4, U i s t a k e n as o u r new universe. Thus, t h e c l a s s { x : $1 r e l a t i v i z e d t o U should be t h e subclass o f U d e f i n e d b y $u , U i.e., { x : $ A x E Ul. The f o l l o w i n g two theorems i n d i c a t e s t h e r e l a t i v i z a t i o n o f d e f i n e d ope r a t i o n s and vbtos. 3.1.4.2
THEOREM,
132
ROLAND0 C H U A Q U I
( v i ) (%B)'
u%B.
=
( v i i ) {a,bIU = {a,b> n U . The p r o o f i s easy. THEOREM SCHEMA,
3.1.4.3
L e t Q be a
60munLLea
and r a tm. Then,
PROOF, I s h a l l prove ( i ) l e a v i n g t h e s i m i l a r p r o o f s o f ( i i ) and ( i i i ) t o t h e reader. L e t Q be a formula, r a term.
PROOF OF ( i ) .
By Def. 2.1.3.4
(i),
Hence, by 3.1.4.1,
..
xo ,. ,x n-1 IT x
=
r
U
: $1
A 9')).
Since t o 3 X0". U QXO' ' * . ,Xn-1
3 Xo
U
= { x : X G U A 3 Xo,
... 3 Xn-l(Xo ,...,Xn-l
3 XnJX [X0 n
=
7
U
xo,
***
u, ..., Xnml
PROOF OF ( i v ) .
We have,
...3 Xn-l(Xo....Y 5U A x
[ X o n U,
=
Xn-I
r
u A +u )
...,X n-1 n U ]
5
A
i s equivalent
A
n U l ), t h e c o n c l u s i o n f o l l o w s .
AXIOMATIC SET THEORY
= { X : 3 ! A(A
133
- U A @') A 3 A(A C- U A @' A x E A ) A c
xEU}
,
- U A 4'3. c
= UIA : A
The n e x t theorem i s an i m p o r t a n t s t r e n g t h e n i n g o f Ax Ref. THEOREM SCHEMA. (B ) . Then
3.1.4.4
A dtee and n o t c o n t a i n i n g u. W A(@
+
L e t @ be a @fmvnuRa od d: WLLh a t monR
3 u(u u c - u A @:
[ A n u l ).
P R O O F I L e t @ be any f o r m u l a o f L s a t i s f y i n g t h e h y p o t h e s i s o f t h e theorem and suppose 4. There i s an e q u i v a l e n t o f d: @*,s a t i s f y i n g t h e P' same c o n d i t i o n s . Then, we have @*. By Ax Ref, t h e r e i s a s e t u, t r a n s i t i v e , such t h a t
@*' [ A
n u]
.
We s h a l l see t h a t I$*' i s e q u i v a l e n t t o QU, p r o v i n g , thus, t h e theorem. Since a l l d e f i n i t i o n s a r e based on t h e c l a s s i f i e r , we have t o worry o n l y about it. I n o r d e r t o o b t a i n I$*,we have t o r e p l a c e p a r t s o f t h e f o r m IIX [ I x : 0) 1 by 3 V ( W x ( x E V @ A 3 U x E U ) A $x [ VI ). Thus, we should
-
prove t h a t ( $ x t { x : @ l l ) u
1LX [ v l
prove f o r $ = X E Y.
We have,
V E Y ) . Also, ( { x : $ ) E VEY))' i s 3v(v c u~
0'1,
i s equivalent t o ( 3 V ( W x ( x E U - @ A 3 U
I t i s enough t o prove t h i s f o r a t o m i c $.
then VCu. -
Y)u
v
{ x : @ I E Y i s e q u i v a l e n t t o 3 U ( V = i x : @}A i s { X : X E U A @ ~ ) E and Y ( 3 V ( V = {x:@}A
= { x : x ~ UA
Thus, we o b t a i n t h a t
equivalence t o 3 V ( V = { x : X E U A
I x : X E U A @'I
E
Y.
-
@'I
4'1
A
v
E
(*)
@ A 3 U x E U ) A V E Y)'.
3V(VcuA Wx(xcuAxEV*@
U
A
Since u i s t r a n r i t i v e , V c and x E V i m p l y Thus, we o b t a i n t h a t (*) 7s e q u i v a l e n t t o
x
u
3 V(V C u A
to 3 V(V =
{x:
v x(xEV
@ }A V E y)'.
-
Y).
But i f
v
= {x: XEU A
( 3 V ( V = { x : @IA V E Y))' is A V E Y ) , which i s e q u i v a l e n t t o
Thus, we j u s t have t o prove t h a t ( 3 V ( V = a l e n t t o 3 V ( Wx(xEV
x€U)A
As an example I s h a l l
I x : $1
A V E Y)'
i s equiv-
This l a s t formula i s :
x 5 u A x E u ) A V E Y). - u, and X E U i m p l i e s x
C
- u. C
Cpu A x E u) A V E Y), which i s e q u i v a l e n t
Wc now proceed t o t h e proof of Ax Un, Ax Pow, Ax Rep, and Ax I n f i n B. F i r s t , a few lemmas t h a t a r e f u r t h e r s t r e n g t h e n i n g s o f Ax Ref. 3.1.4.5
THEOREM SCHEMA,
( B ) L e t @ be a &mvnuRa
06
d: WLLh at m o n t
134
ROLAND0 CHUAQUI
PROOF, L e t 0 be a formula s a t i s f y i n g t h e h y p o t h e s i s and suppose A s a t i s f i e s 0. Then by 2.1.4.3 and 2.1.4.2,
orem.
0 A W x Wy(x,y
.
Applying, now, 3.1.4.4
E
V - . Cx,yl E V ) A 0
t o t h i s formula i n s t e a d o f
V .
6
0 we
o b t a i n t h e the-
3.1.4.6 THEOREM SCHEMA, (B) 1e.Z 9 be a 60munLLea 06 1: w L t h A and 8 dhee and .in which u d o a not VCCWL. Then.
PROOF, L e t $I be a f o r m u l a s a t i s f y i n g t h e h y p o t h e s i s o f t h e theorem. L e t 11, be t h e f o r m u l a 3 A' 3 8'(@ [A',8'1 A,B
n X = [A',B'l)
Suppose @ f o r c e r t a i n c l a s s e s A and 8;then $ w i t h X = [A,B] a p p l y 3.1.4.5 t o t h i s $ and o b t a i n a t r a n s i t i v e s e t u such t h a t
But
1L;
[Xnul
3 A ' 38'(A',B'
.
We now
is C U A ~ : 8[A',B' -
I t i s easy t o see t h a t f o r A',8' [A',8']. Also, s i n c e X = [ A , B l
]
A X n u = [A',8'IU).
C_ u s i n c e u s a t i s f i e s (*) [ A ! 8 ' l U =
Xnu = [ A , B ] n u = (A x 10) u 8 x ( 1 1 ) n u = ((A x 10)) n U)
u ( ( 8 x 111) n u )
.
Now, if x E A and (x,O) E u, then, s i n c e U u C u, XEU. Thus ( A x 1 O l ) n u C ( A n u ) x E O l . On t h e o t h e r hand, i f x E A n C , t h e n (x,O) E A x I 0 1 A l s o 7 s i n c e 0 E u and p a i r s belong t o u, (x,O) E u. Thus, ( A n u ) x 101 5 (A x 10)) n u. T h e r e f o r e (A x IOI) n u = ( A n u ) x 01. S i m i l a r l y , we prove t h a t ( 8 x 111) n u = ( 8 n u ) x 111. Thus,
.
Xnu = [ A n U ,
Therefore, we o b t a i n ,
$I:,~[
B n u ] = [ A ' 8'1
Anu, g n u ]
.
H
.
A X I O M A T I C SET T H E O R Y
135
THEOREM SCHEMA ( B ) . L e A $ be a dohmuRa 3.1.4.7 A , B /~heeand i n wkich u doecr n o t UCCWL. Then, W A WB(B E V A $
-+
Iu(u u c -u A
BEu A
w a h at m w t
0 6 1:
@:,&Anti, B ] )
.
PROOF, L e t (p s a t i s f y t h e h y p o t h e s i s of t h e theorem. Suppose t h a t A , B s a t i s f y 4 w i t h B E Y . Then by 2.1.4.3, CBI E V and 8 E ( 8 ) . We have t h a t t h e f o r m u l a $, d e f i n e d by
(1) 3 B'($*[ 8'1 A V x ( x E a
-
i s s a t i s f i e d by t h e g i v e n A and a =
a t r a n s i t i v e u such t h a t ,
(l),is:
(2) 1 B ' ( B '
'9
A,a
x
= S')),
CBI.
By t h e p r e v i o u s theorem we have
[ A n u , a n ~ l . T h i s formula, a c c o r d i n g t o
w x(x c- u A x E a n u
u A @ i , B [ A n u ,€3'1 A
Since u i s t r a n s i t i v e , x E a n u i m p l i e s implies x C - u. Thus, ( 2 ) i s e q u i v a l e n t t o ,
(3) 3 B'(B' c - u A $'
A, 8
x c u , and s i n c e
[ Anu, B ' ] A W x ( x E a n u
*x
Thus, a n u = {B'I, and, hence a n u c o n t a i n s one element. a = CBI, and, hence, BEu A
-
$t,B
[ A ~ u ,B ]
x c - u A x=B')
8' C - u,
x=B'
= 8')).
Therefore a n u =
.
We say t h a t A i s n u p W a n n U u e i f A i s t r a n s i t i v e and W x W y ( x 5 x € A ) . I f we t a k e uP*A = u IP q : q E A 1 , we can say t h a t A i s supert r a n s i t i v e .if U A U u P * A C It i s c l e a r t h a t - A , (see Problem 5 i n 2.5.1). V i s supertransitive. YEA
-+
3.1.4.8
THEOREM SCHEMA ( B ) .
dhee and i n which u d v u n o t A
WB(B
E
V A
0
+
3
u(U
VCCWL.
u
L e R (p be. a 6omunlLea w a h at m o n t A , B Then,
UUP*u C -
u A
BEu A
$i,B[ Anu,
81).
PROOF, Suppose Q, i s a formula s a t i s f y i n g t h e h y p o t h e s i s o f t h e theorem t h a t h o l d s f o r A and 8 , and, a l s o , B E V . Then, we have, B E V A Q, A
w
X W Y ( XC YE
V-
XE
V).
Applying, t h e p r e v i o u s theorem t o t h i s formula, we o b t a i n a t r a n s i t i v e s e t u such t h a t B E u A $'
A,
[ A n u , B ] A W x Wq(x,y
c -u
A
x c - YEU
-t
xEu).
136
ROLAND0 C H U A Q U I
.
Since u i s t r a n s i t i v e , x C_ q E u i m p l i e s x , q supertransitive. 3.1.4.9
C u.
Hence u i s a l s o
THEOREM,
Suppose U i s t r a n s i t i v e and B E U . We have (u B)' = Since U i s t r a n s i t i v e and B c - U, y E B Thus, i m p l i e s y E U and, hence, q C_U. S i m i l a r l y , x e q E B i m p l i e s X C U . PROOF OF ( i ) .
{x : 3 q(y C - U A x E y E B ) A x E U}. (U
B)'
= U 8.
The p r o o f o f (ii) i s similar. PROOF OF ( i i i ) .
{x: x c -
B A XEU}.
L e t U be s u p e r t r a n s i t i v e and B E U . Since U i s t r a n s i t i v e and B E U , x
The s u p e r t r a n s i t i v i t y i m p l i e s t h a t x
-8
C
.
yields xEU.
The p r o o f o f ( i v ) i s s i m i l a r t o ( i i i ) .
BEu.
-B
U
We have (PB)
iff x C - 8. Hence (P 8)' = PB.
C
.
PROOF, L e t €3 E V . By 3.1.4.8, t h e r e i s a t r a n s i t i v e u such t h a t Therefore, by Ax Sub, U 8 E V C U u C u. Hence, B C - u and U B 3.1.4.11
THEOREM (B
PROOF, L e t 8 E V . t h a t B E u . Hence PB C - u. 3.1.4.12 Wz(zEA+
By 3.1.4.8, Therefore,
THEOREM (B I( ~ € 2 -
B E V + PB
Ax Pow).
V.
t h e r e i s a s u p e r t r a n s i t i v e u such b y Ax Sub, PB E V .
Ax Rep).
z = X I ) ) ) -, A E V .
E
B
E
=
YAWx(xEA +3 g ( g E B A
PROOF, L e t 8 and A be as i n t h e h y p o t h e s i s of t h e theorem and B' = B ~ U A . By Ax Sub 8' E V . Also,
let
AXIOMATIC SET THEORY (I) V x ( x € A
+
3 y(yEB' A W z(zEA + ( q E z
We have, B' E V A 8' C - u A.
-
137
z
=
x)))).
Therefore, by 3.1.4.8,
t h e r e i s a super-
t r a n s i t i v e s e t u such t h a t B' E U A B' C u i s transi(u ( A ~ U ) ) ~ Since . t i v e , by 3.4.1.9, t h i s can be s i m p l i f i F d t o , (2)
B ' E u A 8' c - u (Anu)
-
.
We s h a l l prove t h a t Anu = A. L e t x E A ; t h e n by (I) t h e r e i s a ~ € 2 3 ' such t h a t W z ( z E A (yEz z = x)). But, from ( 2 ) , we o b t a i n t h a t t h e r e i s a z E A n u such t h a t y E z . Hence z = x , and x E A n u . Therefore, A = An u C u E V. Hence, by A x Sub, A E V . +
-
3.1.4.13
THEOREM (B I-
PROOF, By 2.1.4.2, V# 0A
Therefore, by 3.1.4.8
'v By 3.4.1.2
Ax I n f ) .
2.1.4.3
3 a(a
f
and 3.1.4.11,
vcu
V A u P*VC -
0A
a c - LJ a A
LJ P*a
we have
v.
t h e r e i s a s u p e r t r a n s i t i v e u such t h a t ,
# OUA
v u c- (u
V ) U
A ( u P * V yc
v'.
and 3.4.1.9. uf 0 A u C - u u A L J P*u C- u .
Thus we can t a k e our i n f i n i t e s e t 3.1.4.14 PROOF,
METATHEOREM, b y 3.1.4.10,
c(
as u.
MKT A a dubtheoxy ad B .
3.1.4.11,
3.1.4.12,
and 3.1.4.13.
9
c - a).
CHAPTER
3.2
S t r u c t u r e o f well-founded c l a s s e s
WELL-FOUNDED CLASSES I N MKT.
3.2.1
The f i r s t two theorems o f t h i s s e c t i o n a s s e r t t h a t t h e well-founded o f n a t u r a l numbers i s a s e t and t h a t t h e t r a n s i t i v e c l a s s w (Def. 2.4.1.1) c l o s u r e o f a s e t i s a set. 3.2.1.1
THEOREM,
w E
V.
PROOF, L e t Z = n { X : 0 E X A W y ( y E X t h e s e t whose e x i s t e n c e a s s e r t s Ax I n f . i.e. a # 0 A W x(xea
+
{ q }E X ) } and l e t a E V , be a satisfies,
-+
( ly xEyEa A W z(z
We have t h a t t h e r e i s an x E a .
Since 0
f o r a c e r t a i n y ; t h e r e f o r e 1x1 c XEYE~, proved, then, t h a t z C - a. Hence Z E V -
.
Define, now, by r e c u r s i o n (2.4.1.4)
5 x,
2x
+
OEa.
y and, thus
zEa))
.
Also, i f x E a , then E a. We have
{XI
t h e f u n c t i o n F by
( F ) A F'O = 0 A F' S
K
= {F'K}
.
F has t h e f o l l o w i n g p r o p e r t i e s
(1) F'K F'K
E
F'S
K
.
By t h e d e f i n i t i o n o f F, i t i s c l e a r t h a t F ' K {F'KI = F'SK.
E
( 2 ) F'K = F'A
+
K
=
X
E
V, f o r a l l
K;
therefore
.
PROOF OF (2) BY INDUCTION, L e t A = { K : W A ( F ' K = F'X K = A). 0 E A , because F'O = 0 and, by (l), F'S X f 0. Suppose t h a t K E A and F'S K = FIX; then F'X f 0 (by (1)) and, t h e r e f o r e , X = Sp f o r a c e r t a i n p. Thus, Since K E A , ~ = and p { F ' K ~ = F'S K = F ' S p ={F'p}; t h i s i m p l i e s , F'K = F'v. hence, S K = A +
.
From ( 2 ) , we o b t a i n ,
138
139
AXIOMATIC SET THEORY
(3) F
i s a function.
PROOF BY
0
E
F'S
F-l*.Z. K
=
INDUCTION OF ( 4 ) .
Suppose t h a t
F-l*Z,
K E
1 x 1 and { X I E Z ; i.e.
A E V*
THEOREM,
PROOF, By 2.5.1.10 t o prove t h a t T u E V .
x
z.
Thus,
TAE V .
( i ) , i f T A E V , t h e n A E V.
6
Therefore t h e f u n c t i o n
We deduce f r o m 3.1.3.3 'Em}
f o r a certain X E
t o ( 3 ) and ( 4 ) , we o b t a i n w E V .
i s such t h a t d'w = uwa f o r e v e r y w E w .
T a = u (u'a:
= 0, we have
Thus, we j u s t have
I t i s e a s i l y proved by i n d u c t i o n t h a t U ' a E V ,
e v e r y W E U (see 2.5.1.8).
wew)
t h e n F'K =
and F-'O
S K E F"*Z.
Since Z E Y , a p p l y i n g 3.1.3.3
3.2.1.2
z
Since 0 E
and 3.2.1.2
= U d*w.
t h a t 1;*w
Therefore, T a e V .
d e f i n e d by
.
E
V.
for
(u"a :
But, by 2.5.1.9,
We can d e f i n e a f u n c t i o n T by
T = ( T x : x E V ) By 3.2.1.2, we have t h a t f o r x E V , T ' x = T x . Therefore, can i d e n t i fy t h e f u n c t i o n T w i t h T. Thus, t h e f u n c t i o n T w i l l be denoted by t h e same symbol as t h e o p e r a t i o n T a n d we can use t h e n o t a t i o n d e f i n e d f o r f u n c t i o n s w i t h t h i s symbol. For instance, T * A = { T x : x E A 1 .
We now p r o v e t h a t W M K m (2.5.3.13).
3.2.1.3
(Def. 2.5.3.1)
i s a model o f MKT, and hence, o f
THEOREM,
(i) A E W + U A E W .
(ii) A E W+PA E W.
(iii) w
w
(F) A A
(iv) w
E
W.
E
w+
F*A
E
W.
PROOF, A l l these statements a r e e a s i l y proved u s i n g 2.5.3.11 and t h e corresponding axioms o f MKT. ( i ) i s o b t a i n e d u s i n g t h e t r a n s i t i v i t y of W (2.5.3.2). I n o r d e r t o prove ( i i ) , we must show t h a t W i s s u p e r t r a n s i t i v e : - x E W , t h e n q C- W by 2.5.3.2, and y E W by 2.5.3.11. Thus, x E W let q C
140
ROLAND0 C H U A Q U I
implies Px
5W .
Hence b y 2.5.3.11
and Ax Pow we o b t a i n ( i i ) .
I n o r d e r t o prove ( i i i ) , we j u s t n o t i c e t h a t , by 3.1.3.3, A E V Then use 2.5.3.11. ( i v ) i s o b t a i n e d from 2.5.3.12, 3.2.1.1,
F*A E V. 2.5.3.11.
--t
and
From t h i s theorem, we can e a s i l y see t h a t 4 i s a theorem o f M K T R i f
W
and o n l y i f 4 i s a theorem o f M K T . MKT. I n t h i s way, we prove t h a t W i s showing t h e r e l a t i v e c o n s i s t e n c y o f Ax c o n s i s t e n t , then M K T t Ax Reg i s a l s o
Thus, we can s t u d y M K T R i n s i d e an " i n n e r model" o f M K T t Ax Reg, Reg w i t h M K T , i.e. i f M K T i s consistent.
The n e x t theorem, e a s i l y o b t a i n e d from 2.5.3.14, p r i n c i p l e s which w i l l be u s e f u l l a t e r .
(viii)
v x
vV ( H ) 3'
W ! F ( V ( F )A
PROOF OF (i). Let ' V ( H ) .
(1)
H(X,Y) =
Using 2.5.3.16
(2)
H'( Y * ( g l
w x ( x E W + . F'x
g i v e s a few r e c u r s i o n
=
HI ( x , F I T x ) )
D e f i n e t h e b i n a r y o p e r a t i o n H by
:gEX)
.
we o b t a i n a unique o p e r a t i o n F such t h a t
F ( X ) = H(X, [ F ( g ) : q ~ x ) ]
, for
all
x c- W .
D e f i n e t h e f u n c t i o n F , by
F = (F(x) : x
E
W).
We s h a l l prove by i n d u c t i o n on W t h a t ' V ( F ) and W x ( x E W + F'x = L e t A = { x : x E W A x E D F A F'x = H'Flx = F ( x ) l . Suppose x C A. We have, by ( 2 ) , F ( x ) = H ( x , [ F ( g ) : Y E X I ) . Since x C _ A , F ' g = F ( q ) f o r a l l g E x . Hence, by ( l ) ,
H'Flx).
(3)
F ( x ) = H ( x , [ F ' g : g E x ] ) = tl' ( F ' q : q € x ) = H I F I X .
141
A X I O M A T I C SET T H E O R Y
Using 3.1.3.3,
(4)
F l x E V . Since
FIX = F ( x )
and
X ED
V V ( H ) , H ' F l x E V . Therefore,
F.
( 3 ) and ( 4 ) prove t h a t x E A . Hence W C A and t h e e x i s t e n c e o f F i n ( i ) i s proved. The u n i c i t y f o l l o w s by i n d u c t i o n .
-
( v i i i ) can be deduced from ( i ) . I s h a l l o n l y prove t h e e x i s (ii) tence p a r t of some o f them as examples, l e a v i n g t h e r e s t t o t h e reader. The u n i c i t y i s always proved by i n d u c t i o n . PROOF OF ( i i ) .
K'x
L e t ' V ( ff).
D e f i n e t h e f u n c t i o n K by ' V ( K )
-'
= H'Dx-'. By ( i ) , f i n d F such t h a t W V ( F ) and F ' x = K'Flx f o r Hence, F ' x = f f ' D ( F I X ) = H'F*x.
PROOF OF ( i v ) .
Let ""V(
H).
D e f i n e t h e f u n c t i o n K by
x
E
W.
V V ( K ) and
H' ( D x , D x - ' ) . F i n d F, by ( i ) , w i t h W V ( F ) and F ' x = K'Flx. FIX = f f ' ( D ( F l x ) , D ( F l x ) - ' ) = H' ( x , F*x). H
K'x =
and
Then,
( i i ) i s s p e c i a l l y u s e f u l . The n e x t theorem g e n e r a l i z e s ( i i ) somewhat by n o t assuming t h a t DH = V . The o t h e r p r i n c i p l e s can be s i m i l a r l y generalized.
K =
PROOF, L e t D H V ( H ) and P D H - l C D H . V x % D H ) . We have V ( K).
D e f i n e t h e f u n c t i o n K , by
H u (I01
Therefore, t h e r e i s a unique F such t h a t 'V(F) and F ' x = K'F x f o r We show b y i n d u c t i o n t h a t F ' x = H'F*x f o r x E W , i.e., we have t o show t h a t F XEDH f o r X E W . L e t B = { x : x E W A F*xEDff 1 and assume t h a t x C- 8. T h e r e f o r e F l y = K'F*y = H'F*y f o r q E x . Thus, F l y E D H - l f o r xEW.
y E x , and, hence F x C DH-'. By t h e hypothesis, F * x E D H . Thus, we prove - 8 and, hence; t h e e x i s t e n c e o f F. The u n i c i t y i s proved by inducthat W C tion.
PROOF, By 3.2.1.4,since
P WC - W and D F - ' C- DH-'.
=
3.2.1.7 REMARK, Suppose t h a t H i s a u n a r y o p e r a t i o n such t h a t H ( x ) E V f o r X E Y . We can t h e n d e f i n e t h e corresponding f u n c t i o n H by H = ( F ( x ) : W i t h H we o b t a i n r e c u r s i v e l y , F such t h a t W V ( F ) and F ' x = H ' F * x xEV).
142
ROLAND0 C H U A Q U I
for x X by,
E W .
We can t h e n extend F t o an o p e r a t i o n F d e f i n e d f o r a l l c l a s s e s
F ( X ) =H(F*X), for X c -W
,
F(X) = Y
On X C - W, F
.
otherwise
has t h e d e s i r e d e f f e c t as an e x t e n s i o n of F.
PROBLEMS
1.
Show i n t h e t h e o r y w i t h axioms: Ax Class, Ax Ext, Ax Un, Ax Pow, Ax Rep, Ax Ern, and Ax Reg, t h a t Ax I n f i s e q u i v a l e n t t o each o f these formulas:
(1) W x ( x € A ++ x = 0 V 3 y ( y E A A x = { y } ) ) + 3 U (2) 3 A 3 x 3 U ( x E A E U A V y ( y E A 3 x(yExEA))).
AEU
.
+
2.
Show i n t h e t h e o r y w i t h axioms, Ax Class, Ax Ext, Ax Un, Ax Pow, Ax Rep and Ax Em, t h a t Ax I n f i s e q u i v a l e n t t o each o f these formulas: (1) 3 A 3 U ( O E A E U A W y ( y E A -, (y3 E A ) ) . ( 2 ) W B ( O € B A W y ( y E 8 -,Ey) E 8 ) -, A C_ B )
3.
Prove t h a t ,
VV ( F )A ‘ Y ( G ) + H ( F
= HOG A
-,
3
‘ v ( H ) ) - FOG-^
U(AEU).
i s a function).
D e r i v e from t h i s a necessary and s u f f i c i e n t c o n d i t i o n f o r F b e i n g ob(ii)). t a i n a b l e by r e c u r s i o n from a c e r t a i n H ( i n t h e f o r m 3.2.1.4 Show t h a t any b i u n i q u e f u n c t i o n can be r e c u r s i v e l y defined. 4.
F i n d t h e f u n c t i o n r e c u r s i v e l y d e f i n e d from H ( f o r m 3.2.1.4
H is:
(a) H‘x = x n a. ( b ) H’x = (c)
H’x
x u b.
= (X n a)
u b.
(ii)) when
AXIOMATIC S E T T H E O R Y
143
W e now, pass t o the study of the s t r u c t u r e of W. In t h e f i r s t place, we s h a l l d e f i n e a function p on W , such t h a t p ' x w i l l measure t h e complexi t y of x . p ' x will be c a l l e d the m n k of x . Since every element of W i s formed from 0 by the operation I I , this complexity i s measured by t h e "number" of times t h a t { has t o be crossed t o g e t t o 0. Thus, 0 w i l l have t h e minimal r a n k ; then comes { O l , and then ICOII o r (0, I011 which have the same rank. In order t o have an adequate measures, t h e values of p ( i . e . t h e elements of D p - l ) , should be well-ordered. I n f a c t , they will bewell ordered by I N . Once obtained a rank f o r elements of W , we will d i v i d e W i n t o sets t h a t contain the elements whose rank i s l e s s than a given one. This will be done i n t h e next section through the operation R . The d e f i n i t i o n of p will be done by recursion. s i b i l i t i e s f o r doing this. For instance,
There a r e several pos-
,
using 3.2.1.4 ( i i p ' x = p * x u u p * x , using 3.2.1.4 ( i i ) , p'x = p*Tx , u s i n g 3.2.1.4 ( i i i ) . p ' x = Tp*x
¶
The most i n t u i t i v e i s t h e t h i r d . s i n c e i t defines P x a s the c l a s s of the ranks of the elements of T x , i.e. p ' x = { p ' q : y E f x } . Since T x i s t h e c o l l e c t i o n of a l l s e t s of complexity l e s s than x , we would have t h a t p ' x i s the rank immediately above the ranks o f s e t s of l e s s complexity than x . That i s , we have t o cross I one more time. However, f o r technical reasons, we s h a l l adopt t h e second form and t h e o t h e r s will be theorems. We obtain P applying 3.2.1.5 with the function H defined by, H ' x = x u u x , for x E W . 3.2.2.1 DEFINITION, ' W ( p ) A v x ( x ~ W +p ' x = p * x u u p * x ) . The corresponding operation denoted a l s o by p i s defined by ( s e e 3.2.1.5).
p X = p*X U U p*X
, for
X C -W
.
We have, f o r instance, PO = 0
p l = PtOI =
,
I01
= 1
,
pCtO11 = I l l u 1 = C0,l) = 2 , p 2 = p I 0 , ~ O I l = C0,lI u 1 = ( 0 , l I
= 2 .
In t h e r e s t of t h i s s e c t i o n and t h e next one (3.2.3), we understand a17 formulas r e l a t i v i z e d t o W. That i s , a l l c l a s s e s a r e considered a s subclasses of W. Thus, a l l theorems a r e theorems of M K T R .
144
ROLAND0 C H U A Q U I
Next, a l i s t o f p r o p e r t i e s o f p . 3.2.2.2 1.e.
THEOREM,
p
U
x
.
= U P*x
p i s completely additive.
PROOF, 3.2.2.3
H and t h e image o p e r a t i o n p* a r e c o m p l e t e l y a d d i t i v e . p n
THEOREM,
.
x
PROOF, S i n c e by 3.2.2.2, o b t a i n e d from 2.3.2.2.
c - n p*x.
p i s c o m p l e t e l y a d d i t i v e , t h e theorem
PROOF, The f i r s t p a r t (monotony) i s o b t a i n e d from 3.2.2.2 2.3.2.2, and t h e second p a r t from Def. 3.2.2.1. 3.2.2.5
THEOREM,
is
and
P U X = UpX.
PROOF, We have,
up
x
= u (p*(X) u up*X),
by Def. 3.2.2.1,
= u p*(X) u u u p*X,
by t h e a d d i t i v i t y o f
= p ( u X ) u u u p*X = p(u X) u u p V X = p(u
X ) u p u u*X
= p(u
X)
U
~
uUX
= p ( u X ) u u p*u X
But, u p* u X
-
C p
uX
, by
, , , , ,
U,
by 3.2.2.2, by a d d i t i v i t y of p * ,
by 3.2.2.2, s i n c e u u*X
= Uu X
,
by 3.2.2.2.
Def. 3.2.2.1.
Hence
u p x = p u x .
3.2.2.6
THEOREM,
P X = TpX.
PROOF, By 3.2.2.5, 3.2.2.2, and Def. 3.2.2.1, PX = P*X Thus, U p X c - p X and, hence p X i s t r a n s i t i v e .
p*X u u p X .
3.2.2.7
THEOREM,
UUP*X =
m
p X = Tp*X,
PROOF, By Def. 3.2.2.1, p*X c PX; hence, Tp*X C T p X = P X , by On t h e o t h e r hand, p X = p*X U u p X C - Tp*X , by Def. 3.2.2.1 3.2.2.6. and 2.5.1.9.
A X I O M A T I C S E T THEORY
3.2.2.8
PROOFI
Assume
THEOREM,
= P TX.
pX
, using
By i n d u c t i o n on W p*X
= p*PX
p*X
= p*PX
.
Then,
u u
p TX = p (X
.
p Y = p TX
+
We have t o c l i w ,
2.5.3.5.
T*X),
b y 2.5.1.18, b y a d d i t i v i t y o f P , 3.2.2.2
= p X u p(u7*X),
u p * T * X , by a d d i t i v i t y o f P , u u p*X , b y t h e i n d u c t i c n h y p o t h e s i s ,
= p X u = p X
,
= pX
3.2.2.9
PROOF,
p*X
= p**T*X.
THEOREM,
P X = P*TX *
= p*X u u p*X
,
= p*X
u
U p*
*T*X
= p*X
u
p*
b y D e f . 3.2.2.1,
u PX
= p*(X U U T * X )
,
=p*TX
PROOF, 3.2.2.11
PROOF, 3.2.2.12
PROOF,
THEOREM,
T
, ,
by t h e i n d u c t i v e hypothesir-,
,
by a d d i t i v i t y o f images,
by a d d i t i v i t y o f images,
b y 2.5.1.18.
(0p-l)
= DP-’.
By 3.2.2.9.
-
THEOREM,
PPX
= PP*X
By 3.2.2.7,
ppX
= pTp*X;
THEOREM,
PPX
= P X .
b y 3.2.2.8,
By i n d u c t i o n u s i n g 2.5.3.5. ppX
THEOREM,
P X
,
= p*p*X
u u
= p*X u
u p*X,
= P*P X .
PTP*X = Pp*X.
Assume p*p*X
= pp*X
= p x
3.2.2.13
I
L e t us assume as i n d u c t i v e h y p o t h e s i s f o r u s i n g 2.5.3,5, t h a t Then pX
3.2.2.10
b y D e f . 3.2.2.1.
= p*Y.
I
Then,
b y 3.2.2.11, p*p*X
, by
Def,
3.2.2.1,
by t h e i n d u c t i v e hypothesis,
,
b y D e f . 3.2.2.1.
ROLAND0 C H U A Q U I
146
-
T h i s theorem can be f o r m u l l a t e d by, vq(qEpX
PROOF,
p X = PP X
, by , by
= p*pX
THEOREM ,
sx
ZEPX)).
b y 3.2.2.12,
9
= p*TpX
3.2.2.14
3 z ( q = pz A
P
XI
,
3.2.2.9 3.2.2.6. =
sP x
.
R e c a l l t h a t S x , t h e successor o f x , was d e f i n e d i n 2.4.1.1 =xuIx). PROOF,
( i i ) by
By Def. 3.2.2.1.
3.2.2.15
THEOREM,
PROOF,
p X E pq V pq
5P
x.
By i n d u c t i o n , u s i n g 2.5.3.9: V x ( T xc -A
-+
xEA)
-f
W C A .
L e t A = ( x : V y ( y E W - + p x E p y V p q C p x ) } , and assume t h a t Tx 5 A p x f o r a c e r t a i n Y E W . .Then, p q % p x f 0 . Since p q % p x 5 and t h a t p q W , t h e r e is-a z E p q % p x w i t h z n ( p q * p x ) = 0 . Since z e p q , by 3.2.2.6, i.e. z = pq' f o r a z s p y ; hence z c p x . A l s o z E p q = p * T y , by 3.2.2.9, now, t h a t p x p z. Then p x % z + 0. S i n c e p x % c e r t a i n q' € 7 ' ~ ~ Suppose . z C - W , t h e r e i s a u E p x z w i t h u n T p x z) = 0. Now, u E p x , hence u c_ p x and, thus u c z. Also, u E p x = p * T x ; t h e r e f o r e u = p x ' f o r a c e r t a i n x' E T x , T h u y assuming p y C p x we have o b t a i n e d q' E T y w i t h pq' C_ py', we have-obtained x' E T x w i t h p x ' C - pq'. p x ; and assuming p x
-
But p x ' = u $ Since x' E T x , x ' E A . T h e r e f o r e p x ' E p q ' o r pq' C p x ' . Thus, p y ' L p x ' , and, hence pq' = p x ' o r z = u. B u t t h i s c o n t r a d i c t s t h e f a c t s t h a t u E p x and z $ p x . T h e r e f o r e p x ' p q ' = z e p q . H e n c e x E A and W c A . z = pq'.
-
3.2.2.16
THEOREM,
pxEpq
c-+
px
T h i s theorem a s s e r t s t h a t I N 1 D p
C
pq.
= (ELU ID
)I
Dp
-I.
PROOF, Suppose t h a t p x E p q ; t h e n p x C _ p q , by 3.2.2.6. p x , p y E W , we have t h a t p x f p q , and hence p x C pq. Ifp x
C
p q , then,
by 3.2.2.15,
pxepq.
But, s i n c e
147
A X I O M A T I C SET THEORY
PROOF,
By 3.2.2.9,
3.2.2.6,
3.2.2.18
THEOREM,
( P X C_ P q v P q
and 3.2.2.16.
5 ~ x 1A
(PX E P q v P X
= Pq
v
Pg E PX).
T h i s theorem i m p l i e s t h a t IN 1 D p - l ( o r , what i s t h e same,
(EL
U
I D ) 1Dp-l) i s a simple ordering. PROOF, Assume t h a t p x 9 p q and p g and p q c - p x and, hence p g = px.
px.
Then, by 3.2.2.15,
Thus, we have shown t h e second p a r t o f t h e theorem. o b t a i n e d u s i n g 3.2.2.16. 3.2.2.19
pPx
p*Px
The f i r s t p a r t i s
p p x = P{x}.
THEOREM,
p{x} C p P x . On t h e o t h e r hand Since {x} C P x , by 3.2.2.4, Since U P x = x , P x , by 3.2.2.2 and Def. 3.2.2.1.
PROOF, =
pxC_p q
UpU
(1) P P X = p*Px u p x . By 3.2.2.14, px Then q = 3.2.2.16, q E PX V q have shown t h a t p*Px
q
E
p*Px.
I n o r d e r t o prove t h a t p*Px C - ~ 1 x 3 ,assume s p {x}. p z w i t h z f_ x. By 3.2.2.4, q C p x , and, hence, by = PX Therefore, by 3.2.2.14 E P Thus, we S p and, by ( l ) ,p P x C -PIX}.
.
(XI
{XI.
From 3.2.2.14 we deduce t h a t P I X I i s t h e immediate successor of P X i n t h e sense of t h e o r d e r i n g I N . p P x i s a l s o , by 3.2.2.19, t h i s same immediate successor. T h i s i s v e r y n a t u r a l because i n { X I o r P x we have t o cross one { more than i n x. 3.2.2.20
THEOREM,
(i) P u x
C P x CPCUX}.
( i i ) px = P U X V px =
PIUX}.
T h i s theorem shows t h a t p u x i s p x o r t h e immediate predecessor of px. Thus, we can c l a s s i f y ranks (i.e. elements o f D p - ’ ) i n t o t h o s e t h a t a r e o f t h e form p U x and those o f t h e f o r m pC U XI. PROOF,
( i i ) i s o b t a i n e d from (i), u s i n g 3.2.2.14.
Also, by 3.2.2.2
ROLAND0 CHUAQUI
148
.
and Def. PX
By 3.2.2.19,
pux i p x .
3.2.2.1,
Thus, we p r o v e ( i ) .
p{ux}
and 3.2.2.4
=
pPux 3
e
(ii) A C - D ~ - ' A A . ~ - O + ~ I AA E. (iii) X # 0
-+
n
p* X t p* X .
3.2.2.21 implies that I N 1 D p - l beginning o f t h i s section.
i s a well-ordering,
a s I announced a t
I t i s c l e a r t h a t ( i ) , ( i i ) and ( i i i ) a r e e q u i v a l e n t . PROOF, s h a l l prove ( i i i ) .
L e t X # 0 and b =
(-1
p* X.
the
We
We have t o show t h a t b = p u f o r a c e r t a i n
U E X .
Suppose t h a t z c X . Then b C p z = p* T z (by 3.2.2.9). Since p i s a f u n c t i o n , b = p*z' f o r a c e r t a i n - z ' 5 T z . A l s o , s i n c e b i s a n i n t e r s e c t i o n Therefore, b = T b = T p * z ' =
of t r a n s i t i v e s e t s , i t i s i t s e l f t r a n s i t i v e . = p b (by 3.2.2.7 and 3.2.2.11).
p z'=pp*z'
.
We have, t h a t f o r a l l u E X , b C p u Hence, s i n c e b = p b , b E p u or I f we had t h a t f o r a l l u E X I b E p u , we w o u l d g e t t h a t b E b , cont r a d i c t i n g b = p b E W . Hence, t h e r e i s a u E X , such t h a t b = P u , i . e . ,
b=pu. bgp*X.
8
THEOREM,
3.2.2.22
p-'*
x
E WA
W
W (p-'*
1.
PROOF, By i n d u c t i o n we f i r s t show t h a t p-'*px -1* p p xE W 1 . F i r s t , we p r o v e t h a t
(1)
qEA
-+
W . Let A
=
{x :
{y} E A .
Assume t h a t p
E
-1*
qEA;
t h e n by 3.2.2.14,
p t y } = ~ z : p z E p{!/I3
=Fz
: p z E pul
u
= { z : p z E p y V p z = p y l ,
I?
Now, b y h y p o t h e s i s , { z : p z E p [ ( } = p
: p z= pql.
-1*
py E W . L e t
B
= { z : p z = py).
We s h a l l p r o v e t h a t U E c p - l * p y ; suppose t h a t t E U 8 ; hence Z E z E B , f o r a c e r t a i n z. But, since-zEZ3, P Z = ~ q . Thus, p t E p q , i.e., X E p - ' * p q .
AXIOMATIC S E T THEORY Therefore B C - p-’*py tyl E A .
-1*
p
and B E W . Thus, we have proved t h a t Y E A i m p l i e s
Suppose, now, t h a t x C - A.
u
W, p-’*px xEA.
Then, p-’*px
{p{y} : q E x } = u{p-l*p{c{} :
images).
But, i f q E x , t h e n !/€A
Hence, by 3.1.3.4, C -
W.
u
{p-’*~iql
Hence p-’*px
E
PROOF,
By Def.
then X C - p-l*p*XE
3.2.2.24
W
C
and 3.2.2.12,
P XElv,
X
E
q W-
LJ X
{ q :pyEx} c -
p-’*x=
W.
m
E
W .
implies p*XEW.
Finally, i f
XEW.
THEOREM,
=
Thus we p r o v e d t h a t x C-A i m p l i e s for dl1 f t W.
W--+p * X E We+ X
pX
3.2.2.1,
W and
~ { { q :} y E x }
: I ( b y t h e a d d i t i v i t y o f p and i n v e r s e -1* p q E W . By (i),p-’*p{y}E -1*p x E Y . B u t : u! E V , i,n,, p
T h e r e f o r e psl*y
THEOREM,
p
h’ arid u C A .
On t h e o t h e r hand, by 3.2.2.4
{ y : p y E p x } = p-’*px.
-1+
= p
and, hence p
T h e r e f o r e U’c - A and p-’*pr
3.2.2.23
149
= p b’
XEW,
-
Also,
.
i f p*XEW,
then p X , p * X E W .
pX = D p
-1
.
PROOF, F i r s t , by 3 . 2 . 2 . 2 3 , p X = p W i m p l i e s t h a t X $Z W , because X E W i m p l i e s p X E Wand, hence by 3.2.2.4 and 3.2.2.12, p X E p W. Thus, we have shown
(1) p X = p W + X E W . Second, p W = p * W u u p * W , by D e f . 3.2.2.1. B u t i f y E U p* W t h e n y E x E p * W and, hence I I C Y - p 7 f o r a c e r t a i n z E W. By 2.2.2.17,
y = p x ’ f o r a c e r t a i n x’; t h e r e f o r e , and we o b t a i n ,
(2)
pX=pW-+pX=L)p
T h i r d , suppose t h a t X
-1
q W
i/
t iJ* W .
~ n u s ,p
w
=p*
w
= ~p
,
-l,
. We have, t h a t p x
and l e t Y E W .
E
p q
v
py C px for a l l X E X (by 3.2.2.15). I f f o r a l l x E X we had p x E p q , we would o b t a i n p*X C p y , c o n t r a d i c t i n y P * Y 9 W ( u s i n g 3 . 2 . 2 . 2 3 ) . Hence, This t h e r e i s a n x E X - w i t h p y c px. T h e r e f o r e pi4 px C U p X L p X . Hence p r o v e s t h a t f o r e v e r y p E @, p i { c -pX. B u t p b’ = U-{py : y E W l pX C - p W . T h u s we have proved p W C - pX. By 3.2.2.4,
5
(3)
X @ W +pX=pW.
( l ) , ( 2 ) , and ( 3 ) g i v e t h e thporem.
.
From t h i s l a s t theorem, we deduce t h a t pX
E pY
V P Y c_pX
,
.
ROLAND0 CHUAQUI
150
i s a l s o t r u e f o r proper c l a s s e s X, Y, and t h a t pX = pY = Dp-' p a i r o f proper c l a s s e s X, Y .
f o r any
Summarizing, t h e most i m p o r t a n t p r o p e r t i e s o f t h e r a n k f u n c t i o n are:
(1) D p - I i s well ordered by t h e r e l a t i o n s IN -1 c o i n c i d e when r e s t r i c t e d t o D p
.
( 2 ) pX = p* TX = p*p X A
T D p - l = Dp
and EL U Z D
,
which
-'.
T h i s means t h a t t h e rank of a c l a s s X i s a c l a s s o f ranks, i n f a c t ,
it
i s t h e c l a s s o f ranks o f elements of T X .
~p
are transitive (3)
x
E
w++ p-1* x
E
w-
px E
w
I n p a r t i c u l a r any r a n k and
.
F i n a l l y , we s h a l l prove t h a t t h e n a t u r a l numbers and w a r e rank. 3.2.2.25
THEOREM,
PROOF, duction w
OEDp-l .
We have t h a t
a certain Y E W .
5 Dp-l.
Hence,
(and hence, w
w E Dp-'
x
U
x
NOW, i f
{ x } =p{y}.
Thus,
E
x
We a l s o have, pw = p*Tw= p*w = {PK : K E w ) . pK=K
for
K E ~ .
~ h u s ,p w
= IK:KEWI=W.
C Dp
- 1 ).
D P -1
, then x
U
Cr} E D p - 1
.
= pg
But, s i n c e w C D P
SincewEV,
for
By i n -
-1
,
w ~ ~ p - l .
PROBLEM
Determine whether p*X = p*Y
3.2.3
f o r any p a i r of p r o p e r c l a s s e s X and Y.
U N I V E R S A L DOMAINS WITH A G I V E N RANK,
An o p e r a t i o n R s h a l l be d e f i n e d , such t h a t f o r each X C W , i t d e t e r mines t h e c l a s s o f a l l s e t s w i t h rank l e s s t h a n X. T h i s c l a s s has t h e same rank as X. Since a l l elements o f Whave a rank, we s h a l l be able, i n t h i s way, t o decompose W i n t o u n i v e r s e s o f i n c r e a s i n g rank.
151
AXIOMATIC SET THEORY
3.2.3.1
DEF I N I T I ON i
(i) R X = p-'*p (ii) R
x
.
= ( R X : X € W ) .
As usual, t h e same symbol i s used f o r t h e o p e r a t i o n and t h e c o r r e sponding f u n c t i o n .
-
Most o f t h e r e s t o f t h i s s e c t i o n d e a l s w i t h t h e main p r o p e r t i e s o f R. Many o f them a r e e a s i l y deduced from t h e corresponding p r o p e r t i e s o f p . As i n t h e p r e v i o u s s e c t i o n we assume t h a t a l l c l a s s e s a r e subclasses o f W.
3.2.3.2
THEOREM,
W W ( R ) A R ' x = p- '*p x
PROOF, By Def. 3.2.3.1
3.2.3.3
THEOREM,
and 3.2.2.22.
=
{ q :p q
E
pxl
.
9
*RX= W .
X $
.
PROOF, By 3.2.3.2,
RX = W i m p l i e s X $ W On t h e o t h e r hand, assume X $ W ; t h e n p X = D p - l ( b y 3.2.2.24). Thus, f o r every q € W , p q € p X and, hence, b y Def. 3.2.3.1, RX = W . 3.2.3.4
THEOREM,
PROOF, By 3.2.3.3
3.2.3.5
THEOREM,
RX
C
W
-
X E W.
and Def. 3.2.3.1.
R u
x
= UR*X.
PROOF, Complete a d d i t i v i t y o f R i s deduced from t h e d e f i n i t i o n , s i n c e i n v e r s e images and p a r e c o m p l e t e l y a d d i t i v e .
3.2.3.6
THEOREM,
R 0 = 0 A X c -R X .
PROOF, The f i r s t p a r t i s immediate by Def. 3.2.3.1. I n order t o prove t h e second p a r t , assume q E X . Then ~q E P X , b y 3.2.2.4. Hence, by Def. 3.2.3.1, q E R X .
PROOF, By Def. 3.2.3.1
3.2.3.8
THEOREM,
and 3.2.2.8,
RX = T R X .
3.2.2.12
and 3.2.2.11.
ROLAND0 CHUAQUI
152
We have t o p r o v e t h a t u R X c R X . Suppose t h a t y E u RX ; PROOF, and Def. 3.2.3.1, py E theri ~ E ~ E Rf oXr a c e r t a i n z . Thus, b y 3.2.2.4 s i n c e p X i s t r a n s i t i v e (3.2.2.61, p y E P X and, hence, by Def. p ? ~ Yp. 3.2.3.1, !i E R X .
.
3.2.3.9
THEOREM,
= p X.
PRX = P*RX
-l*pX = pX n D p -1 = p X , by D e f . 3.2.3.1 P R O O F , P * R X = P* P 3.2.2.17. P l s o , p R X = p * T R X = p * R X , b y 3.2.2.9 a n d 3.2.3.8.
THEOREM,
3.2.3.10 PROOF,
R R X = RX. = p
RRX = p-l*pRX
THEOREM,
3.2.3.11.
.
and
R
x
-1* pX = R X , b y Def. 3.2.3.1
= U
{Y
:pY=Px).
From t h i s t h e o r e m and 3.2.3.9, w i t h t h e same r a n k as X.
we o b t a i n t h a t RX
PROOF T h e r e f o r e , RX C - u CV : p Y = pX1.
We have b y 3.2.3.9,
I
and 3.2.3.9..
.
i s the largest class
that pRX = p X .
5
I n o r d e r t o p r o v e t h a t u { Y : pY = p X } RX , assume t h a t y E pY = p X } . Then y E Y and pY = pX f o r a c e r t a i n Y. B u t , b y 3.2.2.4, pY = p X , and b y D e f . 3.2.3.1, y E RX. 3-2.3.12
THEOREM,
R
{x}
= RPx =
PROOF,
We have b y Def. 3.2.3.1
3.2.3.13
THEOREM,
u {Y : p
y
E
PRx.
and 3.2.2.19,
that
R
{XI
=
Suppose p-l*p{x]. = p-'*pPx = R P x . We, now, p r o v e t h a t P R x = R {XI. f i r s t t h a t y E P R x ; then g C Rx,and,hence,by 3 . 2 . 2 . 4 and 3.2.3.9, p y C_ pRx = p x . Thus, b y 3 . 2 . 2 . 1 6 , ~ ~ E p x o r p y = p x . T h e r e f o r e , b y 3.2.2.14, p y E p { u j , i.e. b y Def. 3.2.3.1, y E R {x}. Thus, we have shown t h a t P R x c R { x j . On t h e o t h e r hand, assume t h a t y E R { X I . By Def. 3.2.3.1, we h a i e t h a t p i { E p{x}, w h i c h , b y 3.2.2.14 means t h a t p y p x o r py = p x . I f p y E p x , t h e n b y D e f . 3.2.3.1 and t h e t r a n s i t i v i t y o f R (3.2.3.8), y E R x c P R x . i f p y = p x , t h e n b y 3.2.3.11, y C R x and, hence y E P R x . -P R x . Thus: i n a n y c a s e we show t h a t y E P R x . T h e r e f o r e , R { X I C R x = Up* f i x .
T h i s t h e o r e m c o u l d b e used a s a r e c u r s i v e d e f i n i t i o n o f R .
PROOF,
R x= R
U
[{y}:y
E x } = U
=
{ R { y } : y E x } , b y 3.2.3.5,
u { P R Y :y E x }
, by
3.2.3.12,
AXIOMATIC SET THEORY
=up*R*x. 3.2.3.14 PROOF,
(U
px E px
-Z
R x =PR
By Def. 3.2.3.1
U X )A
(u P X B:
and 3.2.2.20.
.
153
. PX
-+
R x =R u
X )
.
Thus, we have shown t h a t t h e two t y p e s o f r a n k s mentioned a f t e r 3.2.2.20give r i s e t o two types o f R .
3.2.3.15
THEOREM ( P R I N C I P L E OF I N D U C T I O N FOR R ) .
VX(RX C A -
-+
xEA)
+
WLA.
T h i s theorem can a l s o be formulated by,
R-1* P A c cA . -A + W Assume R-'*P A C A , l e t By i n d u c t i o n , u s i n g 2.5.3.4. PROOF, B = { x : R x C A I ; and s u p p o s e x L B . We have b y 3 . 2 . 3 . 1 3 , R x = U P * R * x . Assume t h a t F E R x ; then z c Ry f o r a c e r t a i n y E x . By t h e i n d u c t i o n hyp o t h e s i s , R q C A . Hence z-C R q C A and, by t h e monotony o f R ( i m p l i e d by 3.2.3.5) a n d 3.2.3.10, R z - c R f y = R q c A. Thus, z E B and, s i n c e t h i s i s t r u e f o r a l l z E R x , R>C E. But, Ey t h e assumption o f t h e theo- A and Y E B . Thus, we show by 2.5.3.4 t h a t W C rem, B 5 A . Hence, R x C EC -A .
9
3.2.3.16
THEOREM,
R x = Rq
PROOF,
By 3.2.3.7
and 3.2.3.9.
3.2.3.17.
THEOREM,
tf
.
p x = pq.
RxE R y - R x c R y
- P X E
Py.
F i r s t , assume t h a t R x E R y ; then, by 3.2.2.4, pRxE pRy PROOF, and, by 3.2.3.9, P X E p y . Second, assume t h a t p x E p y ; by 3.2.2.16 p x c p y . But, s i n c e p x ~ p y x , E R y 2. R x . Thus, Thus, R x = p - l * p x 5 p-l*py = R y R x c R q . T h i r d , assume t h a t R x c R q . By 3.2.3.9 and 3.2.2.4, p R x C p y . But, by 3.2.3.16, p R x f p y and, hence p R x E p y . By, Def. 3.2.3.1, R x E
.
Ry.
PROOF,
By 3.2.2.4
and 3.2.3.17.
.
154
ROLAND0 C H U A Q U I
PROOF, By 3.2.3.17, 3.2.3.20
THEOREM,
3.2.3.16,
Rx
E
R Y E Rz
PROOF,
By 3.2.3.17.
PROOF,
Assume t h a t X # 0.
n p * X = py f o r a certain yEX.
and 3.2.2.18. -+
Rx
By 3.2.2.21,
Rz
*
n p * X E p*X,
Now, by Def. 3.2.3.1,
and hence
-1* n p * X = p - ’ * p y P -1*
=
Since i n v e r s e images a r e c o m p l e t e l y m u l t i p l i c a t i v e , we have, p -1** P * X = ~ ( -I* p * x = n p P o p ) * x = n mx.
Ry.
These l a s t theorems show t h a t INIDR-’ i s a w e l l o r d e r i n g and LMJDR-’= (EL U ID) 1 D R - l . Thus, we have another c l a s s (besides D p - l ) w e l l ordered by I N . The n e x t d e f i n i t i o n g i v e s an o p e r a t i o n t h a t f o r each c l a s s X s e l e c t s a subset c a l l e d t h e dinLitiMgLLinhed nubneA 06 X. 3.2.3.22
DEFINITION,
-W ,
ds X = X n n @(PX- iO}).
The main p r o p e r t i e s o f ds a r e t.he f o l l o w i n g . 3.2.3.2.3
PROOF,
THEOREM,
( i ) , ( i i ) and ( i i i ) a r e easy.
PROOF OF ( i v ) d Assume t h a t X f 0. Then P X -I01 # 0. By 3.2.3.21, T h i s means t h a t n R*(PX {O}) = & for n R * ( P X - {O)) E R ( P X - { O ) ) . - Ry; thus, y c - dsX and, hence, dsX f 0. a c e r t a i n y, w i t h y f 0. But y c
-
PROOF OF ( v ) . I f y E PX and 3.2.3.9, pds X L p R y =
- {O},
t h e n dsX C - 4. Hence, by 3.2.2.4,
The s i m i l a r p r o o f o f ( v i ) i s l e f t t o t h e reader.
I f R i s an equivalence r e l a t i o n , t h e n
AXIOMATIC S E T T H E O R Y
xRy
for x
-
P { x ) = R*{y) A
DR.
X E
We adopt t h e following d e f i n i t i o n of t h e t y p e E W .
3.2.3.25
155
06 x ~ L t &&5peot h t o R,
THEOREM a
( i ) t R ( x ) E W.
(ii) R = R - ’ ~ R (iii) R =
+
R - ~ O R
-+
-
t R ( x )# 0 ) .
(x t DR
-+
( X R ~
tR(x)
=
rR(y)
# 0).
Thus, t R ( x ) has t h e same property a s R*{x}, w i t h t h e advantage t h a t i t i s a set.
P R O O F , ( i ) and (ii)a r e obtained from 3 . 2 . 3 . 2 3 . PROOF OF ( i i i ) . Let R be a n equivalence r e l a t i o n . By 3 . 2 . 3 . 2 3 , x R y implies t R ( x ) = t R ( y ) # 0. Assume t h a t t R ( x ) = t R ( y ) # 0. B u t tR(x)C - R*{xl and t R ( y ) 5 R*{y). T h u s , R * { x ) n R * { y ) # 0 . Therefore
R*{x)
=
R*{y}
#
0 and t h i s implies xR y .
PROBLEMS
1.
What a r e R R * x a n d R * R x ? . Determine whether R R * x
2.
=
R*R x = R
X .
Let R be a well founded and extensional r e l a t i o n ( s e e Problem 4, sect i o n 2 . 2 . 3 ) . Assume a l s o , t h a t R s a t i s f i e s : (i) (ii)
w a ( a c- D R u R-’*{X}
E
V.
D R - --t ~ 3
ZW
-
x ( x ~ z
X E ~ ) ) ,
( a ) Define by recursion on R a function 6 on DR U D R - ’ R1DS-I i s a well ordering.
such t h a t
( b ) Try t o show f o r 6 theorems s i m i l a r t o those proved f o r p .
( c ) Define t h e operation l$x = 6-’* 6 x . e r t i e s of R .
Try t o show f o r R R t h e prop-
CHAPTER 3 . 3 O r d i n a l s Numbers
3.3.1
G E N E R A L P R O P E R T I E S OF O R D I N A L S ,
Chapter 3.3 i s denoted t o a d e t a i l e d study o f t h e c l a s s D p - l
, which
w i l l be c a l l e d O n . The elements o f D p - l w i l l be c a l l e d o h d i n d w l b m o r , simply, 04dinaRn. By 3.2.24, i t i s c l e a r t h a t O n = Dp-' 4 V . L a t e r , i t w i l l be shown t h a t t o each w e l l - o r d e r i n g t h a t i s a s e t corresponds a unique o r d i n a l as i t s w e l l - o r d e r i n g type. Other w e l l - o r d e r e d c l a s s e s c o u l d be used f o r t h i s However D p - l has t h e a d d i t i o n a l advantage purpose, f o r i n s t a n c e DR-1.
t h a t f o r a E Dp'l i f BEa E Dp-',
, we
have a = {B : f? E D p
-1 A B c a}. T h i s i m p l i e s t h a t
then f?EDp-'.
The idea o f i n t r o d u c i n g these o r d i n a l s as types o f w e l l - o r d e r i n g i s due t o von Neumann. T h i s p a r t i c u l a r way o f i n t r o d u c i n g them as ranks o r i g nated w i t h T a r s k i .
3.3.1.1
DEFINITION,
On = D p -
1
.
3.3.1.2 S T I P U L A T I O N OF V A R I A B L E S , We s h a l l use, Greek l o w e r case l e t t e r s a, 0, 7,s t o refer t o ordinals.
...
Most o f t h e theorems i n t h i s s e c t i o n 3.3.1 a r e immediate consequence o f t h e p r o p e r t i e s o f p discussed i n 3.2.2.
3.3.1.3
THEOREM,
.
PROOF, By 3.2.2.12.
3.3.1.4
THEOREM,
PROOF, By 3.2.2.12
3.3.1.5
THEOREM,
PROOF, By 3.2.2.17.
x
( X E Onu
X=pX
-
= p x ) A (X
c- O n -
X E OnV X = O n .
and 3.2.2.24.
x
.
E
a
x
E On
156
A
x
c a.
X = P*X).
157
A X I O M A T I C SET T H E O R Y
3.3.1.6
PROOF, By 3.2.2.16 3.3.1.7
and 3.2.2.18.
. .
ycxra
THEOREM,
PROOF, By 3.2.2.6. 3.3.1.8
THEOREM,
PROOF, Assume t h a t X 3.2.2.2, u X = u p*X = p UX,
THEOREM, A wx(a =
E
{a,b
O n ; then,
Therefore,
. .
PROOF, By 3.2.2.14. U
a
sx-+x
0 v (0E a A p
1
.
by 3.3.1.2, b y 3.3.1.2, --f
Ca).
X = p*X. Thus, by U X E O n V U X = On..
n X E X.
( V x ( a# s x
E On A
-+
a =
U a )
= u a).
-
PROOF, By 3.2.2.20
and 3.2.2.14.
3.3.1.13 THEOREM,
a =
.
PROOF, By 3.3.1.11.
=
yEa.
X C - On A X f 0
THEOREM,
PROOF, By 3.2.2.21.
3.3.1.12
C
-+
an0
aUP,
PROOF, By 3.2.2.18.
3.3.1.10
.
va
(aep A a c p )
THEOREM,
U
a
U
a
a
-
W[([
E
a + S [ E a)
A c c o r d i n g t o t h e two p r e v i o u s theorems t h e r e a r e two t y p e s o f o r d i n a l s
a: t h o s e w h i c h a r e successor o f U a ( i . e . a = S u a = U a U ( U a } ) and those w h i c h a r e equal t o u a (i.e. a = ua). Those o f t h e f i r s t k i n d w i l l be c a l l e d A U C C ~ A A V J Lvtdin& and t h o s e o f t h e second k i n d , LtjnLt ahdin&. Also, u a i s a o r t h e predecessor o f a . 3.3.1.14
THEOREM,
x
E On
-
x
C On A -
U
x C_ x .
Thus, any t r a n s i t i v e s e t o f o r d i n a l s i s a n o r d i n a l . PROOF,
From l e f t t o r i g h t , t h e theorem i s o b t a i n e d by 3.3.1.5
and
ROLAND0 CHUAQUI
158
and 3.3.1.7. Assume, now, t h a t
x is
a t r a n s i t i v e set o f ordinals.
p*Y f o r a c e r t a i n Y. Thus, s i n c e x i s t r a n s i t i v e , T p * Y = pY. Since x i s a set, x E O n . w
3.3.1.15
THEOREM,
- On
W C
By 3.3.1.2, by 3.2.2.7, x = T
x
x
=
=
A WE On.
Thus, any n a t u r a l number i s an o r d i n a l . I n f a c t , t h e y a r e a l l successor o r d i n a l s , except 0. On t h e o t h e r hand, 0 and u a r e l i m i t o r d i n a l s . PROOF, By 2.4.1.4,
3.3.1.13,
and 3.2.1.1.
The n e x t theorem g i v e s a s u f f i c i e n t c o n d i t i o n f o r t h e u n i o n o f a s e t o f o r d i n a l s t o be a l i m i t o r d i n a l . N o t i c e t h a t s i n c e t h e o r d e r i n g o f O n i s g i v e n by I N , u x f o r x c O n i s t h e l e a s t upper bound o f x. Thus, t h e theorem says t h a t i f t h e l e a s t upper bound o f a s e t o f o r d i n a l s does n o t belong t o t h i s set, i t i s a l i m i t o r d i n a l , i.e. t h a t t h e l e a s t upper bound o f an unbounded s e t o f o r d i n a l s i s a l i m i t o r d i n a l . THEOREM,
3.3.1.16
x C- O n
A U
x9 x
--f
u x = u u x.
PROOF, Assume t h a t x C O n and U x x. By 3.3.1.15, U X E O n and, u u x c U x . Suppose, now, t h a t C; ~ U X i.e. ; E €77 6 x f o r hence by 3.3.1.7, a c e r t a i n q . Hence q c c x . But, s i n c e u x $ x, q # u x . Hence q E U x Therefore-E E U U x and, thus, U x C - U U x. by 3.3.1.6. 3.3.1.17
THEOREM,
x
C - On
-+
x
U
Ux
=
Tx
= n {q : x
Lq}.
T h i s theorem g i v e s two expressions f o r t h e l e a s t o r d i n a l l a r g e r t h a n t h e elements o f a s e t o f o r d i n a l s x . The p r o o f i s l e f t t o t h e reader. 3.3.1.18
THEOREM,
(i) (a C -A
-+
a E A)
( I N D U C T I O N P R I N C I P L E S FOR O R D I N A L S ) . -P
O n C-A .
T h i s p r i n c i p l e can a l s o be w r i t t e n , On n P A CA-+On C -A . ( i i ) tl
E((E
PROOF, 2.5.3.10,
E A -*SEE A) A
(C; C A A C; = u C;
-+
E
E
A))
-P
O nc A .
( i ) i s an immediate consequence o f t h e i n d u c t i o n p r i n c i p l e BcWATBn P A cA-+TB C A ,
n o t i c i n g t h a t O n C_ W and T o n = O n .
(ii) i s deduced from (i):Assume t h e h y p o t h e s i s o f ( i i ) and a
-A C
.
AXIOMATIC SET THEORY
159
I f a = U c i , then a E A , by t h e hypothesis. Suppose, now t h a t CY # U a . Then, by 3.3.1.12, CY = S u a , i.e. U a E a . Hence U a E A . By t h e hypothesis, Q = SU aEA. Thus, we have proved t h a t a C- A i m p l i e s a E A . By ( i ) , 0 n c-A . m
The n e x t two theorems g i v e a few statements t h a t c o u l d serve as d e f i n i t i o n of o r d i n a l s . Some o t h e r w i l l be l e f t as e x e r c i s e s (see Problems).
3.3.1.19
THEOREM
I
( i ) x E O n * T x c-x c-P x A x ~ W ,
-
( i i ) x E On * T x = x = P x A (iii)
x
E
On
X E
w.
y ( y ~ x Tg = y) A x
T x = x A
--L
E
W.
( i v ) x ~ O n- ~ u ~ u ~ y ( u ~ v ( ~ u Ex x -A +( Y E L L - + Y E W ) )A X E W .
PROOF, I t i s c l e a r , by 2.5.1.10 t h a t t h e f o u r f o r m u l a t i o n s on t h e r i g h t s i d e o f t h e equivalence s i g n a r e e q u i v a l e n t . The i m p l i c a t i o n from l e f t t o r i g h t i s e a s i l y o b t a i n e d by 3.3.1.7 3.3.1.4.
and
L e t 8 = { x : x E W A T x = x = P x } . We s h a l l prove t h a t 2'8 then, by t h e i n d u c t i o n p r i n c i p l e 2 . 5 . 3 . 1 0 , t h a t 7B C -On.
and,
= 8
L e t y E x E 8 . Then y E P x , i.e. g = T z f o r a c e r t a i n Z E X . Hence y = Also, s i n c e y E T x , y 5 T x = Px. Thus t h e r e i s a u C_ x such t h a t y = P u . Therefore, by 2.5.1.11, Py = T * T * u = P u = y. Thus we have shown t h a t Y E B. Hence B = T B
Ty.
.
We apply, now, 2.5.3.10. Hence, by 3.3.1.14,
x C On.
3.3.1.20
THEOREM, A
w x(x c- X
x
Suppose x E B and x C O n . Then By 2.5.3.1n, 8 C - On. E On
.
x
= Tx and
On = n {X : W x ( x E X + S X E X ) +
u x
€2
X)}.
S x E X ) A W x(x C_ X + U x E X ) } . By PROOF, L e t A = n { X : V x ( x € X Suppose, now, t h a t X i s one o f t h e classes 3.3.1.11 and 3.3.1.10, A C - On. O nc X I Hence O n c A. whose i n t e r s e c t i o n i s A. By 3.3.1.18 (ii) +
We now pass t o p r i n c i p l e s o f d e f i n i t i o n s by r e c u r s i o n .
3.3.1.21
THEOREM,
ROLAND0 CHUAQUI
160
f such t h a t w V ( F ) and Ion. Thus, On V ( F) and F'[ = HI F*t.
PROOF OF (i). By 3.2.1.4
(ii), t h e r e i s an
Take F = E W-+F ' x = H'F*x). On V ( G) and G ' t = ff'G*E, f o r a l l p. L e t Suppose now t h a t G a l s o s a t i s f i e s A = : F'E = G'E}. Suppose CI C A. Then F*a = G*a. Hence F ' a = G'a and c1 E A. By 3 . 3 . 1 . 1 8 ( i ) O , nc A-and t h e u n i c i t y o f F i s proved.
Wx(x
PROOF OF ( i i ) . of 3.2.1.4.
S i m i l a r as t h e proof o f ( i ) , u s i n g 3.2.1.5
PROOF OF ( i i i ) . Assume ' V (
H)
we use t h e r e c u r s i o n p r i n c i p l e 3.2.1.4 e x i s t e n c e o f an F such t h a t ,
and ' V ( K).
instead
For t h e e x i s t e n c e o f F
(i)t h a t f o r L w i t h
vV ( L )
gives the
The needed L should s a t i s f y , L ' F l x = H'F'
U
x , if Ux
E
, otherwise.
L ' F ~ x= KF*x
x,
Thus , we d e f ine L =(H'y'(UDy)
.
: UDyEDy)
The F t h u s o b t a f n e d s a t i s f i e s t h e theorem. i n d u c t i o n (3.3.1.18 (ii)).
U(K'Dq-l: U D q q D q ) . The u n i c i t y i s proved by
3.3.1.22 THEOREM SCHEMA, Le;t H a n d K be unmy o p e 4 e ~ X o n ~ .Then .ih a unique unahq opum.tLon F buch t h a t ,
thehe
F (SE)= H ( F ( < ) ) , if [ € O n ,
F ( a ) = K ( [ F ( [ ) : t e a ] ),
if
CI
=
u aEOn,
, otherwise.
F(X) = V
which a s s e r t s t h a t f o r a b i n a r y o p e r a t i o n PROOF, We a p p l y 2.5.3.16 G t h e r e i s a unique o p e r a t i o n F ' such t h a t P ( x )= G(x, [ F ( y ) : y ~ x ),] f o r
F'(X) = V G should s a t i s f y ,
, otherwise.
X c -
W
A X I O M A T I C SET THEORY
161
Thus, we d e f i n e , G(X,Y) = H(Y* {U X } ) , i f
U
X
E
DY
otherwise.
G(X,Y) = K ( Y )
A f t e r o b t a i n i n g t h i s F ' , d e f i n e F by F(x) = F'(X)
,
F(X) = V
,
if X
E On
otherwise.
The u n i c i t y i s p r o v e d by i n d u c t i o n , 3.3.1.18
.
(ii).
F i n a l l y , I g i v e some p r o p e r t i e s a b o u t t h e s t r u c t u r e o f W . THEOREM ,
3.3.1.23
(i) (1) R 0 = 0 (2) R S a =P R a (ii)
(3)
~1
x
W+ px
E
= U C .~ .+
Ra =
= n
U
{R
{c; : x c- R
S p x = n {[ : x E R
( i i i ) W = u {R
:aEOn }
PROOF OF ( i ) .By 3.2.3.6,
.
E :C; €a}. E} A
C;}.
3.2.3.7,
3.2.3.12
and 3.2.3.13.
PROOF OF ( i i ) . L e t a = n {[ : x c R C;}. We have, b y 3.2.3.6 and x C R x =Rpx 3.2.3.7, Thus, a 5~x7On t h e o t h e r hand, i f x C -Rt,then by 3.2.3.9 and 3.3.1.2, p x Z p R E = p E = C;. T h e r e f o r e p x C - a, i.e. p x = f15 : x C-R C ; ) .
.
Let = n It : x E R S I. By 3.2.3.6 and 3.2.3.12, x E R S p x . Hence, On t h e o t h e r hand, i f x E R E , we a p p l y 3.2.3.4 and 3.2.3.9 to px. o b t a i n p x C C;. T h e r e f o r e S P X C_ l and, hence, S p x 50. Thus, Spx = n {E : x E RC; I.
CC
C
U
.
PROOF OF ( i i i ) . S i n c e R a 5 W , f o r a l l a, we have, u { R a : a E O n } On t h e o t h e r hand, by ( i i ) , i f x E W, t h e n x ERSp x . Hence, W C { R a : aEOn1.
- W. C
ROLAND0 CHUAQUI
162
PROOF OF ( i ) . Assume t h a t O n € V . Then { R u : a e o n } = R*OnE Y W p V. and, hence, W = U { R a : a E O n 1 E V. But, by 2.5.3.12, qEA,
PROOF OF (ii). Suppose f i r s t , t h a t f o r e v e r y o r d i n a l t; t h e r e i s an Then u A > O n and, hence, by ( i ) , A $ v. with q 2
[.
Suppose now t h a t t h e r e i s a [ such t h a t f o r every Ac t andAEV.
qEA,
2s. Hence
The d e f i n i t i o n s g i v e n here of p , R and On c o u l d t h e r e p l a c e d by t h e f o l l o w i n g procedure which i s more usual. F i r s t we d e f i n e On d i r e c t l y e i t h e r by 3.3.1.19 o r 3.3.1.20. Then we show t h e r e c u r s i o n p r i n c i p l e R i s d e f i n e d r e c u r s i v e l y by 3.3.1.23 ( i ) and f i n a l l y p i s ob3.3.1.21. t a i n e d by 3.3.1.23 ( i i ) . I n M K T i t i s more c o n v e n i e n t t o use 3.3.1.20, whereas i n Z F t h i s d e f i n i t i o n i s n o t p o s s i b l e .
PROBLEMS
1.
Prove 3.3.1.17.
2.
Prove t h a t each o f t h e f o l l o w i n g statements i s e q u i v a l e n t t o x E O n . ( a ) x = T x A X E W A ~ q ~ z ( y E, zx
-
+
y
z V g
E
= z V z E
y).
(b) x = T x A x E W A WyWz(y,zE x - + y C _ z V Z C y). (c) x E W A W g ( g ~ x
y c x ) A v y vz(g,zEx
( d ) Wu(u C -x A u # 0
3 u ( v € u A u n u = 0)
-+
Wu((W u ( u € u A S u E S x
Vw(w c -u A u w
E
Sx
+
+
UW%))
(e) x E W A Wy(u q c -y c x 3.
SuEu)
-+
q
E
-+
g
5z
V z
C -
y).
A
A +
x
E
u).
x).
Prove
On A ( x = 0 V x
( a ) x E w-
x
( b ) x E w-
x E On A Wq(q C -x A y
E
#
LJ
#
x ) A W y(qE x 0
+
UyEy).
+
y = 0 V y#u q)
AXIOMATIC S E T T H E O R Y 3.3.2
163
ORDINAL FUNCTIONS,
I n t h i s s e c t i o n we study functions whose domain i s an ordinal or the c l a s s o f o r d i n a l s and whose range i s included in the o r d i n a l s . F i r s t a notation.
3.3.2.1. STIPULATION OF VARIABLES, Capital Greek l e t t e r s will be used f o r o r d i n a l s o r t h e c l a s s O n , i . e . r E O n or r = O n
.r
,
A
We say t h a t F i s an ofidin& ~ u n c t i v n ,i f rOn ( F ) .
F i s a nequuwe, i f and u r = I'. I f F i s an atrdinak rV ( F ) and u r = r F i s an ordinal sequence, we say t h a t /3 in t h e L i n d 0 6 F dotr an a E D F atr CY = D F , when @ = U F*a = U {FIE :(€a}. p i s a l s o t h e l e a s t upper bound of the sequence F f o r a.
r A ~ ~ U ~ V L C Ii?f , On ( F )
.
Recall t h a t according t o Def. 2.3.2.1, symbols Mo ( F ) , i f $. ,q E D F A t sq
-+
we say t h a t F i s monotone, i n
F't 5 F ' q
.
2.3.2.2 implies t h a t t h i s condition i s equivalent t o each of t h e f o l 1 owing:
(1) x C -D F A
U
x
E
DF
-+
U
F*x C - F' U x
.
( 2 ) x C- D F A n x E D F -,F ' n x c nF*x. ( 3 ) x,q, x u y
6
DF
(4) x,y, x n y
E
DF
+
-+
F'x
U
F'q C - F'(xUy).
F'(xny) c - F ' x n F'y.
I n case of ordinal functions, these conditions can be strengthened.
PROOF, I t i s c l e a r by ( 2 ) , ( 3 ) , and ( 4 ) t h a t the t h r e e conditions on the r i g h t imply t h a t F i s monotone. I s h a l l show t h a t t h e monotony of F implies t h e l a s t condition, leaving the o t h e r implications t o the reader. Assume t h a t F i s a monotone ordinal function and 0 f x C D F . Since x i s a s e t of o r d i n a l s , by 3.3.1.10, n x E x C D F . Hence, by ( Z ) , F ' n x f n F*x. On t h e other hand, since n xE x , n x = t f o r a c e r t a i n t E D F . T h u s , n F x C- F'E = F ' n x . 3.3.2-3 F'x c F'y).
DEFINITION, I n ( F ) * D F Y
I n ( F ) can be read
( F ) A W xW y ( x , y
F Lh d M c L L q LnotLeuALng.
E
DF A x
C
q+
164
ROLAND0 C H U A Q U I
For i n c r e a s i n g o r d i n a l f u n c t i o n s , we have t h e f o l l o w i n g theorem.
The p r o o f i s l e f t t o t h e reader. We say t h a t an o r d i n a l f u n c t i o n F i s adjoivct i f [ Increasing functions are a d j o i n t . 3.3.2.5
rOn ( F ) A I n ( F )
THEOREM,
+
r
WE(tCDF
5 F'E -+
E
for all
E EDF,
C_ F ' O .
PROOF, Suppose t h a t O n ( F ) , I n ( F ) and assume t h a t t h e r e i s a E D F w i t h [ d FIE. Then F'E C E . L e t q = {E : E E D F A F ' E C t } . Then, s i n c e Thus, s i n c e D F = r, F'q E D F and, I N i s awe11 ordering o f O n , F'qCq s i n c e I n ( F ) F ' F ' q c F ' q , c o n t r a d i c t i n g t h e d e f i n i t i o n of q .
.
3.3.2.6
DF-'
THEOREM,
A
c- O n
A
A $ V+
= A).
31 F ( O n o n ( F ) A I n ( F ) A
T h i s theorem a s s e r t s t h a t f o r e v e r y proper c l a s s o f o r d i n a l s t h e r e i s a s t r i c t l y i n c r e a s i n g o r d i n a l f u n c t i o n t h a t enumerates t h e c l a s s . PROOF, We d e f i n e by r e c u r s i o n (3.2.1.4
( i i ) t h e f u n c t i o n F by,
E - F*t), - F*q F- - F*[, F'E F'q - F*q F'E q -
F'E = n ( A
for a l l
E On.
-
We s h a l l f i r s t prove t h a t i s s t r i c t l y i n c r e a s i n g . Assume t h a t [ c q ; Hence, A C A and = n (A F*[) C then F*E C F*q. E A and A F*q ( s i n c e F'F E F*q). n (A F*F) = F ' q . But Therefore, F'E C F ' q . T h i s a l s o proves t h a t F i s biunique, i.e. F-1 i s a
-
f u n c t i o n , and, hence D F - '
= F* O n
We s h a l l now show t h a t D F - '
4 V , by 3.3.1.24.
= A.
I t i s c l e a r from t h e d e f i n i t i o n o f
C A. Assume then, t h a t A P 0 F - l and l e t E E A - D F - ' . Since 4 V , by 3.3.1.24, t h e r e i s a r E D F - ' such t h a t 5 c{. L e t q = = n (A :.f E D F - ' A E C Y I . Then q E D F - ' and E C q . Hence q = F ) for a certain . Suppose, now, t h a t u E F* . Then u E D F-'
F that DF-' DF-' n
(r
-
F* and u c q , s i n c e u = Fa f o r an a E and we have shown t h a t F i s i n c r e a s ing. I f E C u , t h e n qC_u which i s impossible. Thus, a & $ . But u + E , s i n c e
165
A X I O M A T I C SET THEORY
.
F-' and [ $ D F-' and t h i s i m p l i e s t h a t q
u E D
DF-' o f F.
-A 3
Therefore, u ~ t Thus, . we have shown c o n t r a d i c t i n g t h e d e f i n i t i o n o f q.
CC;,
and, hence D F - 1 = A .
, F*
c[
Therefore,
T h i s f i n i s h e s t h e p r o o f of t h e e x i s t e n c e
We proceed t o show t h e u n i c i t y . L e t F and G be i n c r e a s i n g f u n c t i ns t h a t enumerate A. By 3.3.2.4, F-1 and G-1 a l s o i n c r e a s i n g and A O n ( F - ) ,
.
Thus G-' OF and F - l O G a r e s t r i c t l y i n c r e a s i n g .
*On(G-').
3.3.2.4, t C - G-" i.e., F = G.
FIE and [ C - F-"G'[.
Therefore
G't
P
Hence,
by
-
FIE and F'[
C -
C
GI[,
A s l i g h t m o d i f i c a t i o n o f t h i s p r o o f g i v e s t h e f o l l o w i n g theorem. 3.3.2.7 THEOREM, (GI A u D G - ' 4 r 3: ( D F E O n V D F = On ) A I n ( F ) A D F Z D G A U D F - ' = -+
(F) U 0G-l).
A
1
T h i s theorem a s s e r t s t h a t i f h ( = U D G - ) i s t h e l i m i t o f an o r d i n a l sequence t h e n t h e r e a unique s t r i c t l y i n c r e a s i n g subsequence w i t h t h e same limit.
DG-l
PROOF, I f DG-' E
V
, the
g V , t h e theorem i s o b t a i n e d by 3.3.2.6.
proof o f 3.3.2.6
-
i s m o d i f i e d by d e f i n i n g
FIE = n ( D G - ' - ( F * [
F'[
Then F = D G - 1/f d u c t i on.
= U DG-',
ifDG-'
U GI[)
otherwise
.
i s the required function.
v
NOW, if
as
-(f*[UG'()
f
0
U n i c i t y i s shown by i n -
By 3.3.2.2, complete m u l t i p l i c a t i v i t y i s e q u i v a l e n t t o monotony f o r ordinal functions. However, complete a d d i t i v i t y i s s t r o n g e r t h a n monotony.
3.3.2.8
DEFINITION
1
(i)C a d ( F ) + - + r ~ n ( ~ ) A u r = r wA x ( o + -x c r ~ u x + r F' U x =
U
F*x).
(ii) C o n (F) - C a d
F
A W[(tEDF
--L
t
5 F'S).
C a d (F) i s r e a d F i~ a campLeXd!y a d d i t i v e atrdcnd nequence ; Con(F), F cavctinclaw. Since t h e l i m i t o f a s e t o f o r d i n a l s x i s U x , t h e l a s t c l a u s e o f t h e d e f i n i t i o n o f C a d ( F ) can be f o r m u l a t e d by: Thus, t h e name continuous. Continuous f u n c t i o n s F ( 1 i m i t x) = l i m i t F*x. a r e t h o s e c o m p l e t e l y a d d i t i v e and a d j o i n t . I t i s c l e a r by 3.3.2.2 t h a t complete a d d i t i v i t y i m p l i e s monotony and complete m u l t i p l i c a t i v i t y . However, n o t e v e r y c o m p l e t e l y a d d i t i v e funct i o n i s continuous. For i n s t a n c e t h e f u n c t i o n F d e f i n e d by F ' t = U [ i s c o m p l e t e l y a d d i t i v e , b u t n o t a d j o i n t . Also, n o t e v e r y continuous f u n c t i o n
166
ROLAND0 CHUAQUI
t h e f u n c t i o n G defined by G ' t = n {q : t Cq A g = On t h e o t h e r h a d , n o t e v e r y s t r i c t l y i n c r e a s i n g f u n c t i o n i s continuous. The f u n c t i o n S i s s t r i c t l y i n c r e a s i n g b u t n o t continuous. i s s t r i c t l y increasing:
u g I i s continuous b u t n o t s t r i c t l y i n c r e a s i n g .
The n e x t two theorems g i v e p r o p e r t i e s o f c o n t i n u o u s f u n c t i o n s .
3.3.2.9 uDF-~.
THEOREM,
xc DF-l
Cad(F) A 0 #
-+
ux
E
DF-'
V
u x
=
and it x C - DF-' = F ' q ) ). It Y = {q :g E DF A F'q E x A 77 n {C; : F ' t L e t u x # u DF-'. Also, Fly i s b i u n i q u e and F*Y = x. Hence, i s clear that 0 # Y CDF. YEV. PROOF, Assume t h a t F i s c o m p l e t e l y a d d i t i v e , 0
x
we have U x C U D F - l and, hence, U x E U DF-'. Thus, t h e r e i s a t E DF" such t h a t U x E E , i.e. x C t . L e t E = F' 5 ; s i n c e F i s monotone, Y C f and, hence u Y # D F , i.e. U Y E D F . By comp l e t e a d d i t i v i t y , we deduce Since
U
f U DF-l,
Ux=VF*Y=F'UY.
3.3.2.10 FIE
c - F' s t
THEOREM, A
Cad(F)
w E (t Er - { o i A
-
t = u
E
8
(F) A u -+
FIE
=
r
=
r
A
u F* t 1.
w t ( t E r -,
The p r o o f i s l e f t t o t h e reader.
3.3.2.11
Normal ( F ) t + C a d ( F ) A I n ( F ) .
DEFINITION,
The main o p e r a t i o n s on o r d i n a l s a r e normal. f u n c t i o n i s continuous.
3.3.2.12 (i
NOT
-
every normal
THEOREM, ma1 ( F )
vt(C;Er t o ) ~l
=
(ii) Nor ma1 ( F )
u X E D F - ~v u x PROOF,
-
By 3.3.2.5,
=
r = r A w t (t Er u t -+ F" = u ~ * t ) . rOn ( F ) A U r = l? A I n ( F ) A (F) A u
+
FIE
W x(0
uDF-~).
c F'S 5 #
A
x C- D F - l
-+
I s h a l l p r o v e (ii)l e a v i n g (i)t o t h e reader.
From l e f t t o r i g h t t h e p r o o f i s e a s i l y o b t a i n e d by 3.3.2.9 and Def. I n o r d e r t o prove t h e o t h e r d i r e c t i o n , assume t h a t F i s an s t r i c t l y i n c r e a s i n g o r d i n a l sequence and f o r e v e r y non-empty subset x o f
3.3.2.11.
D F - l we have U X E D F - l o r U x = U DF-'. Suppose, now, t h a t q i s a non empty subset of D F and U Y
+ DF
i.e.,
167
AXIOMATIC SET THEORY
UyE D F . I t i s enough t o show t h a t F ' U g C U F * g , s i n c e t h e o t h e r i n c l u s i o n 7s o b t a i n e d by monotony. Since D F i s t h e c l a s s o f o r d i n a l s o r a lim i t o r d i n a l , t h e r e a r e Q , t E DF such t h a t U y E t E q . Then t; C Q and,
hence, F*q c F ' Q , s i n c e F i s s t r i c t l y i n c r e a s i n g .
5
Therefore, u F * g + U D F - . 1
By t h e hypothesis, we have t h a t u F*g = F't; f o r a c e r t a i n t ; . I t i s enough t o show t h a t U g C t ( s i n c e F i s s t r i c t l y i n c r e a s i n g ) . Suppose t h a t t h i s i s n o t t r u e , i . e . , 7 E U y . Then t € 7 E q f o r a c e r t a i n y. Hence F ' t C F l y c u F*q, c o n t r a d i c t i n g u F*g = F ' t . Therefore, u qc - t and, hence, F ' u Y -C 75: = U F*g.
3.3.2.13 (i) A
THEOREM,
q Y
Normal ( F ) A D F - '
A A c - On A W X ( O# x c -A
--f
UXE
A)
+
= A).
3 ! F ( O ~ O( F ~) A
( i i ) r ~ (nG I A u D G - 1 4 D G - 1 A wx(o + x c D G - ' .+ u x E D G - 1 u x = u D G -1 ) + 3 ! F ( N o r m a l ( F ) A D F c D G A U D F - 1 = u D G -1 ). -
v
3.3.2.13 ( i ) a s s e r t s t h a t f o r e v e r y proper c l a s s o f o r d i n a l s c l o s e d under unions, t h e r e i s a normal f u n c t i o n t h a t enumerates i t . PROOF, by 3.3.2.6
3.3.2.14
THEOREM,
o r 3.3.2.7,
and 3.3.2.12
C a d ( F ) A F'O
(ii).
CaC UDF-' -
+
3
!t
F't; E c l C F ' S t ; .
T h i s i m p o r t a n t theorem i s t r u e , i n p a r t i c u l a r , f o r continuous and normal f u n c t i o n s . PROOF, Assume C a d ( F ) and F'O t i o n , t h e r e i s an Q such t h a t c F'v.
S U
.
f
= n Iq
sac
U DF-'. By t h i s l a s t condiThus, t h e r e i s a 5 such t h a t
: u c F'T)).
Since F'O 5 a , 5 # 0. We s h a l l show t h a t 5 # u f . Suppose t h a t f = We have, by t h e complete a d d i t i v i t y o f F , a E F'f = u {F'Q : Q E ~ } n
Hence, E F'Q f o r a c e r t a i n q E b , c o n t r a d i c t i n g o u r c h o i c e o f f o r e we deduce t h a t 5 = S t f o r a c e r t a i n t . Hence,
F't; L a c F ' S t C
There-
.
I n o r d e r t o show t h e u n i c i t y o f t ; , suppose t h a t we a l s o have a cx C F'S [ I . Hence, F't; C F ' S t ; Since F i s monotone, t; C S C g E . Hence, tct;' and t ' S t , i.e. t; = E l .
F't'
t'
.
b.
.
with
t ' and
F i n a l l y , we g i v e a l i s t o f p r o p e r t i e s o f t h e f u n c t i o n s we have been studying.
168
R O L A N D 0 CHUAQUI
3-3.2.15
THEOREM
( i ) I n ( F ) A In(G)
+
In(F-') A I n ( F o G ) .
( i i ) D F ( F~) A ~ (G)~A w Vx(o + x c -D F A u XE D F F I u x F*x) A W x(0 f x C- DGA U x E DG G I U x = u G*x) W x(0 # x c - D(F o G ) +
U
-+
A U X E D (F oG)
-+
(F oG)'
( i i i ) Normal ( F ) u F-l* x ) .
-+
U
x =
W x (0
LJ
#
F*G*x
.
x C DF-' A
-+
UXE
DF-I
+
F-l' Ux
=
( i v ) Normal (F) A Normal ( G ) + N o r m a l ( F O G ) A N o r m a l ( F - 1 o G ) .
(v)
6~
a On A g E BO n A u a = a A u B = B A I n ( g ) A I n ( d ) A a =
~ D g - l u D6-l =
U
-+
PROOF,
(i)
-
D
(6 og)-l.
( i v ) a r e l e f t t o t h e reader.
Assume t h a t 6 and g a r e s t r i c t l y increasing ordinal sequence with P = Dg and a = D d = U g*P. I t i s c l e a r t h a t U D 4 - l = u 6*a and U ( d o g ) " = U ( a o g ) * P . By 3.3.2.4 ( i i ) we have f o r ~ € / 3 , g ' r E a = U g*8. Hence, U ( d o g ) * p C U 6*a , s i n c e 6 i s s t r i c t l y increasing. On the other hand, i f 6 c a, t h e r e i s a r E P such t h a t 6 c g ' y ( s i n c e a = U g*P). Hence, 6 ' 4 ' 7 3 - 6'6 and, t h e r e f o r e , U (6 og)*P-3 - U d*a. PROOF O F ( v ) .
.
PROBLEMS
1.
2.
3. 4. 5.
Prove Prove Prove Prove Prove {a : c1
3.3.2.4. 3.3.2.10. 3.3.2.12 (i). 3.3.2.15 ( i ) - ( i v ) . t h a t t h e r e i s a normal funtion F such t h a t O n O n ( F ) and DF-' = = u a).
A X I O M A T I C S E T THEORY
3.3.3
169
ITERATION O F FUNCTIONS,
I n 2.4.2, f i n i t e i t e r a t i o n o f f u n c t i o n s was i n t r o d u c e d (Def. 2.4.2.7). I t was a l s o shown how a d d i t i o n o f n a t u r a l numbers c o u l d be considered as m u l t i p l i c a t i o n as i t e r a t i o n i t e r a t i o n o f t h e successor f u n c t i o n (2.4.2.9), o f a d d i t i o n (2.4.2.10) and e x p o n e n t i a t i o n as i t e r a t i o n o f m u l t i p l i c a t i o n (2.4.2.14). The same procedure w i l l be used here f o r i n t r o d u c i n g t h e c o r responding o p e r a t i o n s f o r o r d i n a l s . Since t h e f u n c t i o n s F we s h a l l i t e r a t e w i l l have domain On , and hence, be p r o p e r classes, we have t o d e f i n e an o p e r a t i o n t h a t f o r each a w i l l g i v e t h e a - t h i t e r a t i o n o f F , F a . T h i s o p e r a t i o n i s d e f i n e d by r e c u r s i o n u s i n g 3.3.1.22. 3.3.3.1
D E F I N I T I O N BY RECURSION,
Thus, we have d e f i n e d t h e b i n a r y o p e r a t i o n F such t h a t ,
I n o r d e r t o use 3.3.1.22,
t h e o p e r a t i o n H i s g i v e n by, H(X) = FOX
,
and t h e o p e r a t i o n K by, K (x) = I D
I
( D F UDF-'),
if
x
o
=
and, K (X) =
wise.
(U
Y
{X*{q}'x : q
E
D X} :
x
6
V ), other-
The r e s t o f t h i s s e c t i o n w i l l be devoted t o prove some p r o p e r t i e s of it e r a t i on. 3.3.3.2
THEOREM,
ROLAND0 CHUAQUI
170
(iv) u , n E o + F
(v) OnOn (F) (vi)
M O
(F)
( v i i ) Con(F) (viii) u
EWA
+
U S K
F'.
= Fuo
O n O n (F
P ).
+ M O ( F P ) .
-+
III
C o n (+). (F) - + I n ( ? ) .
( i x ) u E w A N o r m a l (F)
+
N o r m a l (F').
= Fo.
( x ) (F0)' (xi) In(F)
-,
W E ( 5 E D F ir
E c_
+
Fh).
The p r o o f i s l e f t t o t h e reader. O n o n ( F ) A Con ( F )
THEOREM,
3.3.3.3.
+
w
6 3 S(E
5'7
A '7 = F V ) .
T h i s i s a v e r y i m p o r t a n t p r o p e r t y o f normal and c o n t i n u o u s f u n c t i o n s d e f i n e d on a l l o r d i n a l s : t h e y have a r b i t r a r i l y l a r g e f i x e d p o i n t s . We a r e a b l e t o prove i t u s i n g i t e r a t i o n . PROOF,
L e t F be a continuous f u n c t i o n d e f i n e d on a l l o r d i n a l s
l e t € be an o r d i n a l . monotone.
By 3.3.3.2
We have, F W [ € O n =D F . (vii),
and
(vi), F w i s
By 3.3.3.2
F and F W a r e a d j o i n t , i.e. E C F''ECF'FWt.
On t h e o t h e r hand, by complete a d d i t i v i t y , F ' F W ' E = F' uV { F h h t : : ~ € w ) =
u { F ' F ~ I E:
U E ~ /=
u {F'~IC; : U E W } c -F ~ ' C ; .
Hence, i f we p u t q = Fa'[,
3.3.3.5
DEFINITION,
F p (F) i s t h e c l a n
3.3.3.6
N o r m a l (G)).
THEOREM ,
we have,
t
C_q and F'q = q
F p ( F ) = { x : x = F'xl
06 dixed
paints
06
.
.
.
F.
O n o n (F) A C o n (F)
-,3
! G ( ( O ~ (F))(G) F ~ A
PROOF, L e t F be a continuous f u n c t i o n d e f i n e d on a l l o r d i n a l .
By
AXIOMATIC SET T H E O R Y
171
and 3.3.1.29 (ii), F p ( F ) 4 V. Also, i f x c F p ( F ) , we have F l u x = F*x = ux and, hence, U X E F p ( F ) . T h e r e f o r e , F p T F ) i s a p r o p e r c l a s s
3.3.3.3 U
o r d i n a l s c l o s e d under unions o f subsets. (i). by 3.3.2.12
Of
The theorem i s t h e n o b t a i n e d
3.3.3.7 REMARK, L e t F be a c o n t i n u o u s f u n c t i o n d e f i n e d f o r a l l ord i n a l s . The normal f u n c t i o n t h a t enumerates i t s f i x e d p o i n t s , F p ( F ) i s sometimes c a l l e d t h e ,+ht de,?ivuLive 06 F. S i m i l a r l y , t h e r e a r e d e r i v a t i v e s o f a l l orders. Now we c o n t i n u e w i t h o u r s t u d y o f i t e r a t i o n . 3.3.3.8
O n O n ( F ) A M ~ ( F A) W C _ P
THEOREM,
+v
c:
F P ' ~2- F P ' F ' c ; .
PROOF, By i n d u c t i o n . L e t B = {P : P E W V V g F P 3 FP'F't.}. It i s c l e a r t h a t OE 8. Suppose t h a t P € 8 . If P E W t h e n S p FW, and hence, S P E B . L e t , now, W C P . Since P E B , F P ' E > F s ' F ' c : . By t h e monotony o f
F, we g e t F s P 1 $ = F ' f P ' ( Suppose, now, t h a t
P
= U
P
2
F'FPIF!g
f
0 and P f 8.
1) Assume 0 = W. We have, F W V E W ] C Uu{Fu'E : u E w } = Fa'[. 2 ) Assume o t h e r w i s e , i.e.,
utlIFq'[
17
for w E q
{ F ~ FVI E : q Therefore, 3.3.3.9
E
C P .
Hence S P E B ,
We have t o c o n s i d e r two cases:
FIE = u v { F u f F f E
:UEW } =
uu{F
su
We have, F P I E = u {FqlE : q E
P.
: w E q c P ) , s i n c e W E P and F ' ' E
F q ' E > _ F'IF'[
u
W C
F ' F'E.
=
C -
F W 1 t for
77
U E W .
Hence, F P I E Z U q { F q r F ' E
C; :
6) =
Since 0 c - 8,
: w c q
C P )
=
pi = F P ' F ~ ~ : .
i n b o t h cases, P E B and t h e i n d u c t i v e p r o o f i s completed..
THEOREM,
OnOn (F) A
-
A
W c
FP'F'E.
MO
(F) A wr;
c: c- ~ ' c :-,
~ P l c : =
Since i n c r e a s i n g , continuous, and normal f u n c t i o n s a r e monotone and a d j o i n t , 3.3.3.9 i s a l s o t r u e f o r these t y p e s o f f u n c t i o n s . PROOFi
Assume t h a t F s a t i s f i e s t h e h y p o t h e s i s o f t h e theorem. Then (vi), F P ' E C FP'F'C;. A p p l y i n g 3.3.3.8, we o b t a i n t h e d e s i r e d conclusion.
C: C_
-
F ' E , and hence by 3.3.3.2
On On F) A
oC_P -+ ( I n ( F ) A
In(FP )
-
F =
3.3.3.10
THEOREM,
PROOF,
Assume t h a t F i s a f u n c t i o n d e f i n e d on a l l o r d i n a l s and
F P = Z D Ion).
172
R O L A N D 0 CHUAQUI
.
3 w I f F = F P = ID I O n , then c l e a r l y F a n d F P a r e s t r i c t l y increasing. Suppose, now, t h a t F a n d F P a r e s t r i c t l y increasing. Then, by 3.3.3.9, F P ' F't = F P ' t B u t F P J. S biunique, hence, t = F ' t . 9
P
.
3.3.3.11
THEOREM,
OnOn (F) A W C
A Con ( F )
+
V 5 FSP15 =
F P f F'5. The proof i s l e f t t o the reader. 3.3.3.12
THEOREM,
O n o n (F) A
2 p A Con ( F )
+
FP
= F
~ .
This theorem shows t h a t f o r normal functions defined on a l l o r d i n a l s the i t e r a t i o n a f t e r wdoes not produce anything new. PROOF, By induction. Let B = CP : P E W V F P = F w } . I t i s clear t h a t O € B and t h a t f o r P E W , S P G B . Suppose t h a t w C - (3 E B . Then by W H e n c e S P E B. 3.3.3.11 and 3.3.3.9, F s P = F 0 F P = F 0 F w = F w o F = F
.
Suppose, now, O + u P = P qEP} =
F0't,
for a l l
[.
- 8.
C
Hence,
Then F P ' 5 = u q { F q ' [ : q € / 3 } = u q { F w $ : PEB.
T h u s , we have completed t h e inductive proof. 3.3.3.13
Fa
DEFINITION,
= q(
Fq"a:77
On).
E
According t o this d e f i n i t i o n , i t i s c l e a r t h a t i f O n O n ( F ) , we have t h a t O n O n ( F ) a n d F ! I ) = F q r a f o r a l l q , a € On a U 3.3.3.14
THEOREM,
OnOn ( F ) A W 5 5 C -
F't
+
Cad(Fa).
Assume t h a t F i s an a d j o i n t function defined on a l l ordinals. Then F q ' a C - F ' F q ' a = F S q l a . Hence Fk q C FA S q , i . e . Fa i s monotone. : ( E q } = u F p. Let 7) = U f 0; then F' q = F"a = U { F t 'a - : ( € 7 ) ) = Ut {F'$ 77 U t a By 3.3.2.10, Fa i s completely a d d i t i v e . PROOF,
From 3.3.3.12,
we obtain t h a t i f F i s normal then F& p
P ~ W . T h u s , Fa i s not of i n t e r e s t when F is normal.
The i n t e r e s t i n g Fa occur when F i s ~%.LcLty a d j a i & ,
=
FAw f o r a l l
i.e., [ C F ' t
f o r a l l 5 . A continuous function cannot be s t r i c t l y a d j o i n t , since in 3.3.3.3 we a c t u a l l y proved t h a t a l l continuous functions have fixed points. 3.3.3.15
THEOREM I
( i ) OnOn (F) A V $ 5
C
F'E + N o r m a l ( F ). U
A X I O M A T I C S E T THEORY
173
PROOF OF (i).Assume t h a t F i s a s t r i c t l y a d j o i n t f u n c t on d e f i n e d on a l l o r d i n a l s . ~
1
~
normal.
.
= 7 F ' 7V I ~ ~
By 3.3.3.14,
Hence FA 7
Fa i s c o m p l e t e l y a d d i t i v e .
c F;
.
SV
The p r o o f o f ( i i ) i s s i m i l a r .
f o r a l l 77.
Also
Fqru C i ) , Fa i s
BY 3.3.2.12
PROBLEMS
1.
Prove 3.3.3.2.
2.
Prove 3.3.3.11.
3.
Suppose t h a t F i s continuous.
3.3.4
Determine whether F B o
Fa
=
F"o F'.
ORDINAL A D D I T I O N I
A d d i t i o n o f o r d i n a l s i s an e x t e n s i o n o f t h e o p e r a t i o n o f a d d i t i o n o f n a t u r a l numbers, d e f i n e d i n 2.4.2.1. As we saw i n 2.4.2.9, addition of n a t u r a l numbers i s t h e f i n i t e i t e r a t i o n o f t h e successor f u n c t i o n . S i m i l a r l y , a d d i t i o n o f o r d i n a l s w i l l be t h e o r d i n a l i t e r a t i o n o f successor. 3.3.4.1
DEF I N I T I ON I
(i) .+=S
(ii) a+ [ (iii) +a =
a' =
(t
a+iE t
.
: [ E On).
We have d e f i n e d f o r each o r d i n a l on t h e l e f t .
CL
t h e o p e r a t i o n a+ o f a d d i t i o n o f a
f a i s the operation o f a d d i t i o n o f
a on t h e r i g h t , which i s
n o t , i n general, t h e same as .+. I t i s c l e a r t h a t t h e f u n c t i o n S i s s t r i c t l y a d j o i n t . Hence by 3.3.3.15, a+ i s normal. The n e x t theorem i s deduced d i r e c t l y from t h e de-
f i n i t i o n and t h e n o r m a l i t y o f a + .
174
ROLAND0 CHUAQUI
3.3.4.2
THEOREM,
(i) a + 0 = a. ( i i ) a + l = S a . ( i i i ) a + SP =S(a+P)
.
x c-O n A x # 0 + a + u x u { a + t : E a + (P u 7 ) = ( a + P ) u (a+r). P = U P # 0 - + a+ P = uCa+t : t E P I .
(iv) (v) (vi)
(vii) a + (viii) a
(P'+r)= ( a + P ) +
+0
= n
E
XI.
y.
{ t : a s t A a+*
ct} -
=
+ a+* u u a+*p
a
=
=auT( a+*P) .
P +
(ix)
0
+
a + P = a+* P u u a+* P = T(,+*
P).
a+PEa+r.
(X)PCY++ ( x i ) P c r -
a + P c a + y .
a+p=a+r.
(xi i )P= r-
( x i i i ) O + X c-O n + c t + n x = n (xiv) a c a +
P
A
(P
# 0
( a + t : t ~ ~ } .
t
+ a c a+P)
( x v ) 0 + a = a.
.
( i ) , ( i i i ) and ( v i ) t o g e t h e r may be taken as a r e c u r s i v e d e f i n i t i o n o f a d d i t i o n . I t i s w o r t h n o t i c i n g t h a t ( x i v ) says t h a t f o r P # 0, t h e f u n c t i o n + i s s t r i c t l y a d j o i n t . Thus, i t w i l l be used t o o b t a i n a new
P
normal f u n c t i o n :
3.3.4.1.
m u l t i p l i c a t i o n on t h e l e f t .
PROOF, ( i ) , (ii), and (iii) a r e immediate consequences o f Def. ( i v ) expresses t h e complete a d d i t i v i t y o f a+. ( v ) and ( v i ) a r e
c o r o l l a r i e s o f complete a d d i t i v i t y . ( i i i ) and ( v i ) . u s i n g (i),
ci U
PROOF OF ( v i i i ) . We s h a l l prove by i n d u c t i o n over P t h a t a + P U u +* P . The o t h e r i d e n t i t i e s a r e o b t a i n e d by 3.3.1.16.
a+*
a
A = { P : a + P = a u
P +P
and
= 7 + 1.
ci
=
a
( v i i ) i s e a s i l y shown by i n d u c t i o n
(a+r) u
u ( a+*y u
+*Puu
a
We have, a
+
+*PI.
a
y =
a
U
OEAsince
a+* y
a+*O=a.
ci
{ a + y l ; hence, a + P = ( a u a+* y u U +*y) a u Ia+'y)) = a U a+* u u .+*P. a
u .+*t : [ € P I
a,u a+* p u U t * P a
.
Thus,
=
P
u u {,+*E E
Wehave,a+P
:E
EP}
=
Let
LetyEA
a+* 7. On t h e o t h e r hand,
{ +'y}) U U ( +*y
L e t , n o w , P = u P + O a n d P c A-. u {a u
U U
P
U
{ci+yI
Thus,
= U { a + E : t E P I =
u u { u cit*t
:T€P
=
A and t h e i n d u c t i v e p r o o f i s completed.
( i x ) i s deduced from ( v i i i ) .
EA.
175
AXIOMATIC SET THEORY
PROOF OF ( x ) ,
( x i ) , and ( x i i ) .
( l ) P S r + . a + P C a + r, and (2)Pcr+a+P
c a+Y,
a r e consequences o f t h e n o r m a l i t y o f (3) a +
P 2a+Y
+.
P 27
i s o b t a i n e d from (2), s i n c e (4) a +
P c a+Y-?
¶
P $ - Y implies
C Y
i s o b t a i n e d from ( l ) , s i n c e
a+ . y C
P.
¶
P $ Y i m p l i e s Y c- P .
( 5 ) P = Y + . a + P = a + Y ,
i s deduced by l o g i c ; and ( 6 ) a + P = a + y - + P = y ,
i s o b t a i n e d from (3). ( x i i i ) i s d e r i v e d from monotony o f
( x v ) i s shown by i n d u c t i o n on a.
quence of (i), ( x ) , and ( x i ) . 3.3.4.3
THEOREM,
.
a+. ( x i v ) i s an immediate conse-
(aC P
+.
a+ Y C - P + Y) A (a + Y
C
P
+
Y
f
ac
0)
The f i r s t p a r t expresses t h e f a c t t h a t + ( a d d i t i o n on t h e r i g h t ) i s Y monotone. The p r o o f by i n d u c t i o n i s l e f t t o t h e reader. 3.3.4.4
THEOREM,
P c -a
+P
.
PROOF, Since a+ i s s t r i c t l y i n c r e a s i n g , i t i s a d j o i n t . 3.3.4.5
THEOREM,
(i)a c P ( i i ) ac P -
(xiv).
PROOF,
I t
a+[ = P
.
.
3 ! t a + t = P .
B o t h ( i ) and ( i i ) from r i g h t t o l e f t a r e o b t a i n e d by3.3.4.2
Assume t h a t a c -P .
Since a+ i s normal and
3~3.2.14 t h a t t h e r e i s a unique
t such t h a t ,
a + t c p c a + s t .
02a
=
at'0, we have by
176
ROLAND0 C H U A Q U I But a
+ S l =S ( a + t ) .
3.3.4.6
DEFINITION,
Hence
0
-
c1 t
a
=
[ =
0.
U {y : P
2.
A
a
t y =
0)
.
-
this difP CY i s t h e h i g k t did6etrencc beheen P and a. By 3.3.4.5, ference has t h e c o r r e c t p r o p e r t y , i.e. i f a 5 P , we have t h a t c1 + ( P -a) =
P .
The problem o f t h e l e f t d i f f e r e n c e i s c o m p l e t e l y d i f f e r e n t and we s h a l l discuss i t l a t e r .
PROOF1 From r i g h t t o l e f t , t h e i m p l i c a t i o n i s o b t a i n e d by 3.3.4.2 ( x i v ) and ( x ) .
a + 6 = P + y and Assume, now, t h a t c1 C _ P t y and P c a . By 3.3.4.5, = a f o r c e r t a i n 6 and t . Hence, P + ( t t 6 ) = P + y and by 3.3.4.2 t c - y. ( x i i ) , t + 6 = y. Thus, by 3.3.4.4,
P +
t
PROOF, S i m i l a r t o t h a t o f 3.3.4.6.
3.3.4.2
PROOF, From r i g h t t o l e f t i s e a s i l y o b t a i n e d by t h e a s s o c i a t i v e law, (vii).
Assume, now, t h a t c1 t P = y + 6 . Then a a r e two cases: a C_y o r c1 = y + t w i t h 4 2 6 .
2y +
( 1 ) a c y. Then c1 + t = y , by 3.3.4.5. by 3.3.4.2 T x i i ) t + 6 = 0.
Hence, a
( 2 ) a = y t t and (xii), l + P = 6.
PROOF, 3.3.4.11
t C- 6 .
By 3.3.4.2
THEOREM,
Then, y +
(x), f
t
6.
By 3.3.4.7
+t +
there
6 = a +
P and,
+ P = y + 6 and, by 3.3.4.2
( x i ) , and 3.3.4.3.
0
+
U
(a+P) =a+
”P .
PROOF, Assume t h a t P # 0. L e t t E U ( a + P ) ; t h e n t E v E a + @ ,f o r we have two cases: e i t h e r q c a o r v = a+( a c e r t a i n 7). Hence, by 3.3.4.8, with t C P
.
A X I O M A T I C SET THEORY
(1) q C a .
C
( 2 ) 7 = c1 + t and t Since t E q , t E u
UP.
P
u t u
and, hence $ E
Then, C P . + U P .
U
u a
= a),
3.3.4.12
THEOREM,
PROOF, now, t h a t ~1 + 3.3.4.2
0
=
0.
.
+ u P. ++
U
q Ca
+
The s i m i l a r p r o o f o f
=
p
#
0 V (p
=
0A
From r i g h t t o l e f t , i t i s o b t a i n e d by 3.3.4.11. Assume, i s a l i m i t o r d i n a l . We have t o c o n s i d e r t wo cases:
(1) P f 0. By 3.3.4.11, (xii), P = UP. (2)
bers.
u (uto) = a+ p
u p.
t
t c u p . Hence,
b y 3.3.1.11,
Thus, we hav e shown t h a t U ( a + P ) a P C- u ( a + B ) i s l e f t t o t h e r e a d e r .
u +
Then u
P
a t
u (c1+0)= u + u P
=
a + P =
c1 = U ( a + P ) =
'Y
.
.
Hence, b y
9
The n e x t f e w t h e o r e m s g i v e some p r o p e r t i e s o f w a n d t h e n a t u r a l num-
3.3.4.13
THEOREM,
{t :a C
U
(a+w) = a
+ w = n {t : aC t
u + w i s a l i m i t ordinal.
By 3.3.4.12,
PROOF,
9
Then q
177
.
=
u [I.
T h e r e f o r e ct i- w z b y 3.3.4.8, [ = b y 3.3.4.12, b is t h a t a + a =n { [ :
= u tl. Suppose t h a t a C t C a f w Then, 0 C K C w . Since K i s a successor o r d i n a l , t o r d i n a l , i.e. t # U t . Thus, we h ave shown
a t K with not a l i m i a c t = u t l . . 3.3.4.14
THEOREM,
+
K
w = w
-
K
E w
.
PROOF, From l e f t t o r i g h t , i t i s o b t a i n e d b y 3.3.4.2 o t h e r hand, assume t h a t K E w Then, b y 3.3.4.4 w cK t w 3.3.4.13, K + wC_ w
.
.
( x i v ) . On t h e Also, by
.
3.3.4.15 REMARKS, The c o m m u t a t i v e l a w and c a n c e l l a t i o n on t h e r i g h t a r e n o t t r u e i n g e n e r a l f o r o r d i n a l a d d i t i o n . F o r i n s t a n c e , we have
, if
(1)
K
+ wf w+
K
(2)
K
+
w = w
w = h
+
0c
,
if
c
K K
( b y 3.3.4.13).
w
C
h
C w
( b y 3.3.4.14).
We s a y t h a t i s a Ledt did6ehence between p and a, i f t t ci = 0. T here m i g h t n o t b e a n y l e f t d i f f e r e n c e . F o r i n s t a n c e , i n t h e case c1 # uci and P = U P , t h e r e i s no [ , s u c h t h a t t + a = p ( b y 3.3.4.12). Also, t h e r e may be many l e f t d i f f e r e n c e s ; a l l n a t u r a l numbers a r e l e f t d i f f e r ences between w a nd w. Anyway i t c a n b e p r o v e d t h a t {a :3 [ [ + ci = ) i s f i n i t e f o r every P
.
3.3.4.16
THEOREM,
K ,
X E w-
K
t A E w .
178
ROLAND0 CHUAOUI
Assume f i r s t t h a t
PROOF,
3.3.4.14,
x cK
+
K
Now, i f K + A we g e t K , X E w .
.
.
-I-W =
E w ,
K ,
X
since by 3.3.4.2
,
E w+
THEOREM,
PROOF,
As f o r the n a t u r a l numbers.
3.3.4.18
THEOREM,
K
E w
-+
+
K
+a
(K
and 3.3.4.3,
(XiV)
3.3.4.17
K
Then, by 3.3.4.2
w .
E
+
=
a
=
-
( x i ) and
EK+
K ,
X,
K.
w~ a V
K
= 0)
.
Assume t h a t a E w a n d K + a = a. Then, by 3.3.4.17, a + K = Q PROOF, and hence by 3.3.4.2 ( x i i ) , K = 0. On t h e o t h e r hand, suppose t h a t w c a. Hence, K + a = K + w + C: = w t t = a, by Then a = w + E , by 3.3.4.5. 3.3.4.14.
.
3.3.4.19
THEOREM,
0
PROOF,
By 3.3.4.17
and 3.3.4.18.
C K C w-)
(a +
K
=
K
+a
c--i
aE w)
.
PROBLEMS
1. 2. 3.
Prove 3.3.4.3. Complete t h e proof o f 3.3.4.11. a) a + a = P + P Prove: a= P . +
b) a + a + a = P + P + P
4. 5. 6.
+
a =P.
C)a+aCP+P-+acP. d) a + a = P + P + P --* 3 ~ ( = aY + Y + Y A P = Y + 7 ) .
Prove: a + P + P + a = P + a + P + a . Prove t h a t f o r each o r d i n a l P t h e s e t {a: 3 C: C: + a = 01 i s f i n i t e . Define by r e c u r s i o n for an a r b i t r a r y s et a E W , W V ( a + ) A V x ( x E W + a + ‘ x = a u a+*x u u +* x ) . a
Study the f u n c t i o n a+. proved f o r w i t h a+
.
In p a r t i c u l a r t r y t o prove the theorems we have
7.
Describe a + w .
Find the f i r s t d e r i v a t i v e o f a+
8.
Determine whether F P o F a = Fa”
for
c1
2w
.
or P
2w
.
AXIOMATIC S E T THEORY
3.3.5
179
ORDINAL M U L T I P L I C A T I O N
The f u n c t i o n a+ i s normal.
Thus i t s i t e r a t i o n i s n o t i n t e r e s t i n g .
However, t h e f u n c t i o n +P i s s t r i c t l y a d j o i n t f o r
P
3
0.
Hence, we can
i t e r a t e i t t o produce a normal f u n c t i o n (3.3.3.15). This d e f i n i t i o n coincide w i t h t h a t given previoesly f o r f i n i t e o r d i n a l s (2.4.2.10).
DEF I N I T 1 ON ,
3.3.5.1
( i ) ax = ( i i ) a * P = ,x'P
-- P ( P * a: P E O n )
( i i i ) xu
x'p = (+ ) ' P =+:I0 a0 Thus a * P i s t h e a d d i t i o n times o f a on t h e r i g h t . Since +a i s s t r i c t l y a d j o i n t f o r a 2 0, X i s normal f o r such a (3.3.2.15). The n e x t a theorem i s deduced from general p r o p e r t i e s o f i t e r a t i o n , e s p e c i a l l y from According t o t h e d e f i n i t i o n , we have, a
the normality o f
=
x 'a = P
o
l
From now on, we assume t h e usual convention f o r r e s -
t o r a t i o n o f parenthesis. 3.3.5.2
P
M u l t i p l i c a t i o n t a k e procedence o v e r a d d i t i o n .
THEOREM,
(i) a 0 0= 0. (ii) a*SP= a-0 f a .
(iii) P =up (iv) 0
*P
O + a * P = wC;
f
= 0.
( v ) x c- o n
-ta.ux=y
(vi) a + O A P C y - a * P (vii)a=OV (viii) a = 0 V
( a = €:C; E P )
(a*€
:t
.
E XI.
c a - y .
P c-y - a - / 3 c a . y .
P
= 0
++
a-P = 0.
( i x ) a 1 = a. (x)
(P
#
O - t a E a * P ) A ( af 0
3
1 + a c a.0).
The p r o o f i s l e f t t o t h e reader. 3.3.5.2 3.3.5.3
(x) implies t h a t xp i s s t r i c t l y adjoint, f o r THEOREM,
3
1.
ROLAND0 C H U A Q U I
180
PROOF OF ( i ) . By i n d u c t i o n . L e t C = I y : a *(P + y ) = a * P + a * y l . I t i s c l e a r t h a t OEC. Assume t h a t y E C; then, a (P + S y ) = a (S(P + y ) ) = Suppose, f i a * ( P + y ) + a = a P + ( a y + a) = a P + a - S y ; h e n c e S y E C. n a l l y , t h a t y = u y f 0 and y c C; then, u s i n g t h e complete a d d i t i v i t y o f + and ax, and t h e i n d u c t i v e hypothesis, we o b t a i n ct (0 + y ) = ct mu,@ +[ :
a
: EEy) = U ( a * P + a * E : E E y I = a * P + u { a - E : E ~ r ) =
r
EEyl=UtkP(P+E)
a - 0 + a * y ..
The s i m i l a r p r o o f o f ( i i ) i s l e f t t o t h e reader. 3.3.5.4
THEOREM (EUCLIDEAN ALGORITHM).
a-y+6 A 6 Ca).
Suppose t h a t a
PROOF,
a$ 0
= 0.
t
.
Hence y E C and t h e i n d u c t i v e p r o o f i s completed.
f
Hence, by 3.3.2.14, a-y c P
0.
Then
ct
f
0
-+
i s normal and
3! 7 I ! 6
axlo
CP,
(0 =
since
t h e r e i s a unique y such t h a t
c
a = s y =a97 + a
.
.
Hence, by 3.3.4.8, = a - y + 6 C a.7 + a. By 3.3.4.2 Since y i s u n i q u e l y determined t h e n so i s 6 by 3.3.4.5.
(xi), 6
C
a.
( i ) i s e a s i l y proved by i n d u c t i o n on B and ( i i ) by i n d u c t i o n
PROOF, on 7 .
.
PROOF OF ( i i i ) . L e t y f U y and a C 0. Then y = 6 + 1 f o r a c e r (xi), a.6 + t a i n 6 . We have, by ( i i ) , a . 6 519 6. Hence, by 3.3.4.2 + a c P - 6 + 0, i.e., a . 7 c y ( i v ) i s o b t a i n e d from ( i ) - ( i i i ) .
.
3.3.5.6 (i) (ii)
THEOREM, K
* hE
K,
(iii) K O PROOF,
W-K,
r( E W + K W =
w-
h
E W V K = 0
ah = 1 * K .
vh = 0 .
K E wA K # 0.
( i ) and (ii)a r e l e f t t o t h e reader.
PROOF OF ( i i i ) .
Assume t h a t
K
o w =
w ; then
K
f
0.
Also, s i n c e
Kx
AXIOMATIC SET THEORY
181
i s s t r i c t l y a d j o i n t (3.3.5.2 (x)), K C K * w = w . proved t h e i m p l i c a t i o n f r o m l e f t t o r i g h t . Assume, now, t h a t
OCKCW.
We have,
Hence
K * W =
K E
w . We have
.
Y{K*k:kEW}
On t h e o t h e r hand, s i n c e K # 0, by 3.3.5.5 (iv), ( b y (i)). Thus, K * W = w and t h e r i g h t t o l e f t i m p l i c a t i o n i s proved.
WCK
O
5 W
W .
3.3.5.7 REMARKS, The commutative l a w i s n o t t r u e i n general f o r o r d i n a l m u l t i p l i c a t i o n . For instance, 2 * w # W * 2, s i n c e 2 * W = W C W. 2 = W t
w .
0, i.e.
The l e f t c a n c e l l a t i o n l a w i s t r u e f o r o r d i n a l m u l t i p l i c a t i o n and a#
a
-, (P
# 0
= y
-
a - 0 = cx - 7 )
(by 3.3.5.2
However, t h e r i g h t c a n c e l l a t i o n law i s n o t t r u e . 0
C K
c h c
W + K
vii)
We have
* W = k * W = W .
PROOF, Assume t h a t a * P = u ( a - 0 ) and P #uVp, Then a c e r t a i n y. Hence a - 0 = a - y t a . Thus, by 3.3.4.12, a = proves t h e i m p l i c a t i o n from l e f t t o r i g h t . Assume, now, t h a t a u (a*PI,. So assume, a l s o
(1) P = 3.3.5.2
UP
(vi), a *
(2)
=
P
= uc1 or c1 # 0 # 0.
and a # 0. E
If a = 0 or We have two cases:
= UP.
Then a - P =
{a * t : [ 7 P I .
c; ( a * [: t
U
Hence, by 3.3.1.16,
u a # O and p = y t 1, f o r a c e r t a i n cx * B = U (a0 0 ) .
Hence, by 3.3.4.12,
.
.
y.
E
PI.
= y
+ 1 for This
Ua.
= 0, a - P = 0 =
B u t , by
a*@ =
U
(a* P ) .
Then a * P = a * y + a .
Thus, i n b o t h cases, we have proved t h a t a - P = U ( a - 1 3 ) and t h e i m p l i c a t i o n from r i g h t t o ? e f t f o l l o w s . 3.3.5.9
THEOREM,
we have t h a t w e t = U ( w . 5 ) . PROOF OF ( i ) . By 3.3.5.8, i m p l i c a t i o n from r i g h t t o l e f t f o l l o w s .
Hence, t h e
Assume, now, t h a t a = Ucx. We have, by 3.3.5.4, a = w * t f K with I f K # 0; then, by 3.3.5.8, a # U a . Hence K = 0 and a = w * C ; Thus, t h e l e f t t o r i g h t i m p l i c a t i o n f o l l o w s .
K
E
W.
.
ROLAND0 C H U A Q U I
182
we o b t a i n a = w * E
PROOF OF ( i i ) . By 3.3.5.4 Hence, b y (i), (ii)f o l l o w s .
+K
with K E w
.
3.3.5.10 REMARK, We can c l a s s i f y o r d i n a l s i n t o even and odd o r d i n a l s u s i g n 3.3.5.9 (ii). a i s even, i f a = b + K w i t h p = U p , K E w , K even. a i s odd, i n case K i s odd. T h i s c l a s s i f i c a t i o n i s u s e f u l f o r many constructions. 3.3.5.11
THEOREM,
a + p = P - a * w CP.
T h i s theorem can be expressed by saying t h a t t h e f u n c t i o n a. + i s t h e f i r s t d e r i v a t i v e o f a+, i.e.
a.
+ i s t h e normal f u n c t i o n t h a t enumer-
ates the f i x e d points o f a+.
.
Q * W +
Then, by 3.3.4.5, p = PROOF, Suppose, f i r s t , t h a t a w c P 7, f o r a c e r t a i n 7. Hence, P = a * w + 7 = a - ( l + w ) + y =
a + (a*w+y)=Ct+P.
Assume, now, t h a t a + B for a l l K E w.
+B
C ~ * K
UK{a* K : K E W)
3.3.5.12
P
0. I t i s easy t o show by i n d u c t i o n t h a t Hence, a . 20 ~ f o r a l l K E w and Q * W = =
c - 8.
DEF I N I T ION ,
The elements of M o a a r e c a l l e d mdin ohd.inaeS 06 a d d i t i o n , and t h e elements o f M o m , main ofidin& 06 muRtipMcation. These a r e o r d i n a l s t h a t a r e c l o s e d under t h e corresponding o p e r a t i o n s . 3.3.5.13
THEOREM,
(i)aEMoa -VE(Ecct+E+a=a)AafO.
E
-+E
( ( i i ) a E Moa *
W E W q(a
(iii) aEMom-
WE(0CE c a + E . a =
=
+q
=
aV q
=
a) A a # 0.
a)AafO.
The p r o o f i s l e f t t o t h e reader. The c o n d i t i o n V E Wq([ * q = a E = a V q = a) A a f 0 i s n o t e q u i v a l e n t t o a b e i n g a main o r d i n a l o f m u l t i p l i c a t i o n . T h i s e x p r e s s i o n d e f i n e t h e prime numbers and i t i s c l e a r t h a t t h e r e a r e f i n i t e prime numbers d i f f e r e n t f r o m 1 t h a t a r e n o t main numbers o f m u l t i p l i c a t i o n . There a r e a l s o i n f i n i t e prime numbers which a r e n o t main numbers. For instance, w + 1, s i n c e w - 0 3 w + 1. I n f i n i t e main numbers o f a d d i t i o n o r r n u l t i p l i c a t i o n a r e always l i m i t o r d i n a l s . The o n l y f i n i t e main o f a d d i t i o n i s 1, and t h e f i n i t e main numbers o f m u l t i p l i c a t i o n a r e 1 and 2. +
3.3.5.14
THEOREM,
a
#
0
+
a
w E Moa
.
AXIOMATIC S E T T H E O R Y
1
t
=
183
PROOF, Assume
01.0
01 it 0. We s h a l l show t h a t i f a - w = o r q = ~ 1 . w 3.3.5.13 . ( i i ) w i l l give t h e theorem.
E
+ q , then
.
Suppose, then, t h a t 01 o w = € + q I f q = 0 o r Q = 01.w the conclusion follows e a s i l y . Thus, assume t h a t q f 0, q f 0 1 . w . Then by 3.3.4.2 we obtain by 3.3.2.14, ( x i v ) , E , q E a * w . Then by t h e normality of t h e r e a r e v and 1.1 such t h a t , a-v
Since t , q contradicting 5
C
+
C (
C 01-Sv
and
01
* p C q C awS1.1.
a . w , w e get I * , v E w .
q = a w.
m
Hence,
+
q c~.(Sv+Sp)c01.w,
Using 3.3.5.14, we obtain t h a t M o a i s a proper c l a s s . I t i s obvious t h a t i t i s closed u n d e r unions. Hence, t h e r e i s a normal function t h a t enumerates i t . However, in order t o describe i t , we need exponentation.
PROBLEMS 1. 2. 3. 4.
Prove 3.3.5.2. Prove 3.3.5.3 ( i i ) . Prove 3.3.5.6 ( i ) and ( i i ) . Prove: a + P = P + a - ~ ~ ~ K ~ X ( K , A E U A ~ == r ~* h. )K. A P
5.
Prove 3.3.5.13.
6.
Prove t h a t x i s n e i t h e r s t r i c t l y increasing nor continuous and t h a t P i t does not s a t i s f y :
7.
Prove:
01#0-+ 3 7 3 6 ( P = r * 0 1 + 6 A S ca).
( a ) a + P + a = P + 0 1 + P - + a =P . ( b ) 0 1 * 2 = / 3 * 3 +3 r ( a = r * 3 A P = 7 * 2 ) .
(c) a
* K
=
p
*?T
-+
oh A K,h p = 1)
-+
E
w A tip t i v t i ? T ( V , p , n E w A
lr(a
=
* h I\ 0 = y
O K ) .
K
= I.1
* V
A h =
184
ROLAND0 CHUAQUI ORDINAL E X P O N E N T I A T I O N ,
3.3.6 We tion.
x
now i t e r a t e m u l t i p l i c a t i o n on t h e r i g h t , xa, t o o b t a i n exponentiac1
i s s t r i c t l y adjoint for a
3
l (3.3.5.2
(x)).
Our p r e s e n t d e f i -
n i t i o n c o i n c i d e s w i t h t h a t g i v e n f o r n a t u r a l numbers (2.4.2.14). DEF I N ITION,
3.3.6.1
( i )expa = (xaIl. (ii) a’
= expip.
We have, a’ = exp$P =(x,),’P
= (xu)P
’ 1.
Thus, a’
i s the i t e r a t i o n
0 times o f t h e m u l t i p l i c a t i o n on t h e r i g h t by u , i.e. a m u l t i p l i e d P times The same symbol w i l l be used f o r e x p o n e n t i a t i o n as f o r i t e -
t o the right.
However t h e r e w i l l be no confusion, r a t i o n a’, F P . i s r e s e r v e d f o r o r d i n a l s (i.e. Greek l e t t e r s ) . Since xa i s a d j o i n t f o r u 2 1, exp,
i s normal f o r such a (3.3.3.15).
The n e x t theorem i s d e r i v e d m a i n l y from t h i s t o t h e reader. 3.3.6.2
since exponentiation
normality; i t s proof i s l e f t
THEOREM,
( i ) ao = I . ( i i ) as’=
a’* a
.
( i i i ) P = u ~ t ~ - a P {a‘: = u ~ [Ep). ( i v ) (P (v) 0
+o C P
+
8=
0) A I ’ = 1 .
C _ Y + a h aY
-
.
(vi) 2 5 c c ~ p c y + a P c a ~ . ( v i i ) (1 ~p (viii)
-,c1 c- a ’ ) ~ ( 2 ~a
o + x c- O n -
aux= u
E
A
2 z p
+
u c a’).
{ c r ‘ : ~ ~ x } .
The second p a r t o f ( v i i ) shows t h a t f o r
0 >_
2 t h e function basep =
Hence we can r e p e a t t h e i t e r a ( a p :a E O n ) i s s t r i c t l y a d j o i n t . t i o n process w i t h t h i s f u n c t i o n and o b t a i n a new normal f u n c t i o n .
185
A X I O M A T I C S E T THEORY
The p r o o f by i n d u c t i o n i s l e f t t o t h e reader. 3.3.6.4 2
5
THEOREM,
+3
A 15
PROOF I expa’ 0 = 1
(EXTENDED EUCLIDEAN ALGORITHM).
!y
Suppose 2
20.
c_ a
and 1 5 P
asy =
aY
.
A OC 6
c aA
E
c ay )
.
We have t h a t expa i s normal and
a.
a y e
x i s a l s o normal and
Hence again by 3.3.2.14,
+E
t h e r e i s a unique y such t h a t
Hence, by 3.3.2.14,
(1) a y CP c The f u n c t i o n
3 ! 6 3 ! E ( P = a’* 6
03 a ’.
0.
t h e r e i s a unique 6 such t h a t
(2) a Y . 6 c p c a y - S6. By 3.3.4.5,
(3)
P
=
t h e r e i s a unique
ay*6+€
E
such t h a t ,
.
.
6 C p c ay.a Hence, by 3.3.5.2 (vi), 6 P 3 a’, 6 1 1 . Also, by ( 3 ) and ( 2 ) aY - 6 + E = p c aY* 6 Hence, by 3.3.4.2 ( x i ) , E c ctY. By (1) and ( 2 ) , ay
and since,
PROOF,
C c1
+ay.
( i ) i s e a s i l y shown by i n d u c t i o n on y.
PROOF OF ( i i ) .Suppose t h a t y # U T and a C P . Then y = S 6 f o r 6 By ( i ) , a 6 C p 6 Hence, by 3.3.5.2 ( v i ) , a’, = a’ CI C 0 * p =
.
a c e r t a i n 6. = p y .
3.3.6.6
THEOREM,
(i) K’EWC-’K,AE (ii)
K x
(iii) y
=w*
+oAo
(K
c
W V K
-IVA c
= 0 .
3
1A A =
W A K E
K
c w-’
owy=
K
The p r o o f i s l e f t t o t h e reader,
w)
my.
v
(K
=
W A X
=
1).
186
ROLAND0 C H U A Q U I
PROOF, Assume f i r s t t h a t
ci
(1) P = S y f o r a c e r t a i n y. i t o r d i n a l by 3.3.5.8.
u a and PZO. We have t h r e e cases:
=
Then a'=
y
( 2 ) P = U P and a f 0. Then a'= ( v i ) , a P 4 C a ' : t E P } , we have by 3.3.1.16
aye a and hence a P i s a l i m -
{ a t : t €01.
Since by 3.3.6.2 that a P i s a l i m i t ordinal.
( 3 ) a = 0. Then a s = 0 and, hence, a l i m i t o r d i n a l . t h a t a Z 1 and P > w . Again we have t h r e e cases. ( 4 ) a = 0.
Then a'=
= U P and
(5)
a
Assume, next,
0 and, hence, a l i m i t o r d i n a l . 0.
f
The p r o o f t h a t a P i s l i m i t i s s i m i l a r t o (2).
(ii), p = y +K f o r certain y K But, by (51, and K w i t h y = u y 2 w and 0 C K C w. Hence, a'= ay.a P Therefore, by 3.3.5.8, ci i s a l s o a l i m i t o r d i n a l . ciy i s a l i m i t o r d i n a l . (6) p # Up and
ci f:
0.
Then, by 3.3.5.9
.
Thus, we have completed t h e p r o o f o f t h e i m p l i c a t i o n from r i g h t t o l e f t . I n o r d e r t o prove t h e r e v e r s e i m p l i c a t i o n we assume t h e n e g a t i o n o f t h e r i g h t hand s i d e , i.e. a # U a and P C w and prove e a s i l y by i n d u c t i o n on
.
P that a Pi s not a l i m i t ordinal. 3.3.6.8
a*b = P
THEOREM,
3t
+ +
= a
T h i s theorem a s s e r t s t h a t t h e f u n c t i o n o f a x , i.e. of
a
W
a
w
*[
wx i s t h e f i r s t d e r i v a t i v e
x i s t h e normal f u n c t i o n t h a t enumerates t h e f i x e d p o i n t s
PROOF, We have, a * a W - 5 = a t i o n from r i g h t t o l e f t i s proved.
+
[ = c1
W
* t . Thus, t h e i m p l i c a -
We s h a l l now show by i n d u c t i o n on P t h a t i f a - p =
8 , then there i s a
t: w i t h P
Suppose, as i n d u c t i v e hypothesis, t h i s statement t r u e = aw a t . f o r o r d i n a l s l e s s t h a n B and t h a t a - P = 8 . By 3.3.6.4, P ='ic 5 +E
w i t h E C cia. I t i s easy t o show by i n d u c t i o n t h a t a " * P = v E w . Hence E C a w C P . On t h e o t h e r hand, a - 0 = a
0
= ci
Since
e
W
E
CaW;
*E
w
(a * [
= a
+ E )
w
-t
+ E , we have, c a n c e l l i n g on t h e l e f t (3.3.4.2
C P , by t h e i n d u c t i o n hypothesis, E =
hence e = 0 and
P
=
a
W
0
P
+CL.E.
for a l l Since a
(xii)), a - E
f o r a certain
v.
a * t . The i n d u c t i v e p r o o f i s completed.
Now, we pass t o t h e t h e o r y o f main o r d i n a l s .
= E
-p
.
.
But
=
187
AXIOMATIC SET THEORY
3.3.6.9
THEOREM,
ci E
Moa
c-).
3
ci = w E .
Thus, t h e normal f u n c t i o n t h a t enumerates t h e c l a s s o f main o r d i n a l s o f addition i s expw.
cases.
P R O O F l F i r s t we prove t h a t w E E M o a f o r every E .
(1) (2) PEU',Y
E = 0.
Then w E = 1 and t h e r e s u l t i s c l e a r .
t!
= UE f 0.
E
u* f o r c e r t a i n T I ,5
Then w E = u
'€El.
u':
Let
E W , E C w
6
, and
6'
y = w * p ' + ~ ' withp'Ew,E'Cw*:and
2 6'.
We may assume t h a t 6 w 6 ! S p . .Therefore, +yEw4
Moa. left.
(3)
fl +
YEW
6
.
E
C
6'CE.
5 w 6 ' = p + e C- w 6 ! S p , and 7 5 + S p ' ) c us'-w = w S s ' C- w'. Thus
Hence, P
-w6'* (Sp c
y
E, i.e.,
P,
Hence, b y 3.3.6.4,
El.
with p
p = w 6 . p + e
P
There a r e t h r e e
and w 5 E M o a .
E = 9 + 1 f o r a c e r t a i n 17.
Then w E = wTI w
. By 3.3.5.14,
wE E
Thus, we have completed t h e p r o o f o f t h e i m p l i c a t i o n from r i g h t t o I n o r d e r t o prove t h e r e v e r s e i m p l i c a t i o n , assume t h a t a# wE f o r a l l
a = w E - p + e w i t h 0 C p C w and E c w E . I f E # 0 l . We have, by 3.3.6.4, W ' * P , E Ec1 and a 4 a ; i.e. a g M o o . Suppose, then, t h a t E = 0. Then p > l , s i n c e 01 # w 4' Hence ci = w E * p = w E * u + u', w i t h 0 C u c u . But, then, w E * u , w E ~ c and i ciqa, i.e. c i @ M o a .
.
3.3.6.10
THEOREM,
ci E
Mom
-
ci
=
1V 3E
T h i s theorem says t h a t t h e normal f u n c t i o n exp
certain
ci 3
2.
ZW
6
.
enumerates t h e c l a s s
Then, by t h i s .theorem,
E which has t o be a t l e a s t 1 s i n c e
uE
Thus, M o m i s a p r o p e r c l a s s .
o f main numbers o f m u l t i p l i c a t i o n except 1. Suppose t h a t a E M o m and
= 2
ci
ci
32.
6
Hence, w6
.
ci
= ZwE
ci =
Zw
for a
(I+6)
for
a c e r t a i n 6. Therefore, c1 = Z w '" = ( 2 w ) w = w Thus, t h e f u n c t i o n e x p w w i s t h e normal f u n c t i o n t h a t enumerates a l l i n f i n i t e main o r d i n a l s o f m u l t i p l i c a t i o n (i.e. M o m - 3 ) . PROOF,
F i r s t , i n o r d e r t o p r o v e t h e i m p l i c a t i o n from r i g h t t o l e f t ,
we s h a l l prove t h a t Z w
E
E
Mom.
Assume t h a t p a y E Z w
E
.
Then, s i n c e w E
188
ROLAND0 C H U A Q U I
i s l i m i t , p E 2',y E 2' with q, 5 E w'. . $ 3.3.6.9, w i s a main o r d i n a l o f a d d i t i o n .
E
y E 2
'
+ { . But, by .$ Thus, q + . $ E W and p *yE2'+' Hence
E . Therefore, 2 M o m . Since i t i s c l e a r t h a t have proved t h e i m p l i c a t i o n from r i g h t t o l e f t .
-2 C
l E M o r n , we
E
Suppose now, t h a t a i s a main o r d i n a l o f m u l t i p l i c a t i o n and a 3 2. We
u
have, by 3.3.6.4,
,
= 2 ' 0 ~+ u
w i t h e = 1 and u c 2
6
CyEa, then p + y C-
Now, a i s a main o r d i n a l o f a d d i t i o n , because i f 6 y * 2 E a , since 2 E a . T h e r e f o r e u = 0 and a = 2
.
2
6
.
6 i s a l s o a main o r d i n a l o f a d d i t i o n , because i f P , r E 6 , t h e n $ , 2 ' E 6 E a = 2 = a and, hence, 2 , i.e., 0 + -yE 6 . Hence, by 3.3.6.9,
.
+'
6 = wE f o r a certain
3.3.6.11
.$.Therefore,
DEFINITION,
Moe
01
= 2 ti = 2
(a : a # 0 A W E w ~ ) ( C ; , q € u
=
-+
Eq€a)}.
Moe i s t h e c l a s s o f main o h d i n a h 04 exponevctiation. A l l main o r d i n a l s o f e x p o n e n t i a t i o n a r e i n f i n i t e . Moe i s a l s o a p r o p e r c l a s s c l o s e d under unions, as we s h a l l see i n t h e n e x t theorem. However, t h e normal f u n c t i o n t h a t enumerates M o e cannot be expressed u s i n g t h e o p e r a t i o n s on o r d i n a l s t h a t we s h a l l study. We need new i t e r a t i o n s . The normal f u n c t i o n t h a t enumerates t h e main o r d i n a l s o f e x p o n e n t i a t i o n except w (i.e. Moe { w 1 ) i s u s u a l l y c a l l e d E and t h e o r d i n a l s i n Moe {w}, E -numbers. Thus, e O i s t h e f i r s t main o r d i n a l o f e x p o n e n t i a t i o n l a r g e r t h a n w .
-
3.3.6.12
a
THEOREM,
E
-
Moe
-
2'
= a.
Thus, Moe c o i n c i d e s w i t h t h e c l a s s o f f i x e d p o i n t s o f t h e normal f u n c t i o n exp2. Thus, i t i s a p r o p e r c l a s s c l o s e d under unions (3.3.3.3).
a 1.w
PROOF, Assume f i r s t , t h a t 2 a = a and l e t 0, y
, since
since 2'
Z K 3 K
, i.e. , by
Also, a =
K
Therefore,
3.3.6.9,
a =
E
2"
a. We have, t h a t
= u E { Z E : t E a } , i.e.,
P +y€Fq uE
E
a i s a l i m i t ordinal,
Thus, by 3.3.6.7,
E w .
Hence, 0, y
i s limit.
f o r a c e r t a i n q E a.
a E Moa
for
5Za=
Z a = Z W E ; hence, by 3.3.6.10,
(Zq)2'= have proved t h e i m p l i c a t i o n from r i g h t t o l e f t .
2q02'€
2';
Assume, now, t h a t a i s a main number of e x p o n e n t i a t i o n . Ua+
0, because, i f 2
P + l & 0P c a .
Thus, 2 " = u
E
50 E
{ Z E : [ Eal.
€ 2 '
a E M o m . Thus, s i n c e
a, we have q - 2 ' ~ a. Therefore,
a=
0, ~
Thus, we have proved a f o r a c e r t a i n .$.
a, then 0 C
0PE
a (3.3.6.2
But, s i n c e aEMoe
, 2 t: E
i.e.,
I), 2'E
we
We have t h a t ( v i i ) ) , hence,
a for all
E
a.
AXIOMATIC S E T T H E O R Y
Therefore, 2
'5 a.
Since
c1
5 2 ',
by 3.3.6.2
189
( v i i ) , we f i n a l l y obtain
c1 =
2".
I t i s possible t o i t e r a t e again t h e s t r i c t l y a d j o i n t function base and obtain a new operation. Repeating i t e r a t i o n i n d e f i n i t e l y , i t i s possible t o obtain a t r a n s f i n i t e sequence of operations, one f o r each ordinal. These operations have been extensively studied i n Donner and Tarski 1965. Many of these normal operations a r e new and with t h e next two of them an a r i t h metical c h a r a c t e r i z a t i o n of the €-function can be obtained. We s h a l l not pursue t h e i r study f u r t h e r . Next we s h a l l study i n f i n i t e addition of o r d i n a l s and obtain a repres e n t a t i o n of them. 3.3.6.13
DEFINITION,
By recursion
(i) C (F'F : t E O ) = 0 ; (ii) C
t
(F't
: t E S p ) = Z
t (iii)P =UP + X
t
t
(F't
:[ E p ) + F r o ;
( F ' [ : t E p ) = U
T
{Z(F'~:[ET):~)EPI.
We could d e f i n e s i m i l a r l y t h e i n f i n i t e product of ordinals. I t i s c l e a r t h a t t h e function defined i s monotone and continuous, f o r every F On with O n ( F ) . Also, i f F'E 2 0 f o r every t , i t i s s t r i c t l y increasing, i .e. , normal * We s h a l l be mainly i n t e r e s t e d in E ( F ' t
t
I n this case we may write, C
t
(F't
: ~ E K =)
F'O + F ' 1 +
: ~ E K f )o r F E
K
O n and
K E W .
... + F ' ( K - 1 ) .
The following i s an extension of t h e Euclidean algorithm. 3.3.6.14 THEOREM, a # O A P 3 2 + 3 ! v 3 ! 631 g ( O C v C w A 6 E ' o n ~ ~~ pr ( p c n c v +6 l p c 6 l ~ ) r \ g ~ ~ ( p - A1 )a =
zpcp
!imp
g ' p : p € V ) ).
This theorem says t h a t given any p 2 2 , every a f 0 can be represented a s a f i n i t e sum of multiples l e s s t h a n P of decreasing powers of 0; i . e . , a = p d'O-g'O t /.?6'1*g'l + + fi6'('g ' ( v - l ) , where 6 ' 0 3 6'1 3
>..>d'(v
-
...
1) and g ' f~ 0,
~ ' I I E P ,f o r p E v .
(1) Existence of t h e representation. Let a f 0 and P 3 2. The proof i s by induction on a. For c1 = 1, take a = p O. Suppose a 2 1 and the representation e x i s t s f o r a l l e c a. By t h e extended Euclidean a l gorithm, 3.3.6.4, PROOF,
a = P Y - 6 t e , w i t h O C 6 C p and e C p y .
ROLAND0 CHUAQUI
190
P
Since 6 2 1, s i s , we o b t a i n , =
E
'5 a .
Hence
Pd" * 9 ' 0 +
E
C
a.
... + P"('
A p p l y i n g t h e i n d u c t i v e hypothe-
... d ' ( v -1) and 0 g ' p c P f o r + ... + / 3 B ' ( u - 1 ) * g ' ( v - 1 ) . We have 6'0 .. d ' ( 7 - 1 ) .
6'1
3
2
-
')g'(u
F E U .
C
OCGCP.
-
, 6'0
1) w i t h
VEW
3
' 0 6 +Pdl0*g'O
Hence, a =
Also, s i n c e 6
CPy,y3
3
2
F i r s t , i t i s easy t o show by f i -
(2) U n i c i t y o f the representation. n i t e i n d u c t i o n on v t h a t
(*) a = Z ( P 6 " J !
-g'p : F E Y ) w i t h v , d,g s a t i s f y i n g t h e c o n d i t i o n s o f
t h e theorem, i m p l i e s t h a t a c OCpEu)
c p 6'0
.
6 ' o - ( g r 0 + 1 ) and, hence, C p (
P
We now prove by i n d u c t i o n t h e u n i c i t y o f r e p r e s e n t a t i o n . t h e r e i s a unique r e p r e s e n t a t i o n f o r a l l p € u ) =
C (0 '"=j'p P
: P E T ) with
E C
a and l e t a =
d'p .glp
:
Assume t h a t
I: ( P 6 " CC
g'p
:
v , n, d,h,g,j s a t i s f y i n g the conditions
o f t h e theorem:
We have by (*), a =
P,
"O,
0 B'o*g'O +
E
=
and E ' C P h ' O . By 3.3.6.4, A p p l y i n g t h e i n d u c t i v e hypothesis, s i n c e clusion. = ECP
0h ' o = j ' O +
f' w i t h 0 c 9'0, j ' O c 6 1 0 = h'O, g ' 0 = j ' 0 , and E = 6'.
E C
a , we o b t a i n t h e d e s i r e d con-
I n o r d e r t o omit. t h e h y p o t h e s i s a # 0, we would have t o a l l o w u = 0.
I f we t a k e P = 2, we can r e p r e s e n t a w i t h g ' p = 1 f o r e v e r y p € u . Thus any o r d i n a l can be represented by a f i n i t e sum of powers o f 2. I f we t a k e P w, t h e n g ' p i s a n a t u r a l number f o r e v e r y p E v . We c a l l t h e r e p r e s e n t a t i o n as f i n i t e sum o f f i n i t e m u l t i p l e s o f decreasing powers of w, t h e c a n o n i d hephe,4eentationY i.e.
a = C (wd"*g'p : p € v ) with 6'0 P
3
and g ' p E w , i s t h i s c a n o n i c a l r e p r e s e n t a t i o n .
PROBLEMS
1.
Prove 3.3.6.2.
2.
Prove 3.3.6.3.
3.
Prove 3.3.6.6.
4.
Prove: a > - a +(Wp W
'(0
'7 =
a
+
p
=
a
v
= a)
6'1
3
...3 d 1 ( u - 1 )
A X I O M A T I C SET T H E O R Y
-
3 t ( a = ww
t
v
c1 =
wE
191
+ 1)).
T h i s theorem g i v e s a c h a r a c t e r i z a t i o n o f t h e i n f i n i t e prime numbers. 5.
( a ) Prove: a-P =
r p
u
c;
Cc1-c;
fa :
tEP1
= u 1 a t . a :t € P I .
E
( b ) Determine whether something s i m i l a r can be shown f o r t h e i n f i n i t e sum and p r o d u c t o f o r d i n a l s . 6.
( a ) Formulate and prove t h e c a n o n i c a l r e p r e s e n t a t i o n o f a + o f those o f c1 and P .
P i n terms
(b) S i m i l a r l y f o r a - P . (c) S i m i l a r l y f o r
c1
.
P
7.
Show t h a t two f i n i t e products o f o r d i n a l s a r e t h e same i f and o n l y i f t h e y d i f f e r o n l y by t h e a s s o c i a t i o n o f t h e o r d i n a l s .
8.
Let
c1 =
with
wyo
KO,...,
+
K~
K
... + w YV
v,hO,...,
Xv
Gw
- K
.
v ’
I n o r d e r t o have t h e same number o f
terms some o f them may be 0. D e f i n e t h e n a t u r a l sum and p r o d u c t o f o r d i n a l s by
a(*)P = w
Y
(KO
ho)
+
... +
0
YV
-(KV*
Xv).
Show: ( a ) The o r d i n a l s w i t h (+) o r groups. ( b ) a (+) P and a ( - )
( a )
form a commutative c a n c e l l a t i o n semi-
P a r e s t r i c t l y i n c r e a s i n g f u n c t i o n on a and P .
( c ) For each o r d i n a l Y, t h e r e i s a t most a f i n i t e number o f p a i r s (a$) such t h a t c1 (+) P = y
.
CHAPTER 3.4
C a r d i n a l Numbers
3,4.1
D E F I N I T I O N OF C A R D I N A L NUMBERS,
I n 2 . 7 . 1 we proved t h a t t h e r e l a t i o n o f e q u i p o l l e n c y between s e t s i s an equivalence r e l a t i o n . Thus, we can use Def. 3.2.3.24 and d e f i n e t h e t y p e o f any s e t x E W w i t h r e s p e c t t o t h i s r e l a t i o n . T h i s t y p e o f x w i l l be c a l l e d t h e cuhdind numbe 06 x, denoted by 1 x 1 . 3.4.1.1
DEFINITIONI
3.4.1.2 S T I P U L A T I O N OF V A R I A B L E S , We s h a l l use b o l d f a c e lowercase l e t t e r s t o r e f e r t o c a r d i n a l numbers, i.e., elements o f C r . The f o l l o w i n g i s an immediate consequence o f Def. 3.4.1.1, ( v i ) - ( v i i i ) , and 3.2.3.25. 3.4.1.3
THEOREM,
a = b
-
3 x 3 q( a = 1x1 A b
=
2.7.1.2
IqI A x = q ) .
We now i n t r o d u c e o p e r a t i o n s and r e l a t i o n s between c a r d i n a l numbers and s t a t e t h e i r main p r o p e r t i e s . Most o f them a r e immediate consequences o f those prove f o r e q u i p o l l e n c y i n 2.7. 3.4.1.4
(i) a < b (ii) a 3.4.1.5
-
DEF I N I T I ON i
3 a 3 b ( ( a ( = a A ( b ( = b A U L b).
++a Q b A a # b .
THEOREM I
(i) a < a . (ii) a < b < c + a < c . (iii) a < b < a +a = b . 192
A X I O M A T I C SET THEORY
193
Thus, G i s a p a r t i a l o r d e r i n g between c a r d i n a l s . i s needed f o r p r o v i n g t h a t G i s a s i m p l e o r d e r i n g . PROOF, By 2.7.1.2 3.4.1.6
( x ) , ( x i ) , and 2.7.1.3.
.
THEOREM (INTERPOLATION THEOREM). < b Q c,c') 3 b(a,a'
.
PROOF, By 2.7.1.8. 3.4.1.7
The axiom o f c h o i c e
a, a'
< c,
c'
+
DEFINITION,
( i ) a + b = U t e : 3 a ] b ( a = la1 A b = I b l A b = U { c : 3 a ] b ( a = la\ A b = ] b l A
(ii) a
c = la+ bl). C c = l a x bl).
The i n f i n i t e sum o r p r o d u c t o f c a r d i n a l numbers cannot be i n t r o d u c e d w i t h o u t t h e axiom o f choice. 3.4.1.8
THEOREM,
PROOF, By Def. 3.4.1.7,
and 2.7.2.3.
.
We use t h e usual c o n v e n t i o n about omission o f p a r e n t h e s i s : c a t i o n t a k e s precedence o v e r a d d i t i o n . 3.4.1.9
multipli-
THEOREM I
(i) a t b = b + a A a .b = b o a .
(ii) a + (b (iii) a (b
+ +
c ) = (a
+
b)
+
c A a
( b - c ) = (a - b )
c) = a * b+ a * c . ( i v ) a + 101 = 101 + a A (a + b = 101 ++a = b = 1OI)A a =
-
101 = 101 =
101. a A (a - b = 101 + + a = 101 V b = 101).
( v ) (a + b = 111
(a = 101 A b = 111) V (a = 111 A b = l 0 l ) ) A
111 = 111 = l a 1
(a
a).
( v i ) a < a ' A b < b ' -+a t b < a ' + b ' A a * b < a ' mb'. (vii) a
c--t
(viii) a + b = a'
A a ' = co PROOF,
a +c = b .
3c
+
+
c
By 2.7.2.4
b ' * 3 c o 3 c 1 3 c 2 3 c (a = c 3 0
2
A b' = c
+
c
1
A b = c
2
+
1 + c3).
(i), ( i i i ) , (v), ( v i i ) ,
( v i i i ) , (ix),
( x ) and
c
3
ROLAND0 C H U A Q U I (xi).
= 3.4.1.10
THEOREM,
PROOF,
By 2.7.3.4.
3.4.1.11
THEOREM, a
(a # 101
+ 111 G b ) .
PROOF,
By 2.7.3.8.
3.4.1.12
THEOREM,
PROOF,
By 2.7.3.9.
3.4.1.12
THEOREM,
PROOF,
V E W A a
By 2.7.3.10
+ IvI
-
+ IvI
= b
111 S a ) A (a
V v ( v F : u - . a # Ivl)
-+
--fa = b .
- a
= b V
w v ( v E ~ +IvI < a ) .
w
(i) and ( i i ) .
REMARK, We t h u s have t h r e e t y p e s o f c a r d i n a l numbers. Finwnbehn a r e those o f t h e form Iv f o r v E w and a r e t h e c a r d i n a l s o f f i n i t e sets. TrraadivLite c a d i n & n u m b m a r e those a w i t h IwI < a. 3.4.1.12 g i v e s o t h e r c h a r a c t e r i z a t i o n s of t r a n s f i n i t e c a r d i n a l s . F i n a l ly, t h e s o - c a l l e d Vedekind indivLite c a d i n & n u m b m a r e t h e r e s t , i.e. The axiom of c h o i c e i s needed those a t h a t a r e n o t f i n i t e b u t a + 111 # a . f o r p r o v i n g t h a t t h e t h i r d c l a s s o f c a r d i n a l s i s empty. 3.4.1.13
I
d e cahdid
PROBLEMS Prove t h e f o l l o w i n g two c h a r a c t e r i z a t i o n s o f f i n i t e s e t s due t o T a r s k i Wa(a -c P c A a # O + 3 x ( x E a A W y ( x c y ~ a + x Y= ) ) ) .
1.
c E F N -
2.
c E F N + + V u ( a-c P c A a + O + 3 x f x E a A t l y ( x-> y E a - x = Y ) ) ) .
3.4.2
CARDINAL EXPONENTIATION 1
Cardinal e x p o n e n t i a t i o n f o r s e t s was a l r e a d y i n t r o d u c e d i n 2.3.1.10 t h e s e t o f f u n c t i o n s w i t h domain b and range i n c l u d e d i n a i s a t o t h e power b. ba, i.e.
AXIOMATIC SET THEORY
195
The f o l l o w i n g theorem, which j u s t i f i e s t h e d e f i n i t i o n o f exponentiat i o n f o r c a r d i n a l numbers, i s easy t o prove. 3.4.2.1
THEOREM,
(i)a = c A b = d + ba = d c . ( i i ) a,b
E
W + ba E W .
The p r o o f i s l e f t t o t h e reader. 3.4.2.2 orem.
ab = U E c : a = la1 A b = I b l A c =
DEFINITION,
I ba ( } .
The elementary p r o p e r t i e s o f t h i s o p e r a t i o n a r e g i v e n i n t h e n e x t t h e 3.4.2.3
THEOREM,
=pa.
( i ) 2‘
( i i ) ‘a = 1 A
1
2
a - a ~a = a
x
a.
(iii) ( b # O + b O = O ) A b 1.1.
b
(iv) ( a = 0 (v) (vi
-
b
a = 0 A b # 0) A ( a = 1
+c ‘a = b x c a a’ c(ba) Y c x
.
( v i i ) ‘ ( a x b ) = ‘ a x b‘ (viii) a # 0 A b
(ix) a 5 b
-+
2 c
‘a
+
-
b = 0 V a
1
1).
.
ba
5
‘a.
2 ‘b.
(x) c ~ o - + u a C a . (xi) 2 < c + a <
U
C -
PROOF OF (i). We use g E a 2 ( P a ) d e f i n e d by 9’6 = Ex: X E U A 6 ’ x = I t i s easy t o show t h a t 2‘ 1} f o r each 6 E ‘2. Pa.
4
The r e s t o f t h e p r o o f o f t h e theorem i s l e f t t o t h e reader. The corresponding theorem on c a r d i n a l s i s as f o l l o w s :
ROLAND0 C H U A Q U I
196
101
( i i i ) (b
101
( i v ) (ab =
10lb = 101) A Illb = 111.
+
-a
111 V b = 101).
= 10IAb+lO1)A(ab= I l I - a =
(v) a b t c = ab * a C . b c b*c ( v i ) (a ) = a
( v i i ) (a * b ) ' = ac * b C ( v i i i ) a # 101 A b (ix) a
+ a
C
.
-+
.
< bC
ab
< a'.
C
( x ) c # O - + a < a . ( x i ) 121 < c + a < c
a
.
We s h a l l n e x t prove a theorem about i n f i n i t e sums and products. The corresponding theorem about c a r d i n a l s cannot be proved ( n o r even s t a t e d ) w i t h o u t t h e axiom o f choice.
3.4.2.5
THEOREM,
( g i : i E b ) , where g'( x , i ) f o r ~€6'4.I t
PROOF OF (i):We d e f i n e t h e f u n c t i o n H by H'g
g . i s t h e f u n c t i o n w i t h domain d'-i g i v e n by g i f x L
=
=
i s easy t o show t h a t D H = B > [ d ' i : i E b ] a a n d D H - l c n L ( d t i a : i E b ) . Now, i f H'g = H ' j we have, gi = ji
for a l l i e b .
Hence, g 1 d ' i x I i ) =
j1dT.i x {i},f o r a l l i e b , and, hence g = j .
-
a
n ( d ' i ~ : i c b ) , then z?[ d'i :i € b l a and H ( U
On t h e o t h e r hand, i f ( g ' i : i E b ) E
gif ( x , i ) = g'i'x, = ( g ' i :L E b ) .
then
U
{gi : i € b }
E
PROOF OF (ii).We know (see remark a f t e r 2.3.1.12),
Hi ( d f i : i
E b ) =
IIx
(ni ( dfi: i E b )
-
f o r ( i , x ) E b x a , we have by 2.7.2.4
%(d'i:iEb)
: xga).
i f we d e f i n e
{Ti: i
E
b3)
that
I f we d e f i n e g'(i,x) = d'i
( i v ) applied twice:
~x(~(g'(i,x):~€b):xE~,
=
197
A X I O M A T I C SET THEORY
=II1, ( n x ( 6 ' i : x E a ) : i E b )
.
=n(ag'i:.iEb)
,
w
We s h a l l now i n t r o d u c e a new n o t i o n o f l e s s t h a n o r equal c a r d i n a l i t y , which i s e q u i v a l e n t w i t h 5 i f t h e axiom o f c h o i c e i s assumed. However, i t may n o t be e q u i v a l e n t i n M K T .
3.4.2.6
DEFINITION,
(i) A S ~ * B + + ~(F)~ A VDF c - B A D F - ~= A . (i i )A < * B *3FAsF*B. (iii) A =
*
5*
8-A
B5 *
8 A
A.
The n e x t theorem, whose easy p r o o f i s l e f t t o t h e reader, i s a consequence o f t h e d e f i n i t i o n and serves t o j u s t i f y t h e i n t r o d u c t i o n o f t h e c o r responding n o t i o n f o r c a r d i n a l numbers.
3.4.2.7
THEOREM,
(i)A
(ii) A r: C A 8 =
D
A A <
(iii) A = C A 8 =
D
A A =
DEFINITION,
3.4.2.8
a
* *
5*
THEOREM,
5*
< * D.
-
8+ C
<*b
As an example o f t h e use o f proved f o r 5 w i t h o u t Ax GC.
3.4.2.9
B+ C
r:
* D. 3 U 3 b ( a = la1 A b = I b l A a
5*
b).
we shall see a theorem that cannot be
D F V (F) A A C - DF
-+
U
{FIX: xEA1
C c [F'x: x E A ] . X
PROOF, We have, C c [ F ' x : x E A ] = u { F ' x x { X I : x E A ) . Thus, i f we X d e f i n e GI( y,x) = q for ( y , x ) E C c [ F ' x : x E A ] , t h e n u { F ' x : x E A ) 5 G* X
C:[F'x:
xEA].
We now s t a t e and prove t h e i m p o r t a n t theorem o f Cantor about t h e c a r d i n a l i t y o f t h e power s e t .
3.4.2.10
THEOREM,
(i)Pb $ * b .
(ii)b < P b A P b # b . ( i i i ) 121b $
*
(iv) b < IZlb.
b .
198
ROLAND0 C H U A Q U I
PROOF, We s h a l l prove ( i ) , s i n c e t h e o t h e r statements a r e immediate consequence of (i), u s i n g 3.4.2.7, Def. 3.4.2.8, and 3.4.2.3 ( i ) , (x), ( x i ) and 3.4.2.4 ( i ) , (x), ( x i ) . I n o r d e r t o o b t a i n a c o n t r a d i c t i o n , assume t h a t P b< i s a function with Dd c - b and D 6 - I = P b.
.
Let c = { x :
6
* b.
That i s ,
A
x $ g'x}.
xEb
d
Then c C b and, hence c E P b Therefore, s i n c e D 6 - I = P b , t h e r e is a d € b such t h a t d ' d = c. But, then, d E 6 ' d i f and o n l y i f d $ d'd, a contradiction. = From C a n t o r ' s theorem, we deduce t h a t f o r each c a r d i n a l number t h e r e i s an i n f i n i t e s t r i c t l y i n c r e a s i n g c h a i n o f c a r d i n a l s . For i n s t a n c e , t h e f o l l o w i n g c h a i n o f i n f i n i t e c a r d i n a l numbers.
F i n a l l y , two theorems and t h e i r c o r o l l a r y about c a r d i n a l s t h a t w i l l be useful l a t e r . 3.4.2.11 PROOF,
THEOREM,
a x b < c t c d + a ~ c V b < * d .
Assume t h a t a x b
w i t h c' n d ' = 0 ; c' two cases.
5
c , and d'
5 5
c
tc d.
Then, by 2.7.2.2,
d for certain c',d'.
(1) There i s a q E b such t h a t a
Cq} c - c'.
x
a x
b =c' ud'
We have t o consider
Then a
<, c'
Q
c.
We d e f i n e , ( 2 ) F o r e v e r y y E b t h e r e i s an x E a such t h a t ( x , y ) E d'. Hence, D d = d' i n t h i s case t h e f u n c t i o n 6 by 6'( x , q ) = q f o r ( x , y ) E d'. and 06-' 3 - b. 3.4.2.12 PROOF,
Therefore, b THEOREM,
5*
c # 0
Suppose t h a t a
orem i s obvious).
Let h = ( g
d'
5d
A a5* b
5 6*b, 06
Also, i f g E a c , t h e n h ' g = g o d €
and so b
c
f
+
'c
5 * d.
5
bc.
0 , and a # 0 ( f o r a = 0, t h e the-
:g E a c ) . h i s a f u n c t i o n w i t h D h ='c. b b c. Hence, D h - I 5 c .
Assume t h a t g , j E a c and h ' g = h ' j . Then g o d = j o d . L e t x E a ; t h e n f o r a c e r t a i n y E b . Hence, g ' x = g ' 6 ' y = j ' 4 ' q = j ' x . Thus, g = j and h i s biunique, i.e. ' c 5 b c.
x
= 6'y
3.4.2.13
THEOREM,
(if a *b
*
d
.
( i i ) c < a A d < b A b $ * d + c + d < a - b . ( i i i ) c f l O I A a < * b + c a < c b
199
AXIOMATIC SET THEORY
(i) and ( i i ) a r e consequences of 3.4.2.11,
PROOF,
( i i i ) of 3.4.2.12.
PROBLEMS
Prove:
(Hint:
Prove:
b2 5
A
3.4.3
*
c).
a t
C
b S * u t c + 3 b j b 2 ( 6 = bl 1
tc
b2 A a t c b l S * a
F I N I T E MULTIPLES OF CARDINALS AND COUNTABLE SUMS,
I n t h i s s e c t i o n we s h a l l s t u d y c e r t a i n theorems o f c a r d i n a l a r i t h m e t i c t h a t can be shown i n M K T w i t h d i f f i c u l t y , b u t t h a t a r e t r i v i a l i n M K T C . The most i m p o r t a n t i s t h e f o l l o w i n g : v € u A a * l v ( =b-lvl + a = b .
I t s p r o o f , as i t w i l l be seen, p r e s e n t s c o n s i d e r a b l e d i f f i c u l t y . Most of t h e development of t h i s s e c t i o n , as i s t h i s theorem, i s due t o T a r s k i . We s h a l l f r e q u e n t l y use f u n c t i o n s ; we s h a l l w r i t e i n t h i s case, A x i n stead of A ' x . of +
C'
A l s o , i n o r d e r t o s i m p l i f y t h e n o t a t i o n , we w r i t e + i n s t e a d
and C ( A x : X E C ) i n s t e a d o f
-
w i t h D A 3 C.
Xi[Ax:
We a l s o w r i t e ZA f o r
w i t h sequences A such t h a t A
E
'V
xECI
, when
Z ( A x :x E D A ) .
or A E
WV, for v
A is a f u n c t i o n We s h a l l deal m o s t l y
EW
.
R O L A N D 0 CHUAQUI
200
I n o r d e r t o be a b l e t o show o u r theorems w i t h o u t t h e use o f t h e axiom o f choice, we must c o n s t r u c t most o f t h e s e t s and f u n c t i o n s whose e x i s t e n c e we a s s e r t . Thus, i n s t e a d o f j u s t showing (2.7.2.4)
a
b .
we have t o s t a t e which a r e t h e c and t h e f u n c t i o n e s t a b l i s h i n g t h e e q u i p o l l e n c e i n v o l v e d . Thus, we s h a l l use t h i s theorem i n t h e f o l l o w i n g form:
(*)
a
5
b
where g ' ( x , O ) = d'x,
-
a
b , 9 and g ' ( y , l ) = q ,
+ (b-d*a)
i f xfa,
=
i f gEb-d*a.
I n o r d e r t o s i m p l i f y somewhat o u r n o t a t i o n , we s h a l l i n t r o d u c e a u n i form system f o r d e n o t i n g t h e s e t s and f u n c t i o n s c o n s t r u c t e d i n t h e theorems. For i n s t a n c e (*) w i l l be:
THEOREM,
3.4.3.1
I n t h e symbol
2
a E
V + (ao < d a l
, ( ~ , d ) ~ t,h e
-
a0 + ,(a,d)o
1
, ( a, d) ,
a,).
number 1 r e f e r s t o t h e number o f t h e theo-
rem i n t h i s s e c t i o n . I n general, we s h a l l g i v e t h e d e f i n i t i o n o f Thus, f o r 3.4.3.1, i n s i d e t h e p r o o f o f Theorem 3.4.3.~.
- d*ao
,(a,d)o
=
,(a,d),
(x,O) = d ' x
,(a,d),
(x,l)
x
=
,(~,6)~
3
, if , if
x€a0 xEa,
,
- d*aO.
I n many o f t h e p r o o f s o f t h e r e s t o f t h i s Chapter we have t o use w
the
c a r d i n a l sum o f a denumerable sequence o f sets; i.e., i f a E Y , we have t o work w i t h Z ( a K : K E o ) . I t i s n o t p o s s i b l e t o prove w i t h o u t Ax C
-
that, f o r a E w V , (t)
Z: a = c
3b(b
E
bK n
O V A c = u {b,
%=
0 A aK
2
:KEW}
A\~KW~(KCXCW --f
bK).
From l e f t t o w r i t e , t h i s statement i s t r u e i n M K T :
If
C a = c, take
d
bK = d*aK. However, i n o r d e r t o prove i t from r i g h t t o l e f t we would have t o choose f o r each K E W , a f u n c t i o n g, such t h a t aK = bK. Thus, f o r 9, working w i t h o u t Ax C, we need a s t r o n g e r n o t i o n o f e q u i p o l l e n c y f o r denumerable sequences o f sets. T h i s i s g i v e n i n t h e n e x t d e f i n i t i o n , where a, 6 should be understood as belonging t o Y , where c E w o r c = w. 3.4.3.2
(i)a
*d
DEFINITION, b
++
6
E
DaVAW
X(X E
D a + ax =
6,
bx).
AXIOMATIC (ii) a(( (iii) a
ii:
6
b
b
tf
tf
6 ED?
201
S E T THEORY
A W X ( X E Da-t a
5
36 a =gb.
bX).
6,
3 b a(( b .
(iv) a ( ( b*
6
We can now e s t a b l i s h t h e theorem corresponding t o (+). 3.4.3.3 (i)
THEOREM,
j b ( b E Y A c = u {bK W K W A ( K E X E W + bKnbh= 0) A a=b). U E
wV
(ii)a,b
E
( i i i ) a,b
E
-t
W
W
( X u = ct-f
V A a=b
-t
wV A a 1 ( b
X +
:KEW}
A
u = Xb.
C a
5
Cb.
The p r o o f i s l e f t t o t h e reader. Using t h e n o t a t i o n i n t r o d u c e d p r e v i o u s l y , we should w r i t e 3.4.3.3 as f o l l o w s : U E W V + K
(Xu
c
-
,(a,6),
E WV
~ ( K E ~ € w (- 3+( a , 6 ) 0 ' ~ )
where 3(UY6), =
(d*(~,
,(a,d),
(
=
61(aK
c =
U
{,(aY6),'K
:K
~
{K)): K E W )
X
I,))
A]
W
n ( ( , ( ~ , d ) ~ ' h ) = 0) A a = 3 (a, 6)1b )
X
:
A
(i)
9
,
K E W ) .
I n g e n e r a l , we s h a l l s t a t e theorems i n t h e s i m p l e r form o f 3.4.3.3, t a c i t l y understanding t h a t we have proved t h e v e r s i o n g i v e n above, i.e. each t i m e we a s s e r t ' t h e r e i s a c ' , we assume t h a t t h e c has been e x p l i c i t l y d e f i n e d ; and each t i m e 'a=b', 'a5 b ' , 'a= b ' , o r ' a ( ( b' i s ass e r t e d we assume t h a t t h e f u n c t i o n t h a t g i v e s these r e l a t i o n s has been a l s o e x p l i c i t y defined. NOW, we r e s t a t e 2.7.2.4
3.4.3.4
(ao+ a1
=6
(xi).
THEOREM (REFINEMENT a2
+
a3
tf
THEOREM).
aO = 4(a,6)4 4(a,6)o
By a s i m i l a r p r o o f as t h a t o f 3.4.3.4,
a E~ Y
' 4(a,6)1
we o b t a i n
A
+
202
ROLAND0 C H U A Q U I
3.4.3.5
~
~
~
XY
( b ) A C a = C b+ 3 u ( u E D b ( D aV )
(a) A
"V
THEOREM,
. q (~ D bx ) X : E( D U )~Ab = ( C ( u *
- x E D a ) : q E D b )
XY
*
The p r o o f i s l e f t t o t h e reader. 3.4.3.6
a,b E wV A v v ( v E w + a, = a, + 1
THEOREM,
~ , + ~ n b= ,O ) +
3c(Wv(vE0+a,
+
+.
bv and a,
VC R E W .
n bV = 0 , f o r e v e r y v
+
+,,
=
a,
+,,
+.
Thus, a,
K E w ) .
x B a,,
u bV + p .
x
5c U u
{b, +.,
3.4.3.7
E
a,,
.
x
av t p
: K E W } .
'
+
w
L e t a,b 5
Y and ( a v + 1 + bv
xEbv+p+lC_U{bv+,:
I
a,b E
wv
A a
: , E d
%
%
(a,,
+
be t h e p a i r i n g f u n c t i o n s d e f i n e d by, P K ' x =
= I D 1 a.
go
4, +. 1 = 9,
0
6,
0
.
a,
= g*,
(3)
6,
= gv* , j * p * b
a, u
1
We s h a l l show t h a t ,
V*
Ew)
.
,f o r x
L e t P,, E
Y.
(Pols, +. 1 ) ' i s biunique, f o r every V E W
Po, and P1 a r e b i u n i q u e f o r e v e r y V E W
(2)
( X,K)
,
.
I t i s easy t o show by i n d u c t i o n t h a t g,
d,,
bv:
a . Then f o r v E w
We d e f i n e by r e c u r s i o n t h e sequence o f f u n c t i o n s ( 4 , : v Then,
for
a, and x $ c. Then,
+3c a = ( c + Z ( b , + . , : ~ E w ) : v E w ) .
T E W ,
since
=
The r e s t o f t h e c o n c l u s i o n s o f
THEOREM ( R E M A I N D E R THEOREM)
PROOF,
for
a,
3 aK 2 b,
L e t cc be t h e l e a s t such K. Hence,
Since
t h e theorem a r e easy t o show.
v E w )
We have,
Ew.
Take c = n {a, : K E w I and suppose t h a t
t h e r e i s a K E W such t h a t xEav
bcl n bv = 0 ) ) .
Assume a and b a r e w-sequences o f s e t s such t h a t
PROOF,
a,
A
= C U U { ~ , + ~ : T € W A}
c n U {bv+, : n E : w ) = 0) A W v W p ( v E p E w
U
b,
U
.
Let,
,
203
AXIOMATIC SET THEORY
5 gv*
+1) n P { b V
6,*((P,*aV
Thus, we have proved (4). By (4) and 3.4.3.5,
(5)
Zv =
CFV +* : K E W I ,
c u u
with c
We now have t o d e f i n e t h e sequence ( h ,
: Y E W )
V E W , we have,
av
IhV
c+C(bv+n:7rEW)
Let t h e n Po'
x
f
uv ; by (21,
gV'xEc -
x
COI.
=cxCO}u(u~b,+K~x(n~:a~oI~CII~. gvfxEZv =
+* , f o r
gV'xEbv
and, hence, P,'(gv
P1'Pr'(gv
+=
0
6,
+K
c u u CKv + n
:KEW
I.
I f gvf X E C -
On t h e o t h e r hand, assume g v f x E u Cbv
t h e n g v t x E bv tK
such t h a t , f o r every
e x a c t l y one n E w
o P1)
-1'
+ * o 6,
gvfxE Z
(
.
By ( 2 1 , ( 9 ,
bv
: ~ E w } ,
+= o 6, +* o
o P I ) -1' g v ' x E b v + n
+K
+A
+= : n E w ) x ClI.
,
x CnI
P1)
-1'
, and
Thus, d e f i n -
ing,
hv
= (Po I c U ( P 1
0 U
CPK
0
(9,
we o b t a i n e d t h e d e s i r e d sequence. As a c o r o l l a r y , we o b t a i n , 3.4.3.8
THEOREM,
a = a + b
6,+,
0
(PIIbv+a)
0
,
: ~ ~ E w } ) g,) o
rn
-
PROOF, Suppose, f i r s t , t h a t b
x
b
x
W<
W Sa a.
.
Hence, by 3.4.3.1,
u=c+
204
b
ROLAND0 C H U A Q U I
x w
Then,
f o r a c e r t a i n c. a+b=c
Hence, a = a
+ (b
b) = c + b x ( w + l ) = c
x w +
+
b
x 0.
+ b.
Suppose, now, t h a t u = a
+ b.
By 3.4.3.7
a l l v € a , we o b t a i n ,
a c + C < b :v
Ew ) =
+ b
c
x
,for
w
w i t h av = a and bv = b
a c e r t a i n c.
for
1
Most o f t h e p r o o f s o f t h e r e m a i n i n g theorems of t h i s Chapter w i l l be g i v e n i n t h e form o f t h i s l a s t p r o o f , i.e. w i t h o u t s t a t i n g e x p l i c i t l y t h e I f necessary ( w i t h t h e i n d i c a t i o n s g i v e n ) f u n c t i o n s o r s e t s constructed. i t w i l l be simple, b u t somewhat cumbersome, f o r t h e r e a d e r t o p r o v i d e t h e required definitions. From 3.4.3.7 and 3.4.3.8 (2.7.1.3) f o r sets. Namely: Let a 5 b A p p l y i n g 3.4.3.7,
2 a.
with
we can deduce C a n t o r - B e r n s t e i n ' s Theorem
Then a
+
= u,
and b b,
c = b
+ d = b2 =
,
a, f o r c e r t a i n c and d. c , and b2 = d , we
,
+
obtain, a = e + c x w + d x w a n d b = el + c x W + d x w , f o r c e r t a i n e and el. Hence, by 3.4.3.8, a + c = a and, thus, b = a. Also, u s i n g t h i s p r o o f o r t h e p r e v i o u s one i t i s p o s s i b l e t o c o n s t r u c t e x p l i c i t l y a f u n c t i o n 6 such t h a t a = b .
b
We now need some p r o p e r t i e s o f w . 3.4.3.9
THEOREM,
wf w= w
.
PROOF, We have, by 2.7.3.7 ( i i i ) , w = w +l. Hence, o = w + 1 + 1. Applying, now 3.4.3.7 w i t h av = w and bv = 1 + 1, f o r e v e r y V E O , we obt a i n W E c + ( 1 + 1 ) x w , f o r a c e r t a i n c. B u t ( l t l ) x w = w + w . Hence On t h e o t h e r hand, o b v i o u s l y , w < w + w . By Cantor-Bernstein w t w< w. 1
w = w + w .
3.4.3.10
THEOREM,
PROOF, f o r every v
-
wx
By 3.4.3.9,
^I
w
.
w = w + w .
A p p l y i n g 3.4.3.7
, we g e t t h a t w = c + w x w But, e a s i l y , w x w. 1
Ew
w.
PROOF, 3.4.3.8.
x
Ifa = a
+
b
x w
, then
, for
w i t h av = w = b,,
a c e r t a i n c.
Hence,
b x a < a, and we o b t a i n a = a + b by
On t h e o t h e r hand, l e t a = a + b. Then b x w < , a, by 3.4.3.8. By b x w = ( b x w ) x w . Hence, ( b x w ) x w s LZ.A p p l y i n g 3.4.3.8,
3.4.3.10,
205
AXIOMATIC SET THEORY
+b
we o b t a i n , a 3.4.3.12
' a v +1:
.
>
V E W
6,
Assume
a
v+l
(
av : v
Ew )
+ b f o r every v
t a i n c and every V E W .
=
(
PROOF, L e t a =
av = av
6,
(av: v E w )
4
0
( a v tl:
a + b v f o r e v e r y v.
foracertainc.
x w : v E w )
V
w i t h t h e a p p r o p r i a t e g. V E W ) ) x u =
p l y i n g 3.4.3.8,
3.4.3.14
j
t
x
w
f o r a cer-
b for a certain g go
6 , , we
have
V E W ).
By 3.4.3.11,
f o r t h e h, c o n s t r u c t e d t h e r e and every V E W . a=c+Cfb
CiV =
We have,
) .
av = c + b
Then i f
+(av : v E w ) =
6:vEw)
6 :VEW
t
+
Then, by 3.4.3.8,
and, hence,
=h, av
uv
By 3.4.3.7,
Ew.
supposed t o be c o n s t r u c t e d i n 3.4.3.9. a,
+
( a v : v E w )= ( a v + l
THEOREM,
PROOF,
av =
.
= a.
x
a=
A p p l y i n g 3.4.3.7
atbv x u we get,
Hence,C(b,xw:vEw)<
4
a
But i t i s easy t o b u i l d a j such t h a t ( C ( bv :
2 ( b V x w : v E w ) . Hence,
( C ( b v : ~ E ~ ) ) ~ ~ ~ gApO j u .
we o b t a i n ,
THEOREM,
a+b=atc
-+
3 b' 3 c ' 3 d ( a = a + b t = u + c ' A
b = b ' + d A c = C' + d ) . PROOF,
y, z , u E
w
xu
Assume t h a t a + b =
V an,d g, h , j ,
=hv x v + l
6 a+c.
We d e f i n e by r e c u r s i o n on w , x,
k E w V such t h a t
+ %+1
'
(4) zv =kv zv + 1 + uv + l ' (5)
xo
= a,
yo = b y
D e f i n e xo = a, yo = b,
zo = c. zo
c, and
uo = 0 ( o r any a r b i t r a r y s e t ) .
ROLAND0 CHUAQUI
206
We have,
xo
go =6
+
xo
+
0'
we define, x = 1 4(a,6)0, 91 = 4 ( ~ 3 d ) 2 ,~1 = 4(a,d),, ~1 = 4(a,6)3, ho = 4(as6)4, 90 = 4 ( u , d ) 5 , j , = 4 ( u , d ) 6 , a n d ko = , ( ~ , 6 ) ~ . We have, t h a t ( 1 ) - ( 4 ) a r e s a t i s f i e d with v = 1, i.e. 1' Yo 91 + U1' xo = j o x1 + 91 t o a. = a , u1 = b,
Applying 3.4.6.4,
and zo
=fro z1 + u1'
xo
^.
ho
a2 = a,
+
a3
= c,
9
Ego
Then, we have,
Hence we can apply t h e same procedure t o this formula and obtain x2, g 2 , z2, u2, h,, g, j , and hl. Proceeding by recursion, we define t h e sequence f o r a l l Y E W
.
Applying, now, 3.4.3.7 ( 6 ) yV = ( 7 ) zY
9.v
t o ( 3 ) and ( 4 ) , we get.
6' + Z ( U ~ + ~ : ~ E fW o r )a l, l Y E W ,
zrnV c '
+ C ( y , + ? l : ~ E ~ f) o, r a l l
V E W .
where l and rn a r e the functions defined in 3.4.3.7. Take d = Z ( u T : ~ E w Then by ( 5 ) , ( 6 ) , and ( 7 ) , b - b ' + d and c = c ' + d . Also, we have, b ' 5 q 1 g v
.
) .
f o r every v E W Then by 3.4.3.3, 6' x w S n Z ( gv + 1 : v E w ) f o r t h e n the function constructed there. Similarly, c' x Z ( z,, v Ew) By 3.4.3.7, ( l ) , (2), and ( 5 ) , Z ( g v + 1 : Y E W ) , Z ( Z ~ + V ~€ :W ) ~ U .Hence, by 3.4.3.8,
u c a + b' - a + c '
asp
..
+
,:
.
Even w i t h Ax C i t i s not t r u e , in general, the Cancellation Law f o r cardinal addition, i.e. a+b=a+c
-+
b-c.
However, we have, even without Ax C: 3.4.3.15
THEOREM (WEAK CANCELLATION LAW).
a+b= a+c
+
3a'(a=a+uf A a'+b=a'+c).
PROOF, Let a + b - a + c . We obtain b',
.
c ' as in 3.4.3.14 and p u t a' = By 3.4.3.11, a + a ' = a . Since b - b ' + d and c = c f + d (with the d of 3.4.3.14), we e a s i l y obtain, a' + b - u ' +c.
b' x u + c ' xu.
AXIOMATIC SET THEORY
3.4.3.16
207
THEOREM,
(1) a t b S a + c - + 3 b ' ~ b l l ( b . b ' + b l l l \ a = a + b f A b " s c ) . ( i i ) a+b
a+c
-+
3 ur ( a c a t a ' A a' t b 5 u f t c ) .
The p r o o f i s l e f t t o t h e reader.
PROOF, L e t V E W . The p r o o f o f t h e i m p l i c a t i o n from l e f t t o r i g h t i s done by i n d u c t i o n on v . For v = 0, we have, o b v i o u s l y ,
a x 0 + b z a x ( 0 + 1) + c - + b L a + c . Suppose t h a t ,
a x v + b ~ a x ( v + l t )c - + b < a + c , and l e t , a x ( v t 1 ) + b 5 a x ( v + 2 ) + c. Then a + u x v + b 5 a + a x ( v + l ) By 3.4.3.15 (i)we o b t a i n a x v + b = b f + b " , a = a + b f , and 6" 5 A x ( v t 1) + c. Hence, u x v + b - b ' + b " 5 b f + a + a x v + c - a x ( v + 1) + c. A p p l y i n g t h e i n d u c t i v e h y p o t h e s i s we o b t a i n , b 5 a + c . The i m p l i c a t i o n i n the other d i r e c t i o n i s t r i v i a l .
+ c.
.
3.4.3.18
THEOREM, t b <, a x v + c (a x v + b = a x v + c
(i) 0 E v Ew+ (a x v (ii) 0 E
VEW+
--
a+ b 5 a+c). a+b
L.
a+c).
The p r o o f i s l e f t t o t h e reader. 3.4.3.19
THEOREM (FUNDAMENTAL THEOREM OF COUNTABLE A D D I T I O N )
( Z ( a ? , : n E v ) :v E w ) PROOF,
6
E
((
(b:vEw)+Z(av:vEw)
Let(Z(an:ntv) :
V E W ) ((
( b : v E w ) . Then we have c ,
wV such t h a t , I: ' a n : a € v ) + cI ,
=d b = -1 V
Hence, Z ( U / E V )
A p p l y i n g 3.4.3.14
+ cv
dv + 1
Z ( a n : n ~ v + l +) cv +1'
I:(an:nEv)
we o b t a i n d , e , k E wV
+ av
t c
vtl'
such t h a t
(1) Z ( U ~ : Z E V Z) (=a n : n E v ) + dv = , Z ( a n : n E v ) + cv '
ROLAND0 CHUAQUI
2 08
( 3 ) av + cv + 1 = ev + kv
.
Since, Z ( a n : r e v ) 5 b, f o r a l l v E w , b y 3.4.3.8 V E W . Hence, by 3.4.3.13,
b t c v , for a l l
and ( l ) , b = b + d v =
.
(4) b - b + C ( e v : v E w )
Define: (5)xv = c V + C ( d n : n E v ) + Z ~ ~ n + v : n ~ w ) , f o r a l l v E ~ .
Hence, b y ( 4 ) , x o = c o t Z ( e n : r E w ) = b + Z ( e n : n E w ) = b . O n t h e o t h e r hand, by ( 2 ) , ( 3 and (5), we o b t a i n xv = a&, + x, + 1 f o r a l l v E w Using 3.4.3.7, we have b = x = q + C ( a v : v E w ) f o r a c e r t a i n q. Hence, 0
.
Z ( a v : v E w ) < 6. 3.4.3.20
THEOREM,
A ( x K : K E W ) ( ( ( xK + l :
X E
H E W )
-+
3 b ( ( x ~ : K E W () ( ( b : K E u ) A V v V d v c ( O E v C- w A ( x K x v + d : K E w ) ((
(c:KEw)
+
b
+ d < c).
v
x
b i s a s o r t o f " l e a s t upper bound" ( i n t h e sense o f c a r d i n a l i t y )
the i n c r e a s i n g sequence x. w
5
of
.
We c o n s t r u c t a q E wV with y o = xo and xK + qK + 1 =x ~ forKEW.By + ~ i n d u c t i o n i t i s easy t o prove t h a t x Z ( qK : K E P + l ) , f o r every P E W By p u t t i n g b = ?: q and P applying 3.4.3.19 we o b t a i n the conclusion. PROOF,
3.4.3.21
-
Let x
E
V and xK
xK
+
f o r every
K E W
.
THEOREM,
(i) a + b < a + c A a < c + b < c . ( i i ) a + b 5 a + c ~ c <, a + b < a . (iii) a
+ a + 1* Vb V
c(a+b
a+c
-+
b=c).
The proof i s l e f t t o t h e reader. We now a r r i v e t o the main theorem of t h i s s e c t i o n . 3.4.3.22
THEOREM,
OE
K
E wA a x K
PROOF, The c a s e K = 1 i s t r i v i a l . make the following i n d u c t i v e hypothesis:
(1) For 0 C X a' + c ' S b ' + c'.
C
K ,
a ' , b ' , c ' and a'
+
c < b x
K
+
c-+ a+c
T h u s , we assume t h a t x
h + c'
5
b'xX+c',
K 2
5
b+c.
1 and
we h a v e ,
209
AXIOMATIC SET THEORY
K
s i o n sequences x , y, z , 5,
( Xv
X K
( 4 ) ' x v + zv
4- Zv
,
+ z +1 : V
(5) z U i ' y v + 1 x 2 and
(yv X 9 =h'xv+l x 2
Zv
K
t
that:
6,
go =
:VE.W) ((
:v€.W)
We s h a l l d e f i n e by r e c u r -
c
yo = b y z o = c,
( 2 ) x o = a,
(3)
+
< b x K t c. -6 w h, j , k E Y such
Suppose, now t h a t a x
,
: V E W )
zv+l : Y E w ) ,
E W ) ,
( 6 ) ( y v + z V : v E a ) ~ h ( ~ u t l + y V t lt l+ :~ v E w ) . Suppose x,, yv , I n o r d e r t o a v o i d n o t a t i o n a l complex-
For 0, t h e sequences a r e d e f i n e d a c c o r d i n g t o (1).
zv
, g, , h, , jv , and
k, d e f i n e d ,
i t i e s on d e f i n i n g them f o r v t 1 , I w i l l o n l y s k e t c h t h e p r o o f l e a v i n g t h e d e t a i l s o f the construction t o the reader. I n order t o g i v e a precise c o n s t r u c t i o n , e v e r y t i m e t h e e x i s t e n c e o f a s e t i s asserted, i t should be specified. We have, by ( 3 ) K + zv + d = y X K + - 2h+1. This {exists
( 7 ) xv x that
xv
x
C K C
+ zv 5 g,
K
x 2
x K
+ z,. (yv x 2) x yv x 2 + z v , i.e.,
+ zv
+ e)X
(K
(yv
x 2 ) x Z
- 1) + X ,
x
(K
X
- 1)
^1
E
Applying, now, 3.4.3.17, zV
- 1 ) + d + xv
+zv.
Let
x,
Then,
AEw
h x 2 +zv
besuch
5
+ z, 5
Hence, xv x '2
2 + z,, f o r a c e r t a i n e.
2
(K
a c e r t a i n d.
by 2.4.2.16. A
+
+
Zv
X
2
yv
X
(K
y,
x
( K - 2 ) t zu
(Xu
'(Yu
x
,for
V
5
By t h e i n d u c t i v e h y p o t h e s i s ( l ) , we o b t a i n , xv + zv
( 8 ) xv + zv + e y y (Zv
5
z
(yv
X(K
+
'2) X ( K
-2)
we o b t a i n , (z, + e ) x
, i.e.,
+
- I),
-
Z,,)XK
-2)
Consequently, we have,
Zv
1
,
-2)
+ yv
+ Zv
X
(K
x
( ~ - 2 j+ x V x ~ + Z v + d
X K
by ( 8 ) ,
+ zYx (K
(K
- 1) 5
- 1) + d ) + xu
yv x
(K
x
K
.
-2) +
( 9 ) y v x ( ~ - ~ ) + z v x ( ~ - 1 ) + d + x Y = ( ~ V + e ) x ~ - l + am , f o r c e r t a i n m. Hence,
210
ROLAND0 C H U A Q U I
= xv Xv
2
( 8 )a
+
X K
= gv
Zv
+ d +qv
( K - 2 ) + Z v x ( K - 2 ) ,
by (91,
( 2 K - 2 ) + zv
x
X
x
(K
- l ) , by
(7).
(ii), we o b t a i n ,
Thus, a p p l y i n g t w i c e 3.4.3.18
.
(10) gv + z,, + m = gv + zv
+ ( z V + e ) x ( ~ - l ) + m, by
x (K-1)
Hence, we have,
(xv + z v ) + (zv + e + m) = ( xv + zv
+
(zv + m),
+ zv + z,, +
gv
x
2
=(Yv
+
zv)
x
2
’(YV
+
zv)
+
(g,,
2
A p p l y i n g 3.4.3.4
e)
+
m
uo
+
u1
by ( 8 ) ,
by (101,
¶
zv).
+
(Refinement) we o b t a i n u E
xv + zv =
,
V , such t h a t ,
,
z v + e + m = uZ f U 3 ’
gv + z,, = uo + u2 = u1 + u3. Again a p p l y i n g 3.4.3.4 t o t h i s l a s t formula we o b t a i n a v E 4 Y such t h a t . uo = v o + v l , u2 = v 2 + v3, u1 v o t v 2 , and u3 = v1 + v 3 . Define, then,
5 +1 =
(11)
Xv
uo, gv + 1 = v3, and zv + 1 = v1
+ zv
and gv + zv =
kV
UhV
xv
2
x
+
+
z,
We have,
u2.
z,, + e + rn =
1,
+
f
9
JlJ
%+lX2
ZV+1’
x f 1 + yv + 1 + zv + 1 , c a l l i n g h,,, j , , , and kv the respec-
t i v e functions. Thus ,
Xvtl
X K
f
f
(YV+1
+ ZY+1)X(K-2)”Xv+1X2+ZV+1 +
5
YV+1
(gv +
Xv
+
Zv
Xv
f
zv + (g,, + zv+
f
(gv
X
( K - 2 )
(Zu
+
c ? ) X ( K - l )
+
Z,,)
+
(Xvt1
+ Z v + 1 ) x ( K - 2 ) ¶
-2)
X (K
m) x
(K
Zv X ( K - 1 )
+ m x
Y
- 2) +
by (111,
,
by ( l o ) ,
Xv) + m x
( K - l ) ,
by (g),
( K - 2 1 ,
+
211
AXIOMATIC S E T T H E O R Y
X
v +1
X
K
This shows ( 2 )
-
t
9" + 1 Y V V + l X K + Zv + l ( 6 ) , and, hence our sequences a r e constructed. v+l
Applying, now, 3.4.3.7 (12) ( x u
5
Z
t o ( 4 ) , we obtain:
+ zv : v ~ w - )( n +
-
C ( X ~ + .~P E+ W~) : V E W ) .
By ( 6 ) we g e t using induction, (13) ( y o
+
zo : v € w ) = ( q v + zv + E
( xP
+l
:
P € V
) :
v€w)
Now, by ( 6 ) and (12), V € W ) ( ( ' q v + zv : v E w ) . +1: Hence, by ( 1 3 ) , ( n + Z ( xP + l : p € v ) : V E W ) ( ( ( y o + zo : V E W ) . From (12), with P E ~ 5 ) yo + z0. Therefore, by 3.4.3.19, n + C (x,, v = 0 we i n f e r t h a t xo + zo 5 yo + zo, i.e., a + c 5 b + c.
( n :
V E W ) ((
( xv t 1
+
+
T h u s , the inductive proof i s complete. 3.4.3.23 (i) 0
COROLLARY E K E
(ii) O E
K E
.
I
wA a x
K t
wA a x
K
c
5 b
2
b x
x K
K
t c + a t c
= b + c.
+ u s b.
(iii) O E ~ E ' ~ r \ u x ~ = b x ~ + u = b .
( i v ) O E K E o A X E w A a x K t c x X 5 b x K + C X \ +ate< b + c . ( v ) O E K E W A ~ E W A ~ X K + C X bX . =x ~ t c x X + a + c =b + c .
PROOF, ( i ) - ( i i i ) a r e immediate consequences o f 3.4.3.22. t o obtain ( i v ) and ( v ) we must a l s o apply 3.4.3.18.
In order
Finally, we show some properties o f cardinal exponentiation t h a t a r e obtained from t h e theorems of t h i s section. 3.4.3.24
THEOREM a
( i ) a + "2
5
a
+
b
t*
'2
5
b
.
(ii) a + b ~ ~ + ~ 2 - b < , ~ 2 .
212
ROLAND0 C H U A Q U I
( i i i ) a t b = a t “2 PROOF,
c--)
b = “2.
The i m p l i c a t i o n s from r i g h t t o l e f t a r e t r i v i a l and ( i i i ) i s
deduced from (i) and ( i i ) . PROOF OF ( i ) . t a i n , ‘2 = c’ t c”, A p p l y i n g 3.4.2.11,
we have, 2‘
rem, 3.4.2.10,
$ * c’.
“2
PROOF OF ( i i ) . ^I
%. 3.4.3.25
5
5 b.
c”
THEOREM,
Thus, a
Using 3.4.3.16
Hence, c ’ t c”
5*
c“ o r “ 2
t
b
5
a t ‘2.
5a
x2 t
( i ) we ob-
2 ‘2 = “ 2
x
‘2.
But, by C a n t o r ’ s Theo-
el.
5 c” 5 b .
Hence “2
Assume t h a t a t b
a2 = a + c f o r a c e r t a i n c. c
5atb.
Assume t h a t a t 2‘ a = u + c’,
Since a
c.
S “ 2 , we have, b 5 at
By 3.4.2.16,
(‘a: 0 E v E w ) = ( a :0 E v E
0 ) .
The i n d u c t i v e p r o o f i s l e f t t o t h e reader. T h i s l a s t theorem g i v e s a counterexample t o : ba = b l a
PROOF,
Let 2
511C W .
b
b’
.
v
It i s clear that w s u { p
2.7.3.12, we o b t a i n f o r P E W , ( ’ p same r e s u l t i s o b t a i n e d f o r p = w
v
+
: 0E V € w )
, by
3.4.3.25.
((
: V E W } .
(w : 0 EVE
Hence,
0).
By
This
V
u { p : V E W } = x ( p : V E W ) ~ ~ ( w : v E W = )w x w = w .
PROBLEMS
1.
Prove 3.4.3.3.
2.
Prove 3.4.3.5.
3.
Prove 3.4.3.15.
4.
Prove 3.4.3.17.
5.
Prove 3.4.3.19.
6.
Prove 3.4.3.24.
7.
I n Problem 2 o f S e c t i o n 2.7.1,
.in a d d i t i o n ,
t a k e A and K s a t i s f y i n g (1)
-
( 9 ) and,
A X I O M A T I C S E T THEORY
213
Prove: ( i ) For each A , K s a t i s f y i n g (1) (10) such t h a t , a d d i t i o n a l l y , ( a ) a, b E A (b)
a, b
E
-f
(a=
K,A
- -
-
( 1 0 ) t h e r e a r e A , K s a t i s f y i n g (1)-
b +.-,a =
K,K b,
?i'+ 3 a ' 3 b' ( a ' , b '
E
(ii) Show t h e f o l l o w i n g theorems w i t h
K A
*I
a' n b'
and
5
= 0 A a=-
K,K
r e p l a c e d by
a' A
=r,K,$,K
K);
;
and r e l a t i v i z i n g t h e q u a n t i f i e r t o members o f ff ( o r 2.7.24 ( x ) , 2.7.2.4 ( x i ) and a l l theorems o f t h i s s e c t i o n except f o r 3.4.3.1, 3.4.3.4, 3.4.3.5, 3.4.3.8, 3.4.3.9, 3.4.3.23, and
3.4.3.24.
Replace a = b
You may use Ax C i n t h e p r o o f s .
aK <, b K ) .
(*)8.LetA = P E
b y V K ( K € W - + aK = b K ) and a
, where E
((
b by
K(K€W+
i s a E u c l i d e a n space o f dimension 2 and K ' t h e
E. L e t K = ~6 : 3 x ] g ( x E 0 A A g ~ gwp *- x+p n gv*xV = 0 = x n x V ) A 6 = u lg
class o f isometries o f t l p V ~ ( p c ~
E
1
.
K '~ A
x :ccfoI).
P I . 1
I.(
( a ) Show t h a t A , K s a t i s f y ( 1 ) - ( 1 0 ) o f Problem 6. ( b ) I n t e r p r e t g e o m e t r i c a l l y t h e theorems o f t h i s s e c t i o n . ( c ) L e t b f A be a c l o s e d r e c t a n g l e and a E A be t h e same r e c t a n g l e w i t h b a s i d e removed. Show a = A,K
.
3.4.4
BRADFORD'S RESULT AND I T S CONSEQUENCES,
We can n o t i c e t h a t most o f t h e theorems o f t h e l a s t s e c t i o n had t o do w i t h f o r m u l a s which,in t h e i r c a r d i n a l form, can be p u t : (E) where
a. K
~
\KO(
+
,
K
...,
... + ap - 1
I K ~
X0'...,
= a.
E
W"
( X o I +...
I f a*,
..., a v - 1 s a t i s f y a f i n i t e ,..., a +?r
*I
Xp-1l-
{ W l .
P - 1 We c a l l formulas o f t h i s form, o f t y p e (E). t i o n can be p u t i n t h e f o l l o w i n g schematic form:
p = v ) , then there are ay
+ ap - 1
Most r e s u l t s o f l a s t sec-
system o f formulas o f t y p e (E) ( w i t h such t h a t ao,...,
another f i n i t e system o f formulas o f t y p e ( E ) ( w i t h p = v +
a
v + x - l satisfy T).
ROLAND0 C H U A Q U I
214
Closures of formulas of this type w i l l be c a l l e d d p e c i a e
beVLteKC63.
Bradford's r e s u l t i s the following.
A special sentence i s provable i n MKT i f an only i f i t i s provable with t h e additional condition t h a t ao, a,, t?r E {la1 : a E w u I w l l .
...,
-
That i s , a special sentence i s t r u e f o r a l l c a r d i n a l s in MKT i f and only i f i t i s t r u e in t h e extended natural numbers (i.e. w U { w l ) . The proof of t h i s theorem, which w i l l be s t a t e d i n a purely mathemat i c a l form l a t e r (not metamathematically, a s above), i s long and d i f f i c u l t , and uses many technical lemmas. However, some of the intermediate r e s u l t s a r e o f i n t e r e s t i n t h e i r own r i g h t and cannot be deduced from the f i n a l result. After p r o v i n g Bradford's theorem, we s h a l l obtain several consequences of it, using elementary p r o p e r t i e s of t h e natural numbers. In order t o make t h e statements more l e g i b l e , o r d i n a l s o r ordinal functions will be denoted by lower case Greek l e t t e r s . In general, the only ordinalsmentioned a r e natural numbers o r w .
PROOF, I t i s obvious t h a t when p = 0, t h e theorem holds. In order t o complete the proof by induction, we s h a l l assume t h a t t h e theorem holds f o r p and show t h a t i t holds f o r p t 1. Suppose, then, t h a t
(1) a
t a
X
p = a
X
(@+I)
1
z ( y K:
Applying 3.4.3.5 we obtain w, (2) a =
all
KEV
.
'V
W I E
,
Z ( W ~ : K E V ) a x p=C(wk
By the induction hypothesis, we g e t such t h a t , ( 3 ) a = Z(zi:XEp') K E U ) ,
for all
,
K E U
wk = C(z\ , X E ~ ' .
Therefore, a = Z ( w K : K E obtain u E "('Y) , such t h a t ,
U )=
( 4 ) w K = Z ( u x K : x E p ' ) and
For obtain,
K ,
K E u ) .
XEv,
l e t O K h = I, i f
(
z\ K
z\
such t h a t , : K E u )
p' E w ,
,
and yK = w K + w;
z ' E p ' V , and a' E ' ( " w
xakx:XEp'),
:XEp').
)
and p = C ( a k x :
Applying 3.4.3.5,
= Z ( u A K : K E V ) , f o r a11
= A,
for
we
KEu,XE~!
and PKh= 0, i f ~ f h . By (4), we
AXIOMATIC S E T T H E O R Y
-
WK
Z ( W h
X P K h :
X E V )
215
= C ( C ( U l r X X P K h : h E V ): 7 1 E p ' ) .
Also, from (Z), ( 3 ) , and ( 4 ) , we g e t : a ? C ( z '71 n E p 0 = C ( C ( u B X : h E v ) : n E p f ) , a n d y K = w K + w W ( K=
Z ( C ( u 7 1 h X P K A :X E v ) : ? r e p ' ) + C ( C ( U l r h X a ; 7 1 : h € v ) Z ( Z ( U ~ ~ X ( P ~ ~: h +E v ) : n E p ' )
yK
c
Z (SKsx
~
(
(
: K € v )
Let
p =
Z
(
= p
U
~
~
X
&
+ 1 for a l l
v - p ' and define zh
~
~
, for a l l
~
Thus,
with ~ E 6" K )7 1 :h--O ~K~ & + E cikz ~ '
:
n ~ p ' X, E V + 7 1=.
K E V .
: nEp') =
.
,) a n d
u n X f o r z E p ' , h E v , and
An elementary argument shows t h a t
a K Y h + , ,.V = 6 K R X , f o r K , X E V , a E p ' .
.
2 , and ci s a t i s f y t h e conclusion of the theorem having (1) a s the hypot h e s i s . This proves t h e theorem f o r t h e case p + 1, and t h e inductive proof i s completed.
p,
PROOF, We a p p l y 3.4.4.1
t o a X P = b + c (with
P E W )
L : ( z h : A E p ) , b - Z ( z h x c x O h : X ~ p ) ,c = Z ( zh x c i l , , : X E p ) cilh = p
and g e t , a = , and a O A +
f o r every h ~ p .
Thus, b = C ( z h x P X : X E p ) and c = C ( ? x ( p - P h ) : h E p ) , whereOc P, C p . Let I = Ch : P A = P I . Then, i f we take d = C ( zh : h E T ) , e = L : ( z h : h + l ) , and 6 = Z ( z h x P A : h q 1 ) , i t i s easy t o v e r i f y t h e conclusions. = 3.4.4.3 a = 2 u A X
THEOREM, KZ
(
XK
+
V E U +
1A x
uK : K E v )).
E
V
V + ( Z x + a = L: x
++
3 u(uEvVA
The proof i s l e f t t o t h e reader,
PROOF,
Suppose t h a t u + Z ( g K x c i K :
K E v )
= u + Z ( g L x
c i ) K : ~ E u ) .
216
R O L A N D 0 CHUAQUI
Then, by 3.4.3.14
and 3.4.3.11,
( 1 ) =YK c
x
aK : K G V
)
A p p l y i n g 3.4.3.5
+
= a"
g'.
w = a, where c = b" t
that,
t h e r e a r e a", b", and
b" , Z
x aK
-
( 3 ) u" = I: x', b " = Z w', and
aK
x
5
c
x w
xK = pK
, for
all
Therefore, C ( pK x a,
x
a;:
K
such t h a t ,
-
+ b",
E v ' ) = a"
x',
E ' V , and
UJ'
= xK + wK, f o r a l l
a;= x > +
aK + L K , q,
LU',
'V
5
, for
E
and a t
"V
K E V ,
such
and
all KEv'.
such t h a t
wK
x
w5
c
w
x
wK +
and
K E V .
Z ( L K :K Ev
: K E Y ) f
)
, such
t h a t t h e r e a r e h, L'E"'V
have by 3.4.3.14
x
and ( 2 ) , we g e t , p, q , L E
( 4 ) qK = pK + 4,,
LK = 9,
y)Kx
yk
x, w
t o ( l ) , we o b t a i n
( 2 ) a" = I: x, b" = C w , and gK
Using 3.4.4.2
(
F'
= C x = C
x'.
Thus, we
that,
( 5 ) C ( p K x a K : ~ E v ) = C k C, L = I : L ' ~ c x o , a n d x \ = h h t L \ ,
for a l l XEv'. Therefore, (6)
y)Kx
by ( 3 ) ,
a\=
xi+
%
= k X + (L\+
Applying, now, 3.4.4.2
w\)
for all X E V ' .
t o ( 6 ) we g e t p ' , q ' , j E
( 7 ) q i = p i + q \ , k x = p \ x a,,+ j , , wl,+ j \ - q \ x a5 5 c x w .
v'
q i 5 (L'x +
and 15'
By r e c u r s i o n we now d e f i n e t e n sequences m, n,
h, b, X E w ( v V ), m', n ' , h', the f o l l o w i n g conditions a r e s a t i s f i e d :
2' such t h a t m, n, ( 8 ) m0 = x, W k : K E V ' ) , fito
I: n',
=
y,
t o=
bt0=
c x w , f o r all
x aK
for a l l
q ' , and
REW
(10) t l r K = Xn+lyK+
t,,
L,
bo =
(11)
nTK
bs+l,K'
+
w\)
JL,h ,
b',
x
w< c x o
.t, m', n ' ,
( " 'V )
X' E
JL',6 ' and
R 0 = p, m b = k, nb = (1' + K
X = ~c
.
= mn + l , K + b T + l , K '
ITEW
q,
tt0= y';
x ( t t B Kx a K : ~ € v= ) ~ m k C,
(9)
5
no =j,
V such t h a t ,
5
n T~ +
h k K= b T K
7 T + 1 , K " t f + l , K
c
x
UK'
w,
and E X , =
bTK5 c
x a K + h ? r + l , K
w,
'
and K E V ;
t k X = "?,+
l,X
+
')?T+
l,h'
x-
C x ~ , b ~ + l , X < C x w , t ~ h X a ' -
n k + l , X "',+ +
'
1,X
"+l,h
+ 1 , A + n ' n + l , h ' and % A
a\
=
AXIOMATIC
S E T THEORY
21 7
We now proceed w i t h t h e d e f i n i t i o n . For K = 0, we ces by (8). By ( 4 ) - ( 7 ) t h e s e sequences s a t i s f y (9) (11) (and t h a t sequences d e f i n e d f o r 71 s a t i s f y i n g (9) s t r u c t e d t h e f u n c t i o n s r e q u i r e d f o r t h e s e statements). quences f o r n + 1:
d e f i n e t h e sequen(11). Assume t h e we have a l s o conWe d e f i n e t h e se-
-
I t f o l l o w s from ( 9 ) and (11) t h a t C
.
(12) C ( t ) l r h x a\:
+ u1
U
f o r every
K
XEV')
EV
' I
.
u3, u4
E
ul
C a)lr
5c
Z h t = C u1
x
o
1
' I
E
, and .tK
'V x aK=
V such t h a t ,
(13) t n K u and
U0'
+
t o t h e l a s t f o r m u l a o f ( 1 2 ) , we have u2,
Therefore, a p p l y i n g 3.4.4.2 V
C
a i : XEV')
and ( l o ) , t h e r e a r e uoy u
Hence, b y 3.4.3.5
C ( . t t B Kx aK : K E v )
(.tkX x
u
K +
~
U
, u+o K
~ u~~ ~
~
~
ctK =~
5
= u q K x aK + u 2 K , u 3 K
cX x w
,for
all
5 u1
K x w< cxw,
K E V .
Therefore, (14) C ( U ~ aK ~ :XK E v )+ C u2 'I Z ( . t f b h x a\ we g e t uI0 , uil
By 3.4.3.5,
(15) C ( U ~ aK ~ : XK uroX t utlh
E v )
C u t 0 , C u2 = C uI1
" V , such t h a t ,
.tkX =
u'4 x +
, and uIlx +
*,
u4, m)?r + 1 = u 1
U13h,
uIzh
We now d e f i n e m =
5
c x w,
and . t ) l r h x a i =
for a l l X E V ' .
u t 2 , u t 3 , ut4 E
c x w
).
such t h a t ,
E
F i n a l l y , a p p l y i n g now 3.4.4.2
(16)
:XEV
R + l
UIOh
t o t h e l a s t f o r m u l a o f (15), we o b t a i n
=
u13X x a\ = '0'
Ut4h
5c
nK+l'ul'
x a,,+ x w
Ut2h'
,for
hn+l'
n ) l r + l = u'1 , /rta + l = u I 2 ,
u13X
5
U j A
x w
5
all XEv'. u2'
A ; + ~ =
= '3'
urn3,and
'n+l=
t ; + l =u'
4'
( 1 2 ) - ( 16) i m p l y t h a t these sequences s a t i s f y (9 ) - ( 1 0 ) . ( I n o r d e r t o g i v e a complete p r o o f f o r t h e e x i s t e n c e o f such sequences we should have c o n s t r u c t e d a l l s e t s i n v o l v e d . For instance, when i t was s a i d ' t h e r e a r e uoyu E 'V I , we s h o u l d have g i v e n an e x p l i c i t d e f i n i t i o n of uo and ul.
1
The r e a d e r can v e r i f y w i t h o u t d i f f i c u l t y t h a t t h i s i s always p o s s i b l e . I n o t h e r s i m i l a r cases t h a t w i l l o c c u r l a t e r , t h e same remarks are pertinent.)
ROLAND0 CHUAqUI
218
Thus, t h e t e n sequences a r e defined. (17) Z z R =
x ( X R K X aK : K E v )
(18)
C
WZa=
n r + l and w
I n t r o d u c e , now, z,
and z 2 n + 1 = Z~ n ' , ++l ;
~
w
V , by
a\:XEv'),
2 ( X i x x
f ~o r a l l
W E
B E W .
Then b y (9), (ll), (17), and (18) we g e t z 2 n = Z m'R = z z n + 1 + W 2 n '
and, by ( 1 7 ) and ( l l ) , z 2 n + 1 = C m ; + l C n\+l = z2n+2 Therefore, we a p p l y 3.4.3.7
K E V )+
+w
~ Thus, + ~ f o. r a l l
~
e
and
K E V ~
E
W
and
w T + zR+r
wR + z R +
such t h a t ,
X R K = n R + l , K +-tR+l,K
R E W ,
%+
+
'Y and a d'
E
"V such t h a t for a17
(20) t O K d K=+ C ( n n + l , K : n e w ) and XIoX = d \ +
(la),
(g), and ( l o ) , C w
--
(by ( 1 0 ) ) and
1 ,A (by (11)) we a l s o g e t
A',X , = A \ + ~ , ~
by 3.4.3.7, that there is a d E and X E v I , we have
Now, by
hR
aK:
~ = e + Z ~ a a n d ~ ~ = ~ + ~ ( ~ ~ + ~ : n ~
Since f o r a l l for a l l X E V '
R E W , zR=
(here we a g a i n assume t h a t zR =
for a definite h E w V ) t o obtain
(19) ~
+ Z n)rr+l = Z ( - t R + l y Kx
K
~
C ( A ~ + ~ n E, w~ ) :
Z ( W ~ ~ : R E WC )( +W ~ ~ + ~ : ? I ~ W ) =
Z ( Z c ( n R + l , K + X ~ + ~ , ~ : K E V ) : R E ZW ()Z=: ( AR + 1 , K : A E W ) x a K : K E V ) ; and, by ( l a ) , (9), and ( l l ) , C ( w r + 1 : R E w ) = Z ( w2 r + l : R E W ) Z ( W ~ ~ + ~ : * E W ~ () = ~
(
n
~
+
Z ( X ( A b + l , h : ~ E w )x a \ : X E v l ) .
l
,
V
A
+
~
;
+
l
,
h
:
~
~
v
~
)
:
+ R
~
W
~
~
AXIOMATIC SET T H E O R Y
219
( 2 ) C x = a " a n d by (1) a " ~ C ( q K ~ a K : ~ E u ) .
C ( ; t O K x a K : ~ E u ) < C x ;by
.
i i
Hence C VJ 5 X ( qK x aK : K u ) A1 so, ( 1) , a" 5 Z ( q x a : h E u ' ) f o r e , by 3.4.3.8 and (21) C ( qK x aK : K E U ) = Z ( q , X aK : K E V ) + ,' = C ( q ' x a' . X E u ' ) + e' + e l l . and C ( q \ x a \ : h E u ' )
- 2 U
+,",
h'
such t h a t ,
we o b t a i n C E ' V and ? E " Y
Applying 3.4.4.3
(22) e' + e0
x
. There-
Z U', qK x
= y, x a, +
ctK
V K for a l l
K
Ev
,
and
q i x a$ = q i x a,,+ iii f o r a l l h ~ u ' .
Now, f o r
K E U
and
d e f i n e u, v
XEu'
E
(23) u K = dK + i i K x w and u i = d'A + (24)
"
K
= Z(nTK
and
: A E W )
vi
'V
iT\
x w
-
for all
a
v'E
'b
, by
, and
= Z ( ~ ) r r + :~7 ,r ~E W ) .
C l e a r l y by (4), (10) and ( l l ) , v K , v i T h u s , by ( 1 ) and 3.4.3.8, (25) a + v, = a + v i
and u:
c
for all
x w
K E V ,
hEu'.
K E U , hEu'.
Noting f o r K E U , h E u ' i f aK = 0, then U K = 0, and i f a i = 0, then LA= 0 (by ( 2 2 ) ) we may apply 3.4.4.3, 3.4.3.13, and 3.4.3.11 t o (22) and o b t a i n , q , 2 q , t U K x w . By ( 4 ) and ( 8 ) q , to,+ h o K . Hence, by (zo), q , - d K t Z ( h T K : I T € w ) + u K x w = u + v K . S i m i l a r l y , q i = q i t ii\x w = d i + Z ( h L + l , h : n E w ) + U'h x w = u'x t v i . T h u s ,
,
(26) y K - u K + v K and y i = u i + v;i
for K E Y , AEu'.
F i n a l l y , we have, C ( u K x aK : K E u ) = Z ( d
K
= Z (dK
-
-
= e
+
x
: K E v ) t
x aK : K E u )
(el
Z(d\x
a,
+ el')
x w
a\:XEu')
C ( d i x a\:hEu')
= Z : ( u i x a\:XEv')
+
C(isiKx w : K E u )
( e l + el') x w
, by ( 2 1 ) , + ( e ' + ,'I) + C ( 2h x
,
, by
, by ( 2 2 )
x W ,
by ( 2 1 ) ,
w : K E u ' )
by (23)
(23),
, by
.
T h u s , t h i s l a s t conclusion, ( 2 5 ) , and ( 2 6 ) prove the theorem. 3.4.4.5 THEOREM, u x W E b c + d e b' d + e A b-b' + (b"+c'td) x w A c=c' + (c"+b'+e)
(22),
b" x w).
C'
C"
9
( a
Y
ROLAND0 CHUAQUI
220
-
-
PROOF, Suppose a x a - b + c . Then b t c + b = b + c = b + c + c , by 3.4.3.8 and 3.4.3.10. Thus, b y 3.4.4.3 we g e t b ' , b", c t , c" such t h a t b = b t + b " , b + b" b y and c + b t = c; and c = c t + c", b t c ' b y and c + c" c. Therefore, by 3.4.3.11, b -- b' + b" x w + c' x w and c = c t + c " x w + b l x w
-.
.
By t h e hypothesis, we g e t a s a x w - ( b ' I + c ' ) x w + ( c " + b ' ) x w . Thus, u = d + e w i t h d 5 ( b " + c ' ) x w and e 5 (c" + b ' ) x w Therefore, by 3.4.3.8, b b t + (br' + c l + d ) x w and c - c t + (c" t b t + e ) x w .
-
.
,
PROOF, When u = 1, t h e theorem c l e a r l y h o l d s ( t a k e 1.1 = 1, zo = a a.
= 1, and
prove i t f o r u + 1. C(yK
: K E U )
+
.
g,
x
Then by 3.4.4.5,
0 + z u , y, + ( x 0 + x1 + x 2 )
P
x w - x ( u
all x
P
K E V
= x w
x
.
P
and C ( E
0,1,2,
v + 2
f*
K h P
+ 60 + 6 1 +
and
zv+2+6
0
zv+2+60
ifX = v
+
2
+
So
t K ;
Pv
by, 0, A = 1, i f A = and A = v
0
and O K , =
+ n;
i f r E 6
2
-
, and
w
C (y,
C
(
:
zK : K E V ) ,
for
= C ( v A p xeKAp :AE6
KP wfor all A66
XESO
,
P
)
p = 0,1,2
P'
t
lil+h = v x 2 '
, '
-61
for
+A = V A 1
A = 0, ifv
ifn € S
2
w
for
.
X E 62
if h = u ; ax= yK0,
if A = v
1) a s f o l l o w s , P u x = w , i f h E u + 2 ; P u A = r K 1 ' ,
define
+
-
u
, for
and a x = 0, otherwise.
DefinePEVtl(l"wt
z v , xo, xl, x 2 , and
y K - z K + U K o + U K l + U K 2
:KEv)
D e f i n e c x E p w + l as f o l l o w s , % = 1 , K ;
= 2 y
and extend z t o 1.1 as f o l l o w s :
z ~ + ~ + ,=,w h o
2 +
1) x
Therefore, b y 3.4.3.5,
f o r p = 0,1,2,
( w i t h t h e r i g h t sequences). Let
we have z u -
(w + z E ,+ z u
t
x w
A p p l y i n g t h e i n d u c t i v e hypothesis, we have
r x p: X E ~ )
K E V ,p =
x1
f o r p = 0,1,2,
: K E v )
and adequate z, u.
= B(vhp
for a l l
KP
v + IY and a
Suppose, then, t h a t y E
w such t h a t a - x K E V ) = u1
-
= a). Thus, we assume t h a t t h e theorem h o l d s f o r u and
Po,
K ;
+
2 C -A#v
p K , = 0, i f
OK,=
E
K T 1 '
6
X E v t 2 and
if ~
a n d A = u + 2 + 6
i2 t
0
€ + 6
1 1
0+
A #
6and h = v
t r .
For
K.
K ;
+
K E V
b K A = EK
2
t
S
0
t
+
, TO;
n;
A X I O M A T I C SET THEORY
.
0, and z
I t i s easy t o check t h a t p , a , theorem f o r u + 1.
PROOF,
0 cu E w
,aE
w
+ 1, q
E
,
V
and a x w = C:
sume w i t h o u t l o s s o f g e n e r a l i t y t h a t f o r a l l
by a> = w i f aK = w
, and
a>
= 1 if aKE w
i t i s easy t o prove t h a t a x w - Z ( y K x
get a = Z ( z X x P X : X E p ' , ) y K x aK Z
(
SK
:
t o P by:
'w+l,
K
Ev) = w
and y, y '
5
y L X = yKX x u
, and
f o r X E p and
for X = p'
yLA =w and aK = w
EW
.
+
K
.
~ L : K E u )
We can as-
).
D e f i n e cx'E'u+l and 3.4.3.18
(ii)
. Hence, by 3.4.4.6
+
Define, a l s o
+
we
K E P ,
u and extend
K E U .
= w f o r X E p ' or A = p' yKh= 6Kh
z ando P' E
with
K
and
K E U
f o r X E p ' and a K E w Y y K h =
y K h = 1 f o r a K = w and X =
C Y ~ E W ;y
and aK = w
0.
By 3.4.3.8
Define, P = p '
Pi
= 0, o t h e r w i s e ;
and a K
yK x aK : K E u
K E P , aK #
= 0, f o r a l l
+K
1) by:
E
Pi
XCp
Ppl
(
Thus, suppose t h a t
C : ( z X xS K h : h ~ p ' )f o r a l l
f o r every X E p ' .
z ~ ' + y, ~ = and
and a K = w , and 0 f o r p'
s a t i s f y the conclusion o f the
I f v = 0, t h e theorem c l e a r l y holds. P
221
p'
+
K ;
K X = S K h x u f o r X E p ' and a K E w ;
, and
0 for X
yLX =
,
# p ' . ~ p' C h c p
I t i s easy t o check t h a t ,
(1) a = B ( z
P,
: h E p )
and a x w - C ( z X xP i : h E p )
( 2 ) y K = Z ( z X x y K XX: E p ) and y, (3)
Pi=P , + P 'Xx w
X U =
,
C ( z X x y L X : h ~ p ,) f o r a l l K E U
= y K X +y i X x w , f o r
and
K E P ,
.
XEp.
We o n l y need t o check t h a t one o f t h e f o l l o w i n g a l t e r n a t i v e s i s s a t i s f i e d , f o r each X E p : or
(4)
Pi =
(5)
P1 =
C ( y K X xaK : K E Y ) w and t h e r e i s a
, and
K E P
aK = w i m p l i e s y i h = 0 f o r
such t h a t
aK = w = y i
K E V
;
222
ROLAND0 C H U A Q U I
We have t o c o n s i d e r two cases: CASE I K
.
Pi#w
I
Then
Pi
and a K E w . Then 7 L h = r K h =0 f o r a l l
K € V )
and i f aK = w
CASE I I subcases :
, t h e n r k h = 0;
For
K E Y ,
Suppose X = p'
(11.1)
K.
K
Pi
and
Pi
Thus,
= 0 = 2 ( - y K hx aK :
, then
7: A = 0.
We have t h r e e
= w f o r a certain K E Y .
Then a = w = -yL A c o n t r a d i c t i n g t h e assumption o f Case 11. K
0 and we a r e i n Case
(11.2)
I.
Suppose X C p ' and
Z((SKhxaK:
K E V )
11, i f a K = w , t h e n
w .
= 2 ( - y K X xaK :
K E v )
Pi
= 0.
We have
, since,
Therefore,
-yKA= 0 = S K X .
XCp' and
(11.3)
Pi
=PI+
(4) i s s a t i s f i e d .
i.e.
we have, i f aK = w
+
for a certain K E V ,h
= 0 and hence,
Hence
Pi =
w = C ( 6 K X : ~ E= ~ )
by t h e h y p o t h e s i s o f C a s e
Pi
= I : ( - y K X xaK : K E v ) .
We a r e a g a i n i n Case I.
T h i s proves t h e l a s t c o n c l u s i o n o f t h e theorem.
PROOF, I f v = 0 o r p = 0, t h e theorem c l e a r l y holds. pose t h a t v # 0 # p and t h a t t h e theorem h o l d s f o r a l l p ' € p . show t h a t t h e theorem h o l d s f o r p . L e t ,
(1)
a
Z(gK
P
X
X
aK : K E V )
Thus, supWe s h a l l
.
Without l o s s o f g e n e r a l i t y we may assume t h a t aK # 0 f o r a l l K E Y . By
+ 1 such t h a t ,
E u c l i d ' s a l g o r i t h m , t h e r e a r e a', 6 ' E
(2) a K =
p x ~ ; + 6 ~ ,S K C p
and i f a K = w , t h e n 6 K = 0 , f o r a l l
Thus, from (1) and ( 2 ) , we get,
(3) a
X
p
z
x ( g K
XU;
:KEV)
X
p
+
2 ( q K X 6K : K E Y )
.
KEY.
AXIOMATIC SET T H E O R Y
223
( i f ) , we have a b such t h a t ,
A p p l y i n g 3.4.3.23
(4) a = z ( q K x
I$ : K E v )
t
b.
By ( 3 ) and ( 4 ) we o b t a i n ,
(5)
z
(
gK
CXk
X
:K
E V )
p t b
X
= 2(9
p
X
:K
X
K
E V )
X
p t 2
(9,
:
XSK
KEY).
q"
we o b t a i n b ' , b", and g',
Using 3.4.4.4,
(6) b = b ' t b " , g K = q l K t g l ~ f o r a l l ~ E v ,b
and Z ( y K
x
a;
:KEv)
Therefore,
+ b"
t
C g" = C ( g
K
x
we have p' E w
by 3.4.4.1,
( 7 ) b ' z Z w , qK x 6 K K E V )= p f o r a l l A E ~ ' .
a' K
,u
K
Thus, e K A E w f o r a l l
KEV,A
t x p = C ( g 1 x 6
: K E v )
E
'I'V,
for a l l
= Z ( U ~ X E -AEp')
A'
such t h a t ,
E 'V
.
K
E E
) so t h a t ,
'("w
K E Y ,
: K E Y ) ,
K
and C ( e K h :
E ~ ' .
We s h a l l , now, d e f i n e by r e c u r s i o n f o r i $Ell cGL ( " a t 1 ) : L E U ) , and 0 E n ( ' " ( ' ' w
EV,
t
$,
,u
1) :
Il ( @c V
E
i Ev)
: L E Y )
,
such t h a t ,
(8) b ' ~ Z ( ~ ~ U ~ ~ ~ $ ~ ~ ~ : f E o r a~ l l~ L E' V ) ;: X E $ ~ ) ,
(9) O
L K X x 6 K = E ( $ l X E x e K 5 :[ E P ' ) ,
(10) q', = Z ( u l
O1
K A
, for
:A€$[)
forAE+L,KEitl; K E L
t1 ;
and
(11) q',
x 6K
= Z(Z ( u L
x E
$l
K E '
t; ~ p ' ):
for
)
.
i EV
K ,
We s h a l l f i r s t d e f i n e these sequences f o r i = 0. By ( 7 ) and (Z), we have, q i x S o = C ( u E x e O E : E E p ' ) and 6 o E p . Thus, by t h e i n d u c t i o n h y p o t h e s i s (i.e.
theorem t r u e f o r 6
f o r a l l C; E p ' and O o
xh0
E p),
0 = Z
(
Therefore, by ( 7 ) b ' = X ( Z ( u O h x $ , , ~ Z ( C ( u O h x
GoXt; xe
K t
*
t a i n e d t h e sequences f o r
!i€p'): A€$*) i
we o b t a i n qb =
$oAt xeOt : E : E Ep') : for a l l
KEV.
~
(
u
~
:
~ p ' )f o r a l l
and q k ~6~
1
Thus, we have ob-
= 0.
Suppose, now, t h a t t h e d e f i n i t i o n has been c a r r i e d o u t f o r 1 and we s h a l l do i t f o r i t 1 E v. By ( l l ) , we have q: x 61 t 1 = 2 ( 2 ( u , A x $, x E( + : E E P ' ) : A W L ) . I f S 1 + = 0, i t i s e a s i l y L t 1; thus, assume 6 c + 0. Since i t f o l l o w s from t h e i n d u c t i o n h y p o t h e s i s ( i c e . theorem
seen how t o d e f i n e t h e sequences f o r
by (2), 6L t 1
E p,
t r u e f o r numbers l e s s t h a n p ) t h a t
~
x
224
ROLAND0 CHUAQUI
(12)
YIL
(13) u
+
1 = z(uL+ 1 , p L +1,1+1,q : " @ L t 1 )
~
~
~
Z
(
U
~
+T ? ~
Z ( U r + l , q ~ t \ q : ~ ~ $ L + lf o )r
Lq=thq+ t xqxw
(141
and
(15) For each
4
Ep')
e L + l y L + 1 ,q x s L + l
and f o r a l l X E Q , Define $'+
implies
, if
C
for
For q E d 1 +
(17)
' L
(18)
oL+ 1
such t h a t C ( $ L , E x ~ L + l , E :
(SXV
0; GLt l y q t
and
+ 1, K 7 )
= z ( C ( S X q x $ L x C ;m : x E
( $ L
qEdJL+
$ L+l , q E e C
t \ 77 --
,
= w
or
(16)
for he@, q E 9 ,
wand t h e r e i s a
= w and
(b)
a l l XE$[,
one of t h e following a l t e r n a t i v e s i s s a t i s f i e d :
qEdL+
(a)
E ,$ ~~ + ~ Xand ) Su L ~ X ~x w : =
and E
w
EP'
a s follows.
~
€ ), 4i f ~ for
x$,
= w
KEC+
1 ,E : C; E P ' ) =
L+
p f + 1 ,c; [
:& P ' )
, then t i q = O . all
X E $ ~ ,
=
, otherwise.
1, l e t O L +
C ( t h,,xOc K h :
be defined by, ) ¶
i f for all X E $ ~ , eL Kx= w
implies 5
= 0; 0 = a , otherwise. l + l , K ? 7 111 I t i s easy t o show t h a t with these d e f i n i t i o n s , ( 8 ) , ( l o ) , and (11) a r e s a t i s f i e d f o r L t 1. We s h a l l prove (9), f o r L + 1, i.e. we have t o prove:
Let q E G L + C AS E I
(1.1).
I
sK = C ( $ ' +
9 K q ~
. K E L
3
77t
xeK
:E
EP')
, for
We have two cases:
+ 1.
There a r e several subcases:
T h e following two conditions are s a t i s f i e d :
qE@'+
K E L +
2.
S E T THEORY
AXIOMATIC
'1+1,Kq
X s K ' I z ( ~ A q X e lK h X ~ K :
225
A E # ~ )
= Z ( Z ( S h q x $ l A E ~ ~ K ~t E: P ' ) :A € @ ,
z ( $ t + l , q t ~ E K~ tE :P
2
€
(5
(16)). Hence, by (9), 0'
g e t by (171, O t + l , K q
el + 1,
x hK
'IX
(
K
w
3
, IL, k c ; = w
, i.e.
BIKA= w .
Also, $ l + l , q t ~ ~ K t w=
=w.
11; + 1, c;
KAx6
E :E
x EK
Ell')
.
We have t h a t 6 K # 0 and
(1.3).
, for
xG1
€ P I .
I n t h i s subcase, we have t h a t f o r a c e r t a i n w (by
by ( 9 ) f o r t ,
(16).
' ) by
$t+l,qt # X
We have e K t # 0 and
(1.2). a certain
),
I n t h i s subcase, we have t h a t
el
, by
, and S A q
(16).
Therefore,
Kh:AE41).
i=(S'XqXer
K h =w
tiq=
and W i t h t h i s we
= w f o r a certain
f K t # 0 and \Il (by ( 1 7 ) ) . Hence, f o r a c e r t a i n t € P I , = w (by ( 9 ) ) . Thus, b y (17) e l + l , K q = w , and b y ( 1 6 ) iL,+ l , K q = w Therefore, t)L+l,Kq X 6
= w'IZ(G
t+1,77tXEKt'
-
.
t €PI).
Thus, we have completed t h e p r o o f o f (18) f o r Case I .
CASE I I ,
(Ir.1).
K
=
i +1.
We have two subcases.
, X ( ICt
For a l l
t i q= 0.
k t x e 1 + l , t :t
I n t h i s subcase we have by (17) t h a t f o r a l l p l i e s \IL+ , q t ' ( ' ~ q*i ~A € : X E 4 ~ *) z ( $ ' + 1 , 7 ) t X e l + 1 , t : c; EP")
.
(11.2).
tiq
There i s a
such t h a t C ( $
1
At
I n t h i s subcase we have t h a t f o r a c e r t a i n that *t
$l
+
$ Kc;
, q c ; = o . Since 6 ( +
+ 1 , 1 + 1, q x 6 c + 1 =
w= C w + l
= w and
EEP',
# 0 im-
Hence by ( ~ 5 ) ( b ) , ~ l + l , r + 1 , q ~ 6 t + l ~
=w.
t h e r e i s a XEG1 such t h a t
= w implies that
€PI)
XE
l+l,t
. $ € P I ,
:(€P
)
2
wand
e l + l,E # 0 and
t i m = a . By (16) t h i s i m p l i e s
# 0 i t f o l l o w s by 1 5 ( a ) t h a t
,qgx
E t + l , t
: < € P I ) .
Thus, we have proved (18) f o r a l l K E L + 2. T h e r e f o r e we have d e f i n e d by r e c u r s i o n t h e r e q u i r e d sequences s a t i s f y i n g ( 8 ) - ( 1 1 ) . By ( 6 ) we o b t a i n , C
(
qK
x
: K E Y )
+
b"
' I
C
(
yK x
CY.~
:K E U )
2
226
ROLAND0 CHUAQUI
~)K:KEv) +
y t i f o r a l l A E v . Hence, by 3.4.4.3, t h e r e a r e u' V and u1 E V ( ' Y ) such t h a t b" = C u and yK x a;+ uK = yK x a: f o r K and q ' k = 2 u i and qK x a: + u ) x u yK X a: f o r a l l K , l e v .
Z(qKx
V
Let p = $v
-
+v + v v
and d e f i n e z
'Y
E
and
Define now
t;: E
+
c ( ev - l K A
y, 7' E v ( c c w + l )f o r
all
-l;
K E V
x
~)K:KEv)
1
'V-
-1
q,,=
for
K A
U E V
for A€#,
;
-l;
a s follows:
for h = a; + w , o r = q5v -1 + v + K + E * V w i t h E E V and a:+ w ; + E w i t h t ; E v and a; = w , o r a;# w and t; + K , a l s o 6, - 1 q + E v w i t h t ; , q E v , E + ~ a n d a ~ = wo ,r E , q # ~ a n d + v + t + K v w i t h t; E v and a: = w, o r a: f f o r A = $v (21) ' K h =
E V ,
+ 1 by:
E
(19) z x = uv -jJ f o r A€$,, - 1 ; z A = uK f o r A = $v -1+ K and and z x = u)FK f o r A = $v - 1 + ~ + t;~ *+V with K , E E V .
(20) P,= c ( IC; and P h = w otherwise.
E
y K x =w
9, - 1 + K w i t h y K A =0 f o r h = +v + f o r A = 9, a;=w; yKA=l w and t; f K
-
.
-
- 1; y L h = w f o r h = $v - + K , o r (22) ev l K A x w f o r h = @v-l + q + t * q with E , q E v , and E = K o r q = K ; y;A= 0 f o r A = $v-l+ E w i t h ~ E and v ~ Z K o,r A = q5v-l + [ + q * v with E , q E v , and E # K o r T#K.
Using 3.4.3.11
and 3.4.3.22
( i i ) i t is easy t o show t h a t
(23) a = Z ( ? x P A : A E p ) a n d W K ( K € V + y K = Z ( z h qK
x
a" Z ( z x A
&:
xyKA: A E ~ ) A
XEp)).
I t i s immediate from the d e f i n i t i o n s (21) and (22) t h a t (24)
y l A ? y K A +y d A x w
for all
K E V
and a l l
A E p .
We s h a l l prove, ( 2 5 ) One of t h e following a l t e r n a t i v e s holds f o r every h E p :
(a) or
P x x p ~ Z ( y K A cx i K : ~ E v ) and ,
(b) there i s a
KEV
Suppose f i r s t t h a t PA
X P
-c
for all
K E V ,
such t h a t aU = P A = y A h = w
AE$v-l.
ciK=~impliesy~A=O,
.
Then we have,
( $ ) - l h E : E E c c ' ) x P + Z ( 6 ' V - 1 K A x a L:
K E v ) x p
= z ( c ( G ~ - ~ : t ;~~ p~l ) Kx : E YE) +~ ~ ~ (
e
~
,
by (ZO), - :~
~
~
~
a
AXIOMATIC SET T H E O R Y
K E V ) x
=
'
P
('Y-1 K
9
by ( 7 1 ,
h x (6K+a\
= C ( y K h x aK :
227
K E v )
X P )
:
K E V )
, by ( 2 ) and
, by
(9),
(21).
From t h i s we a l s o get t h a t i f t h e r e i s a K E V such t h a t aK = w and , t h e n p , = w (by ( 2 1 ) and ( 2 2 ) ) . T h u s , we have proved (25) f o r
Y;~= w
hEGv-l'
In order t o consider t h e other cases, note f i r s t t h a t P
= w for @v-l+77 € where t E v , and aK =
Suppose, now, t h a t X = Gv-l t w implies y j h = 0, f o r a l l K E V . Then by ( 2 1 ) and ( 2 2 ) , C ( y K x x a Then P, X P = C (yK X X a K " Y
q E v + v - V (by ( 2 0 ) ) .
c;,($v-, + t " 9 = w
F i n a l l y , suppose X = $ v - l + v + q + t - v , where t , ~ implies y i x = 0, f o r a l l K E V . Then, by ( 2 1 ) a n d ( 2 2 ) , Y q h X
a - w = P x x P.
77
K : ~ E v ) : K E V ) .
E v and , aK = w
C ( y K x x aK : K E v ) =
.
With t h i s we complete t h e proof of ( 2 5 ) . ( 2 3 ) , (24), and (25) a r e the conclusions of t h e theorem.
PROOF, When v = 0, t h e theorem c l e a r l y holds. Hence, in order t o complete t h e proof by induction we assume t h a t t h e theorem holds f o r v and show t h a t i t then holds f o r v + 1. Suppose a E v + 1 w + 1, P E p w t 1, x E 'V , y E 'V , a,,# 0, and 21 ( xK x 04(: K E v ) + x V x av = Z ( ghz PA: h E I-( ). Then, by 3.4.3.5, t h e r e a r e w, w ' E p Y , such t h a t
'
(1) C ( xK all
XEP.
x aK :K E v ) =
L: w , xv
x
av = X w ' , and
yhx
.,0
w A + w\
for
ROLAND0 CHUAQUI
228
Applying 3.4.4.1 and 3.4.4.6 we obtain r$ X E ~ ), and U E €I A(V : A E p ) , such t h a t (2) y A = Z ( % * x
-
~
for a l l
AEp, v E + A
, where
0
A17
= P , and
0'
i.
, $I,
6'
, 6''
A77
+1:
EII('Aw
C ( u A17 x 8 ~ :q? E f j X )
T?E+A), W A =
6 ' qEcpX) ~ AT?'
=(Uh,l
@E' w
, and
=
implies Jt AT? = 1
Ph#w
Therefore, ( ~ ) z ( X ~ ~ ~ ~ ~ : K E ~ 77) x= eCA ( q C: q( ~U +~ A ) : ~ ~ p ) .
.I,
that,
T h u s , applying t h e inductioy hy o t h e s i s t o (3), we obtain P I E o , E u(p'o+ I ) , y " , y ' " E rI(" w + 1 : h ~ p ,) and z ' E " V such ( 4 ) xK = C
(
zt
: f E p 9 and xK x w = Z: ( z t x a''
a'
x
5
K S
all
UhT? = C ( z ' s X Y x q {
(5)
( 6 ) a'ks = a t
(7)
Yyqf=
K t
+ a"
K3'
Tiq{ t
for for
:{ Ep
for
)
ux7,
{€PI) ;
x w = Z ( z '{ x Y T 9 s :
AEv, qEGX;
K E V , fEp';
xw
1 y * ;;
K f
K E V ;
and
: { € P I )
xw
f
for
A g p , qEGX,
and
SEp';
(8) For each f E p ' one of the following conditions i s s a t i s f i e d : (8.1)
all
K E V , XEpt
implies 7; (8.2) K E U )
AEp)
Or a.
Z
2
X
a K - :K
~
V
=) Z : ( C ( y "
AT? s
x 6'T?X:q/#A):
and q E + h we have t h a t aK = a implies a i r - = 0, and 6'
T?f = 0; There a r e
AEp
and q € d X such t h a t 6'
W ;
(8.3)
= w .
h c p ) and f o r
There i s a
(8.4) There a r e
Define
f EP'w
+
K E V
such t h a t a K = a"
K f
K E V , AEp,
and
qEGA
XT?
=w
=y;;lqf = Z ( a t K t x a K :
=Z(C(y'iqSxBAS:
such t h a t aK = a l l s = B A S
=';if=
1 by,
( 9 ) E ~ = Z ( Z ( ' 6' ~ ~ : ~q ~ rX $ :~h ) ~ p ) i f f o r a l l A E P , A77 implies y ' " = 0; E = w , otherwise.
=w
A77
XqS
5
From (2)-(9) i t i s easy t o obtain t h a t xv f ~ p ' ) . Then, by 3.4.4.7 and 3.4.4.8 ,
x
av = 2
LVI
= C (z'
qEq5X,
sXEs
:
.n
Q
h
W
w
m
v S
m
c 7
UJ
-
Q
4
=*n
=*n
W
3 * n w
II w
2-
-a
?-”
w
+
*”
X
a.
w
11
w
+
Q.
w
X
3
m
L 0 rc
7
7
w
Q
L
0 +
.,
Q
h
W
w
X
w
N v
II
w 3
X -*n N -0 S h
P
W
w * n
W
.*
=*n
w
w .. .. - a a. * Nw
X
v
w I1
3
X
w
.. .. w
-*n U X *uI
N v
w 11
‘*n N
h
N v 4
.*
TI
aJ .Irc v)
m
c,
.r
v)
VI
S
v)
.r
0
-r
c,
. . I
3 11
8” X w
*”
Q.
w
11
S
-0 -A4
.r
0 V
m S .I-
3
0
7
7
0 +
a,
L: c,
0
rc
aJ
S
0 Q
W
w
r V
4
aJ
L
0
.n
3 h
11
-
Q
W
*n
.. *n
u1
X
w
@a
-*n
v
w
11
w
*
-a
a”
II
m m
h
m
4 4
LL
v 4
3 I1
a” w
II
Q W
m
*n
+
-
3 4 V I ”
9
h
c
.
a
?-
J
s *I-
4 a J v n
m p _
i
W aJ L a ? c
.I-
Q -
01 r\l N
>
rn 0 W
I IIW v)
c1
x 4
w
+“ w
N
X
v
w I1
*”
h
$-I
h
v 4
0
r l v
L
0
w
s.
w Y
w
Y
U
-*n
w
.m
*a * * ..
“
Y
w
n
-
Q
W
*n
*n
.. =ay w
X
=*n v
Q.
I1
w
w - Y
*
In
h
4
V
..
n h 4
h
+
3 W
Q -
a
n
-ro 5
*o
S
aJ
S
%
.r
aJ
n
+
m
.r
h
x a UJ
..F
W
Q
h
-
*n
..
F
x
*n
?
F
*
=x X *uI
-.b Q. v
11
v
w w x
*uI rc)
u3
v 4
m
3
I1
w
x
Q
.n
rc)
-
W
*n c 7
m
L
0 + m
I1
0
w Q.
=*n
aJ
v)
.r
c
a E
.r
3 11
h
W
x a
c
c ..
4 X
F
*n
?-
=x v
w
h
x
W
8
..F
h
-
Q
W
*n
*n
..
F
+
Ex
X
w
=*n
@a
v
v
w
I1
W
L
v)
c,
.r
aJ
lt
c,
c
m
0)
u
S
a,
..
.ga;
r
a J ’
v) o
3
x
Zly
-
h
<
h
4
a
+
W
Y
..
Q
..
W
w
w
- Y
*
Q
h
W
w
a
<
..
W
X
v
ro
x
w
+
Lo
-x
w
X
3
w
Y
0
cu
W
v
Lo
-x
w
x .. ..
W
“
+ 1 3
4
Y
x
w
x v .. *
Nw
v
v
W
ro
N w
X
v
v
W
v
- Y
*
U
w
3
-4
La
4
h
h
b
ol
h
v 4
ROLAND0 C H U A Q U I
230
We o n l y need t o prove t h a t f o r e v e r y tion i s satisfied:
t
one o f t h e f o l l o w i n g c o n d i -
Ep
(a) Z ( y K t ~ a K : ~ ~ ~ + l ) - C ( 6 , th ~~ Pp X) :, a K = w i m p l i e s y ' for a l l
KEV
t1
, and P,
6it
implies
=w
= 0
K t
f o r a l l XEp;
=O
( b ) t h e r e i s a A E p such t h a t P , = 6 ' or
= Z ( y K t x a K : ~ ~ v + l ) - w ; AE ( c ) t h e r e i s a % € v + 1 such t h a t aK= 7 L t = C ( t i h c ; x 0:, A E ~ )= 0 ;
(d) there are Let
t
Ep.
CASE I
K E V+
1 and h
~ such p t h a t aK= 7,:
=
P,= 6 '
AS
= w
.
We have t o c o n s i d e r two cases:
+ 1)
Z: ( y K x aK : K E V
yLt= 0.
f w and f o r
KEY
+1,
I n t h i s case we s h a l l show t h a t c o n d i t i o n ( a ) holds. t h e assumption of t h i s case and ( 1 4 ) t h a t , (21) aK # 0 i m p l i e s t h a t 7,
implies
aK = w
I t f o l l o w s from
= C ( P ; t x a ~ S : S ~ p ' f) o, r a l l K E v ,
and, by t h e assumption, ( 1 5 ) , and (6),
5
(22) For a l l
Pi
Ep',
= 0, and a l s o t h a t Z
then
implies that f o r a l l
= w
(a;
xaK :K E v )
# w
K E V ,
.
if a =w K
,
Therefore, by (8) we g e t
'33' (
(23) For a l l { E p ' and q E @ , , and,
Pit=
(24) F o r a l l r e p ' ,
w
= eh q = w
i m p l i e s t h a t 7 i 1 i S= 0
-
implies that C ( a i t x a K : ~ E v )
.
x e h q : qE@,) : A E ~ )
Also, s i n c e y
v t
+w
and av = w i m p l y t h a t y '
v t
by (21) and (13) t h a t , (25) Y, for . f € p ' .
E
av = Z ( P i E X
Suppose, now, t h a t
Sit
fS
= w
:S
Biq
+a
, and
(26) P,=o
and
fS.
= w implies that
f o r a certain h E p .
c e r t a i n q E @ , and S E p ' we have y"I (22) fore,
Ep'),
= 0 (by ( 1 3 ) ) we have
AqS
by ( 9 ) e{+w;thus,
= w and
0;
hy(9),Bxq#w
PI'
SE
= 0
Then, by (17), for
, thus
by (8) and and, by (2)Ph+w.There-
= w
i m p l i e s 6 i t = 0, f o r a l l AEp.
: S Ep') : q E + X ) Now, assume P, f w and 6, f X ( 2(P; x y i q r x f o r a c e r t a i n X E p ; then f o r c e r t a i n q € A , S E p ' we have = w and
\
AXIOMATIC SET THEORY
Hence, by (25)
E
implies S
We have t h u s proved ( s i n c e P,=w (27) all h E p
P,
f 0 implies that S h t =
.
CASE 1 1 ,
XC;
1 s = 0.
= 0,
8, = O (by (2)).
by ( 2 6 ) ) .
C ( C ( P i t x y i t S 5 : t ~ p ' ) r: l ~ $ , ) f o r
+ w,
Z(SXtxPh:XEp)
Sit = o .
Then
and t h u s b y ( 7 ) and ( 9 ) 0 '
+w
5
231
and f o r a l l X E p , p,=w
implies
We s h a l l a g a i n prove t h a t ( a ) h o l d s i n t h i s Case. S i m i l a r l y as f o r Case I,we have t h a t f o r a l l that i f
0,
=w
, then
= O for all
$Iq5
5
Ep',
0;
0,
: 1 E l . c ) j+ w .
qE$,)
SO,,
So, s i n c e O h q
if BAS =w
, then
q q , ) : X E p ) j+
we have t h a t f o r a l l { E p ' ,
p'it
=w
~ € 4 , ; and Z: ( C
7 ! q 5 = 0 f o r a l l XEp,
{ E p ' ,
a K = w implies
z ( z ( q q tx e h q : Hence, s i n c e implies t h a t y
and 0
C
K t
aK C w
=0, f o r a l l
all
ri~4,) :
.
U S X E ~ )
E
, then
Z: (a;
x
aK : K
5
E U)
K E V ;
E
5
+w
Oiq c P , ,
-
Z: ( 2 (y"' x 0 ' hq5 Xv' qE$
Hence, we g e t t h a t xp,:
A77
,Ell)
Pit
= w
*
and C ( a K S x aK :
= w implies
x
and, hence, a'
K 5
;t w .
+w
P" 5t
K E u )
: X E ~ )
+
aK
.
t h a t , f o r a l l ~ E P ' ,O )
, that
Pif
ft
=
+0
=w
=w
W .
- 0, f o r a l l { E p ' and -
z : < p y t x E l . : S E p ) = Z ( Z ( Z ( 0 5' t x q ; 5 ~ e ; q :
Z:(Sht
implies that:
(yiq5x0
= 0 = a" P k t = w = a K i m p l i e s a' we have t h a t K 5 K5' = Z : ( P i t x a'K { .- 5 E p ' ) f o r a l l R E U . Also, i f
I t i s a l l e a s i l y seen, s i n c e
implies that
:
0 .
Therefore, b y ( 8 ) we o b t a i n t h a t under t h e h y p o t h e s i s for
= w implies
and Z: ( C ( y f l q 5 x
X E p , qE$,,
VE@~):XE~):
5
ROLAND0 C H U A Q U I
232
w3;
Therefore, by (13), Y, " a v = [ X € * : . f E p ' ) , and 01 = w V implies Y' = 0. W i t h t h i s we conclude t h a t , aK = w implies y ' = 0 , f o r v t K E all K E v + ~ . yi'q
+0
Now, c l e a r l y , f o r any h € p , P, 5 :{€PI): qE9,). Hence,
PIC( YK
,
"
aK : K E V + 1 )
implies t h a t
z(8(fiitX
: K E V )
QK
+
=
c ( 2 : ( Z (6;
=
Z ( S A E X
txy;:,{
0,
x
Y, c;
:
x
+
{ E p )
x av,
: VE9,)
(BX7,+B;q)
:hEp) :
i- € P I ) ,
.
:xEp)
= Z(Z
The two cases imply ( a ) , ( b ) , ( c ) , or ( d ) and hence we have completed the proof of t h e theorem.
a,
+J
u
r
a, a,
VI
0 c,
a,
ul
3 a
.r
c,
CI
7
m
C
.C
<
a
d
+
h
W
-a
Y ..
x
xy
h
a
pzy
p z v
< w - +
+
w
..
Y
xy
0.
+ -
a
..
Y
W
Y
-+
a
Y
pz X
-+
v
2 w +
1
+
1 W
t
a
W
Y
Y
Y
D
W
Y
Y
rn
v
h h
< > a
W
Y
a
W
..
a"
X
Y
x
i-
Y
w
11
I1
3
x
* Y
ll
i-
11
3
I1
+
4Y
i-
- 3
a
-
<
Y
W
Y
h
v
m
->
-a
Y
W
Y
..
X
@2.
Y
e
v
11
I1
x
W
ay
<
3
a
Y
Y
XI
+
-+a
W
Y
>
v
m
-
I1
0
Y
+
i-
- 3
-a +
..
Y
Y
a
Y
-+
pz X
11
-+
w
h
a
Y
W
Y
..
pz
x
X
Y
i-
v
w
v
Q
t
v
..
Q
Y
W
x .,
x
--Y
W
x Y x
D
..
w v
X
h
h
d
+
v
w
ay
-
-
Y
a W
; <
y
+
+
x
-
-
h
Y
v
W
X
XY
v
11
h
w
-a
W
..
Y
Y
+
3
X
v
a
2:
"
1
+
1
5
-
+
-1 <
I1
a a
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x
a w
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3
x
-1 <
a
< ul
3 -
a ' a
w
a +
a *G w+ w a
v
AXIOMATIC S E T THEORY
"';'
+K,k
233
V 3 ~ 3 t ( ~ E uA E' E u " A Y ; + ~ , ~ -- Y1, + ~ ~ +
:KEP'")) x p / l +' K
). We s h a l l now prove t h e lemma. Suppose t h e h y p o t h e s i s o f t h e lemma i s we g e t X I , x" E PV , and b , b ' E uV such s a t i s f i e d . Then by 3.4.4.4, that,
(1) xK = x k
Z ( x K x a K : ~ E p + u +) X"K
and
+XI;,
K
x w
+K
2
E
and
bK + b k ,
t ~v
),
for all
c ( xK
x aK
K E ~ + v ) ,f
C ( x K x aK :
K E P ;
:K € p
or all
+Y) t
nK'
2
c ( xK " a K
:
K E U ;
c ( b u l+K : K E U ) .
x"
to
get,
A p p l y i n g now 3.4.4.5 cc t Y , so t h a t ,
( Z ) , we o b t a i n , v , L , L'
to
D e f i n e u E " V by uK =LK+,$
Hence, by 3.4.4.9,
( 6 ) x,:
( 7 ) vK
2
and extend
'V
x',
f o r a l l K E U . Then from ( 3 ) and (4) we
we have,
C(y;\x
e K X : h e @ ) and
2( y i x
$K
x;xw
h e @) and uK x K E U
(8)
E
elx=eKh+ 0 L h x
w
for
= c ( qhl
w =C ( y i x
i
K E P + U ,
A € @ , and
X E ~ ) for a l l
xO;h:
$ix:A € @ )
$Lh=
$ihx
w
for a l l
,
for
K E Y ,
( 9 ) For each h G 4 one o f t h e f o l l o w i n g c o n d i t i o n s i s s a t i s f i e d :
(9.1)
z ( e K A x P K : K E p l ) KEP") K E U
;
+ z ( $ K h :K
EU')
+ c ( $ u ~ + K: ,KhE V " ) , and
8'
=. t: (epl +K
+K,X
0 = $ ;
XP,,
+K:
f o r a1 1
ROLAND0 C H U A Q U I
234
There i s a
(9.2)
= 2 : ( e K h x pK
(9.3) There i s a
'
'
W
: K E ~ ' ) ;
such t h a t O ~ + K , h + $ ~ X - w =
KEV'
+K
+K,A x'pl
(9.4) There a r e
t
:KEPI)
+K,A+$iI +K,h
Oj+vl
such t h a t
KEV"
Z($il
:KE/L")
q ~ v such " that
K E Y ' ,
+K,X
:KEV")
;
Oi+K,h+$ih== O ; t v l + 1 7 , x
+
(J
tq,h'
$:I
For n o t a t i o n a l convenience we assume t h a t
(10) 6 = p '
+
p " w i t h $ K h = 0 f o r a l l X E p ' and K E V , and f o r a l l
A E p " t h e r e i s a K E V such t h a t
(11) 2
( XK
X
ctK
:K
E/J
+v) +
hEp".
(12) and
:t:
+ A = Z(J+
$1
F
Ep
and (13) x;1 p + v .
-
E p i u )
and x.$
Z ( t o h t : C ; E p + v > and x
Let p = p '
+
(p" + p +
v)
(
xK x aK:
K
Ep +v)
, for
all
xct
+~ t = ~ . $ ~ c t t y f o r a l l
x a
+ %.$= x t x a F y f o r a l l
K,
t
E
Define y E p Y by y A = y;\ , f o r ( p + v). + . $ + q ( p + v ) w i t h q E p l ' and t E p + u ; and
f o r h = p ' tl: + ( p "
="OK
LJi1+,,= 2:
+v;
A ~ p l q ; x = X q F y for A = p'
qA
0.
t o (11) and (l), we have:
Hence, a p p l y i n g 3.4.4.3
AEp"
+#,
and ( l o ) ,
(41, ( 7 ) ,
Then, by (21,
$K,pl
+
K )
(p + v ) w i t h
K,
t
Ep
+v.
Define, also, y, y ' E p + v ( p ~ + as l ) f,o l l o w s : h. or h = p' ' K
+[+
+a)
(p"
(pII + q ) - ( p + v ) w i t h q ,
(p"
+
q)
+t+q
= 0, f o r : A = p '
(p + v ) w i t h q ,
( p + v ) w i t h q E p " , g E p + v , and at = 0 ;
( p + v ) w i t h .$, q E p + v and ct = 0; o r A = p ' + . $ + .$
E E ~ + v , ~ # K , F+K, andaK#a; o r X = p ' + E + F E p + v , q + ~ ,and aK = w
.
f o r : h = p ' + t + ( p " + ~ ) ( p + v ) w i t h . $ E p + v , ~ Z ,Ka K # w 0; o r X = p ' + E + ( p " + K ) ( p t v ) w i t h F E p + v , at f 0, and
y K X = 1,
and
#
aK = w
.
y K X =w
, for:
A = p' + q + (p"
+
q)
= p'
+ K +
8
( p + v ) w i t h q E p " and 0 E a K E w ; o r
( p + v ) w i t h q E p t v and 0 E a K E u .
r K A€=I K h ,for A E p ' and
K € ~ + u .
,
A X I O M A T I C S E T THEORY
+ t +q
for: X = p '
'i,p
+'I?'
*
235
E
( p + v ) with
E p + v , qEp",
t
#K,
+ t + q . ( p + v ) withEEp+v,qEp',aE+
at+O,andaK+w;orX=p'
0,
and a = a . K
aK
7: A = 0, f o r : X = p ' + t + 77 ( p + v ) w i t h q E p " + p + v , [ E p + v , and = O ; o r A = p ' + t + ( p " + q ) * ( p + v )w i t h q , E E p + v , q f ~ , a n d t f ~ . y i X = w yf o r : h = p ' + ~ + q * ( p + v ) w i t h q E p " + p + v a n d a K f O ; o r
+
X = p' + E y:
(p"
+
, for
= 0;
K
and X E P ' .
Ep
f
t
0.
$:; - p r A ' f o r p C K E ~ + vand A € / ) ' .
8;X+
Y:A"
( p + v ) w i t h E E p + v and a
K )
Y L A " 8K,p' ' qEp", tEp+v,
+q
- p , p ' +q'
"K
E + K , and a
f o r p ~ K E ~ + h v= p, ' + [ + q
9
(p+v),
0.
tf
I t i s easy t o check t h a t
(14) x = ( x ( y X x y K X : X E p ) : ~ E p + v and ) ( x K x w : ~ E p + v )= (z(yxXyiA: and
;
hEp):KEp+V)
(15) y i X = y K h + Y $ x~ w , f o r a l l K E P + V and h ~ p .
I n o r d e r t o complete t h e p r o o f o f t h e lemma we have t o prove t h a t f o r each X E p one o f t h e f o l l o w i n g c o n d i t i o n s i s s a t i s f i e d : (a) for all
~ K E V
:
Y
~
~
x
: K E P I )
-
P
Z~( y p I
+K,X
+K
:
KEPI')
, and
yt
+K,X=
;
(b) there i s a
K
E
such ~ t h a t aK = y i X =w - L: ( y K hX aK : K E p + v ) ;
( c ) t h e r e i s a K E V " such t h a t 7; or
'p'
(d) there a r e
K
and
t
such t h a t
K
fVl
+K,h=~= X(yKhx
E v ' , ~ Eu"
and y;
+K,X-
w .
-
P,:
K E P I ) ;
yp 1 +v
I
+E,X=
We have t o c o n s i d e r two cases:
CASE I l Z ( y K X X a K : K E ~ + v )f a , Z ( Y , ~ X P ~ : aK = w i m p l i e s y L X = 0 f o r a l l K E ~ and , Y ; + ~ , ~ = 0 for a l l We s h a l l show t h a t i n t h i s case subcases:
,
( a ) holds.
K E I I ' ?'a, ) K E v ' .
We have t o c o n s i d e r f i v e
236
ROLAND0 CHUAQUI
(1.1) h = p '
+ K +
[
(ptu)
where
EE~",
K E p t u ,
.
Then from t h e assumption o f Case I , y K h + w t i o n of y K h But, hence,
, aK = a . Thus, rAx = a ,by t h e
(1.2)
X
+
= p'
K'
t
E
definition o f y'.
( p t u ) , where
E Ep",
Then by t h e d e f i n i t i o n s o f y and y ' , the desired r e s u l t . (1.3)
h = p' +
K'
Hence, by t h e d e f i n i -
by t h e h y p o t h e s i s on t h e sequence a ,
I.
assumptions o f Case
and aK+O.
t (p" t
[)
.
T h i s c o n t r a d i c t s one o f t h e
K ' E p t u ,
and aK = 0.
y K h = y i h = 0 and we a r r i v e
( p t v ) , where
Then, y K h f a and, thus, aK = w
K E ~ .
5
K ' ,
Therefore
K
~
at
E p t u and uKl= 0.
and P
= w
, again
c o n t r a d i c t i n g Case I. (1.4)
X
Then y,
= p1 + y;
K'
+
(p" t
= 0 for all
5) K
( p t v ) , where
K ' ,
[ E p + u and a K , = 0.
E p + u and we a r r i v e a t t h e d e s i r e d r e s u l t .
(1.5) X E P ' . Then by ( l o ) ,
IC;tx=
0 f o r a l l K E V . Hence by t h e d e f i n i t i o n o f y and y ' , Z ( e K h X p K : K E p ' ) + Z : ( ~ K hK: E u ' ) = z ( y K h x p K : K E p ' ) Z O . Thus, z ( y K h x p , : K E P I ) = z ( e K hX pK : K E p ' ) 2 '('p'
tK,hX
t K :KEPI')
, and
-
1
yp' t K , h - B p t K , X t
$Lh=
0 for a l l
K E V .
I t can be shown i n t h e same way as f o r Case I t h a t under these assumpt i o n s i t f o l l o w s a l s o t h a t ( a ) holds.
Thus ( a ) , (b), theorem i s proved.
( c ) , o r ( d ) a r e s a t i s f i e d and w i t h (14) and ( 1 5 ) t h e
AXIOMATIC SET T H E O R Y
237
PROOF, I f p = 0 t h e theorem c l e a r l y h o l d s ; so we assume t h a t t h e theorems holds f o r P and show t h a t i t then a l s o holds f o r P + 1. Suppose then t h a t ,
(1) I : ( x x x
E V ) = C ( x X x P K h : h ~ v f o) r a l l ~ ~ p + 1 .
a K X :X
Then, by our i n d u c t i o n hypothesis we have,
( 2 ) x X = 2 ( z X x0 all h g v ; and
Oirl
(3)
"Ohq
(4) f o r a l l
to'
A77
KEP
and
and x A x w = C ( z
7)Ep')
*
X7)'
, for
xw
vEp'
all X E V ,
xx
-
0' q E p ' ) , for Xq'
q ~ p ';
one of the f o l l o w i n g c o n d i t i o n s holds:
(4.1) 2 ( B h r l x a K X : X E V ) = 2 : ( O x Vx f l K X : = 0 for all h c v ;
XEv),
and a K X tO K , =
w
implies 0 '
X7)
(4.2) there i s a X E V such t h a t p K , =
= w = Z ( 6 ' X 7 7 xa K X : X E V ) ; A77 (4.3) t h e r e i s a X E V such t h a t a K h = 0 ' = w = Z ( 0 X 7 7 xP K X : h E v ) ; A77
or
(4.4) there a r e A, Define 6 ,
E Ep'w
= Z ( B X q x a P X :X 6 = a ,o t h e r w i s e . 77
E
E
77
77
=
z
( e X 7 ) xp
v such t h a t a K h = 0 x7) ' = w = p XI -- 0;'f
X'E
+ 1 by:
6
7)
8'
Ev),
if
c1
P X : X E V ) , if p
= w, otherwise.
P A
P A
= w
implies 0 '
= 0, f o r a l l L E V ;
= w
implies 0 '
= 0, f o r a l l X E V ;
A77
Xo
From ( 4 ) , we g e t , (5) 2(z7)x6q:
1)Ep')-
Using, now, 3.4.4.10
Z(z
X €
7 ) 7 7
:7)Ep'),
with ( 5 ) as h y p o t h e s i s we o b t a i n :
( 6 ) ~ ~ = L : ( y ~ ~ I # J ~ ~ x: wt =~ Z ~( y) xI#J' a n d - tz E QEP' ;
and
(7)
r;
7)
$71
r; = I#JT r; + I#J;
(8) f o r a l l t
(8.1) Z( #,(x
Ep
xw
f o r a l l rt
and t
Ep'
77.E-
p ) ,
forall
Ep;
one o f the f o l l o w i n g c o n d i t i o n s holds:
S7):qEp) = Z($
X E
7)t
-
7).
v € p ' ) and 6
+ e = w implies
7 7 7 )
238
$At
ROLAND0 C H U A Q U I
for a l l q E p ' ; (8.2)
(8.3) or
(8.4)
t h e r e i s an q E p ' such t h a t
1=$;5= w = Z ( $ x 6 : qEp'); 715 71 t h e r e i s an q E p ' such t h a t 6 = $ ' = w - " Z ( $ x E : q E p ' ) ; v5 715 71 E7
v
there are
v,
such t h a t 6 = $ '
v'Ep'
D e f i n e now y and y ' E "(' yXE = C ( $ g 5 x
q E p ' ; yx5 =w
e
X71
v5
71
w
W = E ~ ~ =
+ 1) by,
, if
* *
=
OXq= w
implies 4 '
715
otherwise.
T i E = c($;~x
= 0, f o r a l l
eiV:q ~ p l ) .
I t i s easy t o check ( b y ( Z ) , s a t i s f i e d f o r y, 7, and y ' :
( 3 ) , ( 6 ) and ( 7 ) ) t h a t t h e f o l l o w i n g a r e
:5
xxx w - Z ( y x y'
E P )
for a l l
1 X E V and [ E p . We o n l y need t o prove t h a t f o r . f o r a l l f o l l o w i n g conditions holds: ( a ) C ( y X 5 x a K X :X E V ) 1 w implies that y'
aK + O K
-
x5
Z ( yX 5 x
PKX:
+ 1 and 5
E P one o f t h e
X E v ) , and f o r a l l L E V ,
= 0;
( b ) t h e r e i s a X E v such t h a t O K , = or
KEP
-yi5 = w -
( c ) t h e r e i s a X E V such t h a t a, A = 7; (d) t h e r e a r e X , A ' Ev such t h a t p ,
= w = = y i
C ( yX 5 x a K X : X
Z ( yh
E X P K h :
Ev);
X E v ) ;
'l;y
= w = c1,pi'=
We have t o c o n s i d e r f o u r cases.
CASE 1 , a l l X E v ; and
Z ( y X E x a K X :h ~ v # ) w ; a K X =w K
E
and ~ t ~ p .
i m p l i e s y ' = 0, f o r A5
We s h a l l prove t h a t ( a ) h o l d s f o r t h i s K and 5 . From t h e h y p o t h e s i s o f t h i s case and t h e d e f i n i t i o n o f 7 we e a s i l y o b t a i n , P K X + 0 i m p l i e s t h a t
.
.
Suppose, now, t h a t q E p ' and @; Y X E = C ( @ 7 7 E XXe q : v E p ' ) =w Then we have t h a t : aK = w i m p l i e s 0 I = 0, f o r a l l h E v , and C ( BX7) x a, : hq XEV ) # w Hence, by ( 4 ) we g e t under t h e same h y p o t h e s i s (i.e. @; = w)
.
(11) C ( e h q x a K X :h ~ v = ) C ( O X D x O K X :X E V ) f w , a n d P K h = w implies
0, f o r X E V ,
A X I O M A T I C S E T THEORY
@At
i.e., we o b t a i n ,
(12) O K , and (13)
= w i m p l i e s (9).
implies y'
+OJ
h2
0 i m p l i e s yX
8,
x @
- 77Ep') 77t-
CASE 1 1 1 a l l X E V ; and this
from t h e d e f i n i t i o n o f y and y '
= 0 for a l l A E V ,
= z ( @ T ) t xO h V :
we g e t t h a t
From (11) and (13) h € V )
Therefore,
= C(yx2x
~(yhK~P,,,:
239
~ E p l ) f, o r
C ( y A t x a K X :X
BKA:
.
X E V )
h ~ vf )w ; O K , =
EV)
w
and [ E p .
KEP
I n a s i m i l a r way as f o r Case I , i t can be and 2.
2
all XEv. C(C(OXqxaKX:
i m p l i e s y 1 = 0, f o r A2
shown t h a t ( a ) h o l d s f o r
K
CASE 1 1 1 , a l l X E v ; and
C ( y X t x aPX:h€v) f w ;
aPX=w
i m p l i e s y,'(=
0, f o r
EEp.
We s h a l l show t h a t ( a ) h o l d s f o r p and t h i s 2. By t h e h y p o t h e s i s o f t h i s case and t h e d e f i n i t i o n s we o b t a i n : 0 f 0 implies y x t = PA
C ( @ q 2 x O A 7 7 v: E p ' ) j C w ,
f o r a l l h 6 v ; a n d @ 7 ; [ = u implies
z ( eA q X " P A :
for a l l
h ~ v f )w
0 f o r q ~ p and , Z(@ 2
C(yXtx x
= Wqt
w
772
x
6
-
77'
6
77
=
77Ep'. Therefore, 6 = w i m p l i e s @ I = 77 772 q€p')- C ( Z ( @ 7 7 t ~ e h T a P) X ~ :qEp'):XEv)
a P X : X E u ) # w . Thus, by ( 8 ) , we have t h a t Z ( @ x 6 : q E p ' ) 77t q e : q E p l ) , and e = w i m p l i e s 4 ' = 0, f o r q E p ' . Hence, b P X = 71
i m p l i e s y i 2 = 0, f o r a l l X E V ;
@T)t'
VE = w implies e
=C(BXqx
PPX:h~v)
77 C . ' @ q t x e h q : q ~ p ' ) ,f o r
= 0 implies # w , f o r a l l q E p ' ; and 0 PA a l l X E V . Therefore, Z ( y X t x a P X : h E v ) = C ( C ( @ x 0 xPPX:hEv): 772 X77 q E p ' ) = ccg x 6 : qEp')= Z ( @ x E :q€p')= w y X E X P P h : XEV).
77E
CASE I V ,
772
q
77
Z ( Y ~ ~ X P , X, E~V :) # a ;
PP
a l l X E v ; and 2 E p .
= w implies
-yit =0,
for
I n a s i m i l a r way as f o r Case 111, i t can be shown t h a t ( a ) h o l d s f o r
P and t h i s t .
-
Using now Cases I I V Y by elementary l o g i c a l t r a n s f o r m a t i o n s we can o b t a i n (a), (b), ( c ) o r (d), f o r a l l K E P t 1 and 2 E p . Thus, w i t h ( 9 ) and (10) we have proved t h e theorem f o r P + 1 and t h e i n d u c t i o n i s completed.
240
ROLAND0 C H U A Q U I
3.4.4.13 REMARKS, An a n a l y s i s o f t h e p r o o f s o f t h e p r e v i o u s theorems should convince t h e reader t h a t t h e sequences and f u n c t i o n s whose exi s t e n c e i s a s s e r t e d i n t h e i r c o n c l u s i o n s can be e f f e c t i v e l y computed from t h e sequences and f u n c t i o n s i n t h e i r hypothesis. Thus, a l l theorems proved up t o now, w i t h t h e e x c e p t i o n o f 3.4.4.3, can be p u t i n t h e form o f formulas o f t y p e ($) and, hence can be d e r i v e d from B r a d f o r d ' s Theorem, which w i l l be t h e main r e s u l t o f t h i s s e c t i o n . Most o f t h e r e s t o f t h e theorems 3.4.4.18, 3.4.4.21, and needed t o prove B r a d f o r d ' s (3.4.4.14, 3.4.4.17, 3.4.4.22) a r e n o t o f t y p e ($); hence t h e y cannot be d e r i v e d i n t h i s way.
PROOF I
Suppose t h a t ,
(1) xK + y K = xo t yo
, for
all
K E W .
We s h a l l d e f i n e by r e c u r s i o n t h e i n f i n i t e sequences u, w , w, z, s a t i s f y i n g t h e c o n c l u s i o n o f t h e theorem. Hence, we s h a l l have,
( 2 ) xo + yo = u K + w K + w K + z K , f o r a l l
K E W .
1, we g e t x1 = a + b , q1 = c + d , xo = u + c , and yo = b + d . Hence, x + yo = a + b + c + d . D e f i n e uo = u , uo = c , 0 w = b y and zo = d . Thus, x = u + uo, yo = w + zO,and x1 = uo + w o . 0 0 0 0 A p p l y i n g 3.4.3.4
t o (1) w i t h
K
=
Suppose t h a t uK, w K , wK, and zK have been defined.
Then, by ( l ) , and
( Z ) , X ~ t+ ~y K + z = xo + yo = uK + w K + w K + z K . By 3.4.3.4, we g e t Coy C,, C2 , C3 such t h a t , x K + 2 = Lo + C*, qK + 2 = C, + C3, UK + U K +WK = Lo + Cl, and zK = L 2 + 13. D e f i n e uK + 1 = Lo, wK + = Ll, wK + = LZy and zK + =Iy We have, xK = u + 'K+1 ' K + 1 , u K + ' K + w K = u ~ + l K + l + L v K + l * qK+1 wK 1, and zK = wK + l + Z K + l * +
'
+
3.4.4.15 z ( Z K + l X ( K + l )
PROOF, such t h a t ,
THEOREM,
:
+
w ~ ~ u x ~ + ~ c ~ Y zA a( = c z + EL : ~ A b = c x w +
K E W ) ) .
Suppose b
( 1 ) b = C x, and xK
5
a x w
+ yK = a,
.
Then, by 3.4.3.5
f o r every
K EW
.
t h e r e a r e x, y
E
WV
24 1
A X I O M A T I C S E T THEORY
that,
t o (l), t o o b t a i n u ' , u',
We apply, now, 3.4.4.14
xo
(2) and
( 3 ) XK and z,: = w L t l
K E W
u,: +
2
, yK
W,:
.
+
+
v;
+ z' 0'
:h€w)
+h
(5) q0 = u ' , (6)
+ 1
u'tu'
2
0
We s h a l l d e f i n e by r e c u r s i o n 4 , x ,
' K O
l,o,
W,:
u'K
*?
";+1'
+ 1 '
+
";+I
t C ( W ; + ~ + ~ : X E ~ ) ,
Therefore, by i n d u c t i o n we prove, K
+
+
t v:,
K E W .
C ( W ; + ~ : ~ E W ) = u'K
( 4 ) u;+w; t C ( W '
.tK
, u;
vK' + z,:
2
+ z ; + ~ , for a l l
Hence, u i for all
~b
u 0' + Y 0' and yo =
w ' , z' E w V such
for all
/Lo =
0
+
CW'.
.tE
4,
w
v ' , d o = w ' , and t o= z '
and
YKt1,0+4Kt1,0 K Ew ;
(
w
V ) such t h a t ,
;
+ C ( b K h : h E w ) = b K +1,0
hKo
'
and
( 7 ) "i(,q+l 'K+1,77
+ ~ ( b K Y q t l t ~ h:
+ AK+1,77
w
+h K q
+
C(bK
:hEU)
,77 + A
" K t 1
'
Y77
f4K,qt1y
'K,Ilt1
+'Kqt4K7)
" K ,
.tK,77+1y and q K1) for all K , V E W .
E w ) = R.K + l , T j
1 +1
'K,Q
' K V
2,
'K,?
+
q K 0 + a K o+ Z ( b K A : h E w ) ,
Y
Define 4 0 , h o , and tob y ( 5 ) . Suppose t h a t q,, V a r e defined s a t i s f y i n g (5)-(7). We d e f i n e now qK + l y
and .tK E 1y nK 1 y and
hK, b,, hK
.tK + 1 E w V .
A p p l y i n g 3.4.4.14 t o t h e l a s t c l a u s e o f ( 7 ) , we o b t a i n sequences u", w " , w", z l ) E w V such t h a t , 2
' K O
and z'l77 K + 1
that
0
2
+
t y"
0"Ko
'C ( b K , q + l
'K,q+l
h
u"
+ h
:h E w )
w"1 7 + 1t z"7 7 t 1 y
= y"
, bK
= w " , and RK
+
= d
+
L =
+
X E W )
y"
77
z" 77'
W0 "+Z" U''
rl
+
~ E w .Hence,
= z",
=4Kt1+77, 0
qh + K , 0
+
2
0
Y
y"
r)
1 = u"
q,,,
77
+ d' u" 77
2
q+1
i f we t a k e q
K f l
(5)-(7) are satisfied f o r
+
+
w"
77'
y"
q+1'
,
=
u"
K
t 1.
t o t h e f i r s t c l a u s e o f ( 6 ) we o b t a i n a d such
We prove by i n d u c t i o n on % K
2)
for a l l
Applying, now, 3.4.3.7
( 8 ) 4,o
C(AX
+
K ,
'
('K
Therefore, b y ( 9 ) and ( 8 ) ,
*
q g w ) , f o r every K E W .
using (7) t h a t f o r a l l X E w +h -q,V
: q E K ) .
,
242
ROLAND0 C H U A Q U I
Hence,
(11) b = c ( q K O+
A K O
+ C ( d
K
. q e W ) :
77’
K E W ) .
A p p l y i n g 3.4.3.7 t o t h e f o u r t h formula i n ( 7 ) we g e t a z E w V such t h a t , .tK 2 ZK + c ( n K g + l : V E W ) , f o r a l l K E W Thus, u s i n g ( 7 ) w i t h t h i s formula, we o b t a i n , q, + Z(n - 1 ) E w ) - qK+l,o + + aK K 77’
.
+ 1,o * K + 1,o zK +1, f o r a l l K E
‘K
+
W
.
‘K
1,o ^. ‘ K +1,0 “K + 1,o Therefore, a p p l y i n g 3.4.3.7
+
+
’
(bK
+ 1,q
: q E W )
+
once more we o b t a i n a c
such t h a t , (12) 4, 0
ery
+
kK 0
+
Z ( A K q .S q E w ) = c + 2
(
z
~
+ :~ h +g w~)
, for
ev-
K E W .
Then, by ( l l ) , ( 1 3 ) b = E ( ~ + Z ( z ~ +: A~ E+ w~)
2 ( Z K + 1 X ( K + 1) : By ( I ) , + Zo
- C X W
+
(Z), ( 5 ) and ( l l ) , we a l s o get,
(14) a = x 0 + yo ” 4 0 0 ‘ “ 0 0 K E W )
: K E w )
K E W ) .
“00
“00
= 400 ‘“00
= c + c z .
(13) and (14) g i v e t h e c o n c l u s i o n o f t h e theorem.
+‘(‘OK
:
AXIOMATIC S E T T H E O R Y
243
PROOF, When v = 0, t h e theorem c l e a r l y holds, so, we assume t h a t t h e theorem holds f o r v and show t h a t i t them holds f o r v + 1. Suppose, then t h a t
( 1 ) qK
5
a x w , for all
K E V
+I.
By our induction hypothesis, we have a z' E w V such t h a t ( 2 ) u = C z t , and qK =
5
Also, since qv such t h a t
(3) a = Z u
w
E
a
x w
q,,
and
= C ( u
9
Xq'
-
x
E
K E U .
w V and a E " ( w + 1 )
aq : q E w ) , Hence, by 3.4.3.5
, and
q E w ) f o r XEw
w
there i s a
u = C ( w X q : XEw)for q E w .
q
Define z E V by, z A + ( q + l ) .rl= w X q and z A + (71+1) f o r q € w and A C -q
q+l,X
Define, a l s o
.
r
E
-rK,X+(q+l) ,q
=
rih -
.rl
Y,,~+
v(ww
= a,,
-
(17 t 1 ) -
+ 1 ) by,
, for
K E V ,
1
( q + 1 ) -'K,7)+1
'K,Xt(q+1).
and
, by 3.4.4.15 we have a u
From ( 2 ) and (3) we g e t C z' = C u. " ( " V ) such t h a t (4) z i - Z ( w
W
r K h : h € w ) , f o r a17
( z i x
, for
Yv,Xt(q+l).(q+l) ='A
.
r l ~ w , and
for
KEvy
v ~ w , h5
forqEw,
5'7 ; 5' '
s ;
'Eq.
I t i s easy t o show t h a t z and 7 s a t i s f y t h e conclus on of the theorem f o r v + 1. 3.4.4.17
THEOREM,
3 b3c(a=b+c A c 5 b
x w
v E W
A q E
A WK(K€V
vv -+
4
A WK(K€V
5
+
5 q K x U )
+
qK)).
PROOF, I f v = 0, t h e theorem c l e a r l y holds. So assume t h a t t h e theorem holds f o r v and show t h a t i t then holds f o r u + 1. Suppose, then, t h a t a 5 qK x w , f o r a l l K E V + l . By t h e induction hypothesis, t h e r e a r e b ' and c ' such t h a t a = b t + c ' , c ' 5 b ' x w , and b' 5 qK f o r a l l K E V . MoreHence, by 3.4.4.15 we g e t d and z € w V such t h a t over, b ' < a 5 gv X U b'=d x w+ C(z x ( X + 1 ) : X E w ) and qv = d + Z z. I f we d e f i n e b = X+1 : X E w ) and c = c ' t d x w + C ( z A + x (A + 1 ) : X E U ) , i t d + C(zx i s easy t o show t h a t t h e s e elements s a t i s f y t h e theorem f o r v + 1.
.
+
.
244
R O L A N D 0 CHUAQUI
PROOF, When v = 0 t h e theorem h o l d s ; so we assume t h a t the theorem holds f o r v and show t h a t i t then holds f o r v + 1. Suppose, t h e n ,
(1) x A
< c ( x77
x a
~ ~ : ~ ) E v + ~ ) + fw o r ~a l ~l ~x Ewp
Kh77
+ 1.
v
By the i n d u c t i o n h y p o t h e s i s , there a r e x',z' ( 2 ) x,,
and
2
and
qlh, t z i
z i
5
( 3 ) qlh, < C ( y ' X x aK X 7 7 : q E v )
E
a n d h E
'Y such t h a t ,
q i x w for all X E v , f
x ,,x a K x v f w K X , f o r a l l
KEV
,
h E V .
From ( 1 ) - ( 3 ) , we o b t a i n , ( 4 ) xv
<,Z(q)hx
a
K
A77
x w : q E v )
+ xv
x aKvVxW+ WKV
Y
for all
K E P .
+ z ' , , , z\
( 5 ) x v =q\
and z :
t o ( 4 ) , we have q\
Applying, now, 3.4.4.17
x u ,
such t h a t
and q ' , , s Z ( q k x a w K v , for all K E
KV77
: q E V t1)
+
~ .
From (3) and ( 5 ) , we o b t a i n ,
(6) that,
ylh,s2
(
x aK h77X
y;
U :
77 E v
+ 1 ) + wK
x x w
, for
all
Applying t h e i n d u c t i o n hypothesis t o ( 6 ) we have q", z" ( 7 ) y i = q ' i + z ' i and
z l i s q'i x w
, for
all
XEV
h ~ v .
KEP,
such
'V
E
;
and XEV,
(8) y ' i
5
(
q'h x a
KX77
T h e r e f o r e , from ( 2 ) , (9) q'i+ z'i
5
:77Ev) + y \ x
z ( (y'h
+Z'h)
Applying 3.4.4.13
+ (y',,. for all K
x aK A q : 7 7 E v )
wK
5 X ( (q'; +
and
KEP
( 3 ) , ( 7 ) , and (8) we g e t ,
and from ( 5 ) and ( 7 )
(10) y',,
for all
a K X V+ W K , ,
z" ) x aK 77 77
+ q',,
,
x aK v +
t o ( 9 ) and ( l o ) , we o b t a i n ,
wK
2 ; )
E
and ~
, for
x
a K X v+
XEV;
a1 1
K
~p
.
24 5
AXIOMATIC SET THEORY
c ( (P, E +
E
5 )
x a K A V: ? E v + 1 ) + 6
K
, for
At
P,
0, o r P, 5 f 0 and P X E f e h t
check t h a t y A E
and
5
5 Z ( yT E X
f
Ep.
:qEV
t1) +
E
x ( P h E f e h E - ~ X 5 ) : E E p ) +z ) ,
- y, 5 ) :E E p )
+ z\
t h e theorem f o r v
and
PROOF,
.
P,
= 0 and
I t i s easy t o
sK X E , f o r a l l K E ~ X, E V + 1 , E E p ) , f o r A E v + 1; and i f h € v , and
zv = C ( u 5
x(Pv,-
; we can v e r i f y t h a t y and z s a t i s f y t h e c o n c l u s i o n o f
+ 1.
xi
Assume t h a t t h e h y p o t h e s i s o f t h e theorem i s s a t i s f i e d . x l i , x'i< x w , for all X E V ,
3.4.4.18 we have, x h =
xi+
and E
1, i f e i t h e r
0; y A E= 0, o t h e r w i s e .
'Kh77 Hence, i f we d e f i n e yA = C ( u x y X E :
z,=C(uE
K
K E ~ A, E v ,
EP.
D e f i n e t h e sequence y E v f l ( p +~ 1) by y, =
all
A p p l y i n g now 3.4.4.12
and ~ XEV.
xi5
C
we have,
(
xkx
x'
77
2
< Z ( S q E x a K X q* T. ? E v ) , f o r a l l
a l l v E v , and
By
: q E v ) for all A77 Z (u x6 E E p ) for c1
K
5
K E P ,
-
775'
XEv,and
EEp.
x l i S 2 ( u x 6 x w : E € p ) , f o r a l l X E V . Hence, by 3.4.3.5, E AE Z ( Z h E : C ; E p ) and z X E 5 uEx S X E x w f o r a l l A E v and E E p . By 3.4.4.16 we get, u5 = Z ( w E : e E o ) f o r a l l E E p , zX I: = C ( wE e x h g e : B E a ) f o r a l l X E V and E E p , and A X E = 0 i m p l i e s P X E e = 0, f o r a l l h E v , E E P and O E w . Therefore, X'L"
.
Define y
for
XEv,
theorem.
5
+ p .O
Ep,
=
E E P and B E w ; and Y , , ~ + p . e = 'XE+ P A E O C l e a r l y y and y s a t i s f y t h e c o n c l u s i o n - o f t h e
toEe for
and O E w
.
246
tain,
ROLAND0 CHUAQUI
Assume t h e hypothesis o f t h e theorem.
PROOF,
(1) x X = z
(
XEV;
and fied:
(2)
Oiq=
eXq
:qEp)
KEV
and
qEp
z ( e X q x a K X :X E V )
Xq
or
x
and % x w
2.
, for all
:qEp)
Z ( z x0'
' x w , f o r a l l X E V and q E p ; 0 X q t 0 A77
( 3 ) For each
(3.1) implies 6'
I)
Applying 3.11 we ob-
one o f t h e following conditions i s s a t i s =
z(e
xPKA: hEv)
, and
aKh
PK,=
= 0, f o r X E V ;
(3.2) t h e r e i s a
XEV
such t h a t p K , =
(3.3) t h e r e i s a
XEV
such t h a t
(3.4) t h e r e a r e X , h'
Ev
OiV=w = X ( O h q x
a K X =0 '
such t h a t
Xv =
aK A = 0 '
w = X ( 0X q x
hrl
= w =
a K X :X E V ) ;
PKh:
XEv)
@, X I = oil,,
Define 6 E V ( p ( p w ) by: Z i X q E = 1 i f e i t h e r 0 ' = w and B h E f 0, or Since Z ( z x O X q x a : q E p ) = = t ; h X q t = 0, otherwise. 17 X ( z q x O ; , : q E p ) f o r a l l X E v , we have z S Z ( z x 6 x w : t ~ p f)o r 17 t A775 a l l X E V and q E p . Applying 3.4.4.19, we obtain:
Oiq=0 and q
( 4 ) zq = 2 ( y @ x e q 4 : 4 E w ) f o r a l l q E p , a n d € g C ( ~ ~h q~E Xx w :6 174 I E p ) f o r a l l X E v , q E p , and @ E w . v w Define y E ( w + 1) by s t i p u l a t i n g t h a t 7X = ;I: ( E,, x Oh : 7) E p ). Clearly from (1) and ( 4 ) , we g e t t h a t (5) x X = C
(
ytx
r X t: E
Ew)
, for
all X E U .
In order t o prove t h e o t h e r conclusion o f t h e theorem we argue by cont r a d i c t i o n . Suppose t h a t f o r c e r t a i n K E P and l E w we have, Z (-yX x a K X : X E U ) f ,'Z:(rXlxP K h : X E U ) . From the d e f i n i t i o n of 7 , we must have t h a t for a certain q E p , ( 6 ) e P t # 0 and X ( O x q x
a K x :X E V ) f C ( e h q x
PKX: X E V )
Therefore, i t follows from (3) t h a t e i t h e r C ( B X q x
a K X :X
.
EV) # w
AXIOMATIC SET THEORY
o r t h e r e i s a h E v such t h a t aK h E v )-w ,
= w and O{ 17 = w
247
.
I f C ( O h q x aK : On t h e o t h e r
thenclearlyby(6),Z(rhc:xaKh:XEv)=w.
hand, i f h E v , a K x = w , and 0 ' Oh4#OandE
A17 + O (since O i t e
@c:
=w
, then
f o r a c e r t a i n @ E p we have
< C ~ E @ ~ X ~ ~ ~ ~ X W so : we @ E P ) )
vc:
s t i l l have i n t h i s case, 2 ( . y x c :x a K h : X E U ) = w . I t can be shown i n t h e same way t h a t 2 ( y h t x pK
we have t h a t Z ( y x c : x a K h : h ~ v =) C ( yh
c: x
.
:hEv) = w
pK
: hEV W i t h t h i s and (5) t h e p r o o f o f t h e theorem i s completed.
),
.
Thus,
a contradiction.
We now t u r n t o o u r main r e s u l t .
PROOF,
"('a+ 1).
Assume t h a t v , p , p ,
T
E w
, a, p E p ( u w + l),
The i m p l i c a t i o n from l e f t t o r i g h t i s obvious.
v
C ( 7 h x ~ K h h : ~ v = )C ( y h x
(1) For e v e r y 7 E w + 1,
and 6 ,
E E
So assume t h a t :
P,
: h EV
)
for
i m p l i e s t h a t t h e r e i s a 0 t h a t extends 7 t o p and such t h a t C ( O h x S K h : h ~ p =) C ( e h x E * X E ~ f) o r a l l K E T . K h ' every
KEP
Assume, a l s o t h a t x E 'V
(2)
z ( x p x K h :h E u )
A p p l y i n g 3.4.4.20 (3) x x = (4)
( z
m $ h E X
c: x
and
= C ( xh x
0,
:h
we o b t a i n z E
: t
a K h : hEV)
[ E w .
-
E w )
From ( 4 ) and (1) we g e t $
E
,for
all
.
KEP
and @ Ev ( w w + 1) such t h a t ,
,for
X(@
E v )
e v e r y X E u , and
c: x 0,
p ( w w + 1)
: XEv
),
f o r every
K
c p and
such t h a t
( 5 ) J X t = G h t : f o r e v e r y h E v and E E w and C ( % [ X b K h : X E ~ )= Z ( $ , c : x e K h : h E p ) , f o r e v e r y K E ? T and i E w . Let y E 'Y
be t h e e x t e n s i o n o f x d e f i n e d by s t i p u l a t i n g t h a t yh = xh
f o r A E v , and y h = C ( z c : x l a s t formula, we o b t a i n
tc: : t E w ) ,
for v g A E p .
By (3),
( 5 ) andthe
248
ROLAND0 CHUAQUI
xx=
(6)
Z(z
x
$[:
t E w )
, for
every h E p .
From ( 5 ) and ( 6 ) we g e t in an elementary way, ( 7 ) C ( y h x 6 K x : h ~ p =) Z ( q h xK~h
'
*
h ~ p ) f,o r every
KET.
.
T h u s , assuming (1) we have proved from ( 2 ) t h a t t h e r e i s a q t h a t satHence, the implication from r i g h t t o l e f t i s obtained.
isfies (7).
3.4.4.22 REMARK, We can e a s i l y extend 3.4.4.21 t o apply in t h e e can r e p l a c e ( Z ( x h x a K h : A E v ) : K E ~ =) ( Z ( x A x P K X : following case. W X E U ) : K E ~ and ) ( Z ( y x x S K x : h E p ) : K € Z ) = ( Z ( y x x E K k '* X E V ) : K E ~ ) ( a n d t h e corresponding p a r t s with Y' and 0 ) by a conjunction of equations of t h e form X ( U
A
X ~ ~ : A E V ) = Z ( U ~ X ( I ~ : X E V )
and i n e q u a l i t i e s of t h e form, Z ( u x r x : X E V ) ~ Z ( u X x o hX :E h
V )
Statements of these two forms will be c a l l e d of type ( E ) . A special sentence, then, i s a sentence of the form W u ( u € 'V A q5 3u(u E A u = u I v A $) where 4 and $ a r e conjunctions of e q u a l i t i e s and i n e q u a l i t i e s of type ( E ) ( f o r $ replace u by u and v by p ) . --f
Bradford's Theorem can be expressed ( a s a metatheorem) a s follows Let v , p , r , u be given. Then i f 0 i s a special sentence of t h e above form, t h e n 0 i s derivable i n MKT i f f 0 w i t h Y replaced by w t 1 i s derivable i n M K T . We s h a l l show now, by means of an example,how a special sentence can be brought i n t o a form where 3.4.4.21 i s applicable. Let
PEW
; we s h a l l deal with the statement:
( 0 ) For any a, b, i f a x b = d x p t e and e 2 d x 1.1. Let xo = a ,
P
5b5a
x w
, then
t h e r e a r e d, e such t h a t
x1 = b; we have
0 x p + x 2 = x1 (2) b 5 a x w i f and only i f t h e r e i s an x3 such t h a t x1 + x 3 = xox w . (1) a
x
p
5
b i f a n d only i f there i s a n x2 such t h a t x
Let, now, yo = xo, q1 = xl, y2 = x2, q3 = x3, Y4 = d, and q5 = e , Then (3) b = d x p
and
+ e i f and only i f y1 = q4
x p
+ q5
A X I O M A T I C S E T THEORY
(4)
e5d
x p i f and o n l y i f t h e r e i s a
a E
249
q6 such t h a t q5
t
y6 = q 4 x p
.
1 ) by s t i p u l a t i n g a0 = p , a. =al1 = a13 = 1, 2 4 and u K X = 0, otherwise; P E ( w + l ) by Pol = 1, Plo = w, and P , , = 0 , Define:
2(4cd
t
o t h e r w i s e ; 6 E 2 ( 7 w + l ) by 601 = 615 = 616 = 1, and 6 K h = 0, otherwise; E E
2 ( 7 w + 1 ) by eO4 -- € 1 4 = P , € 0 5 -- 1, and e K X = 0, otherwise. I t i s c l e a r t h a t t h e statement ( 0 ) i n q u e s t i o n i s e q u i v a l e n t t o :
Qx(xE4V A ~ Z ( ~ X ~ ~ , : X E ~ ) : K € ~ ~ ~ ~ C ~ ~ ~ € 2 ) ' 3 y ( q E 7V h y 1 4 = X A ~ C ( ~ ~ X ~ , ~ : X E ~ ) ~ ( Z ( ~ ~ X K c 2 ) .
Thus, we can a p p l y 3.4.4.21 and prove t h a t t h i s sentence i s d e r i v a b l e i n MKT i f and o n l y i f i t i s d e r i v a b l e w i t h V r e p l a c e d by w + 1. We see, then, t h a t 3.4.4.21 p r o v i d e s a powerful i n s t r u m e n t f o r p r o v i n g theorems t h a t have t h e form o f s p e c i a l sentences i n MKT ( w i t h o u t A# C). We s h a l l g i v e s e v e r a l examples below o f a p p l i c a t i o n s o f 3.4.4.21 i n combin a t i o n w i t h some well-known r e s u l t s o f t h e c a r d i n a l a r i t h m e t i c o f w t 1. 3.4.4.23 REMARK, I t i s p o s s i b l e t o extend 3.4.4.21 a more general f o r m t h a n s p e c i a l sentences.
t o statements o f
We say t h a t 0 i s an exLended h p e c X newknce i f 0 i s o f t h e form
...
Wu(u E ' V A 4 3w(w E A ' ? u = w ] v A ($, V V $k)))ywhere 4 and $o,..., Gk a r e c o n j u n c t i o n s o f e q u a l i t i e s and i n e q u a l i t i e s o f t y p e ( E ) . A s p e c i a l 3 w ( w E 'VA u = A p a r t o f 0 i s a sentence o f t h e form V u(u E ' V A @ -+
+
$ . ) f o r a c e r t a i n i, i = O,..., 1
k.
WIV
B r a d f o r d ' s extended metatheorem says.
A extended s p e c i a l sentence i s d e r i v a b l e i n M K T ( w i t h o u t Ax C ) i f and o n l y if one of i t s s p e c i a l p a r t s i s d e r i v a b l e i n M K T w i t h V r e p l a c e d by w + 1. The p r o o f o f t h i s r e s u l t i s e s s e n t i a l l y metamathematical, so i t w i l l be o m i t t e d . We now procede w i t h C o r o l l a r i e s o f 3.4.4.21 w i t h easy number t h e o r e t i c a l r e s u l t s .
o b t a i n e d i n combination
THEOREM ( E U C L I D ' S THEOREM). p , v E A 3.4.4.24 W K W A W ~ ( K , A , K E W A R . K = ~ A ~ * X = V + ~ =p ~= )b Ax U v ~+
lc(u=cx v A b = c
x p).
The second c l a u s e o f t h e h y p o t h e s i s says t h a t p and v a r e r e l a t i v e l y prime.
250
ROLAND0 CHUAQUI
3.4.4.25
2 C -v Ew A
THEOREM,
x ( x K x a K : K E V ) = x ( XKX f lK z(yKXflK:KfU)/\
:Key-+
a, p
'w+
E
1A x E 'V A
ly3z(y, z t V V A ~ ( y K X a K : K E v ) =
x = ( qK + Z K : K E V )
A
zo).
Yo 5
This theorem was proved d i r e c t l y by Tarski (Tarski 1949) with u 3.4.4.26
x v
= b
x u
+ c
+
3 d 3 b' ( c + b' = d
(ii)vEw A a x v = b x v + c A 3.4.4.27 x
THEOREM,
v
EW
THEOREM,
A a x v
5
E
WA hEV
b
5a
W : h E K ) =(ZXX
A b
+ b' = b ) .
3 d 3 e(b = d
x W+
- ... - a
E K
b < Z ( % x (p2+l):
5 :h E K ) A ( % X
x u
~ p A= a 0 A K =nl=
KEWAa E K
n2 Z h X
+ 1.
b S c - + 3 d c = d x v .
v).
3.4.4.28
EW
THEOREM,
(i) v Ew A a
A e< d
z1 2
Y1 A
W : X E K )
~ E KA )(
K - 1
x v
+ e
A
no .n2 = au
b : h € ~ =) ( x h
-+
+
).
This theorem i s obtained from the Chinese Remamder Theorem o f Number Theory
.
We now pass t o some theorems t h a t b esid es using Bradford's Theorem, involve i n f i n i t e c o n s t r u c t i o n s . Their proof i s not a s immediate a s the preceding ones.
3.4.4.30 THEOREM , 0 E p C w A x 3 b 3 c ( b + d = c + d = e A ( b : K ~ p ( () x ( (
E (
'V A c :K
(
xK
+d
:K € p
)
= ( e :K
Ep)
+
E ~ ) ) .
PROOF, I f C L E W , the theorem i s an immediate consequence of 3.4.4.21. Y and xK + d = e f o r a l l K E W . We shall c o n s t r u c t So assume i.~= a, x f W three sequences u, w , y E V such t h a t , f o r a l l u E W ,
and
(1) uo =
xo
( 2 ) uv =
Vu
Y
+uv + I
Y
( 3 ) xv + 1 = Yu +uv + I '
251
AXIOMATIC SET THEORY
(4) d = w v + d = y v + d .
,
L e t uo =
go, and u
xo; we have xo + d = e = x1 + d.
A p p l y i n g 3.4.3.14
u 1 y x1 = go
+ ul,
1 such t h a t uo = uo + w u , and qv have been defined. Suppose, now, t h a t uv
wo
+ d - y +d. 0 0 We have, b y ( 3 ) ,
So we a p p l y 3.4.3.14
and (4), xv + 1 + d = y v + u v + 1 + and o b t a i n uv +2, wv 1, and yv +l.
again
Thus, we complete t h e d e f i n i t i o n by
+
r e c u r s i o n o f u, w , and g. By 3.4.3.13,
we o b t a i n
and d = w
(4) gives,
(5) Z u + d = 2 y + d = d .
A p p l y i n g 3.4.3.7 ( 6 ) uv = b t 2
to (
wv
(Z), we o b t a i n a b such t h a t
+K
:K
Ew)
, for
u E W ; whence by ( l ) ,
(7) x o = b + 2 v . F i n a l l y , we put,
(8)c=x0+Zy, and we show i n an elementary way by means o f ( Z ) , (3), and ( 5 ) - ( 8 ) , t h a t b and c s a t i s f y t h e c o n c l u s i o n o f t h e theorem.
’
PROOF,
by 3.4.4.30.
THEOREM, F E U 3.4.4.32 d : K ~ p ) + 3 b ( b + d e A ( b :K E p PROOF,
+ 1, d
5
+
( e : ~ E p ) ( (( x K +
1 A x E ’ V A d S e A
) ((
x).
The theorem being obvious i n case p = 0, we assume t h a t 0
E
e , and e 5 xK + d f o r a l l K E ~ .We have f o r a c e r t a i n c, c + d = e. Hence, f o r e v e r y K E P , c + d 5 xK+d. By means o f 3.4.3.16 (i), we see t h a t t h e r e i s a n x’ w i t h x ’ + ~ = +c d and x; 5 xK f o r e v e r y K E ~ . Since x k + d K e f o r e v e r y K E P , we a p p l y 3.4.4.30 and o b t a i n a b w i t h t h e r e q u i r e d pro-
pEw
perties.
m
3.4.4.33 x
~
3c(x((
p
THEOREM ( I N T E R P O L A T I O N THEOREM) ~ I ~ ~( ~ Ey ~ ( ’~ V W) VA K
( c : K E ~ A ) ( c : A E v )
((q).
(
K~ E
~
~A
I
A
p, v E w E
~6 + K
+
1A
~~ Y ~A ~) +
PROOF, I n case p , V E W , t h e theorem i s o b t a i n e d by 3.4.4.21 (and obvious p r o p e r t i e s o f n a t u r a l numbers). I n o r d e r t o complete t h e p r o o f ,
252
ROLAND0 C H U A Q U I
we c o n s i d e r f i r s t a p a r t i c u l a r case.
CASE I
5 qX f o r
xK
p = w and
I
all
such t h a t w
Ew
.
5
x ( ( w , and wK
K E w ) ,
"V. Assume t h a t
E
,y
for
wV
ce, by t h e f i n i t e case, t h e r e i s a wK
+
and X E V , L e t y X , f o r X E V . Hen-
KEW
-
<
We have, xK +1, wK
Suppose wK i s defined.
w o = xo.
L e t x E wV and y
We d e f i n e by r e c u r s i o n a sequence w
K E w , X E V .
(wKfl:
((
v
such thatxK+l,wK~wK+lsqX
forkv.
Apply now, 3.4.3.20 t o t h e i n c r e a s i n g sequence w and o b t a i n a " l e a s t upper bound" c o f w which s a t i s f i e s t h e c o n c l u s i o n o f t h e theorem.
CASE I I , x
(( K
Assume x , y E
w
Y and xK
w , and w
((
with,
XK,
5
5
ZX
5
We have, xK + wK
f o r every X E w .
,X
E
L e t w o = xo and suppose t h a t
YX
XK+WK,
xK +zX, wK +zX; hence, by 3.4.4.32
z X = xK+wK
and
u,<
we o b t a i n a uh
x K , w K , f o r each h E w .
By Case I , t h e l a t t e r i n e q u a l i t y i m p l i e s t h e e x i s t e n c e o f an
(3)
K
,
WK
(2) u h + that,
K E W )
for all
By t h e f i n i t e case we o b t a i n a sequence z E wV such t h a t
f o r each h € w (1)
yx:
(
i s defined.
5 y,
such t h a t w ( ( C w K +1: K E W ) ,
We d e f i n e by r e c u r s i o n a sequence w E
w.
w
P = w = v.
(LA<
e
5
XK,
wK
,for
every h E w
e such
.
By (1) and (3), we can put,
( 4 ) x K + % = z X and e t d = w K . We e a s i l y d e r i v e from ( 2 ) - ( 4 ) t h a t xK for XEw
.
(5) b
Hence, by a p p l y i n g 3.4.4.32
+ (xK + e ) = d
+ e 5 d + (xK + e ) 5
h X + (xK + e )
again, we g e t a b such t h a t
+ ( x K + e ) , and b
5 5 for
all XEu.
We p u t ( 6 ) v K + l = xK t b .
We have by (3)-(6), 3.4.3.21
xK
5
wK+l
( i ) wK
5
wK
+ e 5 wK + 1 i. e and e 5 wK
w ~ + ~ From . (1) and ( 4 ) - ( 6 )
and w K + l
5 yx f o r
+
1; whence, b y
i t i s a l s o e a s i l y seen t h a t
hEw.
Thus, we have d e f i n e d t h e i n c r e a s i n g sequence w . Using, now, 3.4.3.20 we o b t a i n a " l e a s t upper bound" c o f w such t h a t wK 5 c 5 yX f o r K , h E w T h i s c s a t i s f i e s t h e c o n c l u s i o n o f t h e theorem.
.
AXIOMATIC SET THEORY
253
PROOF, Assume t h e h y p o t h e s i s o f t h e theorem. By 3.4.4.33 (with P = 2, and v = 1.1) t h e r e i s a c such t h a t e 5 c, and d 5 c xK t d f o r K E P .
<
Hence, i f 1.1 # 0, we o b t a i n t h e c o n c l u s i o n by 3.4.4.32,
i f P = 0, we p u t b =
e.
PROOF, Assume t h e hypothesis o f t h e theorem. a c' such t h a t
(1) c' + d = e , and xK
and
there i s
.
5
c' f o r v ~ 1 . 1
We p u t ( 2 ) yh = c'
By 3.4.4.31
yitl=
y A for h E v
.
Then, by (1) and t h e h y p o t h e s i s of t h e theorem, d h e v + 1 (or X E V , i f v = w ) .
<, e 5
q\t d
for
we g e t a c" w i t h
Hence, by 3.4.4.32
( 3 ) c" + d = e, and c"
2qi
f o r h E v t 1.
By p u t t i n g
( 4 ) x b = c", and x :
= xK f o r
K E ~ ,
we have by ( 3 ) and t h e hypothesis, x; P = w) and h E v +l; hence,
(5) x A < c < , y ' ,
by 3.4.4.33,
5
q i for
K E P
+ 1 (or
K
E
~
i, f
we o b t a i n a c such t h a t
f o r K E p t 1 andhEvt1.
By means of ( 1 ) - ( 5 ) we e a s i l y show t h a t c s a t i s f i e s t h e c o n c l u s i o n o f t h e theorem. m 3.4.4.36 THEOREM, VvVdW e.(OEvEwA ( ~ :
X%(XK+b:
x
E
w
V A ( X , + ~:
K € w ) (( x( K
K E W )
x v t d : ~ E m)
( ( x 3 c(c ( ( x A 3b(e< c x v + b A -+
+.
K E W ) ) ) .
T h i s theorem says t h a t a l t h o u g h t h e r e may n o t be a " g r e a t e s t l o w e r bound" f o r a decreasing sequence o f c a r d i n a l s , t h e r e i s an " a l m o s t - g r e a t e s t l o w e r bound" c, i.e. i f e 5 x f o r e v e r y p E w , then e. 2 c + b where b f
cc
254
P
u
ROLAND0 CHUAQUI
x f o r every
P E W .
P
Assume x E
PROOF I t
y, f o r every
xK c such t h a t
KEW
+
(1) xp = c + Z ( g p t K
W
.
V and xK
,< xK
+
f o r every
K EW
.
By t h e Remainder Theorem 3.4.3.7,
: K E W )
L e t x, = there i s a
f o r every P E W .
It i s clear that
(2) c
C, x,,
f o r every P E W .
L e t now, 0 E V E w (3) e
5 xP
x v
+d
, with f o r every P E W .
From (1) and ( 3 ) we conclude t h a t f o r every P E W .
e2 c
x v
+
Thus,
I: ( q P + , :
K E W ) v~ t
d,
( 4 ) ~ ( y h A + K : ~ E ~ ) ~ ~ + e ~ c x v + I : + (d ,g f ho r +e v Ke r y: ~ ~ ~ XCP,
P E W .
we o b t a i n an a such t h a t
By a p p l y i n g 3.4.4.31
(5)
a
t
e=c x v
t C(gh+K:K€W ) x
v+d
and C ( y h t K
: K E P ) x
v
5
a f o r e v e r y p e w and AEp.
t h e Fundamental Law o f Countable A d d i t i o n ,
Thus, by 3.4.3.19,
(6) C ( q k + K
: K € w ) x
v
5 a,
f o r every X E w .
By ( 4 ) - ( 6 ) we have, I : ( y h + K : ~ E +~ e) ~s c~ x v + Z ( Y , + ~ : K E W )
v +d for a l l h E w . we g e t a z, such t h a t x
(7)
~5 c
Using 3.4.3.16,
x v+d+zh
A p p l y i n g now 3.4.4.34
(8) e 2 c x v + b + d
and Z ( y A + ,
3.4.4.3
and 3.4.3.13,
: K E w ) +
f o r each h E w
zh = C ( g h + K : K € w ) + zh'
we o b t a i n a b such t h a t and b
5 zx,
f o r every h e w .
From (2), (7), (8) and ( l ) , u s i n g 3.4.3.11 have t h e r e q u i r e d p r o p e r t i e s .
we conclude t h a t c and
b
PROOF, The case Y E W can be o b t a i n e d u s i n g 3.4.4.21 from an o b v i ous p r o p e r t y of a + 1. So suppose v = a; and assume t h e h y p o t h e s i s o f t h e
255
AXIOMATIC SET T H E O R Y
theorem. Then p e w . The case p = 0 being obvious, assume, a l s o t h a t A p p l y i n g t h e f i n i t e case of t h e theorem, we can d e f i n e a sequence 0 E p 0
q E
.
s qT,
and q T + l
V such t h a t e < q T x p + d
xK f o r a l l
Then,
KCTEO.
t h e r e i s a c such t h a t c 5 qn and a c' such t h a t e 5 c x 1.1 + L e t b = c + c' x w Then e 5 c' + d and c' + yn = yT, f o r a l l T E w Hence b 5 b x p + d and, b y 3.4.3.13 b S yn + c ' x w = y71 f o r e v e r y T E W S xn , f o r every R ~w yT by 3.4.4.36,
.
+
3.4.4.38 ((
THEOREM,
( ~ : K E v ) - +3
.
.
.
.
p , ~ E w +~ A O E V ~ X € ~ V A ( X ~ X ~ + ~
c ( c x p + d S c A x ( (
( ~ : K E v ) ) .
PROOF, Again, t h e case f o r v € u can be o b t a i n e d b y 3.4.4.21. In o r d e r t o prove t h e theorem f o r v = w we reason as i n 3.4.4.37, b u t u s i n g 3.4.3.20 i n s t e a d o f 3.4.4.36.
3.4.4.39 THEOREM, ( xK x p + d : ~ € v =) ( ~ : K (( X ( ( ( C : K E V ) .
p,
v
E v ) -]
E
w A 0 E v A (p+wV v f w ) A x E
'v
A
bgc(b x p + d - - c x p + d = e A ( b : ~ E v )
N o t i c e t h a t c ( b u t n o t n e c e s s a r i l y b ) a l s o e x i s t s i n case p = v = w PROOF,
By 3.4.4.37
and 3.4.4.38.
.
=
F i n a l l y , we prove a r e s u l t due t o F i l l m o r e .
3.4.4.40 THEOREM, ( a x v : v € w ) ( ( ( b x (v T h i s i s a s o r t of Archimedean p r o p e r t y . b x 2.
f
1) : v E w )
.
+
agb.
We have, a 5 a x 2, PROOF , L e t a x v 5 b x (v .t 1) f o r e v e r y v E W Hence, b y 3.4.4.37 t h e r e an e w i t h a S e x 2 and e 5 a,b We now
d e f i n e a sequence q
(1) Yo = e
.
by r e c u r s i o n such t h a t ,
E
9
(2) y v < y v + l
, for
and
( 3 ) cc x (ZV +%)
everyvew,
5 qv
and
x 2'
qv
5 a,b
f o r every V E W
.
We d e f i n e yo by (1). Suppose yv i s defined. By t h e hypothesis, v + 2, b x 2' a x (2' 1) 5 a x 2 A p p l y i n g 3.4.4.37 we g e t a d6a,b w i t h a x ( 2v + 2 1 ) S d x 2 V + 2 By ( 3 ) , q v , d 5 a, b; hence, u s i n g
+'- -
3.4.4.33
.
+' .
( I n t e r p o l a t i o n Theorem) we g e t yv + 1 w i t h qv, d
i s easy t o check t h a t qv + 1 s a t i s f i e s ( 2 ) and ( 3 ) .
5 qv + 1
>
a,b. I t
256
ROLAND0 C H U A Q U I
( 2 ) and (3), t h e r e i s an
By 3.4.3.20,
and Let
( 4 ) gv (5)
56
d5
f o r every v
6
such t h a t ,
EW
a,b.
(6) a = g + d .
d x ( 2 ’ t 1 - l ) = a x
-
3.4.3.17, g x (2’ 1) 5 6 f o r every v E W mental Law o f Countable A d d i t i o n ) g x W Z 4. and ( 6 ) , a = g + d = d 5 b . m 3.4.4.41
b + c.
PROOF,
COROLLARY,
( a x
Assume t h a t a
x
v
v
5
We a p p l y 3.4.4.40
by 3.4.3.19 . FHence, i n a l l y , by 3.4.3.11,
: V E W )( ( ( b x V + C : V € W )-+
b
x
(a+c)xva a x v + c x v < b x v + c x
VEO.
v+1
-1) + ( 2 v + 1 - 1 ) ~ y v x 2 v + 1 ~ 6 x 2 v ’ 1 . Therefore, by
We have, by ( 3 ) , ( 4 ) and ( 6 ) , f o r a l l v E u , g x ( 2
v + c f o r every v (V+l)S(b+C)
EW x (v
and o b t a i n t h e d e s i r e d c o n c l u s i o n .
.
(Funda(5)
a+c<
We have
+ I), for a l l
CHAPTER 3.5
Simple O r d e r i n g s
3.5.1 ings. that
T Y P E S OF S I M P L E ORDERINGS,
T h i s s e c t i o n i s devoted t o t h e i n t r o d u c t i o n o f t y p e s o f s i m p l e o r d e r That i s f o r a s i m p l e o r d e r i n g h , i t s t y p e 7L w i l l be an o b j e c t such
h =
h H h g h .
F i r s t , we p r o v e t h a t o r d i n a l s can be t a k e n as t y p e s o f w e l l - o r d e r i n g s . We need t h e f o l l o w i n g lemma t h a t w i l l a l s o be u s e f u l l a t e r . 3.5.1.1 LEMMA, g E ,TI(x : x c -a A x 'D6 A D 6 6 O n A vt(C; E 06 --t 6't = g ' ( a PROOF, L e t g E n ( x : x c a A x =k o f 6. D e f i n e t h e f u n c t i o n G w i f i D G = and G'x = a, otherwise. We have,
(1) x A =
E
# 0)
+
3
!6(6
E
'6,
A 6-l
E
- d* E ) ) . 0 ) . F i r s t we prove t h e e x i s t e n c e V , b y G'x g ' ( a - x ) i f x 3 -a , =
W+ (G'xEa *x 7 - a).
D e f i n e by r e c u r s i o n t h e f u n c t i o n H, w i t h V ( H ) and H = G OH*. E On A H* t p a l . We have, t h e f o l l o w i n g p r o p e r t i e s of A :
{t : t
(2) A C - On and
Let
H* A C- a .
T h i s i s deduced f r o m (1) and t h e d e f i n i t i o n o f A . (3) u A C - A , i.e.,
EEq E A
+
t
E
A.
( 3 ) i s o b t a i n e d f r o m t h e d e f i n i t i o n o f A.
( 4 ) !i,q
E
A and t stq i m p l y H'E # H'q.
PROOF OF (4). L e t t E q E A. We have, H ' t = G'HY and H'q = G'H*q. A l s o , H'C; E H*q, s i n c e E €77. But, G'H*v E a, s i n c e q E A , and, by t h e d e f i n i t i o n o f 6 , H'v E a-H*q. Hence H't # H'q. (5) A E V . 257
258
ROLAND0 C H U A Q U I
i s a f u n c t i o n and H"*b
T h i s i s c l e a r s i n c e A/H" bc a .
= A for a certain
(6) A E On.
T h i s i s o b t a i n e d f r o m (5), o r d i n a l s i s an o r d i n a l
(2), and ( 3 ) s i n c e any t r a n s i t i v e s e t of
( 7 ) H*A = a . PROOF OF (7). We know by (2) t h a t H*A c a . Suppose t h a t H*A $ a. Then s i n c e A E On (by ( 6 ) ) , by (1) we o b t a i n ' R ' A E a , i.e., by t h e d e f i n i t i o n o f A, A E A , contradicting A e W . Define, now,
6'k = g ' ( a - 6 * € ) .
6=
The u n i c i t y o f 3.5.1.2
6'E
6
6's
E
'A.
i s e a s i l y proved by i n d u c t i o n .
Also f o r
5
E
A,
9
DEFINITIONI
.
It i s clear that 4
A Then, d E a and 6 - l
HIA.
if
( 6,4)
E E N , and
€, q
E
Dd
, then E
~q
t--,
The n e x t theorem shows t h a t every w e l l o r d e r i n g i s isomorphic t o I N l a f o r e x a c t l y one a. That i s , s i n c e D ( E N )
wo
(EN). 3.5.1.3
5W O , t h i s
proves t h a t
THEOREM, 4 E W O - . 3!6 ( 6 , 4 ) 6 E N . Define t h e f u n c t i o n g w i t h D g = P D 4
PROOFl L e t 4 E W O . 4
- CO)
by g ' x = A x , f o r a l l x # 0, x C D h . Since 4 i s a w e l l o r d e r i n g , g ' x E x . By 3 . 5 . 1 . 5 , t h e r e i s an 6 such t h a t Thus, g E II( x : x C D 4 A x # OT D 6 € O n and = g T D 4 6*E) f o r a l l E E D 6 . I n o r d e r t o prove t h a t ( 6 , 4 ) E E N we o n l y have t o prove t h a t € C q E D 6 i m p l y 6'€ G 4 6 ' q . Sup-
.
-
pose € C_q € 0 6 . Then € € q o r € = q . If€ = 17, t h e n c l e a r l y 6'E 5, 6 ' 0 . t E q ; then 6*E C 6*q and, t h u s D h -d*€ 3 D h - d*q. Hence 6'77 =
Let,
( D 4 -d*q)
>h
A D h -6*E 6
Thus, by 2.6.1.3, The u n i c i t y o f
6
=
d'€
.
.
i s an isomorphism between I N 1 06 and 4 .
i s e a s i l y proved b y i n d u c t i o n .
I n a s i m i l a r way, t h e n e x t theorem can be shown. 3.5.1.4
THEOREM,
W O (R) A R
9 Y A W x ( x E DR
-P
OR(x)E Y )
+
259
AXIOMATIC SET T H E O R Y
3!F(INIOn
=F R A D F =
On).
The proof i s l e f t t o the reader. I t i s c l e a r t h a t i f h and A a r e well orderings, then tr = & i f and only i f D (EN ' a ) = D ( E N ' & ) . T h u s , we may use o r d i n a l s a s types of well-orderings. For o t h e r k i n d s of simple orderings we do not have such a s o l u t i o n f o r t h e problem of finding types. However i f h i s a simple ordering and h E W we can use t l- ( x ) a s introduced in Def. 3.2.3.24.
DEFINITION,
3.5.1.5
(i) 7 f = U {x: (heW O A x = D ( E N ' t ) ) V tI
( t E
Y- W O A x =
(&))I. W}.
(ii) Os= (7f:SO(h)
We s h a l l use Greek lower-case l e t t e r f o r elements of 0 s . Next, we proceed with operations on simple orderings. First, the inverse of a type.
DEFINITION,
= U (0 : 3 h ( a = T A P = h - ' ) } . 8 Since h = h, we have a = a . For instance, i f u E W , s i n c e two f i n i t e simple orderings with t h e same number o f elements a r e isomorphic U (2.7.4.2) we have u = u .
3.5.1.6
Now, we define the ordered sum.
3.5.1.6
I:
DEFINITION SCHEMA,
Let
T
be a term.
Then
[ T : x E A ] = { ( ( y , x 1 ) , ( z , x;?) : xl, x2 E A R X A y E D 7 x [ ~ 1 1A z E D r X 1 x 2 1 A ( x 2 5 x1 V
o r Z F I A for C x [ F ' x : R R
I f we have D F Y ( F ) we w r i t e Z ( F ' x : x f A ) xEA1. R Let S = Z [ F ( x ) : x E A ] R r e l a t i o n such t h a t : (1) D S = [ D F ( x ) :
( 2 ) For a, b
a 2
E
xEA1
, where
F i s a unary operation.
Then S i s t h e
=
DS with a =
(
y,xl
)
and b =
(
y, x 2 )
, we
have,
x V (xl = x 2 A y G 2) 2 F(x1) That i s , a t each point x of A ordered according t o R we p u t t h e ordering of F ( x ) .
s
b - x < 1
R
0
260
ROLAND0 C H U A Q U I
I n case X c O n , we p u t 2
3.5.1.7
[T
0
THEOREM SCHEMA,
( i ) PO (R) A A C -D R A
(ii) SO(R) A A
C DR -
A
( i i i ) WO(R) A A C - DR A
:PEX]
instead o f
Z [T :PEX]
INP
.
the^,
L e t T be u Z m .
v
x ( x E A +PO(?))
v v
x(xEA -+SO(?)) S O ( C R x ( x E A -,WO(?)) -+ WO(2 R
P O ( Z x [ ? : x c A ] ). R
-+
+
[?
:xEA]).
[?
:xEA1).
PROOF, L e t F ( x ) = r be t h e unary o p e r a t i o n d e f i n e d by T (we keep t h e f r e e v a r i a b l e s , besides x, f i x e d ) , and l e t S = Z [ F ( x ) : x E A 1 . PROOF OF ( i ) . Assume t h a t R i s a p a r t i a l o r d e r i n g and t h a t F ( x ) i s a p a r t i a l o r d e r i n g f o r each xEA. We must prove t h a t S i s a l s o a p a r t i a l o r d e r i n g . The r e f l e x i v i t y o f S i s obvious
.
Suppose t h a t ( a , x l ) S ( b , x 2 )
x 1 <2 x2
Def. 3.5.1.6,
3.5.1.6,
aG
F(X1)
F(X1)
and and S i s antisymmetric.
and ( b , x 2 ) S ( a , x l ) .
k xl.
and x2
Thus, x1 =
a ; thus a
= b.
x2.
T h i s i m p l i e s by
Hence, a g a i n by
That i s
(
a,xl) =
(
b,x2)
.
Then, x1 Q R x2 b,x2) S ( c , x3 ) and x2 G R x3. I f x1 # x3, t h e n x 1 < R x 3 and by 3.5.1.6, So assume t h a t x1 = x3. We have x1 = x2 = x3, s i n c e R ( a , x ) S ( c,x3) 1 i s antisymmetric. Hence, a G F ( x l ) b and b Q c , i . e . a G F ( X 1 ) C, F (X,) and, we a l s o have ( a , x l ) S ( c , x 3 ) . Thus, S i s t r a n s i t i v e . Hence (i) i s Suppose, now, ( a x
)
S
(
b,x2)
' 1 Thus, x1 Q R x3.
and
(
.
proved. PROOF OF (ii).Assume, now, t h a t R and F ( x ) a r e s i m p l e o r d e r i n g s f o r xEA. L e t ( a , x l ) , ( 6 , ~ E~ D) S . I n o r d e r t o prove t h e c o n n e c t i v i t y F i r s t l e t x1
o f S we have two cases. hence ( a , x l ) S a<
F(xl)
Or
(
b,x2)
or
(
'F (X,) a ,
f x2; t h e n x1 < R ~ 2 o r x2 C R x 1 , and I n case x1 = x2, we have t h a t b,x2) S ( a , x l ) .
and t h e same c o n c l u s i o n f o l l o w s .
Thus, t h e
c o n n e c t i v i t y o f S i s proved and hence (ii). Assume, now, t h a t R and F ( x ) a r e w e l l o r d e r i n g s , PROOF OF (iii). f o r xEA. By ( i ) , s i s a s i m p l e o r d e r i n g . Thus we o n l y need t o prove t h a t Let B c - D S , 8 # 0; we have t h a t 0 D B c - DR. Let S i s well-founded. R x = A D B ; s i n c e R i s a w e l l o r d e r i n g , x E DB. Hence, 0 f B*(xI C D F ( x ) . F lu) L e t a = A D F ( x ) ; s i n c e F ( x ) i s a w e l l o r d e r i n g , a E D F ( x ) . I t i s easy S t o show t h a t ( a , x ) = A B.
+
.
AXIOMATIC SET THEORY
THEOREM SCHEMA, L&
3.5.1.8
SO(R) A V x ( x E D R
x
E
--t
261
F and G be unatry opehation; t h e n
W O ( F ( x ) ) )A R z F S A W x(x
E
D R - + F ( x )p
PROOF, Assume t h e h y p o t h e s i s o f t h e theorem. By 2.6.2.5 f o r each D R , t h e r e i s a unique J such t h a t F ( x ) = G ( F ( x ) ) ; c a l l i t J x . De-
f i n e t h e f u n c t i o n H by H ' ( a , x ) = ( J x ( a ) , F ( x ) ) Z[F(x) : xEDR] 2 Z[G(x) : xEDS], R H S 3.5.1.9 W
Zo[ G
:
(K)
K(KEU
K E U
We &o
]
.
-+
I t i s easy t o prove t h a t
Let F and G be unatry opehation6, then
THEOREM SCHEMA,
u EwA
.
PO ( F ( K ) )A F ( K ) E G ( K ) ) * C o [ F ( K ) :
K E U]
=
have,
U € W AV K ( K € V
-+
F ( K )C -
w ) * zo[F(V) : K E V ]
C
W.
The p r o o f , which i s s i m i l a r t o t h a t o f t h e p r e v i o u s theorem i s l e f t t o t h e reader. I t i s t o be noted t h a t t h e r e s t r i c t i o n t o a f i n i t e superc l a s s i s necessary w i t h o u t t h e axiom o f choice. 3.5.1.10
DEFINITION,
R +oS = Co [ F ( K ): ~ € 2 , 1 where F ( 0 ) = R
and F(1) = S. B e f o r e i n t r o d u c i n g t h e corresponding sum o f o r d e r types, we s h a l l prove t h a t f o r w e l l - o r d e r i n g s t h i s a d d i t i o n c o i n c i d e s w i t h t h a t d e f i n e d previously f o r ordinals.
PROOF,
L e t h,b E W O , ,3 = E N ' h and g = E N ' b . We s h a l l d e f i n e such t h a t D h = D,3 + D g , b y s t i p u l a t i n g ,
(1) h't =
(d'C;,
&,A E
0
-
+o n )
0)
if
( 2 ) h't = ( g ' ( C ; - F ) , 1) h
r+a=
THEOREM,
h = EN'(tr
-
WO-
3.5.1.11
By 3.3.4.8, + 6 and 6 E
C;
n (i.e.
E
E
h,
if
h c-t EX+^.
Z t Z i m p l i e s t h a t C; E Ir or t h e r e i i a-6 such t h a t 6 E Z a n d 6 = C; - z). Thus, D h = h t d .
E
I t i s easy t o show t h a t h = E N ' ( & +o b ) .
3.5.1.12 7' = /r
+o
6
1.
DEFINITION,
a
+0
=
U {y: 3
h 3 6 (a = ? f A
0 = F A
T h i s d e f i n i t i o n i s an e x t e n s i o n o f o r d i n a l a d d i t i o n b y 3.5.1.10.
It
ROLAND0 C H U A Q U I
262
i s j u s t i f i e d by 3.5.1.9. THEOREM,
3.5.1.13
( i ) P O ( R ) A P O ( S ) A P O ( T ) + ( R + O S ) t 0 7 E R to ( S + O T ) A R +
0
O = R z O +
R.
0
+y
( i i ) (a + 0 )
= a
+ (0
t y ) A c1
+ 0
=
a = 0 +
c1
.
The p r o o f i s l e f t t o t h e reader. We a l r e a d y know t h a t c o m m u t a t i v i t y i s n o t t r u e f o r o r d e r t y p e s ( f o r i n s t a n c e w + 1 it 1 + w). We pass now t o t h e p r o d u c t o f o r d e r i n g s . D E F I N I T I O N SCHEMA,
3.5.1.14
3x(g'x If X
5
6 ' x A Wy(y < R x
O n we w r i t e
i s a f u n c t i o n , we w r i t e
Il R
[T
[T
Let
-,d r y
:a
E
Il F a o r R
X ]
=
T
g'y))} instead o f
n ( F'x :x R
be a term; then we d e f i n e ,
E
n
IN^
a) for
i s c a l l e d t h e l e x i c a g h a p k i c ohdehing
:x E a ]
[T
:a E XI
n [ F'x R
06
.
:xEal
IfF
.
II [ D T : x E a 1
a c c o r d i n g t o R. L e t F be a unary o p e r a t i o n , R and F ( x ) s i m p l e o r d e r i n g s f o r x E a . L e t Then S i s t h e r e l a t i o n such t h a t ,
.
S = II[F(x) : x E u ]
R
(1) D S = n [ D q x ) :
and
(2) f o r
6,s
E
X E U I
,
DS we have t h a t ,
6
6 <S g -
=
R g V l x ( x E a A x = A { y : y E u A d'y+g'Y)
A
6 ' <~ F ( x ) g ' X )
That i s , f o r
6
< S g t o be s a t i s f i e d t h e r e must be a f i r s t element x i n
a where t h e y d i f f e r and 6 ' x < F ( x ) g ' x . We have t h e f o l l o w i n g theorem f o r p r o d u c t s o f o r d e r i n g s . 3.5.1-15
THEOREM SCHEMA,
Let F be. a u m y apehation, fithen
(i) S O ( R ) A a C-D R A W x ( x E a + P O ( F ( x ) ) ) - P O ( n [ F ( x ) : x ~ u ] ) ; R
AXIOMATIC S E T T H E O R Y ( i i ) WO(R) A a C -D R A W x(xEa
+
263
SO(F(x)))
+
S O ( ll [F(x): x € a ] ).
R
PROOF, Let S = Il [ F ( x ) : x € a l and assume t h e hypothesis of ( i ) .
R Reflexivity of S i s obvious.
Suppose t h a t 6 S g a n d g S d and l e t U(6,g) = ( x : 6'x f g ' x ) ; assume a l s o , i n order t o argue by contradiction t h a t 6 # 9. Then U(6.g) + 0 a n d , R hence, s i n c e b S g , i t has a f i r s t element x = A U(6,g) E U(6,g). By d S g , 6 ' x G F ( x ) g ' x a n d , by g S 6 , g ' x G F ( x ) 6'x. Since F ( x ) i s a p a r t i a l ordering, / ' x = g ' x , contradicting x € O(t(,g). Thus, 6 = g a n d antisymmetry i s proved.
6
Suppose, now, t h a t 6 S g and g S h . I f S h. So, we may assume t h a t 6 # g a n d g
6
= g
#
h.
or g = h, i t i s c l e a r t h a t R Let y = A D(g,cZ) and x =
R h D ( 6 , g ) . By 6 S g and g S h , x E U ( 6 , g ) and y E U ( g , h ) . since R i s a simple ordering, z = x, i f x Q R y , and z
=
Let z = x R y , i.e. y, i f y G R x . For
x < R y, d ' z < R g ' z = h ' z ; i f y < R x , 6 ' z = g f z < R h t z ; and i f x = q , then 6 ' 2 < R g ' z < R h ' z . In any case, 6 S h . T h u s , t r a n s i t i v i t y and ( i ) a r e shown.
u < R z , i t i s c l e a r t h a t ~ ' u = g ' u = h ' u . Also, i f
Suppose now t h a t t h e hypothesis of ( i i ) i s t r u e , i.e., R i s a well ordering and F ( x ) i s a simple ordering f o r x E a . Let 5 g € D S w i t h d + g . R Then U ( 6 , g ) # 0 and, s i n c e R i s a well ordering, x = A U(6,g) E D(d,g). B u t F ( x ) i s a simple ordering; hence, since 6 ' x # g ' x , 6 ' x < F ( x ) g ' x o r g'x
i.e.
6
c s g or g < S 6 .
Thus connectivity and ( i i ) a r e
I t is n o t t r u e t h a t F ( x ) being well orderings would imply t h a t S i s a well ordering o r t h a t R and F ( x ) simple orderings imply S a simple ordering.
DEFINITION,
3.5.1.16
R .S =
2 [ R :X E D S S
I
R * S i s the product of two orderings. I t i s c l e a r , from the defini t i o n t h a t i f R , S a r e p a r t i a l orderings, then
(1) D(R * S ) = D R
-
(2) i f
(
xl, Y$
9
x (
Y 1 < S Y2" (Y1 = Y2
Thus,
R
S
DS
x2, y 2 ) E D R x D S , then ( x l , y 1 ) < R . S ( ~ 2 , y 2 ) 1 ' "RX2)'
i s not t h e lexicographical ordering b u t , t h e antilexicographical. R*S
= II IN
[ F ( K ): ~ € 2 1, where F ( 0 ) = R and F(1) = S.
-'
ROLAND0 C H U A Q U I
264
T h i s i s d e f i n e d so i n o r d e r f o r t h i s p r o d u c t t o c o i n c i d e w i t h o r d i n a l mu1 t i p l i c a t i o n .
3.5.1.17
R1
Ro * S o
THEOREM, P O ( R O ) A P O ( S o ) A Ro S1
.
= R1
A
So
= S1
--*
The p r o o f i s l e f t t o t h e reader. We now prove t h a t p r o d u c t o f w e l l - o r d e r i n g s c o i n c i d e w i t h o r d i n a l multiplication. 3.5.1.18
THEOREM,
-
WO
h,b E
-+
4.6
--
= &*A.
PROOF, L e t h,b E W O . I f h = 0, t h e r e s u l t i s t r i v i a l , so assume L e t 6 = E N ' h , g = E N ' b . We have t o d e f i n e h, such t h a t h = h # 0. E N ' h - b and D h = D 6 -Dg. By 3.3.5.4 ( E u c l i d ' s a l g o r i t h m ) a E h * ai f and o n l y i f a = /r y + 6 w i t h 6 E h and 7 E 0. Moreover, 7 and 6 a r e 6 is a for a = /r*y+S. unique. D e f i n e f o r a E K * b , h ' a = (6'6, g ' y ) b i u n i q u e f u n c t i o n over D h x Db. Also, i f a ='iT*y + 61 and P = K * y 2 + S 2 , then 1
a CP Hence h
-
is an isomorphism, i.e.
71 E 72
v
(Tl
h = EN'h.n.
.
= 72 A
s1 5 6 * ) .
I t i s c l e a r t h a t i f R and S a r e s i m p l e o r d e r i n g s then R * S a l s o i s a Thus, we may i n t r o d u c e s i m p l e o r d e r i n g ; and i f R,S C W then R * S C W t h e f o l l o w i n g e x t e n s i o n o f o r d i n a l m u l t i p l i c a t i o n t o o r d e r types.
.
3.5.1.19 y =
DEFINITION,
a
*P
3.5.1.20
=a A
THEOREM,
(i) SO(R) A S O ( S ) A S O ( T ) 0 R A R {( 0,O) 3 2 R = {( 0,O) 3 9
a
= U {y : 3 h 3 6 ( a = h A p
=)I.
-
(R*S) T = R R A R (S +o T )
--*
(S-T) AR.0 = 0 = (R 0s) +o (R - T ) .
=
(ii) (a-0) m y = a (P-y)Aa*O= O = O * a A a - l = a = l * a A a - P -I a 7.
(P +7) =
The p r o o f i s l e f t t o t h e reader.
AXIOMATIC SET THEORY
265
PROBLEMS
u
1.
Prove (a+P ) =
2.
Prove 3.5.1.4.
3.
Prove 3.5.1.9.
4.
Prove 3.5.1.12.
5.
Prove 3.5.1.17.
6.
Prove 3.5.1.20. 3.5.2
u
P
3 t a
SCATTERED,
.
DENSE,
AND CONTINUOUS ORDERINGS,
I n 2.7.4 we s t u d i e d denumerable o r d e r i n g s . In p a r t i c u l a r , t h e o r d e r t y p e w w a s c h a r a c t e r i z e d i n 2.7.4.5. Thus, we g e t immediately, 3.5.2.1 R
THEOREM,
sO(R)+
R
( a =w * A D R
D R A V x ( x E DR
E
R
d
+
x < R A { q : x <,q}) A W A(A C DR A A DRE A A V x ( x E A + A { y :
We a l s o c h a r a c t e r i z e d denumerable dense o r d e r i n g s , (Def. 2i7.4.6, and 2 . 7 . 4 . 7 ) and we showed t h a t denumerable dense o r d e r i n g s w i t h n e i t h e r f i r s t n o r l a s t element a r e isomorphic. I n o r d e r t o i n t r o d u c e an o r d e r t y p e f o r o r d e r i n g s we need t o show t h a t t h e r e i s one such i n W . T h i s can be done by a s i m i l a r c o n s t r u c t i o n t o t h a t o f t h e r a t i o n a l number. However, I s h a l l use a d i f f e r e n t c o n s t r u c t i o n , t h a t i s s i m p l e r from o u r p o i n t o f view. 3.5.2.2
3v(vEw A 3.5.2.3
DEFINITION,
x y = 1) A THEOREM,
q3z(q
Q = no[lNI 2 :
~ V ( V € WA
QE
K E W ]
Wp(v&pEa
W ADQ
+
I{x: x
x = 0))) P
E
w2
.
W A D O ( Q ) A V X ( X E DQ
A
+
P R O O F , I t i s c l e a r t h a t Q E W and, b y 2.5.1.14 (ii) that Q i s a s i m p l e o r d e r i n g . We now show t h a t i t i s dense. L e t x E D Q ; d e f i n e L I X = n {V : V E W A W P ( V c p E w xu = 0)). Let, now x < y . D e f i n e +
[ = ~ ' x U ~ ' and y , Z E D Q by z l [ = X I [ = z [ = 1, and Z
P
=o
for P>E.IItiseasy
t o show t h a t x < z i Qy. Thus, Q i s dense. S i m i l a r l y , we show t h a t Q has n e i t h e r f i r s t n o r l a s t element. The mapping 4 g i v e n by d ' x = x l L ' x , f o r x E D Q i s such t h a t
DQ
5
d
u CV2:
VEW
1
5 w , by
3.4.3.26.
266
ROLAND0 C H U A Q U I
Since i t i s easy t o see t h a t W ~ D ,Qwe o b t a i n DO = w . 3.5.2.4
DEFINITION,
By 2.7.4.7
3.5.2.5
3 q3 z
=
?)
Q.
we o b t a i n ,
SO(R)
THEOREM,
q < R x < R z ) A DR
2
--f
w).
(K =
-
77
-
DO(R)
A vx(x
E
DR
+
We now i n t r o d u c e a new d e f i n i t i o n . 3.5.2.6 D O (RI B ) 1.
DEFINITION,
SC(R)
S O ( R ) A 1 3 B(B c -DR A B $ F N A
I f SC(R) we say t h a t R i s bcattehed. I t i s c l e a r t h a t t h e usual o r d e r i n g s o f t h e r e a l s numbers o r t h e r a t i o n a l numbers a r e dense; and w / Z N I w and a l l f i n i t e o r d e r i n g s a r e s c a t tered. There a r e o r d e r i n g s t h a t a r e n e i t h e r dense n o r s c a t t e r e d ; f o r i n stance t h e o r d e r i n g o f t h e r e a l s r e s t r i c t e d t o t h e p o s i t i v e r e a l s and t h e n e g a t i v e i n t e g e r s i s such.
PROOF, Assume t h a t R I A and R I B a r e s c a t t e r e d and t h a t C C A U B i s such t h a t R I C i s dense and C i s i n f i n i t e . We have, C = ( C n A ) C7 ( C n B ) . Hence CnA o r CnB i s i n f i n i t e . Suppose t h a t CnA i s i n f i n i t e . Since R I A i s s c a t t e r e d , R I ( C n A ) i s n o t dense. Hence, t h e r e a r e a, b E C n A such t h a t a < R b and t h e r e i s no c€CnA w i t h a < R b . This implies t h a t i f x € C and ~ < ~ x < then ~ b xEB. ,
Let U = C n
{ x :a < R x < R b l .
R I C i s dense, i f d , e E B and d C R e , t h e n t h e r e i s an
d < R x < R e . Hence i s infinite. scattered.
But
XED
XEC
such t h a t
and t h u s we have proved t h a t R I D i s dense and P
D C- B ,
hence t h i s c o n t r a d i c t s t h e f a c t t h a t R I B
A l l s i m p l e o r d e r i n g s can be c h a r a c t e r i z e d i n terms o f s c a t t e r e d dense o r d e r i n g s , as i s shown by t h e n e x t theorem. 3.5.2.8 W x (x E D S
THEOREM, SC(F'x))
+
Since
SO(R) A R
A R
1
E
Z (F'x :x
S
Y - 3 S 3 F(DO(S) A D S F ' E
D S)).
is and
(F) A
PROOF, L e t R be a simple o r d e r i n g t h a t i s a s e t . Define, f o r x , q E D R , (x,q) = CZ: x Q R t g q V q Q RGt XI. It i s c l e a r from t h i s d e f i n i t i o n that, and
(1) (x,q) = ( q , x ) , f o r a l l x , q E D R , (2) (x,q)
5 (x,z) u
( z , q ) , f o r a l l x,q,z
E
DR.
267
AXIOMATIC SET THEORY
L e t T be t h e r e l a t i o n g i v e n by T = CC x , g
)
: SC(R](x,y))I.
We have,
( 3 ) T i s an equivalence r e l a t i o n w i t h D T = D R . I n o r d e r t o prove ( 3 ) , we n o t i c e t h a t : R l ( x , x ) i s s c a t t e r e d , because ( x , x ) = {XI; R l ( x , y ) s c a t t e r e d i m p l i e s R l ( y , x ) s c a t t e r e d , by (1); and if R I ( x , ~ ), R 1 (y,z)are s c a t t e r e d , then R I ( x , ~ )i s s c a t t e r e d , b y ( 2 ) and 3.5.2.5. We a l s o have, f o r any x
E
( 4 ) i f g , z E T * { x 3 and g (x,.C)
DR G R t G R z , then
tET*{x'l.
c
T h i s i s so, because (y,R) 5 ( y , z ) ( x , y ) u ( x , z ) , by ( Z ) , and, also, 5 ( x , y ) U ( q , t ) ; t h u s R l ( x , t ) i s s c a t t e r e d and t E T * { x I .
Let A {T*{x} : x E D R } and l e t S be t h e r e l a t i o n d e f i n e d by s t i p u g G R x ) ) . We s h a l l prove l a t i n g S = E(a,b) : a,b€A A WxWg(x€a A y E b -f
t h a t S i s a dense o r d e r i n g . I t i s c l e a r t h a t S i s r e f l e x i v e , and t h a t S i s Take antisymmetric. Let, now a S b and b S c and suppose x E a and Z E C . gEb. Then x G R g < R ~ .Thus, x C R z . Hence a S c . Thus S i s t r a n s i t i v e . Suppose, now, t h a t a,b E D S . Then a = T*{xI and b = T * { g } f o r x , y E D R . Hence, a = b o r a n b = 0. I f a = b, t h e n a S b . So assume a n b r 0 . We have, x < R y o r g < R ~ . Suppose x < R y and l e t u E a , v E b . I f v < R u we would have v < R x < R y , hence, by Hence u < ~ w . Thus, a S b .
(4), x ~ c bo n t r a d i c t i n g a n b
= 0.
I n case y < R x we prove s i m i l a r l y t h a t b S a .
Hence S i s connected and S i s a s i m p l e o r d e r i n g . We have s t i l l t o prove t h e d e n s i t y o f S.
Suppose t h a t a
< s b.
Hence,
b y ( 3 ) , a n b = 0. L e t a = T*{xI, b = T * { g } ; t h u s R I ( x , g ) i s n o t s c a t t e r e d and, a l s o , x G R y . Then, t h e r e i s a s e t C c_ ( x , y ) such t h a t C i s i n f i n i t e and R I C i s dense. Thus, t h e r e a r e t , u , v € C w i t h x Q R|
( u , g ) c o n t a i n U , E such t h a t R I D , R I E a r e dense. T h i s shows t h a t S i s dense.
T*EyI.
(x,u) and Hence, T*Ix} <ST*Cu3 < s Also,
Define, now, f o r a E D S , F'a = Rla. We s h a l l show t h a t F ' u i s s c a t t e r e d f o r e v e r y a € DS. We have, D F'a = a = T*{y) f o r a c e r t a i n y € D R . Assume t h a t C C - T * { g } i s an i n f i n i t e s e t w i t h R I C dense. I f xl, x2 E C
5 (x,.y) u (y, x 2 ) . w i t h x1 < x2, t h e n xl, x2 E T*{y3 and, by ( 2 ) (x,,x,) B u t RI (x,,g) and R I ( x 2 , q ) a r e s c a t t e r e d , and RI (x,, x 2 ) i s dense; t h i s conThus, F'a i s s c a t t e r e d .
t r a d i c t s 3.5.2.5. I f we d e f i n e easy t o show t h a t
4
on D R by s t i p u l a t i n g t h a t d ' x = ( x , T*{x}),
Rz6
Z:(F'a:aEDS).
then i t i s
s
We now b e g i n t h e s t u d y o f continuous o r d e r i n g s . o f cuts.
F i r s t the definition
R O L A N D 0 CHUAQUI
268
DEFINITION
3.5.2.9 (i)
x i = { q :q
( i i ) X i = { q :q
E
w
DR A
-+
E DR A W x ( x E X
( i i i ) Cut (a,R) * a
E
x~ , y ) }
~ ( X E X
q GRx)}
-+
+
2 V A ao, al C D R A aOR= al A aiR = a.
.
A few p r o p e r t i e s of c u t s a r e given i n the next lemma.
3.5.2.10
LEMMA,
( ~ ) P o ( R ) A X C- Y C D R - + X +~~ Y ~ A X ~ > Y ~ .
( i i ) PO(R) A Z S D R
+-
( i i i ) P O ( R ) A C u t (a,R) A u
( i v ) PO(R)
+
al = ‘R
+
-t
Z C- 2 R P 2 R
-+
(Cut (a,R)
++
E
uo A
3 Z((a,
t - + -
+
- Z R AZ;
“R
ul
= ZR
u GRv.
LJ
E
=
Z; A al = ZR
-+
-
+-
-+
) V (ao
=
*
Zi- A
)))*
( v ) P O ( R ) A Cut (a,R) A C u t (b,R) ( v i ) S O ( R ) A C u t (a,R)-+ a. n al
+
5
(ao E bo
+ +
bl
5 all.
1.
The proof i s easy and i s l e f t t o t h e reader. 3.5.2.11 bo C_ a,)
.
DEFINITION,
C ( R ) = { ( a , b ) : C u t (a,R) A C u t (b,R) A
C(R) i s c a l l e d the cornpLe;t.ia~ ad R. The reason f o r t h i s name can ‘be seen from t h e next theorem. For d e f i n i t i o n s of t h e predicates and operations see Def. 2.2.3.13 and 2.2.3.14.
3.5.2.12
3 F ( R 2 C(R) A F
THEOREM, P O ( R )
A R E V-. C L O ( C ( R ) ) A C(R)
R a(a C - D R A L u b R ( V a, a )
R
Va(a C - DR A G l b R ( A a, a )
-+
R
+
F’ V a
=
C(R) V
€
V A
F*a) A
R C(R) F’ A a = A F*a)).
Hence, every p a r t i a l ordering can be embedded (or even extended) i n t o a complete l a t t i c e ordering.
PROOF, Suppose t h a t R i s a p a r t i a l ordering t h a t i s a s e t . I t i s c l e a r by 3 . 5 . 2 . 1 0 ( v ) , t h a t C(R) i s a p a r t i a l ordering. By the union and power s e t axioms, C(R) E V . We have s t i l l t o show t h a t C(R) is complete. Suppose t h a t a C - D C(R) and l e t b = U ( x , : x E a } and c = U+ {x, : X E C L ) . We s h a l l show t h a t t h e cut dsuch t h a t do = b;- and dl = bR i s t h e l e a s t upper bound of a. We have t h a t b 5 b i and, hence, x 9 C ( R ) d f o r every
AXIOMATIC SET THEORY
cut
XEU
ul.
.
Assume, now t h a t x G u f o r a l l X E U . We (R) + + uo, f o r a l l X E U , and, hence b C -c uo. T h e r e f o r e b 2 uOR=
(Def. 3.5.2.9
have t h a t
xo C_
269
v), d G
Thus, by 3.5.2.8
l e a s t upper bound o f a.
u.
c (R)
Thus we have proved t h a t d i s t h e
S i m i l a r l y , we can prove t h a t t h e c u t c g i v e n by eo = c- and el = c
-+
i s t h e g r e a t e s t l o w e r bound o f a. Define, now, t h e f u n c t i o n F bx s t i p u l a t i n g , + f o r every x E DR, t h a t F I X i s t h e c u t a g i v e n b y a. = {x}, and al = { x I R . From t h e d e f i n i t i o n s , we o b t a i n ,
x
by
-
c - lyl,
{XI;
* FIX<
C (R)F'y
*
Thus, F i s an isomorphism o f R i n t o C(R). Suppose, now, t h a t A C D R and a i s t h e l e a s t upper bound o f A. I t i s easy t o see t h a t F'x G c (R)7 ' a f o r e v e r y x E A . Assume t h a t y i s a c u t such that F'x < y f o r e v e r y x E A . We have, {XIC - yo f o r x E A . Since x E c (R) {XI- , we have t h a t x E y 0 f o r e v e r y x E A . Thus, { a ) - 5 yo and F'a Gc(R)y; i.e.
F ' a i s t h e l e a s t upper bound o f F*A.
The p r o o f t h a t F preserves g r e a t e s t l o w e r bounds i s s i m i l a r . 3.5.2.13
-
.
DEFINITIONI
(i) C O (R)
S O ( R ) A t ' a ( C u t (a,R) A u
( i i ) C'(R) = C(R)I ( D C ( R )
- {a : C u t (a,R)
+ 0 + al A (ao =
-+
a0 n a1
0 V al
f
0).
= 0))).
I f C O (R) we say t h a t R & a confinuoun oxdetLing, or simply, t h a t R A C'(R) i s t h e eolzt;ilzuoun & V A W L ~ o d R. (see n e x t theorem).
conLLnuoud.
3.5.2.14
THEOREM,
SO(R)
+
C O (C'(R)).
Hence, e v e r y s i m p l e o r d e r i n g can be embedded i n a continuous o r d e r i n g .
PROOF, L e t R be a s i m p l e o r d e r i n g . S i m i l a r l y as i n 3.5.2.12, we show t h a t C'(R) i s a p a r t i a l o r d e r i n g ; we now prove t h a t i t i s a s i m p l e ( R ) b , i.e. a. g bo. L e t o r d e r i n g . L e t a , b E D C ( R ) and assume t h a t a
xEaO
- b o and assume t h a t q E b 0 .
i s a cut, x E b o .
If
x
sCl
G R y , t h e n we would have, s i n c e b
Therefore, s i n c e R i s connected, we must have, y G R x .
Hence y E x We have proved t h a t b C a and thus, b S c , ( R ) a . Hence 0' 0- 0 C ' ( R ) i s connected.
C'(R) i s a l s o continuous, because i f d i s a c u t i n C ' ( R ) w i t h do
#
O#
ROLAND0 C H U A Q U I
270
dl we can prove as i n 3.5.2.12
t h a t t h e l e a s t upper bound o f do e x i s t s and,
then, i t i s e a s i l y shown t h a t do 3.5.2.15
xv
wA
Q c = n o [ I N 12:
DEFINITION,
= 1) A t l u ( v E w
+
n d 1 = C C' v (a1 d o ) .
3pfvEpEw A
K E W ]I
x
8
{x: x
E
= 0))).
P
w 2 A 3 v(v
E
I n Q c we r e s t r i c t t h e p r o d u c t o r d e r i n g t o sequences t h a t a r e n o t a l l 1 a f t e r a c e r t a i n u . We a l s o o m i t t h e sequencte t h a t i s c o n s t a n t l y 0. 3.5.2.16
THEOREM,
Q C lDQ = Q A Q C
A DQc
C'(Q )
2
w2 A
The l a s t c l a u s e can be paraphrased by saying t h a t D Q i s dense, coi n i t i a l , and c o f i n a l i n Q c By 3.5.2.14, we o b t a i n from Q c z C ' ( Q ) , t h a t Q c i s continuous. The l a s t c l a u s e shows t h a t Q c i s dense.
.
PROOF, I t i s c l e a r t h a t Q c lDQ = Q and, by 3.5.1.15 (ii), that Qc L e t a be a c u t i n Q i s a s i m p l e o r d e r i n g . We now prove t h a t C ' ( Q ) = Q c w i t h a. # 0 f al.
.
Q
L e t xf = V { y : y E u O A y I L ' q
u IK2: rcC_vll where L ! x i s as d e f i n e d
E
Since t h e r e a r e o n l y f i n i t e l y many y ' s such t h a t i n t h e p r o o f o f 3.5.2.3. qIL'y E U C K 2 : K C V I , t h i s l e a s t upper bound e x i s t s . Also, L ' x $ C v . We
$ 1 ~
U
IP
x'lcc f o r P C_v. L e t P , V E w , P C_v, and l e t y = xu P U U ( 0 : p CKEw) Then q G Q xv and, hence, q E a o ; a l s o L ' y C- cc, hence Suppose t h a t q Then, from qIL'q E u { K 2 : KCCO. Thus q Sax;. t h e d e f i n i t i o n o f Q , x f < x i , c o n t r a d i c t i n g t h e d e f i n i t i o n o f x," ( s i n c e now prove t h a t
.
E u {K2: xPIL'xP U
d'a =
u
{X;\Y
a v such t h a t
( 6 ' ~ )= ~0.
=
-
KCU)).
:v E m ) .
Define, now,
( 6 ' ~ )= ~1.
Also, s i n c e
Suppose t h a t 6 ' u $2 D Q c .
0 and (d'a), = 1 f o r e v e r y p 3 v .
(6'a)Iv
t
1u
(
1: v
CKC P )
.
by s t i p u l a t i n g t h a t
"(')V
al
f
Since a.
there i s
0, t h e r e i s a v such t h a t
~ aUr e a l l i n ao.
Q
+ 0,
Then t h e r e i s a v such t h a t
Thus t h e sequences z
for P
f o r every p I v i m p l i e s t h a t u 2 w)
6E
I t i s c l e a r t h a t 6'a E O 2 .
w , where w = d ' a l v
P
d e f i n e d by
We have t h a t u U {(
l,v)
}U
( 6 ' ~ =) ~ z
P
=
> Q zP
(0: u
CK
E
Hence s i n c e a i s a c u t , v E a o ; t h e r e f o r e ( d ' a ) , = 1 c o n t r a d i c t i n g o u r
hypothesis.
Thus, 6 ' a E D Q c , i.e.
Suppose, now, t h a t c E D Q c .
D Q A y G Q c cl, al = Cy : y d'a = c.
E
L e t a,b be c u t s i n Q , a
6€
DC
'CQ ) D Qc
L e t u be t h e c u t g i v e n by a. = { y : y E
DQ A c GQc y l . f
b.
Then a.
C
I t i s easy t o show t h a t
bo o r bo C ao.
Assume t h a t
A X I O M A T I C SET THEORY uo c bo.
U
Then, xv
< Q :x
f o r a c e r t a i n v.
Hence, d'a
prove t h a t i s one-one and an isomorphism. p r o o f t h a t C' (Q)= Q c .
6
w
We prove, now, t h a t DQc =
2.
271
< Qc d'b.
Thus, we
W i t h t h i s we complete t h e
We have D Q c =
0
({(O:
2-
K E w ) }
U
A ) where A = {x : ~ V ( V E W A W P ( P > _ xU = 1 ) ) A x E w 2 } . I t i s easy t o P show t h a t A 5 U I y 2 : V E W 1 =I w (by 3.4.3.26). Also, w < D Q c . Hence, w 2.. Q c + w =I B + w + w = B + w - Q c , f o r a c e r t a i n 8. +
C
.
F i n a l l y , we p r o v e t h a t D Q i s dense, c o f i n a l , and c o i n i t i a l i n Q c L e t x,y E D Q c , x < Q c y ; l e t v = n { K : xK c yK) and p = n { K : K > V A xK =
01.
Let
=
2'
xlp
D Q and x < Q c z '
y
'
U
I( 1 , ~1) U K (
<ecy.
0 :
.
K I P ) It
S i m i l a r l y , we f i n d z,
i s easy t o show t h a t z '
z" w i t h z
and
RIA
3.5.2.17 THEOREM, C O ( R ) A C O (S) A 3 A ( A c DR A VxW y ( ~ < ~ ~ + 3 z 3 ]z "z ( z' , z t , z " E A A z k X < R Z ' < R q k Z " ) ) )
D S A = 71 A W x W y( x < y + R E S A D R = W 2.
-+
3
z3 zt 3
z" (z, z '
,z" E 8 A
E
z <sx <Sz'
= 77
A
-
C
<,Y<~Z")))
PROOF, L e t R be a continuous o r d e r i n g , and A C D R be denumerab dense, c o i n i t i a l and c o f i n a l i n R , v = q . We o n l y have t o prove t h a t R C'(Q) We have R I A E Q f o r a c e r t a i n d. For each a E D R Y l e t
.
=
(C(a)),
d
= A n { a ] , and ( C ( a ) )
= A n {u};
bR
.
I t i s easy t o see t h a t C(a
i s a c u t i n RIA. Define g E D C ( Q ) by s t i p u l a t i n g t h a t g ' a i s t h e c u t i n Q g i v e n by (g'a), = 6*(C(a)), and (g'a), = d*(C(a)),. Now, i f a,b E D R w i t h a t h a t a < R b.
g'b.
f
b, t h e n a < R b o r b < R u .
We may assume
Since A i s dense i n R , ( C ( U ) ) C ~ (C(b))o.
Thus, g i s an isomorphism i n t o C ' ( Q ),
Hence, g ' a
we have t h a t 6-'* do, 6 - l * dl i s a c u t i n R I A and, hence determines a c u t e i n R , such t h a t eo n A = 6-l* d and el n A = 6 - l * dl. I f we p u t a =
0
eo n e l , t h e n we can e a s i l y prove t h a t g ' a = d. 3.5.2.18
DEFINITION,
X
Thus, g i s o n t o C ' ( Q ) .
-
= Qc.
X i s t h e t y p e o f a l l continuous o r d e r i n g s R t h a t have a denumerable s e t dense i n R w i t h o u t f i r s t o r l a s t element, c o i n i t i a l and c o f i n a l i n R . X i s t h e o r d e r t y p e o f t h e r e a l numbers w i t h t h e usual o r d e r i n g .
In c o n n e c t i o n with A , S u s l i n ' s problem s h o u l d be mentioned. It cann o t be decided w i t h o u r axioms ( i n c l u d i n g a l l i n B C ) i f T T = X i f and o n l y i f R can be c h a r a c t e r i z e d by: (1) R i s continuous,
272
ROLAND0 C H U A Q U I
(2) R has n e i t h e r f i r s t n o r l a s t element, ( 3 ) Every f a m i l y o f d i s j o i n t i n t e r v a l s i n R i s denumerable.
PROBLEMS
1. 2.
+
Give an example o f s i m p l e o r d e r i n g R such t h a t DR i s i n f i n i t e , R w/ , w i t h f i r s t element, and e v e r y element except t h e f i r s t one has an immediate predecessor.
I N Iw
Show t h a t ifR i s a dense o r d e r i n g and R1, R2 a r e continuous o r d e r i n g s - D R1, such t h a t t h e r e a r e A, 8, A C
B C_D R2, A dense, c o i n i t i a l and B dense, c o i n i t i a l and c o f i n a l i n R2, and R 1 I A IR z
c o f i n a l i n R1,
R21B, t h e n R1
3.
Rp.
I
L e t F be a unary o p e r a t i o n , t h e n
Prove:
SC(R) A W x ( x E D R + S C ( F ( x ) ) ) +SC(C[F(x) : x E D R ] ) .
4.
L e t F be a unary o p e r a t i o n , t h e n
Prove:
SO(R) A 5.
Prove:
W
x(x
E
DR+ DO( F ( x ) ) A F(x)
R
$ 1)
+
L e t F be a unary o p e r a t i o n , then
D O ( C [ F ( x ) : x EDRI). R
DO(R)AR$1At(x(x€DR -SO(F(x))AF(x)+O)
+
1 S C ( C [ F ( x ) : x E D R l ).
R
6.
L e t F be a unary o p e r a t i o n , t h e n
Prove:
C o (Z [ F ( x ) :
R
7.
Prove:
x
DR])
+
DO(R).
L e t F be a unary o p e r a t i o n , t h e n
DR A v z ( z E D R x Q R z GRy)) A C o ( F ( x ) ) ) -+ C o ( 2 [ F ( x ) : x E D R ] ). R
C o (R) A 3 x 3
Vx(x
E
E
DR
(Hausdorff).
-+
g(x,y
E
+
8.
Prove:
DO(R) A
U Z DR
9.
Prove:
C o (R) A
U S DR
10.
Prove:
q t q 7, h +
11.
Prove:
= TI
.
-+
+
~ A ( AcDR A w 2 5 DR.
1 +h
= h,
RIA = q).
and h +
X +h.
AXIOMATIC SET THEORY
12.
Prove:
(w*q)
273
( w - v ) = ( w * g +a) ( w * q +a), w * q # w * v + a .
(Daves-Sierpinski ). 13.
Let R C - w x u be t h e r e l a t i o n g i v e n by, v G R p i f and o n l y i f u has l e s s p r i m e f a c t o r s t h a n p , o r v_ has t h e same number o f prime f a c t o r s t h a n p and u C _ p . Prove t h a t R = W - a .
14.
Prove t h a t i f R = X O X , t h e n t h e r e i s no A C D R such t h a t A = o and, f o r a l l x,y E D R , x < R y implies t h a t t h e r e i s a z E A w i t h x k z k y .
15.
Prove 3.5.2.10.
3.5.3
WELL-ORDERINGS,
Since w e l l o r d e r i n g s t h a t a r e s e t s can be c h a r a c t e r i z e d by o r d i n a l s , what we have proved f o r o r d i n a l s i s v a l i d f o r these w e l l o r d e r i n g s as w e l l . I n t h i s S e c t i o n we s t u d y some theorems t h a t a r e t r u e f o r w e l l - o r d e r i n g s t h a t are proper classes. F i r s t , a d e f i n i t i o n . 3.5.3.1
L e t F be a unary o p e r a t i o n .
D E F I N I T I O N SCHEMA,
Sup(S, [F (x):
x E A ] )
+--f
VT(WO(T) A W x ( x E A * F ( x )
W O ( S ) A V x ( x E A* F ( x )
2 T)
-+
2 S)
A
S 2 T)).
I f S u p ( S , [ F ( x ) : x E A 1 we say t h a t S is a Leabt uppeh bound [F(x) : xEA1
.
3.5.3.2
S zT*
THEOREM SCHEMA,
Then
Lek F be
06
a unatry opehation, then,
V X ( X ~* AW O ( F ( x ) ) ) A S u p ( S , [ F ( x: )x E A ] ) A S u p ( T , [ F ( x :)x € A l ) *
The p r o o f i s l e f t t o t h e reader.
PROOF, Assume t h e h y p o t h e s i s o f t h e theorem and l e t S' = B [ F ( x ) : R x E D R ] . S' i s a w e l l o r d e r i n g . It i s c l e a r t h a t F ( x )
I . For every y
E
7 S'
D S ' , S'
Y
f o r every x E D R .
2 F(x)
f o r some x
11. There i s a y E D S ' such t h a t F ( x ) S S = Si where z = A { y : V x ( xE DR+ F ( x )
7S
a
E
There a r e two cases.
DR.
Then t a k e S' =S.
f o r every x
Y S&) 1.
m
E
DR.
Take
ROLAND0 C H U A Q U I
274
Let R C -AxA
PROOF, Assume t h e h y p o t h e s i s o f t h e theorem. f i n e d by s t i p u l a t i n g t h a t x R y * F ( x ) F ( y ) .
I t i s easy t o show u s i n g 2.6.2.7 and i t s c o r o l l a r i e s , f l e x i v e , t r a n s i t i v e , and connected. By t h e h y p o t h e s i s and a l s o antisymme-tric. We now prove t h a t R i s w e l l founded. L e t z E B . I f f o r a l l uEB, F(z) 7 F f u ) , we have t h a t z l e a s t element. So assume t h a t F ( u ) 7 F(z) and F ( u ) $ F ( z )
=R
F(z)
be de-
t h a t R i s re2.6.2.7, R is L e t 0 # €3 C A . B and B has a for a certain
L e t x = A Cy: E D F ( z ) A ? , u ( u E B A F ( u ) = ( F ( z ) ) y ) l . By 2.6.2.7, t h i s x E DF(z7. L e t v be t h e unique element of B such t h a t
uEB.
F(v) rem.
=
R ( F ( Z ) ) ~ We have, v = A 8 and t h e a s s e r t i o n i s proved.
.
Thus, R i s a w e l l o r d e r i n g .
Using, now, 3.5.3.1,
we o b t a i n t h e theo-
m
3.5.3.5
THEOREM SCHEMA,
L e A F b e a u m y opmration; t h e n ,
- W + I S Sup(S, [ F ( x ) : x E A ] ) . ~ x ( x € A+ W O ( F ( x ) ) ) A A c Thus, w i t h Ax Reg e v e r y superclass o f w e l l o r d e r i n g s has a l e a s t upper bound. PROOF,
-
L e t t h e h y p o t h e s i s o f t h e theorem be s a t i s f i e d .
- A x A , by x R y Define the equivalence r e l a t i o n R C
F(x)
F(y).
L e t T be t h e f u n c t i o n g i v e n by T ' x = t R ( x ) f o r e v e r y x E A (see Def. We have, T ' x 3.2.3.24). d e f i n e d f o r a E 8 by, (1) D G ( a ) =
(6 :6
E
E
Y . L e t , now, B
'V
DF(Y) A ( F ( x ) ) 6 ' x (2)
6
G(a)g
cf
=
T*A and G be t h e o p e r a t i o n
A V x ~ y ( x , y E a+ d ' x
=
E
D F ( x ) A d'y
E
(F(Y))d,yl
3x ( x e a A d'x F(x) g'x).
That i s , G ( a ) i s t h e o r d e r i n g isomorphic t o F ( x ) , f o r X E U , d e f i n e d such t h a t t h e i n i t i a l segon t h e subset o f 'V formed by t h e f u n c t i o n s ment determined by d ' x i n F ( x ) i s isomorphic t o t h e i n i t i a l segment d e t e mined by 6'y i n F(y), f o r x , y E a .
G ( a ) f o r e v e r y x E a . Hence I t i s n o t d i f f i c u l t t o show t h a t F ( x ) G(a) i s a w e l l ordering. I t i s a l s o c l e a r t h a t i f a # b, X E U and y E b F ( y ) . Hence, G ( a ) $ G ( b ) . Thus, we may a p p l y 3.5.3.4 and then F ( x ) o b t a i n an S such t h a t Sup(S, [ G ( a ) : a E B 1 ). I t i s easy t o show t h a t a
+
so Sup(S, [ F ( x ) : x E A ] ).
-
275
A X I O M A T I C S E T THEORY
PROBLEMS 1.
Prove 3.5.3.2.
2.
Prove:
3.
Prove i n M K T R ( w i t h A x Reg) t h e f o l l o w i n g i n d u c t i o n schemas.
vR(WO(R)+lT R s T t lVSup(R, [ R x : x E D R ] ) .
(a) L e t @ be a formula, then S V T( W x ( x E DS
+
x E D S 1 ) +4,(T))A
4, [T21
-,
W T(WO(T)
W O (S*{x)) A 4,
WTlWT2(T1"TpAWO(T1)+ +
4,U I ) .
-
[ S*{x} ] ) A Sup(T, [ S*{x) :
(@,[T1l
(b) L e t @ be a formula; t h e n
w
R(WO(R) A @,
[Rl
-,
4, [ R + 1 1
A W R(WO(R) A Sup(& [Rx :
x E D R 1 ) A W x ( x E D R - + @ , [ R x l ) +@,
[Rl)+WR(WO(R) +@,[RI)).
CHAPTER 3.6
Alephs
3.6.1
A R I T H M E T I C OF ALEPHS,
I t can be proved w i t h the axiom f choice t h a t ever.y e t i s equipoll e n t t o an o r d i n a l . Thus, i n M K T C a l l c a r d i n a l s a r e c a r d i n a l s of o r d i nals. Without Ax C t h e r e may be other c a r d i n a l s . This Chapter i s mainly devoted t o study i n MKT t h e c a r d i n a l s of o r d i n a l s , c a l l e d deph. The reason f o r this name will be c l e a r l a t e r . 3.6.1.1
DEFINITION,
= {la[: aE O n
1.
In this s e c t i o n , the theorem about equipollence of o r d i n a l s w i l l be given f i r s t and, then, the corresponding theorem on alephs. Usually, t h i s l a t t e r theorem i s an immediate consequence of t h e f i r s t one. Our f i r s t theorem i s e a s i l y deduced from p r o p e r t i e s o r ordinals. 3.6.1.2
THEOREM,
(i) l a a = a -
lfi(fiE
WO A a = D f i ) .
(ii) a 5 0 V P 5 a . O # A c-O n + n A E A A W E ( t E A - f n A S I ) . (iii (iv
(v
a, b € A l + a G b V b G a . O # A c-A I - .
3a(a E A A W b ( b E A - + aGb)).
( i v ) a n d ( v ) show t h a t t h e r e l a t i o n G r e s t r i c t e d t o alephs i s a well
orderi ng
.
The proof i s l e f t t o t h e reader. 3.6.1.3
THEOREM,
276
AXIOMATIC
We have, a +
PROOF OF ( i ) . But, s i n c e a n
a +c ( a + E : t € 0 1 .
{a+t
:EEP} =
Also, ( a + E
P
277
S E T THEORY
=
a u {a+[
:E
EPI
= P.
3.3.4.8.
{a+E : E E P 1 = Hence a + P = a +c P .
0, we have t h a t a
: E €01
, by
U
PROOF OF (ii).D e f i n e 6 on 01 x 0, by s t i p u l a t i n g t h a t 6'(E,q ) = Also, d ' ( E v ) = a q + 01*q+E f or ( E , q ) E 01 x 0. Thus, D 6 = 01 x 6 .
E
a
C
(q + 1) z a 9 tt, with
a q+a = a P , then y = a
hence J r ( E , q )= y, i.e. foreaxP = a - 0 .
D
d
6-'=
E a
p.
E
Hence, D 6-l L a P . But, i f y E a, q E P (by E u c l i d ' s a l g o r i t h m 3.3.5.4); 9
P , 6 i s biunique,also by 3.3.5.4.
(iii), ( i v ) , and ( v ) a r e deduced f r o m (i) and ( i i ) .
PROOF, 3.6.1.5 PROOF,
THEOREM, Let w c a .
.
.
By i n d u c t i o n on w
us01
+
.
There-
] y ( y # 0 A a = a').
We have by 3.3.6.4
that a = w y * ~
.
+ w6i t h 0
E
u E w a n d B C w y , y .f 0. Thus, by 3.6.1.3, 3.3.5.14 and 3.3.5.13, and 3.3.5.6, we deduce, a L 6 + u ' - K = 6 + w y + w ? - ~ =( m y + wy u' = W'*K L K = aye
3.6.1.6
-
THEOREM,
(i) wca-+a=ata. (ii) a E A I A Iw1 < a + a = a + a , PROOF, a certain y
Let w z a .
a + a = w7 +,7
By 3.6.1.3,
= 0 7 . 2.2.
"7
3.6.1.5, = ,YL
.
and 3.3.5.6,
a.
we have, f o r
ROLAND0 C H U A Q U I
278
Define 6 f o r C: E 2' a s follows, 6'C: = 0 f o r € = 0; and d'€ = (7, : K E S ) , i f !. = C K ( Z T K : ~ E S )where , 7 and 6 a r e obtained a s in 3.3.6.14. T h u s 6 E w and, s i n c e C: E 2', 7,EP f o r a l l ~ € 6 .Thus, by 3.3.6.14, i t i s easy t o prove t h a t 2' Ix : x c- p A x E F N } . PROOF OF ( i ) .
( i i ) i s obtained from ( i ) . 3.6.1.8
"6
rn
THEOREM,
(i) w c a a = a a . ( i i ) a E A I A 1wI < a + a = a +
.
a .
Let w c a . By 3.6.1.5 we have f o r a c e r t a i n 7 , a = w y = Now, by 3.6.1.7, 3.6.1.3 and 3.3.5.6, ~ 1 . ~ =1 2 w * 7 2 = 2(w*7+w*7) = 2 w . 7 . 2 . 2 2 . w - 7 = 2 w.7 " a . PROOF,
(Zw)'.
3.6.1.9
THEOREM,
( i ) wC_ a V w c P -,a + p = a u P A ( a a+P " a u P ) . (ii) a, b
E
PROOF,
Let a
A1 A
14 < b
U P 3
-
w .
A 0#a
Q
# 0 A
p
# 0
+
a
9
p = a
x
p
b +a t b = a *b= b .
Then,
auPca+Pc(auP)+(auP)"auP.
Thus, a + P = a
U P
.
The rest of the proof i s s i m i l a r . 3.6.1.10
.
( wc a V wc P
THEOREM,
) A a
3
1A P # 0
+
aP
2
a
UP.
The proof i s l e f t t o t h e reader. In o r d e r t o s i m p l i f y t h e statements of a few theorems, we introduce t h e following d e f i n i t i o n . 3.6.1.11
DEFINITION,
3.6.1.12
THEOREM,
a4 b *a 5 b A af b
( i ) a + P q a + r + P < r . ( i i ) a.(P A?O( 6 a + 7 %P + 6 . (iii) a , b, c E A I A a + b < a + c + b < c +
.
.
1
279
AXIOMATIC SET THEORY
( i v ) a, b , c , d E A I A a
P { 7.
PROOF OF (i).Suppose Therefore a t P a +y
.
4
A c
+ a t c
5P;
Then y
d ,
hence a t. y
5 atP.
PROOF OF (ii). Assume t h e h y p o t h e s i s o f (ii).kle have two cases.
P, 6
CASE I ,
clear.
Then a , y a r e a l s o f i n i t e and t h e r e s u l t i s
finite.
CASE I I I P o r 6 i n f i n i t e . Then P + 6 = u 6 , by 3.6.1.9. NOW, if P U 6. On t h e a and y a r e f i n i t e , a t y i s a l s o f i n i t e and hence, c1 t y o t h e r hand, i f a o r y a r e i n f i n i t e , t h e n a + y = a u y , and hence t h e conclusion follows. 3.6.1.13
=
THEOREM,
a
5
++
30: u =
c1 C -
.
The easy p r o o f i s l e f t t o t h e reader. 3.6.1.14
THEOREM,
a
-
5*P
a
5S
.
PROOF, We a l r e a d y know t h a t a 5 P i m p l i e s a s*P (by 3.4.2.7 (i)). - P and So assume t h a t a 6 * P . Hence, t h e r e i s a f u n c t i o n 6 such t h a t 06 C
D 6-l
a.
=
Then g
( d - l * { x } ) n (6-'*{y}) 3.6.1.15
- 6-l.
A l s o g i s b i u n i q u e , s i n c e x +. q i m p l y t h a t
C
for x , q E a .
= 0,
.
D d by s t i p u l a t i n g t h a t g f x = n g - l * { x 1
Define the function g E
f o r every x E a .
Therefore, a < -9
P.
THEOREM I
(i) 6 E P O n + ZC d c Z : , ( d ' a : a E P ) . (ii) Onon (F)
+
zC F 5 O n
6
A (F* O n $ V
.+
I: F = O n C
1,
P O n . We have ZC6
= d'a : a E P I = u {&'a x Ea) :a € P l . On t h e o t h e r hand, s i n c e t h e f u n c t i o n G'E =
PROOF OF (i).L e t
aEE)
TEE
(
c1
E
i s continuous, we a p p l y 3.3.2.14 and 3.3.4.5 : a E P ) unique I and S such t h a t ,
6'.
y = Z
a
( 6 ' a : a ~ E +)
f
with5
E
6't:
.
I t i s easy t o show t h a t
I : a o l ( 6 ' a : a E P ) =g C c 6 .
i s similar. The p r o o f o f (ii)
:
t o o b t a i n f o r each
D e f i n e t h e f u n c t i o n g by s t i p u l a t i n g t h a t g'y = ( S J )
a € b ) and 5 , E as above.
6'13
, for
y E E (d'a: a
280
ROLAND0 C H U A Q U I
3.6.1.16
WO(R) A R C -W+DR
THEOREM1
5 On.
PROOF, L e t R C W be a w e l l o r d e r i n g . D e f i n e t h e f u n c t i o n F by We have t h a t R a ) f o r every a E On s t i p u l a t i n g t h a t F'a RI ( R a F'a E V f o r e v e r y a E O n . By 3.3.1.22, R = u {F'a : a E O n I ; hence D R = u { D F ' a : a E O n I , and D Ffa n D F'P = 0, f o r a + P . Also, F'a E WO. By Define, a l s o 3.5.1 we o b t a i n a Gfa = E N (F'a) and H'a = D(G'a)(= I t i s c l e a r t h a t b y 3.6.1.15, J'x = ( ( G ' a ) ' x , a ) f o r x E D Flu.
-
+
.
m.
DR= C H5On.D J
3.6.1.17 THEOREM1 a' x b' A a', b' 5 t ) .
C;
c
5axb
+
3 a'3 b ' ( a ' C - u A b' C- b A
PROOF, Assume t h a t t 5 a x b. Then C; = c c a x b . - b, A l s o and 6 ' = D c . I t i s c l e a r t h a t a' 5 a and b' c hence E .S A' x b ' . Let and b'
<*
E
E ,<
L e t a' = D c-l = c C - a' x b ' ;
x , y ) = x and g' ( x , y ) = y f o r ( x , y ) E a x b. Then a' < -6 we o b t a i n a' 5 E and b' 5 E. c = l . By 3.6.1.14
*
c = C;
-g T h i s s e c t i o n concludes w i t h some p r o p e r t i e s o f t h e c a r d i n a l s o f t h e s e t s R a f o r a E O n . These c a r d i n a l s a r e c a l l e d beths. R e c a l l t h a t R i s s t u d i e d i n 3.2.3 and 3.3.1.22.
3.6.1.18
3.6.1.19
DEFINITION,
Be
=
{ I H a 1: a E On 1 .
THEOREM,
(i) R a x R a C R (a + 2 ) .
(ii) a = ua
-+
Ra
x
Ra SRa.
l e a v i n g t h e p r o o f o f (i) t o t h e readPROOF, We s h a l l p r o v e (ii), L e t a = U a and z E R a x R a . Then z = ( x , ) w i t h x,y € R a. Thus px = 0, p y = 7 w i t h P , 7 E a. B u t P Z = ( P X U p y y + 2 = ( P U T ) + 2 a Hence z E R a . er.
3.6.1.20
THEOREM1
( i ) R a < R P VRP 5 R a . (ii) ozAc - D R - ~+ n (iii) a, b E B e + a
AEA
A
v
~(XE+ AnA
5
x).
V b Q a.
(iv) 0 Z A C -B e + 3 a(a E A A V b(b E A Thus, beths a r e a l s o w e l l ordered b y 6 The p r o o f i s l e f t t o t h e reader.
.
+
a
Q
b)).
AXIOMATIC S E T THEORY
3.6.1.21
281
THEOREM I
( i ) v E w+ R u
.
(ii) R w= w
E
FN.
PROOF OF (i).Define g E ww by r e c u r s i o n s t i p u l a t i n g t h a t g I 0 = 0 g f v + I = 29'" By 3.6.1.7, we have t h a t 2" ( x : x C- a A x E F N } , where
6a
.
"4 a
6
i s the function
For V E W , we
d e f i n e d i n t h e p r o o f o f 3.6.1.7.
have t h a t 2' = Pv. D e f i n e h E W u { R v w : u E1 ,~ by s t i p u l a t i n g t h a t dv h'O = 0 and Iz'v +l = 0-l o (hlv)*. We have t h a t R'v = g ' u f o r everyl 9'V hv 2 9 '= V E W ; because 0 = R 0 = g'O=O, and R(u t 1 ) = PRv = Pg'v = hO h,* 6,'V 9'(vt1)Ew f o r e v e r y V E W ; ( i ) i s proved. PROOF OF ( i i ) . I t i s c l e a r t h a t w g R w and hence w < R w . o n l y have t o show t h a t R w 5 w We have, by 3.3.1.22 and 3.4.2.9
We
.
R w = u ( R v :u € w ) < * Z ; [ R u : u E w ] .
;C, C
~
By t h e p r o o f o f (i), we have t h a t (Rv V E W ) = ( g ' v : v e w ) .
[Rv: ~
~
Hence, R
V E W ]
'
(ii).B u t X [ g ' v : V E W ] = } <= w ~ ( b y( 3 . 4w. 3 . 1~0 ) . { ~ we g e t R w<, w . =
= C [ g ' v : u E w I , by 3.4.3.3
~
w 5 * w.
3.6.1.22
Hence
~
~
~
~
By 3.6.1.13
:
~
~
~
THEOREM.
(i)a>w-+Ra+cRa = R a x R a = R a . (ii) ( a-> w V P
R(a
U
PI.
(iii) a, b E B e A
3 w ) A a # 0A -
101
P # O+Ra
tc R P
= Rax RP =
< b A O # a < b + a + b = a * b= b .
PROOF OF (i).L e t a
1. w
; then R a
> w > 2.
Hence
R ~ < R c x R+a~ Z R a x R a,. Thus, we s h a l l prove by i n d u c t i o n t h a t R a x R a < R a .
(1)
=
ua. By 3.6.1.19,
( 2 ) a = P + 1 and R
(RP
'c
R P )2
RP
2
PROOF OF ( i i ) . have, by ( i ) .
x RP
R a x R a S R o ; hence R a x R a < R a.
We have,
R
c1 x
R a = RP2 x RP2
R a. Assume t h a t , a
>_ w
or
P 2 w , and a
it 0 f
0.
We
}
:
u
ROLAND0 CHUAQUI
282
PROBLEMS 1. 2. 3. 4.
Prove 3.6.1.2 Prove 3.6.1.12 Prove 3.6.1.13 Define a =n {t : w s t A E # w l Prove: a ) k s n A € # n - € S w .
.
b)wSClAo#n.
c) a 2 n A a f n + a S w . d)Ac_On A O E A A W E ( E E A - E U h * u E A ) + S l
e)t,vEn+t+v, f ) n5 22
.
5.
Show, a) R(o+ a)
. I
+ l € A ) A Wg(dEwA AIn(g)+
C -A .
t-v, t ' ~ n .
.
Z i [ R ( w + t ): t 5 a I
b ) a = u a # 0 + R ( w + a ) = X C [ R ( w + t ) :teal
!.
3.6.2
.
HARTOG'S FUNCTION,
This s e c t i o n i s devoted t o an o p eratio n which p l a y s an important auxi l i a r y r o l e i n s e t theory without choice: H ar t o g ' s o p e r a t i o n . 3.6.2.1
DEFINITION,
H (A)
=
{t : t
E On
T h i s ope ra t i on was introduced by Hartog. 2 3.6.2.2 THEOREM, H ( a ) 5*'2
.
E 5 A)
.
AXIOMATIC SET THEORY
283
2 PROOF, We have t h a t 2' = P ( a x a ) . We d e f i n e 6 E W O , and h'h=h, i f h E W O . s t i p u l a t i n g t h a t : 6'h = 0, i f h
c l e a r t h a t 6"r E On and 6'h 5 a f o r a l l h c a x a . Hence On t h e o t h e r hand, assume t h a t [ E H ( a ) , i.F. E 5 a. Then,
r:
V by It i s
c H(a). = 2 c a
9
-
f o r c e r t a i n g, a ' . L e t h = ((g'v, g ' { ) : { c ct}. Then h c a x a an 6'h =E. Thus we have proved t h a t H ( a ) = D&:Ti. T h e r e f o r e E(a) < *
d
3.6.2.3
*
THEOREM,
( i ) H ( a ) E On. 2 (ii)~ ( a4 ) '22 , ( i i i ) H(a) $ a (iv) (v)
0
o(
.
5
~ ( a-+) P
a.
a< a + H(a).
-
(vi) H(a) 4 H (H(a)). (vii)
P
E
H(a)
P
o(
H(a)
.
I t i s c l e a r t h a t H ( a ) i s t r a n s i t i v e and H ( a ) PROOF OF ( i ) . Also, by 3.6.2.2, H(a) E Y. Hence by 3.3.1.13, H(a) E On.
PROOF OF (ii). By 3.6.2.2,
3.4.2.10
(ii), and 3.4.2.12.
PROOF OF (iii). S i n c e H ( a ) E O n , we have t h a t H ( a ) ply H ( a ) E H ( a ) , c o n t r a d i c t i n g H ( a ) E W . Since H ( a )
PROOF OF ( i v ) . and, hence P 5 a . PROOF OF (v).
E
On,
50".
5 a would im-
P *H(a) implies that P
E
H(a),
We c l e a r l y have a 5 a + H(a). On t h e o t h e r hand, a 2 H ( a ) , c o n t r a d i c t i n g (iii).
a = a + H ( a ) would i m p l y
PROOF OF ( v i ) . By ( i i i ) we o b t a i n H ( H ( a ) ) 8 H ( a ) .
H ( H ( a ) ) E O n , ( v i ) i s o b t a i n e d by 3.6.1.2
(ii).,
Since H ( a ) ,
. , weThen,have 5 a
PROOF OF ( v i i ) . Suppose E H ( a ) and l e t H ( a ) 5 P c o n t r a d i c t i n g ( i i i ) . Hence H ( a ) S P . Since H ( a ) , P E O n P Ma). The i m p l i c a t i o n i n t h e o t h e r d i r e c t i o n i s t r i v i a l .
=
Thus, we o b t a i n a s t r i c t l y i n c r e a s i n g sequence o f i n f i n i t e alephs, namely,
I W I < I H ( 4 ( < IH(H ( a ) ) I <
...a
ROLAND0 CHUAQUI
3.6.2.4
THEOREM,
-
( i ) U E U + H ( U )= v ( i i ) ( a E FN
-
+ 1. E w ) A (a E
H(a)
FN
++
H(a)
(iii) ~ $ € F N A U # U + ~ ~ - H ( U = )w . Recall t h a t i n f i n i t e a's w i t h a a $ w and w $ a (3.4.1.11). PROOF,
(i) is trivial.
(iii) t o t h e reader. (1) a
E
FN +H(a) -a
-
H(a)
2
shown,
a +c 1.
a
+C 1
+
1).
We s h a l l prove ( i i ) l e a v i n g t h e p r o o f o f
+ 1, a +c 1 and t h a t H ( a )
2w
Hence H ( a ) = a, c o n t r a d i c t i n g 2.7.3.7
(2) H ( a ) =
tc
# a +c 1 a r e a l s o c h a r a c t e r i z e d by
i s o b t a i n e d from ( i ) . Suppose t h a t H ( a )
a
H(a)
E w
Suppose now t h a t H ( a ) E w o b t a i n t h a t a E FN, by 2.7.3.9. (3) H ( a ) E w + a E F N
.
Then,
H(a)
tc
1=
(i).Thus we have
.
. Hence, Since H(u) $ finally,
a, by 3.6.2.3
(iii). we
.
( l ) , (2), and (3) g i v e (ii). 3.6.2.5
THEOREM,
(i) w S a + l
3b(a.(b.(a+H(a)). (ii) Wa 3 c ( a .( c A 1 3 b a .( b .( c) Wa 3 c ( a < c A 1 3 b a < b < c ) (iii)
. .
Thus, every c a r d i n a l has an immediate successor. PROOF OF (i).Assume t h a t w < a and a 5 b prove t h a t b = a + H ( a ) o r b = a . We have, b = c ; d We have t o c o n s i d e r two cases.
5
.
We s h a l l a + H(a) 5 a and d 5 H ( a ) .
with c
CASE I , d = H ( u ) . I n t h i s case, b = c +cH(a). Since w < a, we have t h a t w c _ H ( a ) . Hence, c t H ( a ) +c H ( u ) E b +c H ( a ) 2 u +c H ( a ) . b = c + H(u) Thus, we g e t
C
b =a
C tC H(a).
5
a +cd. CASE II, d # H ( a ) . I n t h i s case, we have t h a t b = c +c d By 3.6.1.13, s i n c e H ( a ) E O n , we have t h a t d = c1 f o r a c e r t a i n a. By
AXIOMATIC S E T T H E O R Y
285
5
a, i.e. a = e +C a f o r a c e r t a i n e . I f a ? w , then a = e +c a +c a a tC a 2 b . Also, i f a E w , since w < a, we have a +C d = a and hence b 5 a. Thus, we obtain under t h e hypothesis of t h e case, t h a t b2aandb.a. 3.6.2.3
(iv), a
Y
If a
PROOF OF ( i i ) .
k
w
, we take c
= a +c H ( a ) .
I f w $ a, we take c = a +c 1; c s a t i s f i e s t h e theorem by 3.4.1.11. 3.6.2.6
1 3 b ( H ( a ) .( b.( H ( H ( U ) ) .
COROLLARY,
.
PROOF, I f H ( a ) E F N , then H ( H ( a ) ) = H ( a ) + 1 and the result is c l e a r , I f H ( a ) 4 F N , then by 3.6.2.5, t h e r e i s no b w i t h H ( a ) 4 b 4 H(a) + H ( H ( a ) ) . B u t , s i n c e H ( a ) , H ( H ( a ) ) E O n , we have, H t a ) + H ( H ( a ) ) H ( H(a))
.
2
3.6.2.7
THEOREM,
( i ) a $! F N V b
E
FN
+
H(a
+C
b) = H ( a )
U
H(b)
2
H ( a ) + H(b).
( i i ) a 9 F N + H ( a +c a ) = H ( a ) . PROOF OF ( i ) . Assume t h a t a o r b i s i n f i n i t e . We have t h a t H ( a ) 5 H(b) o r H ( b ) c H ( a ) . Suppose t h a t H ( a ) 5 H ( b ) . We s h a l l prove t h a t H ( a +c b ) = HTb). I t i s c l e a r from Def. 3.6.2.1, t h a t H ( b ) C H ( a +c b). Suppose, now, t h a t E E H ( a +c b). Then E 5 a +c b. We have two cases. CASE I , E t 5 b y by 2.7.3.9.
.
Since b 4 F N (because H ( a ) c H ( b ) ) we have t h a t Hence E E W b ) .
Ew
CASE I I l w c E . T h e n E = n + { w i t h q < a , { < b . A l s o E = q u { , Since-nut = TI o r n u t = P we have t h a t E a o r E 5 b. B u t , by 3.6.1.9. since H ( a ) c H ( b ) , E 5 a implies t 5 b. T h u s , i n any case E E H ( b ) .
( i i ) is obtained from ( i ) . 3.6.2.8
.
<
THEOREM,
(i) ( a $ F N V b $ Z F N ) A a # O A b # = H ( a ) H(b).
O + H ( a x b ) =H(a)UH(b)
( i i ) a $! F N + H ( a x a ) = H ( a ) . PROOF, Assume t h a t a o r b i s i n f i n i t e and t h a t both a r e n o t empty. As i n the previous proof, we may assume t h a t H ( a ) c H ( b ) and we show t h a t H ( a x b ) c H ( b ) . Let t E H ( a x b ) ; thus E 5 a x b . We have two cases.
286
ROLAND0 C H U A Q U I
t
CASE II
Ew
.
Then C;
5 b, because b $ F N .
Thus,
t
E H(b).
By 3.6.1.16, t h e r e a r e a ' , b' such t h a t a t c - a, CASE II, W CC;. t I; a' x b t , and a ' , b' 5 C;. By 3.6.1.12, a' = a c _ E and 6 ' = P c_ E c e r t a i n a , P . Now, ci 5 a and P < b. Since H ( a ) gH(b), we g e t a , P E H(b). Since E 30 and a x P 2 E , we have t h a t a o r P a r e i n f i n t e . Hence, a x P = a Up. Thus E 5 a U P 5 6 and E E H ( b ) .
b' C b,
for
(ii) i s o b t a i n e d from ( i ) . Not much i s known about H('2). F i n a l l y , u s i n g H a r t o g ' s f u n c t i o n , we show a theorem due t o Specker. 3.6.2.9
THEOREM,
a $2 F N A v E w + a2 $ ' a .
PROOF , Assume t h a t a $2 F N , v E u
contradiction.
, and P a S B ' a .
We s h a l l g e t a
I n o r d e r t o do t h i s we s h a l l d e f i n e a b i u n i q u e f u n c t i o n
g E ba , where 6 = { h : 3 a ( h O h - ' ( = Ih'P : P E D h l ) .
E "a A
h-'
Dh-la
E
11,
satisfying
9th
We d e f i n e by r e c u r s i o n t h e f u n c t -
Suppose such a g has been defined.
i o n j g i v e n by j ' t ; = g ' ( j l t ) f o r e v e r y [ E H ( a ) . Hence j E H ( a ) a , and by t h e p r o p e r t i e s o f g, j i s biunique. Thus, H ( a ) 5 . a, c o n t r a d i c t i n g 3.6.2.3
J
(iii).
The r e s t o f t h e proof w i l l be concerned w i t h t h e d e f i n i t i o n o f g. Let p = n
(K
K
:KEWA 2
2 K
V
3.
Thus we have, f o r every
K
z p , zK3~'.
f o r an hE L e t c E ' a be a b i u n i q u e f u n c t i o n . Let, also, h e b , i.e., o r d i n a l a and h i s biunique. I n o r d e r t o d e f i n e g ' h we s h a l l c o n s i d e r several cases. CASE Il D h c p . Thus, l e t
K
= n {rr :
E(t
I t i s c l e a r t h a t Dh" E D h
+
c,
f
h't)).
Dc-' and Dh-' # p . We d e f i n e g ' h = c K . I t i s 5 p =
c l e a r t h a t g ' h $2 D h - ' . Hence = " € a . Since h i s biunique, Dh" = TI. -Dh'l C - a , we have t h a t 6 ' d E v a . P Dh" Z~ 2 " f o r a c e r t a i n k. Since d C and s i n c e 6 i s biunique, we have t h a t ' a 2 6 *PDh'' = A l s o ' D h- 1 = T h i s i m p l i e s t h a t d* P D h-' g V D h - ' , and, hence, t h e r e i s a 2" 3 n L e t e be t h e f i r s t such d ( i n t h e o r d e r d C - Dh-' such t h a t 6 ' d $2 ' D h - ' . p c -D h
CASE 1 1 ,
'.
induced by k-'
6'4
from t h e n a t u r a l o r d e r i n g o f 2
4 Dh-')). CASE 1 1 1 ,
We d e f i n e g ' h =
6'eq 4 D h - ' .
wc D h = a . By 3.6.1.4
" ).
L e t , now q = n {
and 3.6.1.10,
we have t h a t
:
AXIOMATIC SET THEORY
c1
=k
i.e.
V
287
f o r a c e r t a i n k. L e t h ' t ( h ' ( ( k ' E ) ' K ) : K E v ) . Then h E a(va), NOW, is equal t o h a p p l i e d t o t h e K t h element o f k ' t E 'a. (h' E ) ' K c1
i f X E 'a w i t h D
x-' S o h - ' ,
then there i s a
t
E
a such t h a t x = h ' t ;
t = 0, t such t h a t ( k ' t ) ' K = ~ - ' ( X ! K ) f o r each K E V . D e f i n e i f h' [ # D 6 - l , and E = d - l ' h ' t ; , o t h e r w i s e . We have, TIE ' P a . Let d = {h't :h't # El. Since h' 5 C- a we have t h a t d 5 a. Hence, X'dEva. namely t h e
a'
We s h a l l o b t a i n a c o n t r a d i c t i o n from D(T'd)-' c D h - l , showing thus, D(x'd)-' g D h - l . Suppose t h a t D(T'd)-' C- Ohy1. We have, D(7'd)-' = {(d'd) :nEvl. Hence T ' d = h' t o f o r a c e r t a i n E,. Thus, d' t o = d'l'h' E = d. T h a t i s , h ' t O 4 d kt' t o € d, a c o n t r a d i c t i o n . Therefore 0 D(X'd)-' p D h - ' . L e t K = n (1) : (x'd)'~$ Oh-'} and d e f i n e g ' h = ( d ' d ) ' K # D h-'.
-
Thus we have completed t h e d e f i n i t i o n o f g and, hence, t h e proof of t h e theorem. 9 3.6.2.10
PROOF I
COROLLARY,
a
9
FN
--*
a
tC1
4 2'
.
We know by C a n t o r ' s theorem, t h a t a .( '2.
Suppose, now, t h a t a
tc 1
= '2.
Since a
2a.axaZax2.at
4
, we
Hence a +c 1
have,
a ~ a t c l U=2 , C
c o n t r a d i c t i n g 3.6.2.9.
FN
Hence, a +c 1 # "2.
=
PROBLEMS
2.
Deduce C a n t o r ' s theorem, a iC a2 and i t s s t r o n g e r v e r s i o n W a W v ( v ~ ~ A u # F N - + <'2) a ~ v from 3.6.2.9.
5'2.
CHAPTER 3.7
The Local Axiom o f Choice and t h e Generalized Continuum Hypothesis
3.7.1
FORMS OF THE LOCAL A X I O M OF CHOICE1
T h i s c h a p t e r i s concerned w i t h a weaker form o f t h e axiom o f choice, namely, t h e l o c a l axiom o f choice. Since we have t o deal w i t h t h i s sentence v e r y o f t e n , we s h a l l i n t r o d u c e an a b b r e v i a t i o n f o r it. 3.7.1.1
DEFINITION,
AC
++
Wu(Aa * 3 b W x ( x 6 a
-+
x n b = 1)).
R e c a l l t h a t A u means t h a t a i s a s e t o f d i s j o i n t non-empty sets. I t i s c l e a r t h a t AxC i m p l i e s A C . However, i t has been shown t h a t A C can neit h e r be proved n o r d i s p r o v e d i n M T K R and t h a t t h e same i s t r u e f o r AxC i n MTKR t A C . T h i s s e c t i o n i s devoted t o t h e p r o o f t h a t some b a s i c s e t - t h e o r e t i c a l p r i n c i p l e s a r e e q u i v a l e n t t o A C . We a l r e a d y know, by 2.7.3.11, t h a t i f a i s f i n i t e and A u , II ( x : x € u ) f 0. However, o m i t t i n g t h e hypothesis a E F N , t h i s p r i n c i p l e i s equivalent t o A C . 3.7.1.2 ( i ) AC
-
THEOREM,
Wu(A a *
n(x:
x E a ) f 0).
(ii) A C * Wa(O4 a * I I ( x : x E a ) + ( i i i ) A C * V a I I ( x : O # x c a-)
0).
+O.
The a s s e r t i o n on t h e r i g h t o f ( i i ) i s c a l l e d t h e PhincipLe ad Choice.
PROOF,
I t i s obvious t h a t ,
(1) wa ( 0 4 a + I I ( x : x E a ) and
(2) W a ( 0
4
#
0) * w a n
(
x:Of
x c a)+ 0
a * II( x : x 61)# 0) * W a ( A a * II ( x : x E a ) # 0).
Suppose t h a t ,
W a ( A a * n ( x : x E a ) #O), and A a. L e t 6 E Il ( x : x E a ) and b = 6*a. x n 6 = 1. Hence we have proved, (3) W a ( A a * ~ ( x : x E a ) f O ) - t A C .
2 88
I t i s c l e a r t h a t f o r each x E a ,
AXIOMATIC S E T T H E O R Y
289
Suppose, now, A C a n d 0 4 u. Let c Cx x {XI : x E a ) . Then A c . Let b be such t h a t x n b = 1, f o r X E C . T h u s , b E TI ( x : x s a ) . Hence, we have proved, (4)AC-+Va(O+a+TI(x:xEa) f 0 ) .
F i n a l l y , assume t h a t WarI(x: O # x C -a ) # O ,
a n d 0 4 a. Consider t h e t r a n s i t i v e c l o s u r e o f a, T a . We have a C T a and TLen 6 I u E i f x E a , then x C T a and x f 0. Let 6 E II ( x : 0 f x C- T a ) II ( x : x E a ) . T h u s , we have proved,
(5) W a r I ( x :
-
O + X C U )
.
# O +
Wa(Oqa+n(x:xEa)fo).
( I ) , (21, ( 3 ) , ( 4 ) , and ( 5 ) y i e l d our equivalences (i), ( i i ) , and (iii). 3.7.1.3
THEOREM ( W E L L - O R D E R I N G
PRINCIPLE).
PROOF, Assume t h a t I I ( x : 0 $: x C - a ) f 0. t a i n t h a t t h e r e i s an ct such t h a t a = a .
Applying 3.5.1.1
we ob-
Suppose, now, t h a t a = a. We d e f i n e g E II ( x : 0 f x C - a ) by stipulatingthatg'x= d'(nf-'*x) for O i t x c a . H e n c e , I I ( x : O # x C-a ) f 0 . Thus we h a v e proved ( i ) . ( i i ) i s obtained from (i), by 3.7.1.1. We now turn t o t h e generalized d i s t r i b u t i v e laws.
PROOF,
reader.
.
We s h a l l prove ( i ) , leaving t h e s i m i l a r proof of ( i i ) t o t h e
Assume A C and l e t a
E
V.
I t i s easy t o show without A C t h a t
.
b -n{u {a'(i,j):jEcl : i ~ b l (1) u { n { a f ( i , d f , L ) : i ~ b }6: E el c
Suppose, now, t h a t x
E n { U
{ a f ( , L , j ): j E c > : i E b 1 .
Then, f o r ev-
2 90
ROLAND0 C H U A Q U I
e r y i ~ b t ,h e r e i s a j E c such t h a t x E a t ( i , j ) . L e t d ' i = { j : j E c A X E at( i,j ) I , f o r e v e r y i E b. Then, d ' i # 0 f o r e v e r y i E b. Hence by A C and 3.7.1.2, I I ( d ' i : i E b ) # 0. L e t E I I ( d ' i : i E b ) We have t h a t x E a t ( i , d l i )f o r e v e r y i E 6. Hence x E n { u t ( i , d r . L ): i E b } . Since 6 E bc , b we have t h a t x E U I n { u t ( i , j j t i ) : i e b l : 6 E c}. Thus we have proved,
.
(2) n ( u { ~ ~ ( i , j ) : j ~ c } : i € b ) c u{ {~n' ( i , ~ ~ i ) : . L ~ b E) b: c}. , j
-
Hence, t h e i m p l i c a t i o n from l e f t t o r i g h t i s proved, Suppose, now, t h a t t h e g e n e r a l i z e d d i s t r i b u t i v e l a w on t h e r i g h t o f (i) i s s a t i s f i e d and l e t u Z 0 be a s e t such t h a t 0 4 a. Define b E a x u a Y by s t i p u l a t i n g t h a t : b t ( . L , j ) = a, i f j E . L E a ; and b t ( i , j ) = 0, otherwise. Then, we have, n{U { b t ( i , j ) : j E U a l : i E a l
= a#O.
Hence, by t h e d i s t r i b u t i v e law, u = u(n {bt(i,dti)
: i ~ a :ld
E
a u a1
.
Thus, t h e r e i s an 6 E such t h a t n { b t ( i , d t i ) i E a 1 a, i.e. f o r a l l L E U , b t ( i , a t i ) = a . Hence, by t h e d e f i n i t i o n o f b, d ' i e i f o r e v e r y L E U ; i.e. ~ E T I ( X : X E U ) # O . Thus, by 3.7.1.2,
.
we o b t a i n t h e i m p l i c a t i o n from r i g h t t o l e f t .
PROBLEMS
Prove: 1. AC
2.
-
W x(h E P ( YxV)
(a) W R ( R c -w x
w
(b) W R ( R-c W X W
3 B W x(xEA
3.
Prove 3.7.1.4
+
(ii).
+. --t
+
3 d(,j
E
-
DxDx-lA
6 C- x ) .
3 F(F c - R A ~ ~ w ( F ) W) - On 3F(FC -R A D R W ( F ) ) -
x n B = 1)).
W A ( ACW A A A
-+
AXIOMATIC S E T THEORY
3.7.2
291
MAXIMAL PRINCIPLES,
T h i s s e c t i o n i s devoted t o p r o v i n g t h a t c e r t a i n maximal p r i n c i p l e s a r e equivalent t o A C . 3.7.2.1
THEOREM ( M A X I M A L P R I N C I P L E ) .
(i) PO(R) A a c D R A wb(b c - a A SO(R1b)
LubR(x,b)) A 3 a a = a
3x(xEaA
+
y(gEa A x G R y
(ii)v b ( b c -aASO(IN1 b) + u b
v
q ( y E a A x C_ y
+
x = Y)).
(iii)( Z o r n ' s Lemma).
w q(qEa A
3 x(xEa A
x c -y
AC +
-
+
E a)
A 3,
3 X ( X E UA .+
x = q)).
a = a + ~ X ( X E UA
W a ( W b -( b c a A S O ( I N ~ b ) + u b ~ a ) ~
x = Y)).
IfS O ( R l b ) , we say t h a t b i s an R- chain. I f x G R q i m p l i e s x = y, f o r e v e r y y E a , we say t h a t x d R-maximal i n a. PROOF OF (i). Assume t h e h y p o t h e s i s o f (i).Since a = a f o r a c e r t a i n a, by 3.7.1.3 (i), t h e r e i s a g E Il ( x : 0 # x c - a ) . L e t c E - a and
define that,
6 E "V
Define,
by s t i p u l a t i n g t h a t 6 ' 0 = c and h ' x = g ' x f o r x # 0.
now by r e c u r s i o n on o r d i n a l s t h e f u n c t i o n H by s t i p u l a t i n g
H'E = g ' ( C x : x E a A l y ( w P ( P E E + H f P G R y ) A ~ j < ~ x ) l . Since e v e r y R - c h a i n subset o f a has a l e a s t upper bound i n a, we have t h a t H'C; # c i m p l i e s t h a t H'P f c f o r e v e r y P E E and t h e l e a s t upper bound
6{HIP
P
€.!I
:P
+ P I .
exists.
Thus, ifH'E +. c f o r e v e r y
i s i m p o s s i b l e because a E V
x =
>R
Also, H'.!
6 {HIE
6 {H'P 1:
E On
and O n 9 V
.
:P E E I ,
, we Let
Hence H I P
for
$. H ' P l
would have ' O n ( H - ' ) , = n
{E : H f t
= c}.
that
Then
: E € P I i s t h e r e q u i r e d R -maximal element o f a.
PROOF OF ( i i ) : By ( i ) , s i n c e I N i s a p a r t i a l o r d e r i n g and u b i s t h e l e a s t upper bound o f an I N c h a i n b.
-
PROOF OF ( i i i ) : The i m p l i c a t i o n from l e f t t o r i g h t i s deduced from
(ii). Let
Assume, now, t h e r i g h t hand s i d e and l e t a be a s e t such t h a t 0
b = { 6 : 6 E D 6 u a A D 6 c a A Vx(xEDd+d'xEx)}. Recall t h a t f o r f u n c t i o n
dl. d2
we have t h a t ,
4
a.
ROLAND0 CHUAQUI
2 92
-
Suppose, now, t h a t c C b and t h a t c i s an I N chain. I t i s c l e a r t h a t g = U c i s a function-and t h a t Dg = U D * c C- a. Also, i f x E D5 , then x E D f o r an 6 E c and hence, g'x = ~ ' x x.E Therefore g E b. By the maximal p r i n c i p l e , t h e r e i s an IN-maximal element h c b . We s h a l l show t h a t D h = a . In order t o argue by contradiction suppose t h a t x E a - D h f o r a c e r t a i n x. Since 0 4 a, x f 0; hence, t h e r e i s a yEx. Let 77 = Then, h C 7T and 'Ti E b, c o n t r a d i c t i n g the maximality of h. h u (( y,x)). Therefore, hEII(x:xEa)+O.~
THEOREM (TERCHMULLER-TUKEY).
3.7.2.2
X A y g FN-t
Y E
WafW x ( x E a
a)) -i 3 x ( x E a A W y ( y E a A x c -y+ x
=
Y)).
-
W y(y C_
I f x E a i f and only i f every f i n i t e subset of x i s in a, we say t h a t
a LA LndudLve.
The proof i s l e f t t o t h e reader.
PROBLEMS Prove:
-
1. 3.7.2.2. tf h W A ( P O ( h ) A A C- h A S O ( 6 ) 2. AC k! A ' ( S O ( A ' ) A h' c - A ' C- h + h' = A ' ) ) .
3.
AC
++
3.7.3
W a(0
4
a
+
+
3 ht(SO(h')A
3 b(AbA b c - a A W c(Ac A b c - c c- a
4 C - h' C h A
+
b = c)).
C A R D I N A L EQUIVALENCES
In this s e c t i o n we s h a l l study some a s s e r t i o n s about c a r d i n a l i t y t h a t a r e equivalent t o A C . W e s h a l l use extensively Hartog's function H defined i n 3.6.2. 3.7.3.1
THEOREM,
-
(iii) AC
A X I O M A T I C S E T THEORY
v
a'd b V c ( a $Z F N A a = b +c c
+
293
a- b V a=c).
PROOF OF (i).Suppose t h a t a = a. Hence, s i n c e H ( a ) E O n (3.6.2.3 ( i )we ) have by 3.6.1.2 t h a t a S H ( a ) o r H ( a ) 5 a. Thus, t h e i m p l i c a t i o n from r i g h t t o l e f t i s proved.
5
a. By 3.6.2.3 Assume, now, t h a t a 5 H ( a ) o r H ( a ) ( i i i ) , H ( a ) 6 a. Hence, a S H ( a ) . Since H ( a ) E O n , b y 3.6.1.12 a = a f o r a c e r t a i n a. Thus, t h e o t h e r i m p l i c a t i o n i s proved.
PROOF OF (ii).From l e f t t o r i g h t i t i s proved by 3.6.1.2. r i g h t t o l e f t , by (i). PROOF OF (iii). From l e f t t o now show t h e i m p l i c a t i o n from r i g h t of (iii). L e t a be given. If a E So suppose t h a t a $ F N . Then a +c
From
We r i g h t i t i s o b t a i n e d by 3.6.1.9. t o l e f t . Assume t h e r i g h t hand s i d e F N , t h e n c l e a r l y a = u f o r some V E W . H ( a ) 4 F N . Also, a +c H ( a ) = a +cH(a)
Hence, b y t h e hypothesis, a + H ( a ) = A o r a + H ( a ) = H ( a ) ; i.e. o r a 5 H ( a ) . Thus, a p p l y i n g ( i ) ,we o b t a i n an a w i t h a = a
.
H(a) 5 a
THEOREM,
3.7.3.2
- -
(i) 3 a a =
(ii) AC
(iii) AC
-
CY
4
a E F N V H(a) a t H(a). 3 a +c b+a 4 b ) .
W a V b ( a +c a
W aV bW c ( a +c b .(
A
+c c
--+
b .( c ) .
WaWbWcWd(a.(bAc
(iv)AC-
PROOF OF ( i ) . Suppose a = c1 3 w . a <, H ( a ) . Hence a + H ( a ) =-(a), 3.7.3.1, t i o n from l e f t t o r i g h t i s proved.
C
cO(b+Cd).
Then by 3.6.2.3 (iii) and Thus, t h e i m p l i c a b y 3.6.1.9.
.
So assume t h a t a $Z F N and I f a E F N , t h e n a = v f o r some v E W H ( a ) a + H ( a ) . But H ( a ) 5 a + H ( a ) . Hence H ( a ) = a -t H ( a ) . By a = c1 f o r a c e r t a i n a. 3.6.1.13,
A C i m p l i e s t h e r i g h t hand PROOF OF ( i i )AND ( i ii ) . By 3.6.1.12, So s i d e o f ( i i i ) , which, on i t s t u r n , i m p l i e s t h e r i g h t hand s i d e o f (ii). implies AC. we s h a l l prove t h a t t h e r i g h t hand s i d e o f (ii) Assume, then,
b).
W aWb(a+a.(a+b+a.( Thus, we have,
H(a) But,
H(a) $
A
+ H(a) 4 H(a) + (3.6.2.3
(iii)).
a
.
-f
H ( a ) .( a
Hence,
.
H(a) + H ( a ) # a + H ( a )
Thus, assume t h a t a $ F N . Hence H ( a ) = If a f F N , then a = v EO Thus, by (i), we o b t a i n a = a E H ( a ) t H ( a ) , by 3.6.1.9 and 3.6.2.4 (ii).
ROLAND0 C H U A Q U I
2 94
On.
PROOF OF ( i v ) . 3.6.1.11.
The i m p l i c a t i o n f r o m l e f t t o r i g h t i s deduced from
Assume, now, t h e r i g h t hand s i d e o f ( i v ) . So assume t h a t a 9 F N and t h a t
(1) H ( a
x w
)
4a
x w +
I f a E F N , then a = v
Ew.
H(a x w )
(v), a x w 4 a x w + H ( a x w ), we o b t a i n by t h e hypotheSince by 3.6.2.3 s i s and (1) t h a t a x x + H ( a x o ) ~ a x w x 2 + H ( a x w ) x 2 P .
a x w + H ( a x w ),
Thus (1) i s f a l s e ,
a contradiction.
H(a
x w )
since a
5
= a x w + H(a
i.e.
).
Hence, b y 3.6.1.13, a = a E On. a x w , a g a i n by 3.6.1.12, x w
*
a x w
0
P
E
On
Thus
THEOREM,
3.7.3.3 ( i) 3
CI
--
a = a *a
E FN V
H(u)
# a x H(a) .
b). ( i i i ) AC W uW b W c ( a x b o< a . x c b.(c). ( i v ) A C * Wa Wb Wc wd(a < b A c .< d ax c ( i i ) AC
WaWb(axa.(axb*a.(
+
+
.( b x
d).
The p r o o f i s l e f t t o t h e reader. 3.7.3.4
--
THEOREM,
a E F N V a + H(a) = a x H(a). (i) l a a = c1 (ii) 3 a a = c1 a E F N V 2(a + H(a)) = a + H(a). ( f i t ) A C c-) W a W b ( ( a ~ F N V b ~ F N ) A a # O + b - t a + b ~ a x b ) .
(iv)
AC
-
PROOF OF ( i ) o b t a i n e d from 3.6.1.9
w a(a 9
FN
--*
a = 'a).
AND
( i f ) . The i m p l i c a t i o n s from l e f t t o r i h t a r e and 3.6.2.3 ( i ) . Also, s i n c e a x H ( a ) 5 ( a + H ( a ) )
4
we have t h a t ' ( a + H ( a ) ) = a + H(a) i m p l i e s t h a t a + H(a) = a x H(a). Thus, i t i s enough t o prove ( i ) f r o m r i g h t t o l e f t . Assume t h a t a 9 F N and we have t h a t H f a ) 5 a o r a 5 * H ( a ) . a + H ( a ) 5 a x H ( a ) . By 3.4.2.11, we o b t a i n H ( a ) 5 a o r a 5 Hfa). Hence by Since H ( a ) E O n , by 3.6.1.14, 3.7.3.1, a = Q E On. ( i i i ) and ( i v ) a r e e a s i l y o b t a i n e d f r o m 3.6.1.9,
AC.
.
and (ii). and (i)
There a r e many o t h e r statements which have been proved e q u i v a l e n t t o We l e a v e a few as problems.
AXIOMATIC S E T THEORY
295
PROBLEMS
Prove:
1.
2.7.3.3
2.
l! a a =
3.
a $ b A b
4.
AC
+-+
W aWb( 2 a - 2 b
5.
AC
++
Wa3b(aO(bA V c ( a . ( c - + b < c ) ) .
6.
AC
++
7.
AC
-
3.7.4
01
t--,
2
S H(H
a i- H('2)
a
-+
3 c(a 2 c A c
WaWb(a,b
('2))
2
5
V H (H('2))
aA b
2
a t H(%).
c A c $ b).
a=b).
-+
9 FN
+
a s * b V b <*a).
V at/ bW c(a,b,c 4 F N A
b
a
4
'a
+
b < c).
THE GENERALIZED CONTINUUM HYPOTHESIS I M P L I E S A C
.
The continuum hypothesis, f o r m u l a t e d b y Cantor, a s s e r t s t h a t t h e c a r d i n a l o f w 2 i s t h e n e x t c a r d i n a l a f t e r I w I . The g e n e r a l i z a t i o n o f t h i s t o any i n f i n i t e s e t , i s c a l l e d t h e g e n e r a l i z e d continuum hypothesis, namely, C H , where C H i s g i v e n by, 3.7.4.1
(i)C H
DEFINITION,
-
W aW b(a E F N V 1 ( a
b
'2)).
( i i ) H C = C a : l 3ba.(b4"2}.
.
Y
T a r s k i had I t i s c l e a r t h a t C H i s eq i v a l e n t t o V = F N u H C 2 shown, u s i n g 3.4.3.24 t h a t a, 2' E H C i m p l i e s t h a t a = c1 f o r a c e r t a i n a. We s h a l l g i v e Specker's theorem, a l i t t l e s t r o n g e r , based on h i s theorem 3.6.2.9. 3.7.4.2
THEOREM,
a, 2'
E
HC
+
3c1 a
-
c1
.
PROOF, Assume t h a t a, 2' E H C , and s i n c e f o r a f i n i t e t h e r e s u l t i s c l e a r , t h a t a 9 F N . By 3.6.2.10, S i n c e a E H C , a - a tC A 5 a t c l4 a 2 . 2
-
Hence, s i n c e a E H C , a = a Now, if a x 2
'2,
tc a
9 x 2 = a + c 1 2
or a t
C
a
2
-
a ~ i.e., ,
we would have ( s i n c e a x 2
5
a2. a-ax2 or
' a ) t h a t 2'
a
5
2 2 ~ x 2 . 2a, con-
296
ROLAND0 CHUAQUI
tradicting 3.6.2.9.
Hence, a - a x 2. a
-
5
2a
5
a2 x a2
Since a E H C , a 2 a or 2a by 3.6.2.9. Hence a 2a. Also, by 3.6.2.3
I
I
a2,
The second disjunct is impossible,
a x H(a)
a2
E HC,
5
x
H(a)
-
5
“2 x a2 x
As an immediate corollary, we have. COROLLARY,
CH
+
AC
- -
‘2 + H ( a ) , i.e.
Thus, since a E H C , a + H ( a ) = a nally get a 2 a for a certain c1 e O n . 3.7.4.3
a+ca2
(ii), we get,
Therefore, since 2‘
.< a + H ( a ) 5
I
a 2.
%. a
Therefore,
H(a).
H(a)
5 ‘2.
Also,
a2.
Using 3.7.3.4
(i), we fi-
CHAPTER 3.8 Gauging Sizes of S e t s and Classes
We s h a l l study, i n t h i s Chapter, d i f f e r e n t ways of measuring t h e s i z e s of s e t s and c l a s s e s . In t h e f i r s t place, we s h a l l discuss s c a l e s of cardin a l i t i e s , which well-order i n f i n i t e c a r d i n a l s . More p r e c i s e l y , we understand by a s c a l e of c a r d i n a l i t y a sequence F w i t h O n C r ( F ) t h a t s a t i s f i e s : i f a C P then F'a < F ' P . Thus, DF-' i s a c l a s s of cardinal numbers well ordered by t h e r e l a t i o n <. Without A C i t i s impossible t o show t h a t t h e r e i s a s c a l e of c a r d i n a l i t y F t h a t enumerates a l l c a r d i n a l s , i.e. such t h a t D F-' = C r . We s h a l l consider s c a l e s F such t h a t DF-' c o n s i s t s o f i n f i n i t e c a r d i n a l s , s i n c e f i n i t e c a r d i n a l s a r e well-ordered and a r e a l l l e s s than i n f i n i t e c a r d i n a l s . We s h a l l study two s c a l e s of c a r d i n a l i t y : t h e s c a l e corresponding t o i n i t i a l o r d i n a l s (which measures the c a r d i n a l i t i e s of alephs) and the exponential s c a l e (which measures t h e c a r d i n a l i t i e s o f beths). This chapter will a l s o include o t h e r ways of gauging s e t s and c l a s s e s , such a s degree of c o f i n a l i t y t h a t measures t h e "length" of a cardinal (reg u l a r c a r d i n a l s , a r e long and s i n g u l a r c a r d i n a l s a r e s h o r t , i n this sense); inaccesible s e t s , which a r e l a r g e s e t s ; and s t a t i o n a r y s e t s of o r d i n a l s , which a r e t h e "large" subsets of regular uncountable o r d i n a l s . These themes w i l l be d e a l t w i t h again in P a r t 4, w i t h AxC.
3.8.1
INITIAL ORDINALS,
The elements of 0 1 c a l l e d initid ahdin& a r e t h e ordinal numbers not equivalent t o any ordinal t h a t i s l e s s than them. I t i s c l e a r t h a t f o r any ordinal a t h e r e i s an i n i t i a l ordinal P w i t h = n It : t 1 a ) . Thus, t h e r e i s a c l o s e r e l a t i o n s h i p between t h e alephs ( t h e c l a s s A1 C C r ) and t h e i n i t i a l o r d i n a l s ( t h e c l a s s OI C On ): We have t h a t a E A7 i f and only i f t h e r e i s a unique P E 0 1 SUCKt h a t a = I P I . Because o f t h i s correspondence, we could use instead of A1 t h e c l a s s O I and c a l l t h e i n i t i a l ordinal equipollent with the s e t a, the cardinal of u (whenever la( E A 1 ). In t h e theory M K T C , every s e t u i s equipollent t o an ordinal and, hence t o e x a c t l y one i n i t i a l o r d i n a l . Thus, we s h a l l use i n MKTC, 0 1 , instead of C r , a s t h e c l a s s of cardinals. This has t h e advantage t h a t any a E 0 1 has e x a c t l y a elements,whereas i f a E C r , a does not, i n general, have a elements.
P = a, namely P
297
2 98
ROLAND0 CHUAQUI
3.8.1.2
THEOREM,
(i) w c O Z A U E OZ ( i i ) H ( a ) E OI
(iii) x c -OI-. u x ( i v ) OZq V
.
E
.
01.
PROOF OF (i):By 2.7.3.5
and 2.7.3.7
PROOF OF (ii): By 3.6.2.3
(i)and ( v i i ) .
(i).
.
Then, by 3.3.1.8, U X € On. PROOF OF (iii):Assume t h a t x C O I Suppose, now, t h a t t E U x . Then E ECEX f o r a c e r t a i n a. Hence, s i n c e a € O I , [ < aSux, i.e.uxEOZ. PROOF OF ( l v ) : Suppose t h a t O I € Y . Hence, by 3.3.1.8, and O Z c O n we g e t t h a t U O I E O n , and by ( i i i ) U O Z E OZ. By (ii), H( UOZ ) E 07 and, a l s o U
by 3.6.2.3
(v).
OI 4 U OZ
-+
H( UOZ ) = H( U O t )
,
T h i s i s a c o n t r a d i c t i o n , s i n c e by 3.6.2.3,
H ( u O Z ) E H ( H ( U O I ) ) E O I , i.e.
H ( u 0 Z ) EuOZ.
and (ii),
m
From t h i s theorem we deduce t h a t OZ- w i s a p r o p e r c l a s s c l o s e d u n der unions. Hence, by 3.3.2.13 (i), t h e r e i s a unique normal f u n c t i o n enum e r a t i n g it. The n e x t d e f i n i t i o n i n t r o d u c e s t h i s f u n c t i o n
3.8.1.3
H = U {F : O n o n( F ) A Normal (F) A D F - l
DEFINITION,
OI- w
1.
Thus, t h e f u n c t i o n G g i v e n by G'a = IN
f i n i t e alephs (i.e.
3.8.1.4
DG'l
= A1
-{IK~
=
a I enumerates t h e c l a s s o f i n -
: K E w } ) .
THEOREM,
( i ) NO = w .
( i i ) H a + l = H(Ha)= n
(iii) a = u a f 0 -. K
a
{E: €
E
OZ
u {Kc;: €
=
f
AN^ c
[}
.
a).
(iii) c o u l d be t a k e n as a r e c u r s i v e d e f i n i t i o n o f CS. (i), (ii), PROOF OF ( i ) :
By Def. 3.8.1.3.
PROOF OF ( i i ) . By 3.8.1.2
(ii) we have t h a t H(H,) E OZ.
Also, by
299
AXIOMATIC SET THEORY
3.6.2.5
n {t : [
t h e r e i s no b w i t h Ha E
O Z A Ha C
E ) = H"+l'
By t h e c o n t i n u i t y o f N.
PROOF OF ( i i i ) :
3.8.1.5
4 b 4 H ( H a ) + Ha = H(Ha). Hence, H(Ha) =
THEOREM,
-
(i)P4Na+-+PEHa.
(P = Ha
(ii)
Ha
C P c Ha + l)
( i v ) Ha+ 1
- Ha
(0 = Ha
P = Ha.
-, I !
(iii) wcP
A
-
P
E
H" + I
-
")*
= H a t 1'
-
According t o ( i i ) and (iii) we can d i v i d e t h e o r d i n a l s i n t o d i s j o i n t Traditionally the f i n i t e ordinals classes: No, H1 NO, H2 N 1 ' No, i.e. t h e denumerable a r e c a l l e d o f t h e f i r s t c l a s s ; elements o f Hl ( i v ) determines t h e o r d i n a l s , a r e c a l l e d o f t h e second c l a s s , and so on. c a r d i n a l i t y o f each o f t h e s e classes.
-
..* -
-
PROOF, (i) and (ii) a r e o b t a i n e d from t h e d e f i n i t i o n o f i n i t i a l o r (iii) r e s u l t s from t h e n o r m a l i t y o f N u s i n g 3.3.2.14 and (ii). dinals. PROOF OF ( i v ) .
Ha.
But, by 3.6.1.13,
Ha+ 1
= P
+C H"
f o r e , H" + I
= P u N
= P =
3.8.1.6 PROOF,
-
a'
clCP
Assume t h a t a
H
P2 5 PH a
3.8.1.7
Since N
-Ha.
THEOREM,
H
-
We have, H a + l = W"+1 Ha) Hatl Ha = P f o r a c e r t a i n
THEOREM,
5
"c H
+
50.
= (Ha+l+c Hence, by 3.6.1.9,
H a + 1 , we have t h a t P N H0. = ' 2 .
We have, 2
H xH
" Ha
P.
4 Ha
.(
K
"2.
U
Na=P.
There-
Hence, by 3.6.19
H
"2 2 02.
P,y E Ha
+
+y,
P * y , Py
E
Ha
.
Thus, t h e i n f i n i t e i n i t i a l o r d i n a l s a r e main numbers o f a d d i t i o n , mult i p l i c a t i o n and e x p o n e n t i a t i o n (see Def. 3.3.5.12 and 3.3.6.11) PROOF,
By 3.6.1.9
3.8.1.8
THEOREM,
and 3.6.1.10.
3E Ha = w
w
t
A Ha = a:
A (a# 0
+
Ha
H
0").
300
ROLAND0 CHUAQUI
PROOF, The f i r s t two conjuncts are obtained by 3 . 9 . 1 . 7 , (and the remark following i t ) , and 3.3.6.12.
have,
Let now, a Z 0; then, by 3.8.1.7
a n d 3.3.5.13
3.3.6.10
( i i i ) , since w c Ha, we
PROBLEMS
u a
w I: (I: E
1.
a
2.
a = U a # O + N
3.
H a = 2
=
# 0
A
a
d C [ H
a t IH :I: 5 a l .
+
€
HI:4 a)
+
a 9k Ha.
: t E a l .
E I :
DEGREE OF C O F I N A L I T Y
3.8.2
We now introduce the degree of c o f i n a l i t y of a well ordering R . This degree could be considered as a measure of i t s legth. In t h i s section we shall use Greek capital l e t t e r s r, A , when referring t o ordinal or the class On i t s e l f . As usual, Greek lower-case l e t t e r s r e f e r t o ordinals. 3.8.2.1
DEFINITION I
(5)Cof(r,R)-3X 3 A ( X c-D R A W x ( x E D R ~ 3 q ( q E X A < , y ) ) A A S r A R I X IN^ A ) .
( i i ) c f ( R ) = n U ' : C o f (I',R)l. (iii)
cf(r)
= cf(zN1
r ).
are the deC o f (I?, R ) i s read; I? A c o d i n d wiXh R. c f ( R ) and cf((r) gkee od codinaLLty (or, simply, t h e c 0 6 L W y ) 0 6 R and r, respectively. I t i s clear t h a t R = R' implies t h a t C o f ( F , R ) i f and only i f C o f (r ,R'). Thus, cf(R) = c f ( R ' ) . 3.8.2.2
THEOREM,
( i ) cf(R + o = cl T () a + 1) = 1 A c f ( 0 ) = 0. ( i i ) &f(rJNlA)3 X ( XCAA X U U X = A A X S r ) . ( i i i ) C o f (r, I N l r ) .
AXIOMATIC
(iv)
(v)
T ( r )5 r
r
# 0
u r (vi) a =
-+
1. U c1
( v i i ) c1 = u c1 Thus, w i t h S.
301
THEORY
.
v(r)= n {A
+0 f
SET
: 3 F(In(F) A D F O n (F) A DF = AA u DF-'=
A In(d) A D 6 =
0
+
cf(a)
=
~1
A U D 6-l = p
-+
T(a) =
v(p).
T(H~).
(i) says t h a t i f an o r d e r i n g S has a l a s t element, 1 i s c o f i n a l
( v ) g i v e s another c h a r a c t e r i z a t i o n o f t h e degree o f c o f i n a l i t y , espec i a l l y u s e f u l i n case r i s a l i m i t o r d i n a l . Thus, i n t h i s case, c f ( r ) i s t h e l e a s t o r d i n a l t h a t i s t h e domain of a s t r i c t l y i n c r e a s i n g sequence of o r d i n a l s whose l i m i t i s r . Thus, (i)and ( v ) a r e a h e l p t o determine degrees o f c o f i n a l i t y . PROOF OF (i):By t h e d e f i n i t i o n s . PROOF OF (ii): Assume t h a t F i s c o f i n a l w i t h A . By Def. 3.8.2.1 t h e r e a r e X and I" c r such t h a t I N I X z I N I I", and f o r e v e r y a E A I t i s c l e a r t h a t X u u X = A and t h a t X = E X wit?;a 5 6 . there i s a I" c r, i.e., X 5 r. Thus, t h e i m p l i c a t i o n f r o m l e f t t o r i g h t i s proved. Assume, on t h e o t h e r hand t h a t t h e r e i s an X A and X 2 I?. We have t o c o n s i d e r two cases.
A = c1 t 1. S i n c e X u u X = A CASE I. hence I'3 - 1. By (i) I' i s c o f i n a l w i t h A
.
-A
C
, we
. .
such t h a t X
U U
X =
have t h a t X # 0 and,
CAS E I I , A = U A We have, U X U U U X = U A , hence, s i n c e U X r" C I' where I" = O n o r We have X = i s transitive, u X = u A = A G-l
.
I" E O n Since X 3.3.2.7 and o b t a i n
C
A
CW
and
-
U
X = A
,U
X
9 X.
Hence, we may a p p l y
a s t r i c t l y i n cHr ee ansci en gD sequence o f o r d i n a l s F such t h a t a n d D F-C D G-c r . F = r "-c r a n d I N I A r F Z N I r " , i.e.
F=UG=A F i s c o f i n a l w i t h A.
U
(iii) and ( i v ) a r e o b t a i n e d from t h e d e f i n i t i o n s . PROOF OF (v).
Let
r#
0 and l e t F be a s t r i c t l y i n c r e a s i n g o r d i n a l
f u n c t i o n w i t h DF = A and U DF-' = u r. I f we t a k e X = DF" t h e n we have t h a t X c r, U X U X = r and X 5 * A ; by 3.6.1.14, X 5 A. A p p l y i n g ( i i ) we obtaTn (v). ( v i ) and ( v i i ) a r e l e f t t o t h e r e a d e r .
3.8.2.3
THEOREM,
.
302
ROLAND0 C H U A Q U I
(ii) TtOn) = On. (iii) a = ua # 0
+
cf(a) E 01
T ( T (r ) ) = TW).
(iv)
-w .
From (i), we deduce t h a t t h e f u n c t i o n ( v ( a ) a E O n also c a l l
F, has
the property OnOI
i s easy t o see t h a t D T t h a t i t i s even smaller.
-
(3).From
C (01- w ) u
{O,l}.
),
t h a t we s h a l l
iii)and 3.8.2.2
(ii) it
L a t e r we s h a l l prove
PROOF OF ( i ) . Assume t h a t P i s c o f i n a l w i h a and 13 4 O I . Then (ii), t h e r e i s an x C - a such t h a t t h e r e i s a y C P w i t h 7 = P . By 3.8.2.2 x uuX = a and x = P . Thus, x = 7 and y i s c o f i n a l w i t h a. T h a t i s cf(a)
c P.
PROOF OF (ii).Suppose t h a t c f ( O n ) = P E O n . By 3.8.2.2 ( i i ) , the t h e r e i s an X c On such t h a t X u U X = O n and X = 0. Hence X E Y , and so X u u X E V implying t h a t On E V , a contradiction. PROOF OF ( i i i ) . L e t a = U a # 0. By (i), cf(a) E O I . Suppose t h a t 0 f P E w and d i s a s t r i c t l y i n c r e a s i n g sequence o f o r d i n a l s w i t h
D d = P and D 6 - l C a . Then i t i s easy t o see, t h a t U D 6 - l = u d * P = d u p . But u P E 0; hence ( v ) we see t h a t P + v ( a ) . U P € a. A p p l y i n g now 3.8.2.2 T h e r e f o r e c T ( a ) E O I a.
-
It i s e v i d e n t t h a t T ( 0 ) = 0 and T(1)= 1. Hence PROOF OF ( i v ) . Suppose t h a t fc ( a ) 3 1. f o r a = 0 o r a = 1, we have q ( v ( a ) ) = cf(a). Then T ( a ) E 0 1 - w , s i n c e cf(y + 1) = 1, f o r e v e r y 7 . A l s o a = U a Z 0 and U c T ( a ) = v ( a ) . By 3.8.2.2 ( i v ) v ( q ( a ) ) c c f ' ( a ) . I n o r d e r t o show t h e i n c l u s i o n i n t h e o t h e r d i r e c t i o n we a p p l y 3.8.2.2 (v) and o b t a i n a
s t r i c t l y i n c r e a s i n g sequence o f o r d i n a l s 6 E T ( a ) a such t h a t U d * ( T ( a ) ) = A 1 i n g again 3.8.2.2 ( v ) we f i n d a n o t h e r s t r i c t l y i n c r e a s i n g sequence E (' ( a ) ) cf(u) w i t h u g * ( T ( T ( a ) ) = q ( a ) . It i s c l e a r t h a t 4 o g i s 9 a s t r i c t l y i n c r e a s i n g sequence o f o r d i n a l s w i t h D(d o g ) = cf(cT(a)) and
a.
-Y+
u DYOg)-'
T ( c
(a)).
. =
a.
A p p l y i n g 3.8.2.2
( v ) f o r t h e l a s t time, we o b t a i n c f ( a ) 5
I I
We now t u r n t o w e l l - o r d e r i n g s t h a t a r e " l o n g e r " t h a n I N O n , i.e. w e l l o r d e r i n g s R such t h a t I N ] O n i s isomorphic t o a p r o p e r i n i t i a l segment o f R Such w e l l o r d e r i n g s are, f o r instance, I N On +o I N a , I N I On +o I N l O n , I N J O n - I N l O n . However, i f we assume Ax Reg (i.e. i n MKTR) On i s c o f i n a l w i t h a l l such w e l l o r d e r i n g s ; hence t h e y a r e n o t much longe r t h a n I N I O n (Marek and Z b i e r s k y ) .
.
I
3.8.2.4
THEOREM,
R c - W A WO(R)
+
cf(R)
On
.
PROOF, I t i s c l e a r t h a t i t i s enough t o show t h a t :
AXIOMATIC SET THEORY
r
(1) There i s a f u n c t i o n F w i t h D R(F) where such t h a t ,
x(x
(b) W
E
DR
+
r
3P(P E
Assume, f i r s t , t h a t R c On x O n and 0 sequences A and G such t h a t -
and,
G'a =
r
n
y
A'a, i f
+
r
i f t h i s i s satisfied,
.
€
or
r
= O n and
x GRF'P)).
A
According t o o u r Def. 3.8.2.1. R and, hence, cf ( R ) 5 O n
A ' a = {y : V P ( P
r E On
,
(a) w a w P ( a , P E r + ( a c P - F t a < R F t P ) and
303
G'P
€
i s cofinal with
4 DR. D e f i n e by r e c u r s i o n t h e
Y A G ' P < R )~3
A ' a # 0; G ' a = 0 , otherwise.
Since CY C p and A'a = 0 i m p l y t h a t A'P = 0, we have # 03. O n . From CY E r we deduce t h a t A'a # 0, and hence, G ' a E A ' a . Also, f r o m t h e d e f i n i t i o n of A ' a we o b t a i n t h a t f o r P E a E r we have t h a t G'P < R G ' a . Hence, Let
r
:G ' a
= (a E On o r
r
=
(2) p E a E r
+
G'P
' A~ G'P E G ' E~ ~ ' a
We have t o c o n s i d e r two cases: CASE I A'a = 0.
r
E On.
Then t h e r e i s an a such t h a t G'e = 0 and, hence, K We have t h a t f o r a l l y, V ( G ' P : E a ) < R y i m p l i e s t h a t y 5 I
u { G ' P : P € a ) , because i f G'P < R y f o r a l l E a, we must have t h a t ' y E G ' B f o r a c e r t a i n P E a ( s i n c e A'a = 0). Also, { G ' P : @ € a )E Y and, hence, R (3) h = Rl{y : v { G ' P : P E a )
I f k = 0, we t a k e F = Glr. s i n c e h = 0, F s a t i s f i e s (b). I f h # 0, h
F s a t i s f i e s ( a ) and (b). a
CASE I I On. Take F
r
I
E
that,
= G.
(Z), On q V
We have,
(5) a
E
F
= EN'h.
By ( 3 ) ,
.
I n t h i s case, G ' a # 0 and A'a # 0 f o r e v e r y By ( 2 ) we o b t a i n t h a t t h i s F s a t i s f i e s (a).
= On
( 4 ) G'P < R a f o r a l l P G-'*
I n t h i s case, l e t
Assume t h a t G does n o t s a t i s f y (b).
By
v.
By ( 2 ) t h i s F s a t i s f i e s (a) and by ( 3 ) ,
a w e l l ordering,
i 5
E
E
Then, t h e r e i s an
c1 E
O n such
On.
G i s s t r i c t l y i n c r e a s i n g a s an o r d i n a l f u n c t i o n . Hence, Thus, t h e r e i s a P such t h a t a E G ' P . L e t 6 = n ( p : a E G ' P ) .
.
G'S
.
304
ROLAND0 CHUAQUI
Also, i f
Y E & , then
G'r
G ' 7 E a ( f o r ~ € 6 ) . By
C
S i n c e ct
a.
T4),G'r < R a .
f
G'P for e v e r y 8 E O n , we have t h a t Using t h e d e f i n i t i o n o f G, we ob-
t a i n t h a t G ' S C a , c o n t r a d i c t i n g ( 5 ) . Hence G s a t i s f i e s (b). have completed-the p r o o f f o r R C - O n x O n and 0 $Z D R .
Thus,
we
Consider, now, an R C O n x On w i t h 0 E D R . D e f i n e S b y s t i p u l a t i n g t h a t S = I ( a + l , P + 1 ) : Ta P ) E R ) . Then S i s isomorphic t o R , S C O n x O n , and 0 9 D S . S i n c e (1) i s t r u e f o r S, u s i n g t h e isomorphTsm we prove (1) f o r R. Let, f i n a l l y , R
-W
C
.
A p p l y i n g 3.6.1.16
we g e t an H such t h a t D R L,,
O n . L e t S = { ( H ' x , tl' ) : ( x , q ) E R). Then, S C O n x O n and S i s i s o morphic t o R. Since (IT i s t r u e f o r S, we e a s i l y p r o v e i t f o r R. 3.8.2.5
DEFINITION.
( i ) SN(X)
-
3 Y(X = u Y A Y 4 X A W z ( z € Y
(ii) S N = Ix : S N ( x ) )
( i i i ) RG(X) * 1S N ( X )
. .
-,
z4X)l
.
(iv) RG = Y - S N . We c a l l t h e elements o f S N , . b A&, ~ and t h e ~elements~ o f R G ,~ k e g d m A&. S i n g u l a r s e t s x a r e t h o s e e q u i p o l l e n t t o t h e u n i o n o f a fami l y o f c a r d i n a l i t y l e s s than t h a t - o f x , o f sets o f c a r d i n a l i t y less than t h a t o f x ; i.e. t h e i r c a r d i n a l i t y i s a c c e s s i b l e t h r o u g h t h e u n i o n o f a fami l y of l e s s c a r d i n a l i t y t h a t c o n t a i n s s e t s o f l e s s c a r d i n a l i t y . Regular s e t s a r e i n a c c e s s i b l e i n t h i s sense. The n e x t theorem g i v e s some p r o p e r t i e s o f these t y p e s o f s e t s provable i n MKT. 3.8.2.6
THEOREM,
(i) ( S N ( X ) A Z = X + S N ( Z ) ) A (RG(X) A Z = X (ii) 0, 1, 2, w € R G (lid) 2 E K E W+KE S N .
.
S N * 3 q(a = U y A q 4 a A tl z ( z e y E 0 1 %w - + ( a E S N * 3 q(a = U y A q 4 ( v i ) a E OI- w + ( a E S N - c T ( a ) c a ) . ( v i i ) a E OI- w + ( a E R G * F ( a ) = a). (iv) a (v) a
E
+
( v i i i ) RG(0n PROOF,
+
RG(Z)).
z 4 a)).
~1
-
A g C a)).
1.
(i), (ii), and (iii) a r e o b t a i n e d from Def. 3.8.2.5.
PROOF OF ( i v ) . I t i s easy t o see t h a t t h e r i g h t hand s i d e i m p l i e s t h a t a E S N . So, i n o r d e r t o prove t h e converse i m p l i c a t i o n , assume t h a t a E S N . By Def. 3.8.2.5 we have t h a t
~
305
A X I O M A T I C S E T THEORY
u g'
a A y' 4 a A W z ( z € y '
z5
L e t y = Cd*z : z E y ' ) .
It i s clear that g
Wz(zEy'
Finally,
U
y
{d*
= U
5 *d* q ' .
But y t 4 a
and,
g < y'o< a. Also, & * z - z - < u. Hence,
and 3.6.1.13,
by 3.6.1.14,
z < a ) , f o r c e r t a i n y ' and d.
-+
z - ( a).
--*
A*
z :zEg') =
u {z :zEg') =
d*
u y' = a .
- .
PROOF OF (v). Assume t h a t a E O I w The i m p l i c a t i o n from r i g h t t o l e f t i s o b t a i n e d by Def. 3.8.2.5 and t h e f a c t t h a t g c - a implies that every z g y s a t i s f i e s z 4 a .
I n o r d e r t o show t h e converse i m p l i c a t i o n suppose t h a t a 6 S N ; i.e.,
by ( i v ) ,
a = uu A v
.(
NOW, i f Z E V , t h e n c o n s i d e r two cases.
CASE 1.
aA
z
There i s a
Wz(z~w
+
c - u and, hence,
C -u v
Z E U
U
z C - U a = a.
We have t o
with u z = a .
I n t h i s case, we have u z =a, z the conclusion i s satisfied.
CASE 2.
z o( a ) , f o r a c e r t a i n w .
.(
u, and z C - a. Thus, t a k i n g y = z,
F o r e v e r y Z E g, u z C a.
I n t h i s case t a k e y = CU 2 : z E v 1 . Then y C a. Also, g < * u by t h e f u n c t i o n u . S i n c e u d a , by3.6.1.4and3.6.1.13we have, q < v ~ ( a . Bes i d e s t h i s , s i n c e g C a, U g C U 'Y = a. I n o r d e r t o prove t h e i n c l u s i o n a =-u u. Then C; E Z E u f o r a c e r t a i n 2 . Hence C; E U z a c u y, l e t C; - U q + 1. or? = U z and, thus, C; E yU U g . We o b t a i n , U g C a C y U U y C B u t u = U a and so a + U y f 1. T h i s i m p l i e s that-a =-Uy. ( v i ) and ( v i i ) a r e deduced from ( v ) . l e f t t o t h e reader. 3.8.2.7
THEOREM,
( j ) a = u a # 0 A a 4 H a
+ H
a
E S N .
(ii) a =U a# 0 A H ERG + a = N ( i i i ) Ha
The s i m i l a r p r o o f o f ( v i i i ) i s
a
.
HN E S N a (iv) a = u a 4 0 1 - + H a ~ S N . E
S N
-+
(v) a # U a + N H PROOF OF ( i ) .
a
a'
E S N .
Let a = u a
+
0 and a 4 Ha.
We have, Ha
=
cNC;:
306
E
ROLAND0 CHUAQUI
a], because H i s normal. 3.8.2.5 ( v ) , Ha E S N E
.
Take q = {H : €
E
t
a1 = a 4 Ha.
( i i ) , ( i i i ) , and ( i v ) a r e deduced from ( i ) .
T h u s , by
.
Suppose t h a t NH E R G . Then, by ( i i ) , HN = Na , a c1 Since H i s normal Ha=a and this implies t h a t a = u a. PROOF OF (v).
In MKT t h e only i n i t i a l o r d i n a l s t h a t can be proved regular a r e 0, 1, 2, and w. On t h e other hand, i t possible t o prove t h a t many i n i t i a l ordinals a r e singular. For instance, H w , H w + w , H 2 , H w w , NN ,HH a r e singular. w w 1 We s h a l l see l a t e r t h a t u s i n g A C we can prove t h a t H a + l i s regular f o r every a. However, i t i s c o n s i s t e n t w i t h MKT t o assume t h a t N1 i s s i n gular. The problem of t h e existence of a regular Ha f o r l i m i t a # 0 i s not solvable even i n M T K t A C . An i n i t i a l ordinal of t h i s form i s c a l l e d weakly inaccessible. We s h a l l study them l a t e r . By 3.8.2.7 ( v ) , we know t h a t i f Ha i s weakly i n a c c e s s i b l e then a i s a fixed point of t h e function H ( i . e . Ha="). However t h e f i r s t fixed point l a r g e r than w is H W w , i.e. u{H'w:v€wl = wu H u which i s obviously singular.
...,
3.8.2.8
THEOREM I
PROOF1 BY 3.8.2.3
3.8.2.9
THEOREM,
and 3.8.2.6 (JIKONIG).
(vii).
.
0 f a
5
U
x
A Vy(qE
x
+
qda)
+ U X < X U X .
u x and y 4 a f o r every y E x . In order t o obtain a contradiction suppose, a l s o , t h a t ' U x U x , i.e. 06 = U x a n d 6 D 5 - I = 'UX. Thus, f o r y E U x a n d Z E X we have t h a t ( 6 ' ~ ) ' Es U x . Def i n e the function k f o r Z E X by s t i p u l a t i n g t h a t k ' z = ( ( 6 ' q ) ' z : ~ € 2 ) . Then h ' z C- u x. Also, k l z < * z by t h e function ( ( 6 ' y ) ' z : q E z ) . B u t , Y k ' z # 0, f o r since z 4 a, by 3.6.1.14 and 3.6.1.13, h ' z 4 a. Hence, g* a every Z E x (since g* a = a). PROOF I Assume t h a t a
z*
4
-
Define, now, f o r h'z =
Z E X the 41
(n
function h by s t i p u l a t i n g t h a t ,
it : E
E ~ - I *(g* a
-
k l z ) ~ ),
AXIOMATIC S E T T H E O R Y
3 07
i.e. h ' z = gf.$ f o r a c e r t a i n w i t h gf.$ $ k ' z . Also, D h = x and D h - l 2 Therefore, t h e r e i s a y E U x such t h a t h = 6'y. We u x. Hence, h E 'UX. have y € z € x f o r a c e r t a i n z. Thus, h ' z = ( 6 ' y ) ' z E k f z , by t h e d e f i n i t i o n T h u s , we obtain a contrao f k. B u t , by the d e f i n i t i o n of 12, h ' z ($ k ' z . d i c t i o n from t h e assumption 'Ux < * u x.
I f we had Suppose, now, t h a t we a l s o have a # 0. Then, u x s 'Ux. u x 2 'ux, we would g e t ' U X ~ * U X which we know t h a t i t leads t o contrad i c t i o n . Hence, U x .( 'Ux. = 3.8.2.10
COROLLARY
8
This c o r o l l a r y shows us the only i n i t i a l o r d i n a l s t h a t can be proved not t o be equivalent t o w 2 (and in general t o % 2 w i t h Ha r e g u l a r ) . That N i s , i f T ( N P ) = o , then O 2 $ No. We may r e c a l l t h a t I w 2 1 i s the cardinal number o f t h e s e t o f real numbers; thus t h i s r e s u l t i s very important.
I t can be shown c o n s i s t e n t w i t h MKT (even w i t h B q t h a t " 2 t o any aleph except those excluded by t h i s c o r o l l a r y . Let a
PROOF OF ( i ) .
E
01- w .
We have, by 3.8.2.2
x c- a , and x = c f ( a ) f o r a c e r t a i n x. Hence by 3.8.2.9,
-+
PROOF OF ( i i
Then, "(')('2)
c o n t r a d i c t i n g "2 = P
.
.
= c (''0
.
i s equipollent (ii), a
Ux,
a 4 a' = m a ) ,
.
Assume t h a t a E O I - w , 2' = p , and F(p)5 a . > P , by ( i ) . B u t , $(@)(a2) ^. c f ( P ) X " 2 a ,
( i i i ) i s a reformulation of ( i i ) and ( i v ) i s a p a r t i c u l a r case of (iii).
PROBLEMS
1.
Let F be t h e normal function t h a t enumerates t h e fixed points o f N, i.e.
F p (H) (Def. 3.3.3.5).
Show, a ) FYa +
1)E S N .
3 08
ROLAND0 CHUAQUI b) a =
2.
U
a
0 A F'a E R G
#
+.
F'a = a.
Define by recursion, F P o = FP
"1.
= F p ( G ) , where G i s the normal function t h a t enumerates F p a . FPa+ 1 a = u a ;it: 0 F p a = n { F p p : 0 €a}. -+
Show,
a) Fpa 4 V
, for
every a
E On.
b ) F p a i s closed under unions of s e t s , for every 3.
E
.
b) a =
f o r every t
1)E S N ,
U a#
0 A ts
01
E
RG
V A W y (ye x
+
y
AC * W a 3 b W x ( x E b
++
E
= { a :
-+
E On.
(0 5
c)a=Ua#OAH ERG+ a E F p a Prove:
6
A
Show,
Fpa).
a ) F$S +
5.
On
Let Fa be the normal function t h a t enumerates F p c , , and F p a
4.
CI E
O(
a s 6 ' y)
+
A
-+
a E F p P ).
. u x 4 II ( 6' y : q E x
)
.
Prove (Tarski):
3.8.3
x c b A a $ x).
THE E X P O N E N T I A L SCALE
We should now recall the function R defined in 3.2.3, a n d the cardinal The c l a s s of the carproperties of DR" studied in 3.6.1.18 3.6.1.23. dinals o f elements of DR-' is B e . The i n f i n i t e beths, i.e. elements of B e - w can be p u t in a new scale o f cardinality the exponential scale ( IR(w+a)l: a E On). This section i s devoted t o the study o f t h i s scale. a
-
3.8.3.1
THEOREM
( i ) Ka < * R ( w + a ) . ( i i ) a = u a +.
tsa
5R
(w
+a)
.
AXIOMATIC SET THEORY
309
W i t h o u t A C i t i m p o s s i b l e t o show t h a t t h e i n f i n i t e beths a r e alephs, i.e. t h a t R (w t a ) i s e q u i p o l l e n t t o an i n i t i a l o r d i n a l . Thus, i t c o u l d happen t h a t t h e e x p o n e n t i a l s c a l e m i g h t be incomparable, i n t h e sense o f , w i t h t h e s c a l e o f i n i t i a l o r d i n a l s . However, t h i s theorem shows t h a t i n t h e sense o f t h e e x p o n e n t i a l s c a l e grows a t l e a s t as f a s t as t h e scale o f i n i t i a l ordinals.
s*
H
€
5
PROOF,
We s h a l l d e f i n e by r e c u r s i o n t h e f u n c t i o n F w i t h R ( w + € ) ,f o r t = uE, and N 5 * R ( w + t ) f o r t = a t 1 .
Ft
(2)F = a + l . We have, H
€
H(Ha)<,
=
.
*
ga
R w= w = H
Take Fo = g where g i s t h e 0’ with o = R ’ w .
9
H
a2
, where
g,
i s o b t a i n e d from t h e f u n c t i o n s
Ha used i n S i n c e we assume F d e f i n e d f o r a, we have 2 R ( w t a t 1 ) . We d e f i n e , Fr; = 9, o F; (3) 5 =
’a
U
E
ka.
w i t h kilP=
a+P
; and NO =g R w , w i t h g as i n (1).
Then, we haveKatl-Ha
( Ha’ p , a Ug.
H
i.e.,Natl-Ha%
);
ThenH
<
€ -Ft
3.8.3.2 (i) CH
a a
R f w t a t 2).
R ( w t a t 2 )
sF* a
.
We have, H = U E H a : a E € 1 = u { H a t l H a : a E € 1 u NO. € H “2 p u t t i n g PJ; = {( 1 , p ) 1 u ( 0 : y E H ~ - { P ) ) ; H~~~
< -Ja
Hat1
.
Assume Fa d e f i n e d f o r a E €
Z 0.
V (F),
F€
€
(1) $. = 0. By 3.4.1.21, f u n c t i o n o b t a i n e d i n 3.4.1.21
On
- K~
Eh
a
L e t Ha=F;oJao
Define, n o w p i
x(a).
Also,
P
=
.
LetFt=U{qa:aEE}
U t 3 ( w t a t 2 ) x C a l : a E € ) u R w C-R ( w + € ) .
THEOREM,
-,V a
Ha = R ( w t a ) .
(ii)V a Ha = R ( w +a) -+ W c -HC ( i i i ) W = V - . (Wa K
a
= R ( w +a)++
. CH).
Thus, assuming C H , t h e g e n e r a l i z e d continuum hypothesis, t h e s c a l e o f i n i t i a l o r d i n a l s and t h e e x p o n e n t i a l s c a l e c o i n c i d e . I t i s known t h a t C H i s c o n s i s t e n t w i t h MKT and independent o f MKT. The a s s e r t i o n Wa N a = R ( w + a )
,
we see t h a t i n M K T R t h i s i s known as t h e a l e p h hypothesis. From (iii), h y p o t h e s i s i s e q u i v a l e n t t o t h e g e n e r a l i z e d continuum hypothesis.
310
on a.
ROLAND0 C H U A Q U I
PROOF OF (i).Assume C H , we prove t h a t Ha = R ( w +a) by i n d u c t i o n
(1) No = R w
, by
3.4.1.21.
( 2 ) Suppose t h a t HP = R ( w + P ) and l e t a = P H R ( w + P ) 2 P 2 = Hp t l . (3) L e t a =
Uci
+0
+ 1.
Then, R ( w +a) =
and suppose t h a t Hp = R ( w + P ) f o r e v e r y
P E ~ .By
C H i m p l i e s A C . So we have A C . Also, R ( w + € + l )= ( R ( w + C ; + l ) R ( w t t ) ) u R ( w + t ) f o r t E a. Thus, b y t h e i n d u c t i o n hypothesis, R ( w + € + 1 ) - R (a+[)= H H Since we have A C , choose d f o r each E F + l F* F = u such t h a t R ( w t € + l ) - R ( w t t ) = d t H t + l Ht ' L e t : E €a)U g + .
-
where Ho =4 R w .
-
-
{H H :E [+I F R ( w + [ ) :E €a} ~ R w = R ( w + a ) . Then Ha
=U
E
a} u HO
=d u
{R (w t E
Assume t h a t Ha = R ( w +a) f o r e v e r y a and PPOOF OF (ii).
x
-
+ 1)
C -
W
.
Then, x E R ( w + a ) f o r a c e r t a i n a . Hence, t h e r e i s a P w i t h x = P ; i.e. i f x 9 F N , t h e r e i s a 7 w i t h x =.'H Thus, i t i s enough t o show t h a t H '2 = H f o r a l l 7. B u t t h i s i s o b t a i n e d from,
r+l
( i i i ) i s deduced from (i)and ( i i ) . We s h a l l now s t u d y Ax t h a t Ax C i m p l i e s A C . We ordered, and we would hope well-ordered. However, we 3.8.3.3
C, t h e axiom o f c h o i c e f o r classes. I t i s c l e a r know t h a t A C i m p l i e s t h a t e v e r y s e t can b e w e l l t h a t Ax C would i m p l y t h a t e v e r y c l a s s can be need Ax C p l u s Ax Reg f o r t h i s purpose
THEOREM I
(i) AxC+AC.
(ii) AxC
+
W = On.
(iii) V = W+(A x C PROOF,
++
V = O n ).
(i) i s trivial.
- -
PROOF OF ( i i ) . Assume A x C . We know t h a t W = U { R a : a E On I (by 3.3.1.22 ( i i i ) ) . Hence W = u { R ( a + l ) R a : a E On 1. S i n c e we have A C , f o r e v e r y a t h e r e i s a P such t h a t R ( a + 1) R a = 0. L e t g ' a = f~(0 : R ( a t 1 ) - R a = P I . We have t h a t I d : R ( a + l ) - R a = g ' a } E g ' a x ( R ( a + l ) R a ) E Y. Hence we can d e f i n e a f u n c t i o n F by F'ad= :R(a+l) R a = g'al. L e t A = { F ' a : a E On I . Since 0 F'a E V and ( F ' a ) n ( F ' 7 ) = 0
-
+
-
f o r e v e r y a + y, we have t h a t F'a E A , f o r e v e r y a and A A . Hence, we can a p p l y A x C and f i n d a B s u c h t h a t 8 n x = 1 f o r every x E A . L e t d a E B n F ' a .
AXIOMATIC SET THEORY
- R a =6ag'a.
D e f i n e ha x =
311
( d i x,
a) and H = U { h : a Hence, W =+, u { g ' a x (a): a E On 1 5 On x On = O n . Thus, c1 E On I. Therefore W - O n . W < O n . But i t i s c l e a r t h a t O n < W Thus, R ( a + l )
.
The p r o o f o f ( i i i ) , which i n one d i r e c t i o n i s o b t a i n e d from ( i i ) , i s l e f t t o the reader. 9
PROBLEM
Prove 3.8.3.3
3.8.4
(iii).
INACCESSIBLE SETS,
We have t h a t R w s a t i s f i e s t h e f o l l o w i n g two c o n d i t i o n s .
(1)l 3 x
x 4 R w
( 2 ) R w E R G , i.e.,
1 3 x ( x 4 R w A W y(yEx
--t
y"< R w ) A R w = Ux).
Thus, we cannot r e a c h t h e c a r d i n a l i t y o f R w n e i t h e r by t h e o p e r a t i o n
P n o r by unions o f f a m i l i e s o f l e s s c a r d i n a l i t y c o n s i s t i n g o f s e t s o f l e s s cardinality. C l e a r l y w , which i s e q u i p o l l e n t t o R w , a l s o has t h e s e p r o p e r t i e s . We s h a l l c a l l inacced&b& d e A those posessing these p r o p e r ties. 3.8.4.1 an a
DEFINITIONl
I A
{x: x
#
0 A 1 3 y q"( x
5Py
A
XE
RG)
.
I t i s i m p o s s i b l e t o show i n MKT o r even MKTR + A x C t h a t t h e r e i s w with R a E I A However, t h i s i s p o s s i b l e i n BC.
2
PROOF1
(i) and (ii) a r e easy. We have t h a t Q and R a E I A . Sincea=ua#O,Ra=~{R[:[Ea).
Assume t h a t w C PROOF OF (iii).
a c-R a ; hence, a 4 R a o r a - R a .
312
ROLAND0 C H U A Q U I
But R a Also, a W - WC
i s r e g u l a r , hence a
5 Ha 2 R ( w + a ) ,
a.
3.8.4.3
R a and, thus, w P < a (because w = Kg
by 3.8.3.1
Hence, by 3.3.5.11, LEMMA,
-
(ii).
w t a = a.
-
.
Rw).
Since w . w = w O ( a , we have t h a t Thus, a
5 Ha 5 R,
= a .
a = { y : gc a A q < a 3 + 313 a = P .
PROOF, Assume t h a t a = { y : g c a A g < a } . I f g E a we have t h a t a; hence T a = a . Thus, O n n a T ( O n n a ) . B u t O n n a € V . Hence, by3.3.1.13, O n n u € O n . We have t h a t O n n a 5 a. Suppose, i n o r d e r t o argue by c o n t r a d i c t i o n , t h a t O n n a < a. Then, we would have t h a t O n n a E a. But we a l r e a d y know t h a t O n n u € O n . Hence, we would g e t t h a t O n n a Therefore, P = O n n a = E O n n a, c o n t r a d i c t i n g t h e f a c t t h a t O n n a E W
y
C
.
a.
.
PROOF, Assume t h a t R a E I A . By 3.8.4.2 ( t i ) , a = U a f 0. Suppose t h a t g € R a . Since R a i s t r a n s i t i v e g L R a . On t h e o t h e r hand, y E Rt; f o r a c e r t a i n E E a (because R a = u {RE : { E a}). Hence, g 5 R t ; d R a . Thus, we have shown t h a t R a C -{g:ycRa A y 4 R a J . Suppose, now, t h a t g C R a and y 4 R a . We have t h a t i f x E g , then, x + 1) w i t h p x + 1-C a. Thus , q C U { R ( p x + 1) : X E g} 2 R P , u { p x + 1 : x E g 3 . I t i s c l e a r t h a t P c a . I n o r d e r t o argue by c o n t r a d i c t i o n , assume t h a t 0 = a. Then R a = U { R(p x + 1) : x E y } . We have IR(p x t 1) : x E g 1 < * g O ( R a . By 3.8.4.2 R a = a , i f a 3 w and R w Thus, by 3.6.1.13 and 3.6.1.14, {R(px + 1) : x E y 1 5 g 4 R a . Since w R(p x + 1) R a y f o r x E g , we a r r i v e a t a c o n t r a d i c t i o n w i t h t h e f a c t t h a t R a i s r e g u l a r . Thus, P E a and g E R(P t 1 ) C R a . I n t h i s way we have proved t h a t {y : g C R a A y o ( R a 3 5 R a and Kence, completed t h e p r o o f o f t h e i m p l i c a t i o n from l e f t t o r i g h t .
x
E R (p with P =
.
Assume, now, t h a t a = u a f 0 and R a = { y : g L R c l A g 4 R a ) . I t i s e v i d e n t t h a t R a f 0. Suppose t h a t q 4 R a . Then t h e r e i s a y' such t h a t y = y ' c R a , i.e. g f c R a and g' 4 R a and, hence, g f E R a . Thus, g' c R f o r a c e r t a i n €-a. Hence P g' c R(E + 1) 4 R a Therefore R a $ PyT= Pg. Thus, we have proved t h a t t h e r e i s no g w i t h y 4 R a < P y .
.
F i n a l l y , suppose t h a t R a i s s i n g u l a r . By 3.8.4.3, R a y P f o r a cert a i n P . Thus, by 3.8.2.6, t h e r e i s a g 4 R a such t h a t R a = U y , and 2 4 R a , f o r every z E g , NOW, i f z E y , z c - u q = R a and, hence z E R a . Theref o r e y c R a . Since g o ( R a , y E R a y i.e. y E R { f o r a c e r t a i n t; E a Hence, U y C Rt; C R a , c o n t r a d i c t i n g u q = R a . Therefore, o u r hypothesis t h a t R a i s s i n g u l a r i s f a l s e , i.e., R a i s r e g u l a r .
.
.
Thus, we have completed t h e p r o o f of t h e i m p l i c a t i o n from r i g h t t o l e f t and, hence, o f t h e theorem.
AXIOMATIC SET THEORY
3.8.4.5
313
THEOREM,
(i)a
= U u #
(ii) R u E ZA
0 -
-+
W x ( x E R a - + ux, P x € X u ) .
V F W x(F
-R a A C
x
E
Ra
+
F*x
E
Ra).
PROOF OF ( i ) . Assume t h a t 0: = u a + 0 and t h a t x E R a = u { a]. Hence, x E R F f o r a c e r t a i n F_ E u, i . e . x f R F and, t h u s On t h e o t h e r haTd, l e t g c x. R 5 . T h e r e f o r e u X E R(F + 1) c R a q c R t and hence, g € R(F + 1): Thus, we have proved t h a t P x c-R ( F and, hence P X E R ( [ t 2 ) c Ra
F
.
E
.
R
5 :
u x c Then1)
t
PROOF O F ( i i ) . Assume t h a t R a E I A and l e t F c R a and x E R a . We have, F*x 5 * x 4 R a , by 3.8.4.4. NOW, by 3.8.4.4 and 3.8.4.3 using 3.6.1.13 and 3.6.1.14, we o b t a i n t h a t P x 5 x 4 R a . A p p l y i n g 3.8.4.4, again, we g e t t h a t F*x E R a . 9 3.8.4.6 REMARKS, From t h e p r e v i o u s theorem i t i s e a s i l y shown t h a t i f R u i s i n a c c e s s i b l e , w i t h u 2 w , t h e n R ( u t 1 ) i s a model o f M K T R Using 3.8.4.4 and 3.8.4.3, i t can a l s o be shown t h a t i t s a t i s f i e s Y - O n , i.e., AxC.
.
I t can a l s o be shown, u s i n g V = O n , t h a t i f R a t 1 s a t i s f i e s M K T R then R a i s i n a c c e s s i b l e . The use o f V = O n i n t h i s p r o o f i s e s s e n t i a l (see t h e problem a t t h e end o f t h i s s e c t i o n ) .
Since R(r+ 1 ) i s a model o f M K T R + A x C i f R a i s i n a c c e s s i b l e and u 2 w , t h e e x i s t e n c e o f an i n a c c e s s i b l e s e t R u w i t h a 3 w i m p l i e s t h e c o n s i s t e n c y of M K T R t A x C . By t h e w e l l known theorem of G'ddel, t h e c o n s i s t e n c y o f M K T R + A x C cannot be proved i n t h i s theory. Hence, t h e e x i s t e n c e o f an i n a c c e s s i b l e R u w i t h a 2 w , cannot we proved i n M K T R t AxC.
PROBLEM Prove:
3.8.5
RG(W)
f--f
W = On.
CLOSED UNBOUNDED AND S T A T I O N A R Y C L A S S E S ,
The degree o f c o f i n a l i t y o f an i n i t i a l o r d i n a l measures, i n a way, i t s l e n g t h . Thus, f o r i n s t a n c e , N u i s a r e l a t i v e l y s h o r t i n i t i a l o r d i n a l , s i n c e i t i s c o f i n a l w i t h w . The purpose o f t h i s s e c t i o n i s t o study subs e t s o f l o n g i n i t i a l o r d i n a l s and subclasses o f t h e o r d i n a l s . L e t a be an Among t h e subsets o f u t h a t a r e e q u i p o l i n i t i a l o r d i n a l w i t h cf(a) 3 w
.
ROLAND0 C H U A Q U I
314
lent with a r y ones.
01
As i n 3.8.2, itself. 3.8.5.1
t h e Greek c a p i t a l l e t t e r s
r,
A r e f e r t o o r d i n a l s or O n
DEFINITION,
r) Cub(X, r )
(i) cl(x, (if)
the station-
we d i s t i n g u i s h some t h a t a r e p a r t i c u l a r l y ' l a r g e " : The same i s done f o r p r o p e r subclasses o f O n .
t--,
y 6 ( S ~ Ar 6 = u (X n 6 ) + S
-Cl(X,
r)
A u X =
E
x)
A x
r.
.
c -r
I f Cl(X, r ) we say t h a t X i~ CLoned in r; and i f Cub (X, r ) , t h a t X h d o o e d unbounded in r. Examples o f c l o s e d c l a s s e s a r e {y :y E T A y = u 7 ) and { y : 7 E r A 3 x ( x c r A V S ( ~ E X - , U ~ = S ) A ~ ~ X A ~ = U X ) ) ( t h i s l a s t class i s the class o f l i m i t s o f l i m i t ordinals). I f r E OI ( a +1) or r = O n , then these c l a s s e s a r e a l s o unbounded. I f r = Ha f o r
-
a l i m i t , cx 3 0, t h e n O I n Ha i s c l o s e d unbounded i n Ha. t o see (by subclass X as r. One subclasses
I t i s a l s o easy
3.8.2.6 ( v ) ) t h a t i f r i s r e g u l a r (or = O n ) t h e n any unbounded o f r i s e q u i p o l l e n t t o r, and, thus, have t h e same c a r d i n a l i t y of t h e purposes o f t h i s s e c t i o n i s t o d i s t i n g u i s h among these some t h a t a r e considered l a r g e .
F i r s t we i n t r o d u c e some f u n c t i o n s a s s o c i a t e d w i t h c l a s s e s o f o r d i n a l s . 3.8.5.2
DEFINITION,
(i) Cx = U { F :
771))
'On(F) A
v[(E
E
DF
-,
F I E = n {q : q
E
X A
5 c-
*
( i i ) E X = U { F : O ~ ( D F ) A D =F X- ~ A I N I D F = ~ I N I X ) .
I f X i s a c l a s s o f o r d i n a l s , and f u n c t i o n such t h a t C i E
t h e enumerator o f X; i.e. On
(see Def. 3.5.1.2
UX= UUX,
= E i f and o n l y i f
[ E X,
then Cx i s an o r d i n a l i.e.
F p (C,)
= X
.
Ex i s
i f X i s a set, t h e n Ex = E N ' ( I N / X ) and D ( E X )E
and if X i s a p r o p e r c l a s s , then Ex i s
and 3.5.1.3);
t h e unique s t r i c t l y i n c r e a s i n g f u n c t i o n t h a t enumerates X and D ( E x ) = O n (see 3.3.2.6). U
r
A Cub(X,
PROOF, Assume f i r s t t h a t i s easy t o see t h a t D ( C X ) = r =u
r r
.
3.8.5.3
THEOREM,
r=
r)
c--t
Con (C ) A D X
Let 0 #
i t i s enough t o show t h a t i f 0 # [ = u
E=
U
5
E
r;
.
= u r and X i s c l o s e d unbounded i n I'. I t Now, we prove t h a t Cx i s continuous.
I t i s c l e a r from t h e d e f i n i t i o n t h a t CX i s monotone and a d j o i n t . 3.3.2.10,
(c,) = r
E
E
r
Thus, by
we have C i [ =uC;'.
the r e s u l t i s c l e a r i f E C - Cx 7 f o r some 7
E
E
be-
AXIOMATIC SET THEORY
315
cause i n t h i s case C ' € = C ' y = u C t € . So assume t h a t C i r c t f o r a l l X X , have t h a t U C?E = U ( ( C , * € ) n E ) = € . ThereY E € . Since y C C ~ Y we
( X n € ) = 5 , because CZE C -X.
(i), E € X ,
Thus, b y Def. 3.8.5.1
fore
U
i.e.
C' € = € , and, hence, C i t = u C * € X
.
r.
Assume, now, t h a t Cx i s c o n t i n u o u s and D ( C x ) =
Let 6 E
r
and
.
Since Cx i s continuous, C i 6 = u CX* ( 6 n X ) = u (6 n X ) = 6 Thus, 6 E X and, hence, X i s c l o s e d i n r . X i s unbounded i n r , because
u (6 n X ) = 6 .
u x = D(c,)
r
=
3.8.5.4
.
THEOREM,
u X
=
Note t h a t u X = r and C 1 (X, By 3.3.2.12
PROOF,
isomorphism and i t s range 3.8.5.5
THEOREM,
r
u uX
r)
=
r
A CI ( X ,
r)
-
Normal ( E x ) .
i f and o n l y i f C u b ( X ,
r).
(ii) ( s i n c e D ( E x ) i s l i m i t because Ex i s an i s limit).
'F(F) A Con(F) A
T ( r )3
w+ Cub(Fp
(F),
r).
PROOF, The p r o o f i s s i m i l a r t o a combination o f those o f 3.3.3.3 and 3.3.3.4. L e t F be a continuous f u n c t i o n f r o m I' t o r and l e t c f ( r ) I w . We f i r s t prove t h a t F p ( F ) i s unbounded i n r. L e t -$ Er. By i n d u c t i o n we prove t h a t F' ( u s i n g 3.8.2.2
6.
'E
Er
(v)),
f o r every v E w
B s i n 3.3.3.3,
.
Since F a ' €
E r , because
we prove t h a t
.
5 C- F w ' t
I n o r d e r t o prove t h a t F p ( F ) i s closed, l e t 6 E r w i t h Then F'6 = u F* (6 n F p ( F ) ) = 6, i.e. 6 E F p ( F ) . 3.8.5.6
U
Fp
(F).
w
(6 n F p ( F ) ) =
DEFINITION,
(i) limR (ii)lim
F ( r )3
E
A
= ( U
CR*{aI'E : a E D R 3 : €
E U
{DR*(aI:aEDRI)
.
R=(u{R*{cr)'€:aEE):€ EU{DR*(aI:aEDR)).
These d e f i n i t i ns have i n t e r e s t when we have a sequence of f u n c t i o n s Fa f o r a E r, w i t h r ( F a ) . Then, i f we t a k e a 0 C cf(F), we have t h a t
p.
r
l i m [ F : a E p ] i s a f u n c t i o n such t h a t r ( l i m [ F a : a E p ] ) and ( l i m [ F a : a a E p ] ) I € = u (Fa'E : a E p ) f o r e v e r y E E r , Also, i f r i s r e g u l a r , then l i m A [ F a : a E r ] i s a f u n c t i o n w i t h rr ( l i m A [ F a : a E r ] ) and ( l i m A [ F a :
a ~ r ] ) '=tu ( F d E : a E € 1 . t o p.
lirn [ F a : a ~ p i ]s t h e L h L t
lirn A [ F a : a € r ] i s t h e diagonal UnLt
l o w i n g theorem.
06
Fa i n
06 Fa
r.
when
OL
ten&
We have t h e f o l -
ROLAND0 CHUAQUI
316 3.8.5.7
THEOREM a(a E p
(i)
-+
I
rr(R*cai) A Con(R*CaI)) A P c
~ E P ] )A C o n ( l i m [R*{a) : a E P ] ).
.S (r)
f
'r(fim [ R * { ~ I
a ( a E r rr( R*{a))A C o n (R*{al)) A R G (r) -+ rr'(li m A t R*{a} (i i) a ~ r ) ]A C o n ( l i m A [ R*{u} : a E r J 1 . -+
:
r f o r P C cf(r) i s continr e g u l a r ; t h e diagonal l i m i t of continuous.
T h u s t h e l i m i t of continuous functions i n
uous i n
r,
and, f o r
r
PROOF OF ( i ) . Let R*IaI be a continuous function from r t o r f o r We have t h a t every a E P w i t h P c cT(F). Let G = l i m [R*Ca) : a € P 1 G'€ = U {R*CaI'E : a E P l f o r every t Er. I t i s c l e a r t h a t G i s monotone and a d j o i n t . Let E = u € E r , then
.
G'E =
u CR*(a}'€ : a E PI = u U {R*(~)'v : v € € I : a E P I u { u {R*{aI'q : a 6 P I : v E E I = u G*F
.
By 3.3.2.10,
=
G i s continuous.
PROOF OF ( i i ) . Let r be regular and R*{aI a continuous function Then H I € = from r t o r f o r every a E r. Let H = Iim A [ R*{a) : a € r ] I t i s c l e a r t h a t tt i s a function U { R {al'E :a E € I f o r every € E r. from r t o r ( s i n c e r i s r e g u l a r ) , and t h a t i t i s monotone and a d j o i n t . Let 0 # E = U € E r. Then H I E = u {R*{aI'E : a E El = u{u IR*{~I'v : q € E I : a E € 3 . On the o t h e r hand, u H I E = U t u {R*{aI'q : a E q l : v E € 3 . I t is Let y C H I E ; then t h e r e a r e a, G E E such t h a t obvious t h a t U H I € C - HI€. Let S = (a U q ) + 1. Then a E 6 E $, and R*{a} q C y C R*{a}'q 5 H I [ . Therefore, y C-U H*€ C R*{a}'6. B u t R*{a}'S 5 H I S , hence R*{a}' 6 5 U H*[. HI€. Thus, we have proved, H'C; = U H*E, a n d by 3.3.2.10, H i s continuous.m
.
3.8.5.8
COROLLARY
T ( r )2 r) .
(i)
a E PI,
(ii) RG(I') A
n
{s*{aI
:
E
A Va(a E p
0,/3
r
3
w A Wa(a E
T I } , r).
+
r
Cub(S*{aI,
-+
r))
-+
C u b ( n {S*{al :
C u b ( S * { a } ,r)) + C u b ( { y :y
E
(i) says t h a t i f t h e c o f i n a l i t y of r i s l a r g e r than a, then the i n t e r s e c t i o n of any family of l e s s than T(r) closed unbounded c l a s s e s i s closed unbounded. ( i t ) says t h a t i f r i s regular a n d uncountable, and S *{ a } f o r (Y E r i s a family of closed unbounded c l a s s e s , then i t s d i a g o n d intehd e c t i o n Q = (7 : wa(a E y -,y E S*{aI)l i s a l s o closed unbounded.
.
PROOF OF ( i ) . Let G = f i m [ C : aE P I Then, by 3.8.5.3 and S*{al 3.8.5.7, G i s continuous function from I' t o r. I t i s c l e a r t h a t F p ( G ) =
AXIOMATIC S E T T H E O R Y
n {S*{a} :aE
.
PI
By 3.8.5.5,
3.8.5.7, 3.8.5.5,
ff i s a continuous function from r t o F p ( f f ) i s closed unbounded in r .
3.8.5.9
DEFINITION,
r.
F p ( G ) i s closed unbounded in
Let ff = l i m A [Cs*{a} : ~1
PROOF OF ( i i ) .
317
S t a t (X,
r)
-
E
r.
v
r].
By 3.8.5.3
and
Also, F p ( H ) = D.
Y ( C u b (Y,
r)
+
Y nX
By
f
0).
S t a t (X, r ) i s read X L4 A L u X o n a h y in r . I f r i s regular then a l l unbounded subclasses X of r have t h e property t h a t X .-. r. I t i s c l e a r t h a t s t a t i o n a r y c l a s s e s a r e unbounded, s i n c e r a i s closed unbounded f o r every cx E r. Among these c l a s s e s equipollent t o r, s t a t i o n a r y c l a s s e s a r e large.
-
3.8.5.10
THEOREM,
(i)cf(r) 30,pA a
E
PI,
r)).
( i i ) cf(F)
3
w c i ( a ~ p + l S t a t ( s * I a } , r ) +lStat(u{S*{a)
w A Cub(X,
r) A
r)
Stat (Y,
--f
Stat (XnY,
:
r).
( i ) says t h a t the union of l e s s t h a t cf(r) non-stationary c l a s s e s i s ( i i ) says t h a t t h e i n t e r s e c t i o n o f a closed unbounded non-stationary. c l a s s w i t h a s t a t i o n a r y c l a s s i s s t a t i o n a r y . In p a r t i c u l a r , every closed unbounded c l a s s i s s t a t i o n a r y . PROOF,
By 3.8.5.8.
3.8.5.11
THEOREM,
(i)
r
= u
rz
0
+
(ii) r = U r # O +
x
f
PROOF, and 3.8.5.4. 3.8.5.12
( S t a t (X,
r)
-
Stat(X,r)-
v
F(rI'(F)
A Con(F)
+
DF-'nX#O)).
vF(rr'(F)ANormal(F)+DF-ln
0)).
( i ) i s obtained by 3.3.2.9 THEOREM,
Y E r A m y ) = a},
r).
z ( r ) 3 ci A
and 3.8.5.3. cx
E
R G n (01
(ii), by 3.3.2.13
-w)
+
S t a t {y :
Thus, i f cf(F) 3 w , t h e r e i s a s t a t i o n a r y subclass o f r t h a t i s not closed, namely {y : y E F A F ( y ) = u}. Without A x C we cannot prove t h a t o t h e r i n f i n i t e i n i t i a l o r d i n a l s a r e r e g u l a r , s o t h i s theorem does not provide us w i t h many s t a t i o n a r y s e t s . PROOF, Let T ( r ) 3 c1 and ci a regular i n f i n i t e i n i t i a l o r d i n a l , and l e t F be any normal function from r t o r . We have t h a t F ' a . = u ( F ' P : E a}. Since a i s regular and a C_ F'a, F ( F ' a ) = a. T h u s , F ' a E { y :
318
ROLAND0 CHUAQUI
.
By 3.8.5.11
rEI'AcT(r) = a). orem.
3.8.5.13
DEF IN IT I O N
(i) d F = E
FP(F
( i i ) , we obtain the conclusion of the the-
i
1-
( i i ) (By recursion). d0 F = E DF-~' datlF =dda F , o + = ~u a + d a F = d ( l i r n [ dP F : P E a
I).
( i i i ) F A = d (lirn * ( [ d P F : P E D F] )). d F i s the derivative of F , i.e. the function t h a t enumerates the fixed points of F . The fixed points of l i m [ d o F : P Ecxl are the common fixed points of a l l d P F f o r P E a ; thus f o r a l i m i t , d " F enumerates the common fixed p o i n t s of a l l d P F with P E a. F A , the diagon&zation 06 F , enumerates the diagonal intersection of the fixed points of d P F , i.e. the class Cr : W P ( P E r d P F'r = 7 ) ) . +
3.8.5.14 (i)
THEOREM,
F ( r )3
u A 'I?(F) A C o n ( F ) + v a ( a E
( i i ) RG(r) A
r
3 wA
rl'(F)
A Con ( F )
The proof i s l e f t t o the reader.
PROBLEM
Prove:
3.8.5.14.
+
V(r)
+
Normal ( F
A
Normal ( d a F ) ) .
).
PART 4 Morse-Kelley-Tarski
Class Theory w i t h Choice
CHAPTER 4.1 The Global Axiom o f Choice
4.1.1
AXIOMS
OF CHOICE,
What we would l i k e as new a x i o n i s t h e g l o b a l w e l l - o r d e r i n g p r i n c i p l e V = O n . However, t h i s i s v e r y l o n g t o s t a t e i n 1: Thus, we s h a l l P' adopt as a d d i t i o n a l axiom Ax GC. I t would be s i m p l e r t o adopt AxC. However, w i t h o u t Ax Reg, i t does n o t i m p l y V = O n .
The f o l l o w i n g i s a r e s t a t e m e n t o f Ax GC:
WA(W x W g ( x , g E A A x # y + 3 z Z E X A 1 3 z ( z E x A z E g ) ) 3BW x ( x E A + 3 ! y ( g E x A g e B ) ) ) ) A 3 C ( W D(W x ( x E D xeC) A 3x XED Ax GC.
-+
+
3 x ( x E U A Vg(yED
+
W
Z ( Z € X
--*
z E y ) ) ) A W x ( 3 U X E U '39
-+
xEgGC)).
The t h e o r y MKT w i t h Ax GC added w i l l be c a l l e d M K T C . 4, a l l theorems except f o r 4.1.1.3 a r e theorems o f M K T C .
I n t h i s Part
The f o l l o w i n g theorem i s j u s t a r e w r i t i n g o f Ax GC i n d e f i n e d n o t a t i o n .
I l e a v e t o t h e r e a d e r t h e p r o o f t h a t t h i s i s so. 4.1.1.1 A V = LJ C ) .
THEOREM,
WA(A A
+
3 BVx(xEA
-+
x n B = 1) A 3 C ( W O ( I N I C )
The f i r s t c l a u s e of t h i s theorem i s e x a c t l y AxC. 4.1.1.2
C
THEOREM,
Y = On.
PROOF, L e t C be such t h a t V = Y and, hence, I N I C 5Z V
.
U
C and I N I C i s a w e l l - o r d e r i n g .
L e t x € C ; then O I N l , ( x ) = { y : g C x A q E C ) S P x E V . V . Thus, a l l t h e hypotheses o f 3.5.1.4 i s a unique R such t h a t I N ]On zRI N I C .
E
are satisfied.
The r e s t o f t h e p r o o f i s t h e same as t h a t o f 3.8.3.3 i n s t e a d o f R a and V i n s t e a d o f W 9
.
I t i s easy t o see t h a t V - On i m p l i e s i n MKT forms a r e e q u i v a l e n t i n M K T . 32 1
,
Then
Hence, O I N l c ( x ) Therefore, t h e r e
( i i ) p u t t i n g R'cr
Ax GC.
Thus, b o t h
322
ROLAND0 CHUAQUI
4.1.1.3
METATHEOREM,
A
MKTC
u nubtheohy
06
BC.
PROOF, By 3.1.4.14.
4.1.2
I M M E D I A T E CONSEQUENCES I
I n t h i s s e c t i o n , we s t a t e some consequences o f Ax GC. Most o f them a r e o b t a i n e d from t h e equivalences o f A C proved i n Chapter 3.7. 4.1.2.1
PROOF, L e t R
x)
E
RI,
WR(R C V x V -
THEOREM,
5
.
V and suppose On zG V . L e t F I X = G I (n { a : ( G ' a , I t i s c l e a r t h a t D R V ( F ) and F 5 R.
V
for x E DR.
3 F ( D R V ( F )A F c - R)).
x
T h i s theorem i s e q u i v a l e n t t o Ax GC i n M K T R . 4.1.2.2
THEOREM,
(i) AC. (if) Wa3a a a. ( i i i ) W a ( A a + ~ ( x : x E#~ 0). ) ( i v ) Wa n ( x : x c -a A x # 0 ( v ) w a(0 4 a n ( x : X E U ) -+
)# f
0.
0).
.
PROOF, (i)i s an immediate consequence o f Ax GC. (ii)i s o b t a i n e d and 3.7.1.3.(iii), ( i v ) , and ( v ) a r e o b t a i n e d from (i) and from (i) 3.7.1.2.
L e t On =G V a n d assume t h a t F ( x ) # 0, f o r e v e r y
PROOF,
xEA.
f i n e F f o r x E A by s t i p u l a t i n g t h a t F I X = G'(n {a : G'a E F ( x ) I ) . I t i s easy t o show t h a t F s a t i s f i e s t h e c o n c l u s i o n o f t h e theorem. 4.1.2.4
De-
THEOREM ( G E N E R A L I Z E D D I S T R I B U T I V E L A W S ) ,
(i) " V ( A ) + n { u { A 1 ( , L , j ) : j E C } : i € B ) = u { n { A ' ( , L , F ' i ) : B i E B 3 : C(F)}
.
(ill
' " Y ( A ) + u { n { A V L , ~ j) :E c ) : i E ~ }= n { u { A ~ ( , L , F ~ , L ) : :' C ( F ) ) .
i E B )
PROOF,
The p r o o f i s s i m i l a r t o t h a t o f 3.7.1.4
from l e f t t o r i g h t ,
A X I O M A T I C S E T THEORY
u s i n g 4.1.2.3
i n s t e a d o f 3.7.1.2.
4.1.2.5 (i)
THEOREM (MAXIMAL P R I N C I P L E S )
W b(6 c - u A S O(1N I b )
-,
- + x = Y)).
(ii)
323
Wu( W X ( X E U
u be u )
Wq(q c -x A q
t--f
3 x ( x € u A W q ( q u~ A x
-+
-,q
FN
E
4.1.2.6
By 4.1.2.2,
3.7.2.1,
and 3.7.2.2.
-+
c_
q
3 x ( x ~ uA
THEOREM,
(i)V A ( A $! Y + A 2 O n ) . (ii)W A V B ( A , B $! Y + A = 8 ) PROOF
E a))
.
Wq(qEaA x c -q+ x = Y)). PROOF,
I
L e t A $! V OF (i).
f o r c e r t a i n B and G. A = G-'*8
.
.
Since V
; thus 8
On
2
9 Y.
, we
have t h a t A
By 3.3.2.6,
s t r i c t l y i n c r e a s i n g f u n c t i o n F such t h a t DF = O n and DF-' i s b i u n i q u e and O n e F 8. Therefore, On = A .
zG
B
C - On
there i s a = B.
Hence, F
(ii)i s o b t a i n e d f r o m (i). 4.1.2.7
THEOREM,
(i) WAWB(A 5 B v B 5 A ) . ( i i ) W A W B W C ( A $! F N A A = 8 +c C ( i i i ) WA W B ( A +c A 4 A ( i v ) W A W B W C ( A +c
t c B -,
84 A +c
-,
A = B V A=C).
A d 8).
C
-+
B 4 C )
(v) W A W B W C W D ( A 4 B A C o ( D + A t c C O(B ( v i ) W A W B ( A $! F N V B $! F N
( v i i ) WA(A q FN PROOF, 4.1.2.8
-,
A = 'A)
-+
.
A
tc B
=AxB).
By t h e theorems o f S e c t i o n 3.7.3, THEOREM,
A 5.8
++
A
<
D).
tC
4.1.2.2,
and 4.1.2.6.
.
8.
PROOF, The i m p l i c a t i o n from l e f t t o r i g h t i s o b t a i n e d by 3.4.2.7 The o t h e r m u l t i p l i c a t i o n i s o b t a i n e d b y 3.6.1.14 f o r B E Y and by a (i). s i m i l a r p r o o f a p p l y i n g 4.1.2.6 for B Y .
324
ROLAND0 C H U A Q U I
4.1.2.9
THEOREM,
(i)aEFN+-+w$a. ( i i ) ~ E F N - a + ~ l # a . (iii)
U E
4.1.2.9 nite.
a=b c a .
FN - 1 j b
(iii) g i v e s Dedekind's d e f i n i t i o n o f f i n i t e sets.
PROOF, We have by AC cds a o r a 4 w . Hence, i f w $ a, a i s f i The r e s t o f t h e theorem i s o b t a i n e d from 2.7.3.10.
R C -Y THEOREM, 4.1.2.10 D x - l L ~ AW t ( t E w - + xE t 1 R x t ) ) .
x
Y
Assume t h a t R c -V x V
PROOF,
-,
.
(WF (R)
E
t+1
and x
1 3 x(x
E
D R A x-'
E
Suppose, f i r s t , t h a t t h e r e i s a
b i u n i q u e sequence x E O D R such t h a t x
Ox";
++
Rx
t
E
f o r every
EW
.
Let A =
i t i s clear that 0 # A C - DR. L e t Y E A ; then y = xE f o r a c e r t a i n R x w i t h xE +. xEt + 1 E A. T h u s R i s n o t well-founded.
p
E
t+l
be Suppose, now t h a t R i s n o t well-founded and l e t A C D R U DR-' such t h a t A it 0 and f o r e v e r y Y E A t h e r e i s an x E A w i t h x + y and x R y . we L e t S = R n A x A . We have t h a t Os(y) # 0 f o r e v e r y Y E A . By 4.1.2.3, o b t a i n an F such t h a t
A V(F) and F ' y
r e c u r s i o n t h e sequence x
E
A"
E
Os(y) f o r every
YEA.
We d e f i n e by
as f o l l o w s .
L e t y be a f i x e d element o f A ; t h e n
X
t+1
=
F'x
E. R x f o r every t+1 t
I t i s easy t o show t h a t x i s b i u n i q u e and t h a t x
t E w .
4.1.2.12
6'v
A+O A
THEOREM ( P R I N C I P L E OF DEPENDENT C H O I C E S ) ,
w
X ( X E A-* 3
Rd'vtl)).
Y ~ E A
X R Y ) ) -, 3
616
E #AA
V(VEW-'
PROOF, Assume t h e hypothesis and l e t rA(F), A I'(F-') w i t h r E On x and 6'v t 1 = F'(n I t : o r r = O n . L e t xEA. D e f i n e by r e c u r s i o n , 6 ' 0 b'v R F'EI).
CHAPTER 4.2 Cardinality Theory
4.2.1
INITIAL O R D I N A L S A S C A R D I N A L S ,
In M K T C , every set is equipollent with a certain ordinal and, hence, with a certain initial ordinal. This is why we can take initial ordinals as cardinal numbers. Thus, we shall define a new "cardinal number" for each set a, namely the initial ordinal equipollent with it.
i = u {r: (r E or v r = o n ) A r
DEFINITION,
4.2.1.1
=- A }
.
We clearly have, 4.2.1.2
THEOREM, -
(i) A $ Y + i (ii) A
Y-,X
E
(iii)
=
(iv)
A =
-
u
=On E
A
.
or.
II
B
i.
can be used as a type of equipollency, just as [ A ( was By (iii), used. It has two main advantages over this older cardinal number: it can be defined for every class and, by (iv), it is equipollent to A. (i) a+'p
4.2.1.4
(i) (ij)
-
DEFINITION,
4.2.1.3
=a+p.
THEOREM, =
axb
=
i
i Sc
(iii) a,b,cE W +
E.
' +
6 .
(la1
+ Ibl
=
1.
325
c*
'+
6
=
z).
ROLAND0 CHUAQUI
326
,c
( i v ) u,b
E
W + (la1
J b l = IcJ
-
u
b = c)
.
Thus, t h e new d e f i n i t i o n o f sum and m u l t i p l i c a t i o n o f c a r d i n a l s , coi n c i d e s w i t h t h e o l d one. PROOF, By 3.1.1.3
and 4.2.1.2.
4.2.1.5 S T I P U L A T I O N O F VARIABLES, From now on, we r e s e r v e t h e Greek lower-case l e t t e r s p r v, K , and n w i t h or w i t h o u t s u b i n d i c e s and primes f o r i n i t i a l o r d i n a l s . The n e x t theorem g i v e s some p r o p e r t i e s of o u r new c a r d i n a l s which a r e e a s i l y deduced from p r o p e r t i e s o f alephs (3.6.1) and i n i t i a l o r d i n a l s
(3.8.1).
4.2.1.6
THEOREM,
(i) W O ( I N 1 0 1 ).
(ii) W C -p (iii) wc p (iv) p
+'
+
v o c- v v c p
(V) p c V A PROOF,
from 3.6.1.9.
p = p
+'
+'
p = 1.1
C
v = p u v A (p + o +v
+
p
+'
A
-t
v c n
.
ACV
+'
BCK + p
.
p
-,p
='
v =puv).
K
(i)i s o b t a i n e d f r o m 3.6.1.2. (ii), from 3.6.1.6. ( i v ) and ( v ) , from 3.6.1.12.
=
(iii),
The general p r o p e r t i e s o f c a r d i n a l a d d i t i o n and m u l t i p l i c a t i o n a r e a l so v a l i d f o r t h e new o p e r a t i o n s :
327
AXIOMATIC SET THEORY
n2
+'
n A 3
= no
p'
n2 A u' = n1
By 4.2.1.2
PROOF,
4.2.2
+'
+C
and 2.7.2.4
n3). ( i ) , (iii),
(v)
-
(ix).
I N F I N I T E SUMS AND PRODUCTS,
D e f i n i t i o n 2.7.2.2 i n t r o d u c e d t h e a r b i t r a r y c a r d i n a l sum o f classes. However, w i t h o u t an axiom o f c h o i c e i t was i m p o s s i b l e t o work w i t h a r b i t r a r y sums o f c a r d i n a l numbers. Also, t h e analogue o f 2.7.2.2 c o u l d n o t be proved. Now we can do it. We s h a l l use t h e n o t a t i o n i n t r o d u c e d a t t h e b e g i n i n g of 3.4.3.
i
#
j
-
I v ( F ) --* (c F = c 3 G ( I v ( G ) A w i ;~( i , ; r A~ 4.2.2.1 THEOREM, + G ' i n G ' j 0 ) A C = u G*I A W i ( i E 1 --* F ' i 2 G ' i ) ) . Assume I V ( F ) .
PROOF,
G ' i = H*(F'i)
Suppose, f i r s t , t h a t C
F =H
C.
Then p u t t i n g
f o r L E I , we o b t a i n t h e r e q u i r e d G.
Suppose now t h a t G i s a f u n c t i o n w i t h domain 7 such t h a t G ' i n G ' j = 0 , f o r i # j , i , j E I ; C = U G * l ; and G ' i = F ' i f o r e v e r y L E I . Now, F ' i , G ' i E V . And, hence, i f G ' i = 6 F ' i , then tj E V. L e t On =I V and 6c. =
L e t h.' x = F'i}). c u (hi : i ~ l } .Then, C = H C F.
J ' (n ( a : G'i =
4.2.2.2
J'cl
THEOREM,
(
dix, i ) f o r
D F V ( F )A A C - DF
+
U
x E G ' i and i E
I and H =
{F'x :x E A } 5 C ( F ' x :
xEA)
The p r o o f i s l e f t t o t h e reader. Thus, we a r e j u s t i f i e d i n i n t r o d u c i n g t h e i n f i n i t e c a r d i n a l sums and products o f i n i t i a l ordinals.
4.2.2.4
DEFINITIONI
(i) Z
( K ' x :
C
(ii) n
x€A) = C X
X ( K ' X : X E U )
C
We have:
=
(K'X
:x E A )
ZI X ( K ' X
: X E U )
. .
ROLAND0 C H U A Q U I
328 4.2.2.5
THEOREM,
( v i ) 'OI ( F ) A 'I01 (G)
-+
Z C
(
F'i
C t
G ' i :LE7
) =
C C
,
4
(
F ' i : i~ 7 ) tc
The proof is l e f t t o the reader. The following theorem due t o J . Konig i s a strengthening i n M K T C o f 3.8.2.9. Unfortunately, i t i s n o t possible t o deduce i t from 3.8.2.9 so we s h a l l have t o repeat t h e proof. 4.2.2.6 THEOREM, XEU) c n (M'X : X € U )
c x
.
m, n
E
'01 A Wx(x€a
+
m'x
C
n'x)
-+
2 (m'x :
c x
PROOF, Assume t h a t m, n E OI and m'x C n'x f o r every X E U . I t i s enough t o show t h a t Z (m'x : X E U ) Il (n'x x { X I : x e u ) We show, f i r s t t h a t 2: (m'x : X E U ) 5 ll( M'X x Ex} : x E a ) . We have t h a t m'x x {x) C n!x x {XI f o r xfu. Hence, (M'X x Ix))-(m'x x {XI)+ 0 and, t h u s , U
.
32 9
AXIOMATIC S E T THEORY
-
] I ( (n'x x {x})-(m'x x {x}) : x E a ) # 0. L e t 5 E II ( (M'X x { X I ) (mlx x {x}) : xea). F o r y E C (nr'x:x E a ) and x E a d e f i n e dd'x= I'x , i f q 9 m'x x 4x1; and dg'x = q , otherwise. I t i s e v i d e n t t h a t dg E n (n'x x { x ) : x e a ) f o r every q E C (m'x : xEu). Also, i f y#z, then + 6:. I n o r d e r t o prove t h i s we assume t h a t y#z and we c o n s i d e r two cases. F i r s t , l e t y E m ' u x {u} and z E m ' w x { w } f o r u f v . Then $ ~ ' u = y E m ' u x { u } (m'u x { u } ) ; thus d g f d z . Second, l e t y,z E and d z ' u 6 u E ( M ' U x {u}) m'x x { x } ; then d y ' x = y # z=dz'x; i.e. d q + d z . Hence, i f we p u t g ' q = d y f o r every q E 2: (m'x : x e a ) we have t h a t C (m'x : x E a ) SgII ( n'x x {x) : x E
-
a)
.
I n o r d e r t o argue by c o n t r a d i c t i o n , assume t h a t C (m'x : x E a )
{XI : x € a ) . Thus, f o r y E I: (m'x : xEu), we have t h a t (hly)'xE n'x x Ex). Define, f o r X E C L , k'x = {(IL'IJ)'~ : yEm'x x {x)}. Thus, h'x C n'x x {XI. I t i s c l e a r t h a t h'x <, tu'x x Cxl M'X x {XI. Therefore, h'xc x { x ) ) - k'x # o f o r e v e r y X E C I . L e t j E n(nfxx{X)) M'X x 1x1, i.e. ( M I X - k'x: XECL). Then t h e r e i s a y such t h a t j = ii'g w i t h qEm'x x {XI f o r
n (MIX
ch
x
.
a c e r t a i n x E a . Hence j'x = (h'y)'xEl2'x, by t h e d e f i n i t i o n o f k . But, j i s such t h a t j'x 4 l z ' x . Thus we o b t a i n a c o n t r a d i c t i o n and we prove t h a t C ( m ' x : x ~ a f.) ~ ( M ' xx {x} : x ~ a ) . 4.2.2.6
i m p l i e s t h e f o l l o w i n g s t r e n g t h e n i n g o f 3.8.2.9: y(yEx
--*
y 4 u x)
+
ux4 x u x .
T h i s i s so, b e c a u s e u x < C ( y : y E x ) and 'Ux = n ( u x : q € x ) .
PROOF, we have,
$ [(m't:
L e t t h e h y p o t h e s i s o f t h e theorem be s a t i s f i e d .
: [ € a )c
nc t ( m ' ( [ t
1 : [
E C ~ c)
nc
t
By 4.2.2.6,
( m ' ~: [
We i n t r o d u c e d i n 3.8.1.3 t h e normal f u n c t i o n H t h a t enumerates a l l i n f i n i t e s i n i t i a l o r d i n a l s . From 3.8.1.5 we immediately o b t a i n .
U
F i n a l l y , we s h a l l prove a few theorems about i n f i n i t e sums o f i n i t i a l ordinals.
ROLAND0 CHUAQUI
330
4.2.2.10
THEOREM,
PROOF OF (i).Let a = = Hp
,
p
by 4.2.2.9.
PROOF
OF (ii). Let a
,
ZC ' H E : t g a ) = Ha 4.2.2.11
c(Kd,t :I C
t
Then Z ( H t : t E a )
1.
C
= Z ( K t :.$ C
CP)=
ucr Z 0. Then Ha = "{HE : t E 0.1 & by 4.2.2.2 and 4.2.2.9. =
c1 =
THEOREM,
E a) = H
U a + @ A I n ( d )A
4 E 'On
A
P
= U
6%
+
P '
The proof i s l e f t o t t h e r e a d e r .
PROBLEMS
1.
2. 3.
Prove 4.2.2.3. Prove 4.2.2.5. Prove 4.2.2.11.
4.2.3
DEGREE OF C O F I N A L I T Y W I T H C H O I C E ,
We s t u d i e d i n S e c t i o n 3.8.2 the d e g r e e of c o f i n a l t i y of o r d i n a l s a . Here we s h a l l be mainly i n t e r e s t e d i n the d e g r e e o f c o f i n a l i t y o f i n i t i a l o r d i n a l s . As b e f o r e we r e s e r v e the Greek l e t t e r s K , T , p , u f o r i n i t i a l ordinals.
From 3.8.2.2
( v ) , we o b t a i n ,
c ~ ( K )
= n
~p : j d @ n ( d ) A 6
On the o t h e r hand, Def. 3.8.2.1 Cf(K) =
n (P :
E
PO, A u 0 6 - 1 = K I I .
implies
cOf(p,
INIK)}
,
AXIOMATIC SET THEORY
33 1
I n M K T C we can a l s o show. 4.2.3.1
THEOREM,
T ( K= )n {P
n(n E ' ( O I n
:
K )
A
= Z
K
C
a E p ) A p C _ ~ l .
P
PROOF, Assume f i r s t t h a t
i s cofinal with IN
IK
and l e t
s t r i c t l y i n c r e a s i n g sequence o f o r d i n a l s w i t h 06 = P and u 6 i s b i u n i q u e , we have t h a t P 5 K . Thus, by 4.2.2.2, K
-
: ~ E P ) < Z ( K : ~ E P- K)
=U[6'E:EEP}<,Z(6'1
( E: E € 0 )
Therefore K = Z we have shown t h a t
C ~ ( K ) > ~ { P3 :n ( n P~( Assume,
6't
and, s i n c e
EK
and
K E
d =
< n'a
be K .
:
a Since
X ~ = K .
01,
6'E
CK.
Thus
= x ( ~ ' ~ : ~ E P ) A P ~ K ) I .
0 I n K ) A K
C
i n o r d e r t o prove t h e converse i n c l u s i o n , t h a t a i s t h e l e a s t
P PC K such t h a t t h e r e i s an n E OI w i t h K = Z ( n ' E : [ € 0 ) and M ' E C K f o r e v e r y E E P . By 4.2.2.1, K = u Iu'E : E Eal w i t h a'[ = n'E and u ' t n a'{ = O We d e f i n e 6'E = U [a'S : 5 c t } . Then f o r E +S. = Z ( n ' S :S 56 ). C
O I , because i f we had ci 3 y = a we would have K = d C ( n ' d ' E : E € 7 ) c o n t r a d i c t i n g t h e c h o i c e o f a. Hence, = Z ( p ' S :5 € E t 1 ) w i t h [ + l e a . Therefore, ~ ~ ~ ( ~ ' ~ : ~ E E + ~ ) = We have t h a t a
E
- C
f o r a l l 5 € a . L e t g'E = I N 1 b ' t . B u t U [ b ' t E E a I = K and thus, U
S i n c e b'E 4 K , g ' t E K f o r e v e r y 5 E a. g*a=K. Using, now, 3.3.2.7 we o b t a i n a S t r i c t l y i n c r e a s i n g sequence o f o r d i n a l s Dh w i t h DDh = Y 5 a and u D h - l = K . Suppose t h a t y C a ; then u s i n g t h e f i r s t p a r t o f t h i s p r o o f we g e t t h a t K =
Z(
:E € 7 ) f o r a c e r t a i n p E ' ( 0 1n K Hence Y = a and 01 i s c o f i n a l w i t h I N \ K . C
n (6 : 3M(n f2 '(0Il-l 3.8.2.6
A
K )
K
c o n t r a d i c t i n g o u r c h o i c e o f a.
) ,
Thus, we have shown,
= C
(M'a : a € P ) A
P
CK))
ST(K).
We now t u r n t o s i n g u l a r and r e g u l a r i n i t i a l o r d i n a l s . W e know, by ( v i ) and ( v i i ) t h a t , K
3 W-'
(K
E RG
-
(K
E
K 2 W-'
s
++
CT(K)
N H S ( K )
= K )
,
c
,
K )
Without t h e axiom o f c h o i c e we have o n l y been a b l e t o prove r e g u l a r t h e i n i t i a l 0, 1, 2, and w. On t h e o t h e r hand, we c o u l d p r o v e t h a t many i n i t i a l o r d i n a l s were s i n g u l a r (3.8.2.7). I n M K T C , we can show t h a t many i n i t i a l o r d i n a l s a r e r e g u l a r , namely:
~
'
E
ROLAND0 C H U A Q U I
332
PROOF, Assume t h a t X M'C; c Hence, - Ha.
(
M'C; : [ €0 ) = Ha
+
w i t h n'C; c H a t l .
Then
CARDINAL EXPONENTIATION OF INITIAL ORDINALS, CIJ r 4.2.4.1 DEFINITION, K = K 4.2.4
.
We p u t a c o v e r K i n o r d e r t o d i s t i n g u i s h c a r d i n a l e x p o n e n t i a t i o n from o r d i n a l e x p o n e n t i a t i o n . When t h e r e i s no danger of c o f u s i o n we s h a l l omit it.
THEOREM,
4.2.4.2
.
PROOF, 3.8.2.10.
(F
THEOREM,
4.2.4.3 (
(i) is o b t a i n e d from 3.8.1.10.
m ' x :x € u ) ) K = PROOF,
c (m'x:
c m
K
n ( (m'x)K C
:X E U )
(ii)-(iv) XEU)
.
=
a r e o b t a i n e d from
n(K C
m'x:X E U ) A
By 3.4.2.5.
We n e x t prove a few c l a s s i c a l r e c u r s i v e formulas f o r e x p o n e n t i a t i o n . 4.2.4.4
THEOREM,
%
(i)Ha+1= a (ii)
V E W +
Na + v
*c H
a+l'
a
(Hausdorf f ' s Formul a).
a
+
.
(Generalized H a u s d o r f f ' s Formula).
333
AXIOMATIC SET T H E O R Y
(iii) VEW+
V
H
"=
mC
Hv
( B e r n s t e i n ' s Formula).
PROOF OF (i).We have t o c o n s i d e r two cases: c HP
a + 1 50. Then by 3.4.2.4 H
CASE I ,
H C H a+l- P
".
C
CASE 1 1 1
Hence,
P ca
ka p = 5c1
1.
t
0'
=
(i),H a t
HP =
K
ka O .
But,
Katl.
Then,
I n o r d e r t o prove t h e converse i n e q u a l i t y , we s h a l l show t h a t ,
Let that Thus,
6
U 0 d - l
6
E
u{
H
P H a t 1. Since T ( H 1)~ = Hat i.e. U D 6 l = t E Hatl C H "0 y 1 '
E
t;.
E
-
E
Ha
PROOF OF ( i i ) .
+
and K
P
and hence
1} and we have shown ( * ) .
we have
6E '(t
+ 1).
By (*) and 4.2.2.2,
Hat"
H P , a
= &
NCt+V'
H
H P.CH a t v t 1' a PROOF OF (iii).By (ii)w i t h a = 0, and 4.2.4.2.
(7) a
Ha+1,
By i n d u c t i o n on v , f o r V E W . W i t h v = 0, t h e formu-
l a i s obvious. Assume, then, t h a t h y p o t h e s i s we o b t a i n ,
4.2.4.5
C
By ( i ) and t h i s
.
THEOREM ( T A R S K I ' S FORMULA). = u
# 0 A
6E
OlOn A I n ( 6 ) A y = u
6 *a A HP c T ( a )
-+
c *P =
Hy
3 34
ROLAND0 CHUAQUI
Assume t h a t 6 i s an i n c r e a s i n g sequence o f o r d i n a l s PROOF O F (i). w i t h 06 = a = u a f 0, y = u 6% and Hp C F ( a ) . We s h a l l show f i r s t ,
H
(1)
H y C U i P H 6'E
Let g
E
Hence g
HP
E
6 ' € ' - C:
Since Hp C T ( a )
Y'
with
E
PROOF O F ( i i ) . 4.2.4.6
PROOF, L e t b = For
5
dlEl.
By ( i ) .
THEOREM,
=
Then,
/J
-+
{x: x
C V
C a A F=v> = p
We s h a l l prove t h a t
6
< *9
V
a and, hence, b
On t h e o t h e r hand, l e t 6 E ' a . Then 6 c - a x v and { x : % = v A x C- a x v ) . But, a x v = p o c v = p . Hence, 4.2.4.7
7
V
= v.
a
5
DEFINITION,
( i ) Pv(A) =
ix x c - A A i c v}.
(ii) pz = z ( p K C
4.2.4.8
A'€' 3 5.
such t h a t
.
{ x : x C- a A % = v } .
6%.
E l
: K E v ) .
COROLLARY,
x
= p A w
5v C p
PROBLEMS
1.
Prove 4.2.4.8.
2.
Prove:
C H
xa+al
CHa
= 2
, for
.
3
5 -
E
Hence
T h e r e f o r e (1) i s proved and, by 4.2.2.2,
wC_v z p A
E ' a , l e t g'd =
we have t h a t u D 9 - l . Hy
It is c l e a r t h a t y = u y ; thus there i s H
Y'
6
and, thus,
2
M
2 $(y),
Also, s i n c e U 6 * a = y , t h e r e i s a
with SEy.
H
H
:EECd.
e v e r y a.
-
-+
Pv ( a ) = p2
b.
b
.u
. 'a.
L a. V
Hence ' a c -
AXIOMATIC SET THEORY
4.2.5
335
THE EXPONENTIAL SCALE OF I N I T I A L O R D I N A L S ,
I n S e c t i o n 3.8.3 we s t u d i e d t h e e x p o n e n t i a l s c a l e o f c a r d i n a l i t y , i.e. t h e beths o r c a r d i n a l s o f t h e s e t s R ( w t a ) . W i t h Ax GC (even w i t h AC), each R ( w t a) i s e q u i p o l l e n t w i t h an i n i t i a l o r d i n a l . We now i n t r o d u c e t h e s c a l e o f i n i t i a l o r d i n a l s corresponding t o beths.
We immediately o b t a i n from 3.6.1.21,
4.2.5.2
and 3.8.3.1:
THEOREM m
As we see, normal.
4.2.5.3
3.6.1.22
3
i s s t r i c t l y increasing.
The n e x t theorem i m p l i e s i t i s
THEOREM,
( i ) a = u a # o + ~= L J C Z L ~ : ~ E ~ I = Z : ( + : ~ E ~ I .
(ii) Normal ( 3 ) .
a
PROOF OF (i). Assume a =
C
u a f 0.
By 4.2.2.2,
R ( w + a ) = u { R ( w + t ) : t E a ) ~ C ( R ( w +: t[ )€ a ) .
336
ROLAND0 CHUAQUI
PROOF OF ( i i ) .
(iii). =
By (i) and 4.2.5.2
Our n e x t d e f i n i t i o n g i v e s us a f u n c t i o n t h a t f i x e s a correspondence between t h e exponencial s c a l e and t h e s c a l e o f i n i t i a l o r d i n a l s .
4.2.5.4
C = H-l 0
DEFINITION,
Thus , we have,
la= 4.2.5.5
.
PROOF,
4.2.5.3).
4.2.5.6
1.
H~ a f o r e v e r y
ci E
On
.
THEOREM,
Normal ( C ) .
By 3.3.2.15
( i v ) , s i n c e N and l a r e normal (Def. 3.8.1.3, = Ua
THEOREM,
PROOF, L e t a =
U a #
0.
f
0
+ -
F(Ha) = ?(a)
By 3.8.2.2
cc'a
(vii), q ( N a )
-
= .f(a)
6
.
and
.
Let, now, c f ( a ) = 6 and 6 E a be a s t r i c t l y i n c r e a s i n g sequence such t h a t u 6*6 = a . Then, s i n c e C i s normal ( 3 . 2 . 5 . 5 ) , we have, C ' a = C ' ( U 5*6) = u (C 0 6)*6. But CO 6 i s s t r i c t l y i n c r e a s i n g , thus C ' a 3 6
-
4.2.5.7
.
THEOREM,
PROOF OF (i):s i n c e C i s s t r i c t l y i n c r e a s i n g (4.2.5.2). PROOF OF ( i i ) . Assume t h a t C'(y + 1) i s a l i m i t o r d i n a l and l e t 6 = $(C'(y Thus,
+ 1)).
Suppose 6
which i s a c o n t r a d i c t i o n .
27;
t h e n by 4.2.4.2
Thus, y c 6 .
=
( i i ) , NC ' ( Y + l )
Nc'(y+l)*
33 7
A X I O M A T I C SET THEORY
PROOF, (i), we have,
Let a = u a
Z" c
#
0.
Assume, f i r s t t h a t P C c f ( a ) . By 4.2.4.5
"aI+ *
: & € a )C - la .c; H laHP t lo,, i.e. la =1
=la
- C
51.
Therefore,
Assume, now, t h a t c T ( a ) C - # c- C'a and l e t creasing, and u 6*(cf(a))= a. Then, by 4.2.5.2
la
H
P
c?5
On t h e o t h e r hand,
a
6
.
E cT(a)a,
(iv),
6
s t r i c t l y in-
ROLAND0 CHUAQUI
338
Therefore, 4.2.5.8
'.
H latl = 2 a
a
COROLLARY. 1
=
ua
0
#
( T ( c i ) ~ C ' P c C ' a + 2' =~ I a + , ) A
-+
(C'P C T ( a ) + 2
(aCP-Ila
la
=
1 , ) A
=lPtl).
1 The main d i f f i c u l t y f o r computing la i s t h e i n d e t e r m i n a t i o n o f C .
I f we f i x t h i s f u n c t i o n , f o r i n s t a n c e b y C H (i.e. C ' a = a ) we o b t a i n more d e f i n i t e r e s u l t s . We now study some consequences o f C H .
4.2.5.9
THEOREM,
CH
-
We a l r e a d y saw i n 3.8.3.2
MKT R.
H
==.
t h a t t h i s equivalence i s a l s o p r o v a b l e i n
PROOF, The i m p l i c a t i o n from l e f t t o r i g h t i s o b t a i n e d from 3.8.3.2 ( i ) . So assume t h a t H =1.L e t a be an i n f i n i t e set. Then a 2 Ha f o r a c e r t a i n a. Hence,
Thus, t h e r e i s no b such t h a t a
PROOF,
By 4.2.5.9
4 b.( '2.
and 4.2.5.8.
.
fi
9
T h i s s e c t i o n c l o s e s w i t h a few remarks about l i m i t and s t r o n g l i m i t c a r d i n a l s . F i r s t , a few d e f i n i t i o n s . 4.2.5.11
DEFINITIONa
(i) K + = n ~p : ~ E O Z A 2 K~ ) . (ii)LOZ={H
(iii) S LOZ =
a
: a = u a ~ O n ) .
{l :a a
=
ua
E
O n }.
L O I i s t h e Cease 06 LimLt cahdiCahdim. I t i s c l e a r from t h e d e f i n i t i o n s t h a t L O Z and S L O Z a r e c l o s e d unbounded i n O n . We have t h e f o l l o w i n g easy theorem whose p r o o f i s l e f t t o t h e reader. t
K i s t h e CahdCrZae nucce6noh 06 K . Maen, and S L O Z , t h e 06 n h o n g
A X I O M A T I C S E T THEORY
(ii)
K
E
sLor
-
w'(~(EK
-+
C Z'E
339
K ) .
T h u s , limit c a r d i n a l s a r e those which cannot be reached by the successor operation, and strong l i m i t c a r d i n a l s cannot be reached by exponentiac SLOI tion. I t i s clear that LO1 -
.
PROBLEMS
1.
Prove: (a) C H (b) C H
2.
+
--*
v
p(p
c
T(K)
kj
K(K
E
RG
+
-P
Ep=
Z( C
K':
K )
~ E K =) K ) .
Prove t h a t C H i s equivalent t o each of t h e following statements:
CHAPTER 4.3 F i l t e r s , Ideals, and Trees
4.301 FILTERS AND IDEALS, I n t h i s section we introduce the notions, of f i l t e r and ideal over a given set.
4.3.1.1
DEFINITION,
( i ) FI ( a ) = { x : x c P a A U E X A 0 4 x A w u v u ( u , A W u W w ( u ~ xA u c -u c - a -, V E X ) ) .
V E X -,
unu~x)
( i i ) I d (a) = { x : x-c P a A O ~x A a q ~ A W ~ ~ U ( ~ , L J E X ~ U U U E A \luwu(uEx A v c - u + VEX)). ( i i i ) du(x,a) = {a-y :Y E X } .
id&
F l (a) i s t h e n e t v b p h o p e h ~.LUemv w e h a . Z d ( a ) , t h e b e Z phupeh uueh a. I t i s clear t h a t i f x i s a f i l t e r over a, then d u ( x , a ) i s
an ideal (called the dud dideae) and vice-versa.
A MwhL b.LUeh over a i s {a). A f i l t e r x i s p&LncipuL i f n x E x . Equivalently, x i s principal i f there i s a y C a such t h a t x = { u : q c -u C a). PhinCip7.X id& are the duals of principal f i l t e r s .
The notion of ideal extrapolates the concept of small s e t . Given an ideal x over a, a s e t q 5 a considered small, i f y E x . Dually, the members of a f i l t e r are the large sets. This i s related t o the notion o f measure. If we have a probability measure p defined on subsets of a , { y : p ( y ) = 0) i s an ideal and { y : p ( q ) = 1) i s a f i l t e r . A useful f i l t e r i s the F&Echet &LLteh. Let a be an i n f i n i t e s e t and l e t x be the ideal of f i n i t e subsets of a. The dual f i l t e r ,
du(x,a) = {a-y :Y E X I ,
i s called the Frgchet f i l t e r over a.
4.3.1.2
DEF I NIT I O N
This f i l t e r i s not principal.
I
( i ) S a t ( ~ , ~ , at .) v b ( b c-P a A b
= K
340
A ~ u ~ ~ ( u # u Au Eu b, + u n u € x )
341
AXIOMATIC SET THEORY
+
b n x + 0).
(ii) sat(x,a) = n (iii) Comid
(K,x)
( i v ) Comfl
(K,x)
(v) Pr(x,a)
-
(K
:Sat ( K , x , u ) }
++
W b(b c -x A
++
W b(b c x A
t/ u(u c -a *
UEX
6 6
. cK
-+
u 6~ x).
cK
+
n b E x).
V a-UEX).
An i d e a l x o v e r a i s c a l l e d ~ - b a t U n a t e d i f Sat ( K , x , u ) . s a t ( x , a ) i s t h e saturatl'on o f t h e i d e a l x over a. An , & f e d x oveh a i s K-cornp&Le i f C o m i d ( K , x ) . A &LLteh x oveh a i s ~ - c o r n p k t ei f C o m f l ( K , x ) . An i d e a l x o v e r a i s r.]hime i f Pr(x,a). A f i l t e r x o v e r a i s an LLet)ra6.ieteh ifP r ( x , a ) . x A a p h e ,&fed uveh a 4 and onLq .i6 d u ( x , a ) ( t h e d u d &LLteh) .& an dLta&LLteh o v a a. I t i s a l s o easy t o see t h a t doh any ,&fd x o v a a, x .h p h e 4 and o n l y 4sat ( x , a ) = 2.
4.3.1.3
THEOREM,
(i) b c Fl(a)
-+
n b E Fl (a).
(ii)b c -Fl(a) A S O ( I N Ib)
-+
u b E Fl (a).
+
(iii x c w-, n q 0) c ~ ( aA) y q ( y c -x A wA n q c u)} = n ( v : x C - v E FI ( a ) } . ter.
+
-
(u . 3 Y ( Y
cx A i c
(i)says t h a t t h e i n t e r s e c t i o n o f a f a m i l y o f f i l t e r s o v e r a i s a f i l (ii), t h a t t h e u n i o n o f an I N - c h a i n o f f i l t e r s i s a f i l t e r .
x
s a t i s f i e s t h e h y p o t h e s i s o f (iii), we say t h a t x has t h e d U e e v e r y f i n i t e subset o f x has a non empty i n t e r f o r every x w i t h t h e f i n i t e i n t e r s e c t i o n property s e c t i o n . Thus, b y (iii), there i s a smallest f i l t e r containing x, indicated i n the conclusion o f T h i s s m a l l e s t f i l t e r c o n t a i n i n g x i s c a l l e d t h e @Lteh g e n m d d (iii). by x. If
Memec-tAon p o p e h t q i.e.,
The p r o o f i s l e f t t o t h e reader.
4.3.1.4 THEOREM, x q'x = Y)). Thus, a f i l t e r i n FI ( a ) .
x
E
F l ( a ) -+ ( P r (x,a 1.
-
W Y(Y
E
F i ( a ) A x C_
i s an u l t r a f i l t e r i f and o n l y i f i t i s I N - m a x i m a l
PROOF, Assume x i s an u l t r a f i l t e r , x C y, q a f i l t e r and u E y Then U - U E X C - y. Hence u, a - u E q, a c o n t y a d i c t i o n .
- x.
Let, now, x be a f i l t e r t h a t i s n o t an u l t r a f i l t e r ; we s h a l l f i n d a f i l t e r q 3 x . L e t u be such t h a t n e i t h e r u n o r a - U E X . Consider t h e fami l y v = x u (ul. We now prove t h a t v has t h e f i n i t e i n t e r s e c t i o n property.
342
ROLAND0 CHUAQUI
L e t t E x ; t h e n t n u # 0 s i n c e o t h e r w i s e we would have a - u 3.t and, thus, Therefore, j f z i s a f i n i t e subset o f x, we have n Z E X and so a-u€x. u n n z f 0. Hence, v has t h e f i n i t e i n t e r s e c t i o n p r o p e r t y and, by4.3.1.3 Thus, x i s n o t I N - m a x i m a l i n F l ( a ) . 9 t h e r e i s a f i l t e r g >_ v 3 x. 4.3.1.5
THEOREM (TARSKI I S ULTRAFILTER THEOREM)
x E Ff (a)
+
3 y(x C - y E Fl ( a ) A Pr(y,a)).
Thus, e v e r y f i l t e r o v e r a can be extended t o an u l t r a f i l t e r . PROOF, L e t x be a f i l t e r over a and l e t 6 = { y : Y E F l ( a ) A g > x } . Hence, a p p l y i n g I f c i s an I N - c h a i n i n b, then, by 4.3.1.3, U C E b. Z o r n ' s lemma 4.1.2.5 (i), t h e r e i s an I N - m a x i m a l yE6. y is By 4.3.1.5, an u l t r a f i l t e r . 4.3.1.6 THEOREM, x ( S a t ( ~ , x , a )* W b ( b c P a A
-
I d ( a ) A C o m i d ( ~ , x )A K E OZ- w + ~ = AK ~ u ~ v ( u , v ~ b A u ~ v + u 0 )n+ v
E
b n x # 0). PROOF, I t i s c l e a r t h a t i f x i s K-saturated, the r i g h t i s satisfied.
t h e n t h e c o n d i t i o n on
So assume t h i s l a s t c o n d i t i o n t o be s a t i s f i e d and l e t b c P a with b = K
u
and u,vEb,
# w
i m p l y i n g unwEx.
d
Let
-
E
K 6 w i t h 6-l
E
b ~ . Define
u d*C;, f o r t: E K . Then g'!i n g ' q = 0 f o r [ # q . Thus, g E Y by g't: = d ' F t h e r e i s a t: E K such t h a t 9'5 E x . But,
d't:
=g'€
u u Id'F n 6'17 : o E E l
E x
,
by K-completeness. Thus, b n x # 0. 4.3.1.7 v6(6 E H
THEOREM (ULAM) ~
+
,
x
E
Zd(Ha
+
1) A Cornid (Ha + l,x)
A
+{ 6 } ~E x ) -+lS a t ( H a + l , X y H ~ + ~ ) .
Since a K-complete i d e a l on K i s n o n p r i n c i p a l i f a l l s i n g l e t o n s belong t o i t , t h i s theorem says t h a t a n o n p r i n c i p a l Ha+l-complete i d e a l on Na+l i s n o t Ka+l-~aturated,
i t s s a t u r a t i o n i s l a r g e r t h a n Ha+l.
i.e.,
such t h a t a l l L e t x be an Ha l-complete i d e a l on Ha PROOF , s i n g l e t o n s a r e i n x. Note t h a t e v e r y subset o f N o f cardinality ' I N a i s +
i n x.
L e t g = d u (x, Ha+1).
f u n c t i o n f r o m 6 i n t o Ha.
h'(P,t)
=
+
F o r each 6
F o r each
IS
:PC6
A
P
E
E
a+l Ha+1, l e t
and
C;
E
d6
be a b i u n i q u e
Ha, l e t
6; P = € I rNa+l
AXIOMATIC S E T T H E O R Y
343
Since each d6 i s biunique, we have t h a t P + . r implies h ' ( P , E ) n h ' ( r , E )= Furthermore, for each P E H a + 1 , 0, for any 5 E Ha. w {h'(P,(, :1,
E
= (6 : P € S E
KJ
Ka+ll
E
y.
Since x i s Ha+l-complete we cannot have t h a t h ' ( p , [ ) E x for every Hence, we have g ' P E t E Ha. Let g'P = n { t : h ' ( P , E ) @ x}, for P E % + 1 Since H a + 1 i s regular, there Ha and h ' ( p , 4 ' 0 ) 9 x. B u t g E Ha* For t h i s ,$',{ h ' ( P t ) : g ' P = [ I i s a i s a [ E Ha with g-'*{E} = H a + l . family of d i s j o i n t subsets of of cardinality in x. Therefore, x i s n o t Ha+l-saturated (by 4.3.1.6).
with no member
PROBLEMS
1.
Prove 4.3.1.3.
2.
Prove t h a t every f i l t e r over a f i n i t e s e t i s principal.
3.
Prove:
4.
E
FI (a) A y
E
x
-+
( P q ) n x E Fl ( q ) .
Prove: K E
5.
x
01- w A i 3 - K A x = { q : q C- a/\$
C K ] -*U
X q
X E
Id(U).
Prove: > w ~ x {= y : y-c a ~ z c W )A;= { q : q ~ x ~ p c q ) ~{ qw : = (i) a q 5 x A 3 p ( p ~ Ax y 3 - ;)I -,w E F ~ ( x A) n v 9 U.
( i f ) The f i l t e r v in ( i ) i s generated by
((21
:tEaI.
(*) 6. Prove t h a t i f p i s a o-additive probability measure on P a , then the ideal of s e t s of measure 0 i s H1-cornplete and H1-saturated.
7.
Prove t h a t i f Ha i s singular, then there i s no nonprincipal Ha-complete ideal over Ha.
8.
Prove:
Every principal f i l t e r i s Ha-complete f o r every a.
ROLAND0 C H U A Q U I
344
9.
Prove: I f K i s a r e g u l a r i n i t i a l o r d i n a l and x an i d e a l on sur(x,K) i s a regular i n i t i a l ordinal.
4.3.2
then
K ,
THE CLOSED UNBOUNDED F I L T E R AND THE T H I N IDEAL.,
The d o b e d unbounded 6.ieten on an i n i t i a l o r d i n a l K w i t h T ( K 3 ) w is t h e f i l t e r generated by t h e c l o s e d unbounded s e t s on K . There i s such a f i l t e r s i n c e by 3.8.5.8 t h e f a m i l y o f c l o s e d unbounded s e t s on K has t h e f i n i t e i n t e r s e c t i o n p r o p e r t y . The dual i d e a l i s t h e f a m i l y o f non-stat i o n a r y s e t s o r t h i n s e t s and i s c a l l e d t h e t k i n dud. Thus, small s e t s a r e t h e n o n - s t a t i o n a r y ones; medium s e t s a r e t h e s t a t i o n a r y sets; and l a r ge s e t s a r e those i n t h e c l o s e d unbounded f i l t e r . 4.3.2.1
DEFINITION,
4.3.2.2
THEOREM, C U ~ ( K E )
(i) c ~ ( K 3) w + (ii)
T ( K )
=
K 3 W-t
(iii) F ( K= )K
= (X :
CUb(K)
Fl
COmfl
(K
(K
). ,cub(K)).
a + sur (du(cub(K),
3
3 q(Cub(y,K) A y c c K)). -x -
K
),
K
)
3 K.
Thus, i f K i s a r e g u l a r i n i t i a l o r d i n a l , t h e n t h e t h i n i d e a l i n K-complete and n o t K-saturated.
is
K
(i) and (ii) a r e o b t a i n e d by 3.8.5.8.
PROOF,
PROOF OF (iii).L e t K be a r e g u l a r uncountable i n i t i a l o r d i n a l . We have t o d i s t i n g u i s h two cases. 4.3.1.7.
CASE I
,
CASE I 1 ,
K
= Ha + l f o r a c e r t a i n a.
K
= H
The r e s u l t i s o b t a i n e d by
a = Ha' A l s o , by I i s s t a t i o n a r y i n a f o r every regular
cc w i t h a = u a .
By 3.8.2.7,
3.8.5.12, (7 : 7 E a A r ( 7 ) = R i n i t i a l o r d i n a l II. L e t F be t h e i n c r e a s i n g f u n c t i o n t h a t enumerates t h e r e g u l a r i n f i n i t e i n i t i a l o r d i n a l s . We have,. cc = F'p f o r a c e r t a i n 0 La. I f we had 0 C a, t h e n a would be t h e l i m i t o f on i n c r e a s i n g sequence of l e n g t h l e s s t h a n a , which i s i m p o s s i b l e by t h e r e g u l a r i t y of a. Therefore, a = F'a and R G n (01 o )n a has c a r d i n a l i t y a. Hence, t h e f a m i l y b o f
-
.
s e t s (7 : 7 E a A T ( 7 ) = R 1 i s a f a m i l y o f d i s j o i n t s e t , w i t h b = a b n du(cub(K), K ) = 0. Thus, t h e t h i n i d e a l i s n o t K-saturated.
and
AXIOMATIC S E T THEORY
Thus,
C U ~ ( K )i
PROOF,
4.3.2.4 K
.+ 3
s a l s o c l o s e d under d i a g o n a l i n t e r s e c t i o n s .
By 3.8.5.8
From 4.3.2.3,
(ii).
we o b t a i n .
THEOREM ( F O D O R ' S PRESSING-DOWN
= c ~ ( K 2 ) w A Stat
a(a E
K
34 5
A stat(d-'*{a),
(x,K) A x c K A K ) ) .
6E
'K
A
LEMMA).
v a(0 #
a
x
E
+
6'aca)
An o r d i n a l f u n c t i o n d t h a t s a t i s f i e s d ' a c a f o r e v e r y non z e r o a i n i t s domain, i s c a l l e d mg.gheshLwive. Thus, t h e theorem s t a t e s t h a t i f K i s a r e g u l a r uncountable i n i t i a l o r d i n a l and 6 i s a r e g r e s s i v e f u n c t i o n on a s t a t i o n a r y subset x o f K and i n t o K , t h e n 6 i s c o n s t a n t i n a s t a t i o n a r y set. PROOF, Assume t h e h y p o t h e s i s o f t h e theorem and, i n o r d e r t o argue by c o n t r a d i c t i o n , suppose d-'*(a) i s t h i n f o r every a E K . Choose a closed
unbounded s e t ca such t h a t c
cl
n 6-'*{a}
diagonal i n t e r s e c t i o n o f t h e ca' s; i.e.
= 0 f o r every
c = {y : y
E
~1
E
L e t c be t h e
K.
n {ca :
E y]}.
By
c i s c l o s e d unbounded. B u t 7 E c i m p l i e s t h a t 6'y f a f o r e v e r y hence 6'737. Thus, c n x = 0, c o n t r a d i c t i n g t h e h y p o t h e s i s t h a t x i s stationary. =
4.3.2.3,
c1 E y;
-
L e t x be a f i l t e r o v e r a c a r d i n a l K ; we say t h a t x i s m m d i f and o n l y i f x i s c l o s e d under diagonal i n t e r s e c t i o n s . An nalund i f and o n l y i f i t s dual f i l t e r i s normal. Thus, we have proved t h a t C U ~ ( K ) i s normal f o r r e g u l a r uncountable i n i t i a l o r d i n a l s K . I t can be proved (see Problem 3) t h a t a K-complete i d e a l x over a r e g u l a r uncountable i n i t i a l o r d i n a l K i s normal if and o n l y i f f o r any r e g r e s i v e f u n c t i o n 6 i n t o K w i t h 06 P x , t h e r e i s a y c - 06, q $ x such t h a t i s c o n s t a n t on q. The c l o s e d unbounded f i l t e r over a r e g u l a r uncountable i n i t i a l o r d i n a l K i s t h e s m a l l e s t f i l t e r o v e r K t h a t i s K-complete and normal, and cont a i n s a l l complements o f bounded sets:
L e t R = {a : 0 # CI = U @ EK } . Then R is t h e diagonal i n t e r PROOF, ( a + 2 ) f o r a € K , which a r e i n x. Thus, L € x . section o f the sets K
-
L e t c be any c l o s e d unbounded stubset o f K and l e t d E x be t h e diagonal i n t e r s e c t i o n o f t h e sets K (Ec a + l ) whichare i n x ; i.e. d = (y : y E n {K
- @,)a+'):
-
E y}}.
Then c 3 - co n
d. Thus,
CEX.
346
ROLAND0 CHUAQUI
PROBLEMS
c K i s t h i n ( K a regular uncountable i n i t i a l o r d i n a l ) then 1. Prove: i f y there e x i s t s a regressive function 6 on y such t h a t d-l*{y} i s bounded f o r every y E K . 2.
Prove: f o r every s t a t i o n a r y x 5 N1 and even a = U a E N1 there i s a normal function d such t h a t D d = a and D 6 - l c x.
3.
Prove: A K-complete ideal x over a regular uncountable i n i t i a l o r d i nal K i s normal i f and only i f f o r every y 9 x and every regressive 6 from y t o K t h e r e i s a z c - y such t h a t z r? x and 6 i s constant on 2.
4.
Prove t h a t t h e r e i s no normal nonprincipal f i l t e r over a.
5.
Prove t h a t i f K i s a s i n g u l a r i n f i n i t e i n i t i a l o r d i n a l , then t h e r e i s no normal ideal over K t h a t contains a l l bounded subsets of K .
6.
Let a be a s e t of i n f i n i t e i n i t i a l o r d i n a l s such t h a t f o r a l l regular i n f i n i t e i n i t i a l o r d i n a l s K , a n K i s not s t a t i o n a r y in K . Prove t h a t t h e r e i s a biunique r e g r e s s i v e function g on a, i.e. g E ' O n , g-' E -1 D g a , and W a(a E a -+ g ' a c a). Hint. Use induction on y = U a and d i s t i n g u i s h t h r e e cases: s o r c a r d i n a l , y s i n g u l a r , and y regular 1imit.
4.3.3
succes-
TREES,
Trees a r e t i a l orderings T i s a a3~e.ei f oT (y) = ( x : x
p a r t i a l orderings of a special kind. Recall t h a t f o r parR , we w r i t e x G R y f o r x R y , and x < q~ f o r x R y A x # y. and only i f T i s a p a r t i a l ordering such t h a t every segment < y)~ i s well-ordered by T and i s a s e t .
Formally:
For t h e next d e f i n i t i o n , which gives some c h a r a c t e r i s t i c s important f o r t r e e s , r e c a l l t h a t i f h i s a well-ordering, K i s t h e ordinal t h a t i s t h e type o f h. 4.3.3.2
DEFINITION,
( i ) hr(x,T) = TIOT(x).
AXIOMATIC S E T T H E O R Y ( i i ) L e v r ( T ) = {x : x
-
E DT
A ht ( x , T ) =
(iii) H f (T) = n {r:Levr(T) ( i v ) Stree(T',T)
0)
.
347
r).
v x v y ( y < T E~ D T ' + y < T , x ) .
T' C -T A
h t ( x , T ) i s read t h e heLgkt a6 x i n T ; i t i s t h e ordinal associated with T I O T ( x ) . L e v a ( T ) i s t h e a-Zh Level 06 T. H f ( T ) is t h e height 06 T; H t ( T ) = u { h t ( x , T ) + 1: x E D T ) , when T i s a t r e e . S t r e e ( T ' , T ) i s read T' i s a subtree of T . I f T ' i s a subtree of T and x E DT' , then h f ( x , T ' ) = ht ( x , T ) .
Examples of t r e e s a r e I N I 6 f o r any ordinal 6. In this case, hf ( I N IS) = a, L.eva(ZN16) = {a), f o r CL E 6 , and H t ( I N 1 6 ) = 6 . {
CL
,
For each a and s e t a , t h e compL&e a-my &thee ad h&gh;t a i s I N I ( U t h e t r e e of l e s s than a sequences from a. When a = 2, I N l u $ 2 : €a} i s referred t o a s t h e compLeLe b W q &ee ad h e i g M a. I n general , when S c u { E 2 : E €a), we c a l l I N I S a binahq ;DLee 06 h e k t a. In t h e sense t o 3 e spelled out in t h e next theorem, any t r e e whose height r i s a regular i n f i n i t e cardinal o r On , i s isomorphic t o a binary t r e e (not n e c e s s a r i l y complete) of height r.
E a : 5 €a)), i.e.
4.3.3.2
F(r ) = r 2
THEOREM,
r
w A Tree(T) A H t ( T ) =
A
w a ( a ~ r - + ~ v a ( ~- )~ 4G r~) F ~ S ( ~ T ( G ) A I ~ -( G ) A S C U ( A~ Z : E E ~ TS(F) A SOT ( ~ - 1 )A F * ( L P ~ ~ ( cT )G ' a 2 A T = I~N I S ) .
o(
r
Let T be a t r e e of height r w i t h r = T (r ) 3 - w and Lev a( T ) Assume x E g ' a 4 r with g ' a E O I f o r each a E r. Let L e v a ( T ) PROOF,
Leva(T), U
.
6,
a. Let G'a = Define F'x by,
F defined f o r a l l q
{ G ' P : P €a) + g ' a .
r
F'x = u { F ' q : q
d', x
Taking S = F* 4.3.3.3
r
f
{(
4 XI
U
{( 0,
E
r.
u { G ' P : €a} + [
1, u IG'P : P e a l + d',x)).
-
DEFINITION,
K-
:
)
E
S O ( T I 8 ) A W a(a
c Ht (T)
A r o n ( T ) * Tree(T) A H t ( T ) =
K
-,
-
.
Eg'a A
i t i s easy t o check t h e conclusion of t h e theorem.
( i ) Path(8,T)
(ii)
[I u
i s r e g u l a r , G'a
E
B n L e v a ( T ) = 1).
A W a Leva(T)
C K
A
1 3 8 Path(8,T). If Path(B,T), then 8 A a p&h t h o u g h T , i.e. a c l a s s 8 simply o r d e r ed by T t h a t contains exactly a r e element from each non zero level o f T .
348
ROLAND0 CHUAQUI
- A r o n ( T ) i s read, T LAa K -A.rru~ozajnh e ( ? , i.e. a t r e e o f h e i g h t which e v e r y l e v e l has c a r d i n a l i t y l e s s t h a n K and which has no paths.
K
K
in
I f T i s a t r e e o f h e i g h t c1 which i s a s u b t r e e o f t h e complet a - a r y s u b t r e e o f h e i g h t a , we can i d e n t i f y a p a t h b t h r o u g h T w i t h t h e f u n c t i o n
such t h a t b = {61t : € € a } . Complete a - a r y t r e e s have paths and t h e 6E same i s t r u e i n general f o r t r e e s whose h e i g h t i s a successor c a r d i n a l . An example o f ~ ( v ~ Then T has o f natural
a t r e e o f h e i g h t w w i t h no paths i s t h e f o l l o w i n g . L e t S = { x : w A x x ~ ~w1 (A1 r + I E v ~ x ' p ~ x ' ( l * + land ) ) }l e t T = I N I S . h e i g h t a b u t no paths, s i n c e t h e r e a r e no decreasingw-sequences number.
4.3.3.4
T H E O R E M ( D , K O N I G ' S T R E E THEOREM)
,
1 3 T w - A r on(T).
PROOF, Assume T i s a t r e e o f h e i g h t w i n which every l e v e l i s f i n i t e . L e t Tx = ( y : y E DT A x < y~l . D e f i n e t h e r e l a t i o n R between elements o f
D T by,
xRy Let A = i s an
x
E
-
3 v ( x E L e v v ( T ) A y E Lev,, + l ( T ) ) A x < y). ~
{ x :Tx
3w ) .
Since L e v o ( T ) i s f i n i t e and DT i n f i n i t e , t h e r e
Suppose, now, t h a t x E A . nite.
x
L e v o ( T ) w i t h Tx i n f i n i t e , i.e.
But Tx n Levv t l ( T )
Then
x
E
E
L e v o ( T ) n A.
k v , , ( T ) f o r some o and T, i s i n f i -
i s f i n i t e and non empty.
Thus, t h e r e i s a y i n
Tx n Lev,, + l ( T ) w i t h T
i n f i n i t e . Therefore, y E A and x R y . Y We now a p p l y t h e P r i n c i p l e o f Dependent Choices 4.1.2.12, and o b t a i n a sequence 4 E w A w i t h 6'v R 6 ' v t 1 f o r every v E W , I f we t a k e b = d * w , b i s a p a t h through T. Thus, T i s n o t w-Aronszajn. The g e n e r a l i z a t i o n t o H
ol+l
4.3.3.5
THEOREM,
PROOF, Let s
=
.
fails:
3 T H1 - A r o n ( T ) .
{x: 3 t(t
E H~
x-1
A x E
E
Dx
-1
t )I.
S i s the
s e t o f b i u n i q u e f u n c t i o n from c o u n t a b l e o r d i n a l i n t o o. For x , y E S d e f i n e an equivalence r e l a t i o n R by
x R y i f and o n l y i f D x = D y and { t
:XI[
#
We s h a l l d e f i n e by r e c u r s i o n a sequence A E (1) if c1 c
and
0, t h e n
(2)
ha E a w
(3)
w
-
~
ha R ( d O l a ) ,
¶
h
i s ~i n f i n- i t e . ~
y't) i s finite.
S such t h a t
AXIOMATIC S E T THEORY
Let Of
ba t o
Suppose A
0.
b0 =
a+l.
a
i s defined.
Suppose, now, y = U y and A
a for
we d e f i n e a sequence g E
'5
=
t'l
'dfu
=
t'6'2 v
b
and
b
Then A
Y
Y
Y
satisfies
any b i u n i q u e extension
a r e given.
u
6*w =y).
Let
U
for
(g'u : u € w ) .
C; ~y for
- {d'v : v
6E
6 ' ~
Then t € Y w . E
wy
be an
By r e c u r s i o n on w
such t h a t g ' u E 6 ' u w , g ' u R A
L e t ,t =
implies g'/.i=g'uld'u. by
Take
c1 E y
i n c r e a s i n g sequence whose l i m i t i s y (i.e.
349
and, /.i C u Define
w}
u E w .
(l), ( 2 ) , and ( 3 ) .
Thus, we have d e f i n e d t h e sequence a. L e t n o w ~ = I N \ C x: ~ ~ ( X E S ~ ~ ~ A X R A ~ ) } . I t i s n o t d i f f i c u l t t o prove t h a t T i s an N1- Aronszajn t r e e ,
PROBLEMS
3 T Na+l-Aron(T).
1.
Prove:
2.
Complete t h e p r o o f o f 4.3.3.2.
rn
6
Y
PART 5 Bernays Class Theory
CHAPTER 5 . 1 lnaccessib i l i t y
INACCESSIBLE SETS,
5.1.1
T h i s l a s t p a r t o f t h e book w i l l be devoted t o t h e s t u d y o f l a r g e c a r d i n a l s whose e x i s t e n c e i s assured by t h e R e f l e c t i o n P r i n c i p l e , Ax Ref. T h i s s t u d y w i l l be done i n B C , i.e., t h e t h e o r y g i v e n by Ax Ext, Ax Class, Ax Sub, Ax Ref and Ax GC, a l t h o u g h most o f t h e theorems c o u l d be proved i n M K T C . However, most o f these theorems c o u l d be v o i d i n M K T C , s i n c e t h e I n B C , we a r e sure t h a t t h e y a r e c l a s s e s t h e y t a l k about m i g h t be empty. n o t empty. Anyway, theorems t h a t depend on B C s h a l l be marked (+). We f i r s t r e c a l l some p r o p e r t i e s o f i n a c c e s s i b l e s e t s from another ang l e ( c f . S e c t i o n 3.8.4). L e t F be a u n a r y o p e r a t i o n , we say t h a t a c l a s s A i s F - i n a c c e s s i b l e i f and o n l y i f 1 8 (0
.f
8
o(
A A 'd q ( y € B
--t
q 4 A) A A
5 F(8)).
For i n s t a n c e , l e t F1(X) = u X , F 2 ( X ) = n P*X, and F3(X) = n G*X, where G ' u = a + 1, i f w d a, and G'a = a f CH f a ) , i f w < a. F o r F2 and F3, t h e C
i n t e r e s t i n g cases a r e X = 1x1; we have, t h e immediate c a r d i n a l successor o f x.
F2(1x))=Px and F 3 ( C x I ) =G'x, i.e.
I t i s easy t o show t h a t
(1) X i s F 1 - i n a c c e s s i b l e i f and o n l y i f X i s r e g u l a r . (2) I f X i s F2-inaccessible,
then X i s F3-inaccessible.
Sets X t h a t a r e Fly F 2 - i n a c c e s s i b l e we have c a l l e d i n a c c e s s i b l e (see Def. 3.8.4.1;
t h e c l a s s o f i n a c c e s s i b l e s e t s i s I A ) . Fly F - i n a c c e s s i b l e
c l a s s e s may be c a l l ed weakeq ~nacce.hAibLe.
3
We proved i n 3.8.4 some p r o p e r t i e s o f R a , when R a (and hence a ) i s i n a c c e s s i b l e . I n p a r t i c u l a r , R a i n a c c e s s i b l e i m p l i e s t h a t R ( a + 1) s a t i s f i e s a l l t h e axioms of M K T C (3.8.4.5). F o r p r o v i n g t h e converse, i.e. if a l l axioms o f M K T C h o l d i n P , t h e n P = R ( a t 1) w i t h a i n a c c e s s i b l e , Ax Reg i s e s s e n t i a l . However, even w i t h o u t Ax Reg, we need models o f MmC a l t h o u g h n o t n e c e s s a r i l y o f t h e form R a . Thus, we i n t r o d u c e t h e f o l l o w i n g 353
3 54
ROLAND0 C H U A Q U I
definition. 5.1.1.1 X E U -+
u
E
M o d = (u :
DEFINITION,
F*xEu) A
UuU P
*u C u A W FW x ( F C u A
u}.
W E
Mod i s t h e c l a s s o f s e t s u such t h a t P u i s a model o f M K T C , i.e. Mod i f and o n l y i f a l l axioms o f M K T C a r e s a t i s f i e d i n Pu. The p r o o f o f t h e f o l l o w i n g theorem i s l e f t t o t h e reader.
5.1.1.2
THEOREM ,
5.1.1.3
REMARKS ON ABSOLUTENESS,
If u
..., ' n - 1 - C u,
,..., xn - 1 i f and o n l y i f
E
M o d , and F i s any de-
f i n e d o p e r a t i o n , we have t h a t x C u imp1 i e s F ( x ) 5 u and F '(x) = F(x). S i m i l a r l y f o r n-ary o p e r a t i o n s azd n o t i o n s . Thus, i f R i s a d e f i n e d n-ary n o t i o n and xo,
R xo,
..., 'n
- 1'
then R
U
xo, x1
When t h i s happens, we say t h a t
R or F i s u-abbaUe.
The d e t a i l e d p r o o f o f these f a c t s c o n s i s t s i n checking t h r o u g h t h e I t i s an easy b u t l o n g j o b , so we s h a l l n o t do i t i n d e t a i l . definitions. As an example, we s h a l l do i t f o r t h e d e f i n i t i o n o f e q u i p o l l e n c y . We have, This i s a ternary notion
(aZdb)'
*
6
E ( a b )u A
I t i s easy t o check t h a t (6-l)' We assume t h a t a , b E u . a
( b)
u
=
E Cba)'
(6-l)'
i-', for
~ E u .
a u L e t us see ( b )
= {g : g ~ Au g c - ( b x ~ A) ~V x ( x c- u A X E U
+
3 ! y(y c - u A ( ( x,y) )' E g ) ) .
We have:
b x a e u , since b x a c -P P ( u u b ) . Hence,
(bxa)'
= {(( x,y))'
:
XEUA YEb}
.
I t i s c l e a r t h a t , s i n c e u i s t r a n s i t i v e , (( x,y))'
(bxa)' a
( b)
u
= (x,y)
.
Hence
= bxu.
Now, x E a i m p l i e s x Cu and ( x , y ) E g E u , i m p l i e s y C u. a = 'b. Also, b C_ b x a ; t h u s ' b E u .
Hence,
.
AXIOMATIC
- -
Thus, we have proved t h a t f o r d , a , b ~ u , ( a Now, a - b i f and o n l y i f 3
-
6
355
SET T H E O R Y
a=
6
-
zi
b)'
b. Thus, ( a - 6 ) '
a=
6
b.
i f and o n l y i f
-
36(6 c Suppose, a = b w i t h a , b E u . Then, 6 E ' ~ E u , and, - u A ( a =6 b)'). 6 hence, 6 5 u. Thus, ( a b)'. I f ( a b)', t h e n c l e a r l y a 6. Therefore, we have ( a = b ) ' i f and o n l y i f a = b f o r a , b E u .
As an example o f o p e r a t i o n s d e f i n e d by r e c u r s i o n , we s h a l l t a k e p , t h e rank function: pX = p*X u u p*X, f o r a l l X c -W
.
We s h a l l i n d i c a t e t h e p r o o f o f , for all
= p XEU
(px)'
x
T h i s has t o be done by i n d u c t i o n . We have,
p y E u for a l l y E x n u .
By 3.1.4.9,
UuX
Since x n u = pothesis,
x,
w n u.
E
Let
x
W n u and assume (py)'
E
=
= uX .
because u i s t r a n s i t i v e , we have by t h e i n d u c t i o n hyp*%
=p*x
.
Hence p'x = p x. Since P u i s a model o f M K T C , p l x E u and XEU, I t is alp * x € u and hence, p x E u . Thus t h e i n d u c t i v e p r o o f i s completed. so easy t o show u s i n g t h e r e s u l t we have j u s t proved, t h a t pux = p x c -u for all x C Wn u. 5.1.1.4
THEOREM,
LL
E
W n Mod
-+
3a
= Ra
Thus, t h e o n l y well-founded models o f M K T C a r e R ( a + 1) f o r c e s s i b l e and uncountable. PROOF, L e t u be a well-founded model know t h a t u C R a . Since P u i s a model, by i.e., E E u T o r e v e r y t E a. Thus, a g a i n by - u, and, hence, R e = t E a . Therefore, R c1 c
c1
inac-
o f M K T C and l e t a = p u. We 5.1.1.3 p x EU f o r every X E U 5.1.1.3, R [ E u for every
u.
.
F i n a l l y , a s t r e n g t h e n i n g o f 3.1.4.8.
(+) 5.1.1.5 THEOREM SCHEMA, L d 4 be a 6ohmuLk U h at m v A t A,B 6kee and i n which c1 dues n o t o c c ~ ~ Then, . V A VB(A
C W A -
B
E
W A I$
-+
3 a(a
E
IA n 01
- (o + 1) A B
E
Ra A
ROLAND0 CHUAQUI
356
PROOF,
have,
Assume Q i s a formula s a t i s f y i n g t h e hypothesis.
Then, we
A c W A B E W A Q A W x ( x E W ~ P x € W ) A ~ F- ( F c W A x E W ~
F*x
W ) A
E
W E
W
.
t o t h i s formula, we g e t a super t r a n s i -
Applying 3.1.4.8 a n d 3.1.4.9 tive s e t u such t h a t ,
A n u c-W n u A B G I V n u A Q [Anu, 81 A W n u E M o d . A, B By 5.1.1.5, W n u = R a f o r a c e r t a i n a. Also, A n u = A n W n u =
A n a.
T h u s , we have t h e theorem.
PROBLEMS
1.
Prove 5.1.1.2.
2.
Prove t h a t i f u
5.1.2
E
Mod
, then
R
and 0 1 a r e u-absolute.
INACCESSIBLE CARD1 NALS m
In t h i s and t h e following sections we s h a l l concentrate on i n i t i a l ord i n a l s t h a t a r e i n a c c e s s i b l e o r weakly inaccessible. As usual, P , v , K , and R a r e reserved f o r i n i t i a l ordinals. 5.1.2.1
DEF I N ITION
(i) WCIA
=
I
RG n L O I .
(ii) C I A = RG n S L O I . K E W C I A i f and only i f K i s F l y F3-inaccesT h u s , by 4.2.5.12 s i b l e ; K E C I A i f and only i f K i s F l y F2-inaccessible. The elements of W C I A , i.e. the regular l i m i t c a r d i n a l s , a r e t h e weakly i n a c c e s s i b l e card i n a l s ; elements o f C I A , i.e. t h e regular strong l i m i t c a r d i n a l s , a r e the ( s t r o n g l y ) inaccessible c a r d i n a l s .
AXIOMATIC SET THEORY
It i s also clear, that
E
K
357
C I A i f and o n l y i f t h e r e i s an x E I A such
t h a t E = K and K 2. a. Thus, C I A = I A n ( 0 1 - a). Also, every inaccess i b l e c a r d i n a l i s weakly i n a c c e s s i b l e . I n B C we can prove t h a t t h e r e a r e many i n a c c e s s i b l e c a r d i n a l s , i.e.
(+) 5.1.2.2
v
CIAQ
THEOREM,
.
.
- a # 0 for
L e t c1 E e v e r y a E On By t h e r e f l e c t i o n p r i n c i p l e t h e r e i s a y uncountable i n a c c e s s i b l e such t h a t ,
PROOF, We s h a l l prove t h a t C I A
On be g i v e n and l e t @ be t h e formula a 5.1.1.5,
E On.
a E O n n R y = y .
A f t e r t h u s p r o v i n g t h e e x i s t e n c e o f i n a c c e s s i b l e , and, hence, of weakl y i n a c c e s s i b l e c a r d i n a l s , we c o n t i n u e w i t h t h e s t u d y o f t h e i r p r o p e r t i e s . We have a l r e a d y proved (3.8.4.2), that
R a E I A + a = U a + O and
a>wA
RaEIA-tazH
a
= R
5.1.2.3
THEOREM,
a
2
0
-+
a '
-
We a l s o have:
(Na E W C I A
a = Ha = v ( a ) ) .
PROOF, Assume t h a t a 2 0. The i m p l i c a t i o n from l e f t t o r i g h t i s deduced f r o m 3.8.2.7 ( i i ) and 3.8.2.6 (vii).
-
Suppose, now, t h a t a = Ha = c f ( a ) .
s i n c e Ha E 0 1 , a = 5.1.2.4
uc1.
9
THEOREM,
a
0
-,(Ha E
( v i i ) , Ha E R G .
By 3.8.2.6
CIA
-
?(a)
= a =
Also,
Ha =I,).
The i m p l i c a t i o n from l e f t t o r i g h t i s o b t a i n e d PROOF, L e t a 2 w. Assume, now, T ( a ) = a = Ba = R a . Then, from 3.8.2.6 ( v i i ) and 3.8.4.2. by 3.8.2.6
(vii),
iaE R G .
x d H f o r a certain E
3.
c:
c: + 1
=
c%.
5.1.2.5 va(z c K A
6
E a.
Suppose x 4 Ha.
Also, a = u a # 0 . By 4.2.5.2
(v), H c 1
E -
E'
v
p(p C K
K E W C I A t--, W C K A THEOREM, ( 6 ' a :r E a ) c E K ( ' nOr) C c = -+
Hence,
-+
3v
X
Then 2
p C v C I C )A
K).
Thus, weakly i n a c c e s s i b l e c a r d i n a l s a r e a l s o i n a c c e s s i b l e w i t h r e s p e c t t o sums o f c a r d i n a l s . PROOF,
Easy, u s i n g 4.2.3.1.
3 58
ROLAND0 C H U A Q U I
5.1.2.6
PROOF, L e t i s strong l i m i t .
K
In[
U
d
?T
K
'€2
C
K
C K.
c C - c = Assume, now, t h a t ( K = : R E K )
K
(KnOI) with
c
c -K
2 C K.
C p
K
E
n
Cz
( g c':=
-cr Therefore =
* K
K
K
K
E
SLOI A 2 C
.$EK).
K
.
R ( E R : ~ C =~ K)
t , REK, ( t = : . $ E K )C K But f o r
C
n ( ~ ':R E K ) =
.
R x t c K
= K .
O K
i s strong l i m i t and 2
K.
Let
Thus,
4
f
We have, b y K u n i g ' s Theorem 4.2.2.6, --
( d ' p : p e a ) c II
Thus, by 4.2.3.1,
-
CIA
be i n a c c e s s i b l e . I t i s c l e a r from t h e d e f i n i t i o n t h a t Assume, t h a t R C K . Then, s i n c e K i s r e g u l a r , R~ =
Hence
: t E K ) .
Hence
a
THEOREM,
C V
K
( K :
i s regular.
p ~ a =) K a
c Z~CK': C
~
E
K =) K
=
F i n a l l y , a theorem t h a t w i l l be u s e f u l l a t e r . 5.1.2.7
THEOREM,
K E
CIA
-f
Cub(SL0In
K,K).
Thus, t h e s t r o n g l i m i t c a r d i n a l s a r e c l o s e d unbounded i n an inaccess ib le cardinal
.
The p r o o f i s l e f t t o t h e reader.
PROBLEMS
1.
Prove 5.1.2.7.
2.
Let
K
= n C I A (i.e.
K
A = {h: 3'(a
i s the f i r s t inaccessible cardinal). E K
A hE
' K )
A V S ( B E D h +h'B
W P W ~ ( P , ~ E S L nO DZ h A r z P Prove t h a t I N [ A i s a K-Aronszajn t r e e . Hint:
Use 4.3.2,
Problem 6.
h'rzh'p)).
C
Let
1 + 0) A
AXIOMATIC S E T T H E O R Y 5.1.3
359
CLASSIFICATION OF INACCESSIBLE CARDINALS,
The cl s s i f i c ati on of inaccessible and weaklv inaccessible cardinal discussed in t h i s section i s due t o Mahlo. The Gresentation here i s based on Gaifman 1967. F i r s t we must consider theorems on operations on classes of ordinals. DEF I N I TI ON
5.1.3.1
( i f Pf(X)
8
= Fp(EX), =
V
(ii) D(R,Y)( i i i ) CD(R,Y) P ECLl).
-
if X 5 O n ;
, otherwise. V a V P ( a , P E Y A a c P + R * CallR*CP l). D ( R , Y ) A V a(0
#
a
= U C L EY
+
R*{cL} = n (R*(P} :
P f ( X ) i s the class of fixed points of the enumerator of X. D(R,Y) i s read, the ~ e c j u w c eo d c l a A A L b [R*Ca) : a € Y ] LA d m e a s i n g . C D ( R , Y ) means continuoudLy d m u i n g . th at t h i s same berjuence
A binary operation F ( X , Y ) will a l so be written F y ( X ) . 5.1.3.2
by :
DEFINITION SCHEMA,
( i ) Let F be a binary operation.
We define the operation n I F L : i € l l
fo r every X, (n IFi : L E 13) (X) = n {FL(X) : L E I } . ( i i ) Let F , G be unary operations. Define the operation F O G by, for every X, ( F O G ) ( X ) = F ( G ( X ) ) .
D E F I N I T I O N , R:
5.1.3.3
R F i s called the [
R*M : a E
5.1.3.4 operation
rI
$
.
: a E R * { ~ I u) n { R * ( ~ }: a E r ) .
=
go^ M e h o e c t i o n
DEFINITION SCHEMA, by, f o r every X, Fr D (X)
06
-the Ae.quence
56 & A A U
Let F a binary operation. D = [Fa(X): a E r ]
.
Define the
F: (X) i s the diagonal intersection of the sequence [Fa[ XI : a E r I .
i s called the diagona4-izdion
F: E
r.
5.1.3.5
06
DEFINITION SCHEMA,
.the ~ e c j u e n c e06
0pehtraA;ian~Fa
Let F be a unary operation.
f or Then,
( i ) For each class X, define by recursion the operation F ( a ) ( X ) , for
ROLAND0 C H U A Q U I
360
a
E On,
by
,
F(O)(X) = X
F ( a 1) (X) =
,
(X))
+
i f a = U a # 0, t h e n F ( " ) ( X ) = n { F ( @ ) ( X :) /3 (ii) Define the operation F
f o r e v e r y X, F
A
A
by,
( X I = I F ( ~ ) ( x ): a
E
E
a}.
D
.
On 10n
Thus, i f we t a k e G t o be t h e b i n a r y o p e r a t i o n C(a,X) = F ( a ) ( X ) ,t h e n F A = D GOn q(X) = X I n XI, i f X c O n 5.1.3.6 DEFINITIONl
-
q(X) = V
, otherwise.
I t i s c l e a r t h a t q ( 0 ) = 0 and q ( X ) = X
5.1.3.7
THEOREM,
Pf
= 9
A
- {Ex' 0)
-
for 0 # X
-O n .
C
.
The p r o o f i s l e f t t o t h e reader.
LocuUy hedudng i f and o n l y i f We say t h a t t h e unary o p e r a t i o n F f o r a l l X c O n we have t h a t F(X) c X , and F ( X n a ) = F(X) n a when a E O n . I t i s e a s y t o show t h a t t h e l a s t crause can be r e p l a c e d by: f o r e v e r y 0, P E F(X) i f and o n l y i f p E F(X n (p cl)). The p r o o f s o f t h e n e x t two theorems a r e l e f t t o t h e reader.
5.1.3.8
let F a n d G b e u m y ape/rationh.
(i)
Then,
- O n + F ( X ) u G ( X ) c- X A Wa(F(Xna) = F(X) n a A G(Xna) = -,Wx(x c- On F(G(X) c X A W C ~F ( G ( X n a ) ) = F(G(X)) n a ) .
W X(X C ( x ) n 01)
THEOREM SCHEMA,
C
+
(ii)L e t F b e a b i n u t y opetration.
Then,
WiWX(i€'I A X c - O n + Fi(X) c- X A Wa FL(Xna) = F.(X) n a ) L
-*
t l X (-x ~ O n + n { F ~ :( Xi )~c~X)A W a n { F . : i ~ I ) ( X n a=) n { F i : 4.
i f l I ( X ) n a).
(i)says t h a t i f F and G a r e l o c a l l y reducing, t h e n so i s F O G . (ii) says t h a t i f f o r a l l L E I , FL i s l o c a l l y reducing, t h e n so i s n ( F i : i e l ) . 5.1.3.9
THEOREM SCHEMA,
let
F be a b L m y o p W o n .
(i)W ~ V X W Y ( ~ E-~Y A- FX, ( xC)
Then,
> _ F ~ ( Y+) )W X W Y ( X-C Y + F ; ( X )
AXIOMATIC SET T H E O R Y ( i i ) Wa WX WY(a
F; ( X I
w
E
r
A X
2 $(Y) 1.
( i i i ) Wa WX(a
X(X con
+
E
r
-Y c
+.
Fa(X) c F a ( Y ) )
361
+.
w Xw
+. Fa(X) c X A WP Fa(Xnp) W P F,D(xnp) - = F~D (x) n p ) .
A X c - On
F,O(X) c x A
Y
Y(X
+
= Fa(X) n
P)
+.
( i ) says t h a t i f Fa i s decreasing f o r every a , then F; i s a l s o ( i i ) and ( i i i ) a r e s i m i l a r w i t h monotone and l o c a l l y reducing instead of decreasing. 5.1.3.10
THEOREM,
( i ) WXWY(X c -Y
+.
q ( X ) c q ( Y ) ) A WX(q(X) E X A wa q ( x n a ) =
4 ( X I n a).
cpf(Y))
( i i ) W X W Y ( Xc Y-Pf(X) P f ( X ) n a).
A wXPf(X) c X A waPf(Xna) =
Thus, q and Pf a r e monotone and l o c a l l y reducing
I t i s easy t o see t h a t q i s monotone and l o c a l l y reducing. By induction we can prove t h a t q ( ' ) i s monotone and l o c a l l y reducing f o r every a. Hence, by 5.1.3.7, Pf = q A , and by 5.1.3.9, Pf i s monotone and l o c a l l y reducing. PROOF,
DEFINITION,
5.1.3.11
L ( X ) = {U q : q C -X A U q
If X L ( X ) i s t h e c l a s s of l i m i t s of subsets o f X. i s a closed c l a s s o f o r d i n a l s , c a l l e d khe ~ o h u t ead X.
(i) X c - On
x
0 A Uyf? ql.
C On , then
XuL(X)
THEOREM,
5.1.3.12
(ii)
+
c RG
+.
Pf(X) n L ( O n ) c L ( X ) .
n OI-. P ~ ( B + ' ) ( X )=
x
n L ( P ~ ( ~ ) ( x= )~) f ( a ) ( x ) n
L(Pf ( a ) ( X ) ) * PROOF OF ( i ) . Assume t h a t X c o n and a E Pf(X) n L ( 0 n ) . Then a = EX' a and, hence, X n a = E;a. Since CL = u a and Ex i s increasing, a = U
{P
U a
:PEa)
=a.
cu
{E#
:PEa) =
Thus, a c u ( X n a ) .
uE;a.
Therefore, a = u ( X n a ) and
c1 E
L(X).
PROOF OF (ii).Assume X Z R G n O I .
P f b + l )(x)
B u t U(Xna) c
By ( i )
S P ~ ( ~ )n( LX( )P ~ ( ~ ) ( xc-)x) n L ( P ~ ( ~ ) ( x ) ) .
Suppose, now, t h a t P
E
X n15@f(~)(X)).
Then P
E
X, and
P
=
u y for
ROLAND0 C H U A Q U I
362
a certain q C - Pf ("'(X)
P 4 q.
with
cause P E R G n OZ and, hence
5.1.3.13
Let
v(P)= P
z = y u {PI. (3.8.2.6
We have
(vii)).
=
E:P
be-
Also,
DEFINITIONi
(i) C 1 y(X,r)
-
c-f
(ii) Cuby(X,r)
3 Z ( X n Y = Z n Y A C I (z,r)).
3 Z(XnY = ZnY A C u b ( 2 , I ' ) ) .
f o r Cuby(X,I').
r
w i t h & a p e & Zo Y. Obviously, C l ( X , r ) if and o n l y if C I X ( X , r ) .
C l Y ( X , r ) means t h a t X
Ceoded h
We c l e a r l y have, Cl(X, UX) i f and o n l y i f L(X)
i f and o n l y i f L ( X n
r ) n Y c- X .
C X.
Similarly,
Also, Cly(X,
r
)
PROOF, Assume t h a t X i s a c l a s s o f r e g u l a r i n i t i a l o r d i n a l s , closed w i t h r e s p e c t t o R G n O I . I t i s c l e a r t h a t Pf (X) c Pf (X u L ( X ) ) n R G n O I . t h a t Pf (X) = Let, now, cx E f l ( X u L ( X ) ) n RG n 01. We have, b y 5.1.2.12, X n L(X). Hence we have t o show t h a t cx E X n L ( X ) . From t h e h y p o t h e s i s we g e t t h a t cx = EXuL(X)'cx and cx E(X u L ( X ) ) n RG n O I . closed, EX
L ( X ) i s normal.
We have two cases.
C A S E I, a E L(X) n RC n O I . r e s p e c t t o RG nor. C A S E 1 1 , a E X n R G n 0 1 , i.e. Since cx E 0 1 , a = u a L e t Y = (E;uL(X)cx)
.
Since X U L ( X ) i s
Then a E X , because X i s c l o s e d w i t h
EX.
Thus, a = u EX*u L(x) a.
n L ( X ) and z = (E;uL(X)cx)
n X.
363
AXIOMATIC SET THEORY
L e t Y ' = C y : q c X A 3 y ( y E Y A y ~ q A ~ = u y ) } Then, . X, a $ z LJ U Y ' , and% = U ( z U LJ Y ' ) . Thus, ci E L ( X ) .
Therefore, i n b o t h cases, a
5.1 3.15 a
E
-
z U U Y ' C
X n L(X) = P f ( X ) .
F I R ST CLASS I F I CAT I ON OF I NACCES S I B L E S
I t i s c l e a r t h a t P f ( R C n O I ) = WCIA
8
(since P f ( R G n O I ) = RGn O I n
L ( R G n O I ) . ) We c a l l t h e c l a s s Pf(")(WCIA ) f o r a 1 1 , t h e c l a s s o f weakl y h y p e r i n a c c e s s i b l e c a r d i n a l s o f t y p e a. S i m i l a r l y , t h e c l a s s P f ( a ) ( C I d ) f o r a 3 1 i s t h e c l a s s o f hyperinacc e s s i b l e c a r d i n a l s o f t y p e a. Thus, a i s an h y p e r i n a c c e s s i b l e o f t y p e a t 1 i f and o n l y i f i s a r e g u l a r i n i t i a l o r d i n a l such t h a t i s a l i m i t of h y p e r i n a c c e s s i b l e s o f t y p e a. A s i m i l a r i n t e r p r e t a t i o n can be g i v e n t o
Pf (a+ 1 ) ( W C I A ) . We now pass t o a new o p e r a t i o n which produces even l a r g e r c a r d i n a l s .
5.1.3.16
M ( X ) = {a : a = U a E X A S t a t ( X n a , a ) } .
DEFINITION,
N o t i c e t h a t P f ( X ) = {a : 0 # a = ua E X A u ( X n a ) = a ) . u ( X n a ) = can be read, Xna i s unbounded i n a. The o p e r a t i o n M , t h e Maheo u p e m t i a n r e q u i r e s something s t r o n g e r f o r a E M ( X ) : X n a must be s t a t i o n a r y i n a .
5.1.3.17 THEOREM, V X ( X-c O n c _ X A V a M ( X n a ) = M ( X ) n a).
+
V y ( xc y+M(X) c M ( Y ) )AM(X)
Thus, M i s monotone and l o c a l l y reducing. From t h e d e f i n i t i o n we immediately o b t a i n t h a t M i s monotone PROOF, and M ( X ) C - X f o r X c- On. L e t X c- O n ; then M ( X n 0 ) = {a : a = u a E X n = {a : a =ua E X n = M(x) n
5.1.3.18
LEMMA,
p
A Stat ( x n p n u , a ) )
p A
Stat ( X n a ,
a)]
p. X CRG nOI+M(X) CPf(x).
PROOF, L e t X C R G n O I . By 5.1.3.17, M ( X ) 5 X . L e t a E M(X) , s i n c e X n a i s s t a t i o n a r y i n a , i t i s unbounded i n a. Hence a = U ( X n a : and a E L ( X ) . Therefore, M ( X ) c X n L ( X ) = P f ( X ) ( b y 5 . 1 . 3 . 1 2 ) .
-
364
ROLAND0 CHUAQUI
PROOF OF ( i ) . Suppose a E F(Z) w i t h Z C Y, Z closed with respect e have, F(F(2))n L(Y) = L ( F ( Z ) ) n F(Z7. T h u s , we just have t o t o Y. W prove t h a t a E L ( F ( Z ) ) . We a l s o have, F(Z) n L(Y) = L(Z) n 2 ; hence F(Z)> L ( Z ) 7 Z and so L ( F ( Z ) ) > L ( L ( Z ) n Z ) . Therefore, i t i s enough t o show t h a t a E L ( L ( Z ) n Z ) . Thus, we must show t h a t a n L ( Z ) P Z is unbounded i n 01. Since Z i s closed w i t h r e s p e c t t o Y , we have Z = YnA w i t h A closed. B u t a E F(Z) and a E M(Y) s L ( Y ) . Hence a E F(Z) f- L(Y) = L ( Z ) n Z C L ( Z ) . T h u s , f o r every p E a, A n L ( Z ) n (a 8 ) i s closed unbounded i n a. Theref o r e , since O! n Y i s s t a t i o n a r y i n a, A n L ( Z ) n (a 0 ) n Y # 0; i.e. t h e r e i s a y E Z n LfZ) 0 01 w i t h y>P, f o r every p E a.
-
-
PROOF OF ( i i ) . Assume the hypothesis o f ( i i ) . Since R*{yl i s closed r e l a t i v e t o Y f o r every T E ~ ,then R*{r} = A*{y) n Y w i t h A * { r I closed, f o r every ~ € 0 . Since a E F(R*Cr}) n L(Y) = L(R*{yl) n R*{rI, a E L ( R * { y } ) and A*{r) n a i s closed unbounded i n 01. T h u s , by 3.8.5.8, n { A * { r ) : y e p ) n a i s closed unbounded i n a. Therefore, f o r every 6 E a, n {A*{rI : ~ E P nYn I (a 6 ) f 0, i . e . n iR*IrI : r e p } n (a 6 ) f 0 f o r Thus, a E L (n {R*{-rl:rEP}) every 6 E a. Hence, a E L ( n {R*irl: ? € P I ) . n n {R*CrI : 7 € 0 l c F ( n ER*{rI : ~ € 0 1 ) .
-
-
PROOF OF ( i i i ) . . Assume t h e hypothesis of ( i i i ) . We have t h a t R*{y) is closed r e l a t i v e t o Y and a E F(R*Ir})nL y). =L(R*{r))nR*{y) f o r every* 6 n a. Hence a E and R*{r) i s unbounde i n a. We have, R-17) = A*{yI n Y w i t h A * { ? ) closed unbounded i n a. T h u s , =R: A: n Y , and, by 3.8.5.8, D :A i s closed unbounded i n a. Thus, Ag n (a p ) n Y f 0 f o r every 8 E a. Therefore ~1 E L ( RD6 ) . Hence, a 6 L(R:) n :R CF(R:). 9
6
Rt
-
M(Y).
5.1.3.20
COROLLARY,
Y C -RG nOf
-+
M(Y) C- PfA(Y) A n Pf*(Y) 9
a E PROOF, Assume t h a t Y C R G nOf and l e t ct E M(Y). By 5.1.3.18, Pf(Y).By 5.1.3.12 pf(''l)(Y) i s c l o s e d r e l a t i v e t o Y. Hence, using 5.1.3.19(i) and ( i i ) , weprove by induction thatol4PP@)(Y) f o r a l l PEa.Hence a E ~ f ( ~ ) ( y )
and so aEPfA(Y). BY 5.1.3.19 5.1.3.21
( i i i ) , aEPf(PfA(Y)). Hence, a f n PfA(Y). =
SECOND CLASSIFICATION OF INACCESSIBLE CARDINALS,
The i n i t i a l o r d i n a l s i n M ( " ) ( R G n 0 1 ) f o r a 2 - 1 a r e c a l l e d weakly Mahlo of type a. Similarly, those in M ( " ) ( C Z A ) a r e t h e ( s t r o n g l y ) Mahlo
AXIOMATIC S E T T H E O R Y
365
of type a. I t i s c l e a r from 5.1.3.21 t h a t i f P E M ( R G n O I ) then, P i s weakly hyperinaccessible of type y f o r a l l Y E P . Similarly f o r p E M ( C I A ) . I t i s a l s o easy t o show t h a t : 5.1.3.22
THEOREM,
cx
2 1 -,M ( n ) ( C I A ) = M ( " ) ( R G
n01) n CIA.
c M ( a ) ( R Gn O I ) n C I A . PROOF, I t i s c l e a r t h a t M ( " ) ( C I A ) l e t P E M ( " ) ( R G n O I ) n C I A . We prove by induction on u t h a t P E M ( ")(CIA).
So
F i r s t n o t i c e t h a t by 5.1.2.7, t h e s e t X = S L O Z n P i s closed unbounded in 0, i f P E C I A . X i s t h e s e t of strong l i m i t i n i t i a l o r d i n a l s below 0. NOW, we proceed with t h e induction. t h e r e s u l t i s c l e a r . Assume i t t o be t r u e f o r ci and l e t p E M('+ ( R G n O I ) n C I A . Then p n M ( " ) ( R G n O I ) i s s t a t i o n a r y i n 0. Hence X n M ( " ) ( R G n O Z ) i s a l s o s t a t i o n a r y i n 0. B u t X n M ( Q ) ( R G n 0 1 ) = p n C I A n M ( a ) ( R G n O I ) = p n M ( a ) ( C I A ) . T h u s p E M (a-b ')(CIA). For
ci
= 0,
0 and t h e r e s u l t t r u e f o r 6 E a. Then M ( a ) ( R G n O Z ) n C I A = n ( M ( 6 ) ( R G n O I ) n C I A: 6 E =M(~)(czA). Let now a = u a
f
PROBLEMS
1.
Prove 5.1.3.7.
2.
Prove 5.1.3.8..
3.
Prove 5.1.3.9.
4.
Prove:
5.
Prove in M K T C :
(i) V u E
Stat ( C I A , O n ) Stat ( C I A , O n )
Pf (")(X)
--t
W
I
= d c i ( I D X);
Pf ( " ) ( C I A ) $! V .
i s L&vy's axiom.
(li) E A
= ( I D ) X)*.
Pf ( 4
CHAPTER 5.2
Weakly Compact and L a r g e r C a r d i n a l s
5.2.1
WEAKLY COMPACT CARD1 NALS m
Weakly compact c a r d i n a l s a r e v e r y l a r g e i n i t i a l o r d i n a l s . There a r e many e q u i v a l e n t d e f i n i t i o n s f o r them (see Drake 1974 o r Kunen 1977). The one adopted here i s s u i t a b l e f o r o u r purpose. 5.2.1.1
Wc =
DEFINITION,
w A K E C I A A 1 3 T K-Aron ( T ) } .
{K : K 3
Weahey cornpad ca/ulindL a r e t h e uncountable i n a c c e s s i b l e c a r d i n a l s
w i t h no K-Aronszajn t r e e s .
(+) 5.2.1.2
Wc9 v
THEOREM,
.
PROOF, We s h a l l f i r s t prove t h a t t h e r e i s no O n - A r o n s z a j n . Assume t h a t t h e r e i s such a t r e e T ’ . By 4.3.3.2, t h e r e i s a b i n a r y t r e e T isomorp h i c w i t h T ’ . Hence T i s a l s o an O n - A r o n s z a j n t r e e . T s a t i s f i e s : T r e e ( T ) A H t ( T ) = O n A 1 1 8 Path(B,T). Since T i s a b i n a r y t r e e , T C W . By t h e r e f l e c t i o n p r i n c i p l e 5.1.1.5, t h e r e i s an a E On such t h a t , R Z E Mod and
(*) Tree(T n R a ) A H t ( T n R a ) = O n n R a A 1 3 B ( 8 E R a A Path( B , T n R a ) ) .
Let
x
We have, t h a t O n n R a = a. E
D T n R a and q f -
C; c u u x and thus DTn R a . Let, now,
x
E
C;
E
x.
a. Now, q
Leva(T).
which i s a subset o f R a .
We c l a i m t h a t T n R a i s a s u b t r e e o f T :
We have t h a t E
x
E
€2 for a certain
‘2 f o r ‘CC;.
Then { q : q < T x
1
C;.
Hence
Hence, ‘2 E R a and q E
i s a p a t h through T n R a
,
T h i s c o n t r a d i c t s (*).
Thus, we have proved t h a t t h e r e i s no O n - Aronszajn t r e e . s a t i s f i e s , f o r any a E O n ,
Thus, O n
1 3 T O n - A r on(T) A a E O n .
Using a g a i n 5.1.1.5
we g e t an uncountable i n a c c e s s i b l e p such t h a t ,
366
A X I O M A T I C SET THEORY
1 3 T(T C -R p A O n n Rp-Aron(T)) A a
367
E
On n R p
.
But, s i n c e O n n R p = p , we o b t a i n t h a t p i s an uncountable inaccess i b l e w i t h no p-Aronszajn t r e e T C R p . But, by 4.3.3.2, any p-Aronszajn t r e e T ' i s isomorphic t o a binary-tree T and b i n a r y t r e e s T a r e subsets o f R p . Hence, t h e r e a r e no p-Aronszajn t r e e s and, t h e r e f o r e , p i s weakly compact. We now proceed w i t h t h e p r o p e r t i e s o f weakly compact c a r d i n a l s , By 5.1.2, Problem 2 i f K i s weakly compact, t h e n K i s l a r g e r t h a n t h e f i r s t i n a c c e s s i b l e c a r d i n a l . We s h a l l see a p r o o f due t o Shelah 1979 t h a t t h e y a r e indeed much l a r g e r . The f i r s t s t e p i n t h i s p r o o f i s t o g i v e another c h a r a c t e r i z a t i o n o f weakly compact c a r d i n a l s .
The theorem can be rephrased as f o l l o w s :
6E
p i s weakly compact i f and o n l y i f p - i s i n a c c e s s i b l e and f o r e v e r y
-
II ('a
- p w i t h X = p such t h a t d l X i s coherent. : a ~ p )t h e r e i s an X c
P R O O F ) I s h a l l prove t h e i m p l i c a t i o n f r o m l e f t t o r i g h t l e a v i n g t h e o t h e r d i r e c t i o n (which we s h a l l n o t need) t o t h e reader. So assume t h a t p i s weakly compact and 6 i s a f a m i l y o f f u n c t i o n s such t h a t 6,eac1
f o r every CI E p . We s h a l l c o n s t r u c t a p - t r e e T such t h a t a p a t h t h r o u g h i t w i l l g i v e us t h e f u n c t i o n 8. Since p i s weakly compact, every p - t r e e has a p a t h t h r o u g h it. Thus, we s h a l l o b t a i n 8.
L e t g E n ( P p : @ € a )f o r some a € p . We can l o o k a t g as sequence of approximations t o t h e d e s i r e d L . We c a l l -y a f a i l u r e o f g i f t h e f o l l o w ing three conditions are satisfied:
(1) Y
E
a.
( 2 ) One o f t h e f o l l o w i n g a l t e r n a t i v e s holds: EEqEr
(g,'6
(2.1) f o r e v e r y 6 ~ y f ,o r e v e r y e w i t h 6 E e C Y t h e r e i s an q w i t h w i t h g '6 f g '6 (we can say t h i s by: f o r a r b i t r a r i l y l a r g e S E T , 17
Y
: d c e ~ y i) s n o t e v e n t u a l l y g ' 6 ) ;
Y
(2.2) t h e r e i s no (3 E p s a t i s f y i n g : (2.2.1)
Bplr
and g,
a r e e v e n t u a l l y equal, i.e.
368
ROLAND0 CHUAQUI
%(EEy A
dpIh-0
= 9,I(YE))
and
dP maps bounded subsets of
(2.2.2)
i n t o bounded subsets of
y
y,
i.e. 3E(EEy
+
3T(SE-Y A
6*PtCS)).
(3) y i s an uncountable strong l i m i t cardinal. F o r a E p , l e t L a = ( ( g , h ) : g E n ( Pp : P E a ) A D h = ( y : y i s a f a i l ure of g ) A h i s biunique and regressive). (g,h)
Q
hl)
T ('1'
Define the t r e e T by,
i f and only i f
We have t h a t T i s a t r e e with L a being i t s a t h l e v e l . I t i s a l s o easy t o see t h a t g = g l / a and y E a, imply t h a t y i s a f a i l u r e of g i f and only i f y i s a f a i l u r e of gl. We f i r s t prove: (4) T i s a p-tree.
Clearly L~
5 n(Pp
x
.
"a4 p
T h u s , we just have t o prove t h a t L a # 0 f o r every a E p . The f i r s t uncountable strong limit cardinal i s I ,. T h u s , t h e r e a r e no f a i l u r e s before 1 , and, hence, L a # 0 f o r every a E lo. Therefore, i t s u f f i c e s
t o prove by induction on S f o r P (5) If ( g , h )
E
that, (5.1) 7
E
L a + 1 and a c p , then t h e r e i s
D q l and a c 7 c P imply
<-,- ( P I ,
(5.2) ( g , h ) and
(5.3) a c Let
0
= 6
CASE ( a )
31 , .
y c
+
41)
ell'
(
P1, q l ) E L o such
3 a,
a
6 implies p l y la
v w i t h 6 = u6 and v
=
g,
EW
.
.
We have several cases.
6 not strong l i m i t c a r d i n a l .
Choose y + 1 C 6 such t h a t t h e r e i s no strong limit E w i t h y c e c By t h e hypothesis t h e r e i s a ( p2! q l ) E LY + l ' s a t i s f y i n g (5). ChTse+ l L P = p2. Then ( p l , q l ) E Lp, s a t i s any p 1 E Il ( ' t :E E P ) such t h a t pl f i e s ( 5 ) s i n c e t h e r e a r e no f a i l u r e s between y + 1 and P .
6.
AXIOMATIC SET T H E O R Y
369
S i s strong l i m i t singular.
CASE ( b )
Choose u C 6 w i t h 0 2 ~ 1 c, T ( S ) ; u n o t s t r o n g l i m i t . L e t ff be t h e normal f u n c t i o n t h a t enumerates t h e s t r o n g l i m i t c a r d i n a l s ; l e t ff't c u c
ff't +l. L e t j E Ef(a)S, ff'(t + l ) , j ' l + u C j ' 2 j'l and j ' 2 .
-
C
j normal w i t h u 0 ; - I = S, j ' 0 = a , u c j ' l c ff'(t + l ) . Thus, t h e r e i s no s t r o n g l i m i t between
L e t j ' ( c T ( S ) ) = 6.
We now d e f i n e ( p 2 , q 2 ) E
LS + 1 by c o n s t r u c t i n g by r e c u r s i o n on q 2 q2\j'(q)+1)€L. increasing i n
cf(6) the sequence(p2\j'(q)+l, t h e t r e e T, so t h a t D q;'n(jl(v) +I) n
For 7 'jl(q)+ I
+ 1 and
- (,,(I) + 7))
o
=
for q
c'E~(s).
l e t ( p 2 \ j ' O t l , q l j ' O t 1 ) = cg,h). 2
= 0,
+ 1#2.
Suppose q = ;I(,)
( j l 2
J ' ( 7 ) + 1'
By t h e i n d u c t i o n h y p o t h e s i s t h e r e i s a ( p 3 , q 3 )
such t h a t : ( p 2 J j ' ( t )+1, ( z , l j l ( t ) + 1 ) Q T ( p 3 , q 3 ) , E
E
1 j ' ( t ) + 1 and n(j'Z-(j'(l) +
D q 3 i m p l y q3e
p 3 f l j ' € = p2,jlt., Thus, D q 3 (P21j1(d+ 1 , Ci21J'(S) +I) = ( P 3 , 'i3) Suppose now, q = 2.
-
j't
CE
q ) ) = 0.
Hence t a k e
Proceed as i n Case ( a ) t a k i n g y =
D(q21 j ( 2 ) + 1) 2 j ' l and t h e c o n d i t i o n i s s a t i s f i e d .
c
j ' t c E
c j ' ( v ) + 1 implies
Hence
j l l .
F i n a l l y , l e t q = u?# 0. Since j(q) = u { j ' t : t E v ] ,we have a l r e a d y defined p 2 ) j ( v ) and q 2 1 j ( v ) P ~ i s any , f u~ n c t i o~n i n ~ -!V j ' v e x t e n d i n g
g,
+ q i f j l q i s a f a i l u r e o f p2.
q 2 ' j ' q =j'(l)
Thus q 2 i s s t i l l b i -
unique and r e g r e s s i v e . Let, now, p1 be an e x t e n s i o n o f p2 t o and q1 = q 2 .
P.
P such t h a t each pl0 extends g,,
Thus, ( p l , q l ) E LP s i n c e t h e r e a r e no f a i l u r e s between S and
CASE ( c ) .
S
i s uncountable, s t r o n g l i m i t , and i n a c c e s s i b l e .
L e t S = {u : u E S A V T ( q E u
-,6;
qEu)}.
S i s c l o s e d unbounded i n 6.
.
I t i s c l e a r l y c l o s e d . We now prove t h a t i t i s unbounded. L e t E E S . Def i n e h'O = CI and h ' v + 1 = (u {6,'0 : 0 c - k'v} u h'v) + 1 for v EW Let u =
I t i s easy t o show t h a t u € S .
u hew.
L e t j be a normal enumeration o f t h e l i m i t p o i n t s o f S(i.e. L ( S ) which i s a l s o c l o s e d unbounded), w i t h c1 C j " 0 . As i n Case ( c ) , we d e f i n e ( p 2 , ( z 2 ) , E LS + 1 by a r e c u r s i o n on v c 6 , c o n s t r u c t i n g (p21j'(q) + 2 , 4 . ; ( j 1 ( q ) +
2 ) such'that f o r 0 = 0.
f y i n g (5).
3 j(v)
and c1
cy c
j ' v we have p2
y = &'y.
By t h e i n d u c t i o n h y p o t h e s i s t h e r e i s ( p 3 , q 3 ) E L j ( o )
+
D e f i n e pZy = pgy f o r y c j ' ( 0 ) + 2 and q 2 1 j ( 0 ) + 2 = q 3 .
satis-
370
ROLAND0 C H U A Q U I
for
E
dslf
for
a 5E c j'q
0
for
E =
Define q 2 1 j ' q = q4 and q 2 ' ( j ' q ) = j ' t
1 i s biunique, s i n c e q41y failure.
Thus,
(
2 j ' ( t )+ 1
, if
j'q
j l q i s a failure.
q2\j1(q)+
f o r y 3 j ' ( t ) + 1 and j l ( t ) + 1 i s n o t
p21 j ( q ) + 2 , q21 j ( q ) + 2 )
F i n a l l y , assume q = U q # , O . have d e f i n e d p for all y E j ' q .
c a
gale
L. J(V) + 2 ' We have j l v = U { j ' t : t € q l . Thus, E
we
2Y
Define p
2Y
f o r y = j ' i , j l ( q ) + 1 by
Let, now, pZ6 be d e f i n e d by
L e t pl be an e x t e n s i o n of p2 t o L e t q1 = q2. Then ( p l y yl) none between S and p .
6
P such t h a t p1
extends g,
for 8 €0
LP s i n c e S i s n o t a f a i l u r e and t h e r e a r e
.
Thus, we have completed t h e i n d u c t i v e p r o o f o f ( 5 ) and, hence o f (4)
37 1
A X I O M A T I C SET THEORY
i.e., T i s a p - t r e e . S i n c e p i s weakly compact, T has a p a t h 8. L e t (g, h ) E B n L f o r e v e r y a E p . L e t G = u {g : a E p } and H = u {ha : a E p } .
a
a
a
-
C l e a r l y G = ( G : P E P ) w i t h G E P 0, Dff i s t h e s e t o f f a i l u r e s o f G, ti i s
P
P
L e t S ' = Ca : Wy(y E D H n a
b i u n i q u e and r e g r e s s i v e .
H'y E a ) ] . Then,
( 6 ) S ' i s c l o s e d unbounded i n p . S ' i s c l e a r l y closed. I n o r d e r t o prove t h a t i s unbounded, l e t P E P be g i v e n and suppose t h a t H ' y c P w i t h y 3 P . D e f i n e by r e c u r s i o n k ' o = y ,
-1
and k ' v t 1 = n {H-lS : 6 13 k'v A 6 E D h - } f o r V E W . L e t a = u k * w and assume t h a t k'e E a. Then H'e E k'v f o r a c e r t a i n v E W . Hence, ff-" H ' E C ff-l6 f o r 6 3 S ' i s unbzunded.
k'v.
Thus, e
5 k'v
t
1 E a.
Hence
5a
and E =
E S ' and
L e t S = S ' n p n S L O I . Then S i s a l s o c l o s e d unbounded. NOW, S n D H = 0, s i n c e a E Dff i m p l i e s H'aEa. Hence, i f a E S, t h e n a i s n o t a f a i l u r e of G. By c o n d i t i o n (2.1):
( 7 ) There i s a f u n c t i o n
j oE
S
1.1
such t h a t j o ' a C a and j o ( a )
i m p l i e s t h a t t h e r e i s a 6 such t h a t GE'y = Ga'y f o r e v e r y (2.2)
'
t h e r e a r e f u n c t i o n s i, j l y j 2E
57
(8) i f jl'a ( 9 ) ify
and
(10)
5 j,', j2'a
jl'cl,
C
a , then dita'y
, then d . c , a ' ~ c c a,
c_
jo'a
'
j1
1
a
S
1.1
=
a
5y
- 6.
C
a By
such t h a t
Ga'y
j2'a
E E
,
,
.
By F o d o r ' s Pressing-Down Lemma 4.3.2.4,
since j o ' a c a f o r a
E
S, t h e r e
0c - S such t h a t j o ' a = y o f o r a l l a E S O . S i m i l a r l y , we o b t a i n s t a t i o n a r y s e t s S c S and S2 5 S 1 w i t h jl'a = y1 f o r a E S 1 and 1- 0 j 2 ' a = y 2 f o r a E S 2' i s a stationary set S
c p.
Thus, we have 6rla y C y f o r e v e r y a E S2. Therefore, {6i~aly1:afS2} 1 7 2 Since t h e t h i n i d e a l i s p-complete, t h e r e i s a s t a t i o n a r y s e t S 3 C S 2
such t h a t ditalyl = Let, now, y o
6. .cI P
56
E
Iyl
f o r e v e r y a,@ E S 3 .
a E S3.
We have by ( 7 ) t h a t t h e r e i s an
-
E
such
t h a t G 'y = Ga'y f o r e v e r y 77 E a E. 77 Then i 6 ' a c a. So f o r L e t i 'a = n { E : Wv(v E cx E -,G 'y = G 'y)}. 6 77 a some s t a t i o n a r y S6 cS3, i 6 ' a = k'6 i f a E S 6 . Thus, s i n c e S6 i s unbounded, G ' 6 = GkI6'6
P
Define
eE
f o r every 0
2
k'6.
' p as f o l l o w s .
For 6
- yl,
J I
l e t L'6 = G k I 6 ' 6 ; f o r 6 c y l , l e t
372
CHUAQUI
ROLAND0
L'6 = dipa'6
f o r any a E S3.
If a
L e t a E p.
Lla.
cyl
L
This
s a t i s f i e s our conclusion:
Assume y1 C a E p. L e t 6 ' = {k'S :S i ' 6 ' '3 a . We have, f o r E E a , E 37 - 1'
diiSIl'~= For e E yl,
dit6,,'e
3 a.
t h e n t a k e any 6 E S 3 w i t h i'6
=
Take 6 ' c - 6"
a}.
E
G S l l ' ~ = Gkt
E
'E =
L ' E , because 6" E S 3 .
L'E
E
Then 6i,61a= S3 with
.
Thus, d i r 6 , , I a = [la.
We i n t r o d u c e another measure of smallness f o r subsets o f p:
l y c o m p a c t id&.
THEOREM,
5.2.1.5
WC
p E
+
t h e weak-
wci ( p ) E I d ( p ) A C o m i d (p, wci ( p ) )A
W d ( C u b ( A , p ) - ' p - b E w c i ( p ) ) A ~ a : ~ ( aC )a E p } E w c i ( p ) .
Thus, ifp i s weakly compact, t h e n w c i ( p ) i s a p-complete i d e a l t h a t p r o p e r l y i n c l u d e s t h e t h i n i d e a l , s i n c e Ca :cf(a) c a E p } i s s t a t i o n a r y i n p and t h e t h i n i d e a l i s {A : A C p A p b does n o t c o n t a i n a c l o s e d unbounded s e t )
-
-
.
PROOF, (1) I t i s t r i v i a l t h a t 6' C - d E w c i ( p ) i m p l i e s 6' E w c i ( p ) . We now prove t h a t w c i ( p ) i s p-complete. Let Let 6 E wci(p) f o r [ € 6 Cp
.
E
: q E p ) e x e m p l i f y A E w c i ( p ) ; i.e. t h e r e i s no e E p p such t h a t = ( dt I: E f o r a l l a E p t h e r e i s a P E A~ w i t h a C P and 4, la = $la. Define k = 17 if q E d u (A5 : 5 E [ } and k = 6 6I: 1) E v, 077' otherwise. Assume A = u { A E *'
E E ~ 4} a
5 j'a
u j'l*
-
w c i ( p ) . Then t h e r e a r e L, j E p p such t h a t f o r each CL E p, j ' a E A , and k j l a l a = Lla. We have t h a t p = u { j - l * ( n [ - u cog: 3 E I : I ) : I : E 6 1
do.
Since p i s r e g u l a r , j-'*(A[-
ed f o r a c e r t a i n [ € 6 . ed. thus
For
d
j-l*h0
E
c1 E P ,
Assume
l e t m'u=j(nB
does n o t e x e m p l i f y A
t
U
{Ag : 5 € [ } ) o r j - l * h 0 a r e
: 5 € [ } i s unbound5 Then, k m t C L l a = L l ab;u t k m , a = 6 [ , m l a ;
t h a t 8 = j-'*(hE-
-
a).
unbound-
E wci(p),
U {b
a contradiction.
The p r o o f f o r
unbounded i s s i m i l a r .
Since P i d e a l on p.
4
w c i ( p ) (5.2.1.3)
we have proved t h a t w c i ( p
( 2 ) L e t 6 be 'closed unbounded i n p.
Define f o r o
i s a p-complete
~ p
A X I O M A T I C S E T THEORY
Then f o r a,P
Suppose t h a t L
E 'p.
and
- y, u
-
3
( 3 ) L e t S = {u : F ( u ) c U
D go1 = u.
gu' y -C a) f o r a
E
u.
E Kp.
y E K , h'y = a c E
.
Take
P
EP with
E
L e t k! E 'p
d
{TI f?
u {u : u ~p
duly
-n
6,'
0 = T ( u ) , and
w i t h b'0 =
For
K.
6,'
~
A du'a =
=Lly. g,
Thus
d
normal
(1+a) =-u (7 :
dKe f i n e h'y = n {a:
E
K .
Hence u = u {g,)y : ~
S E wci(p).
5e
E K }
=
contradict-
LEMMA, E
wc
A S E pwci(p)
( i i ) A E Pp- wei(P) A -1,
3
such t h a t
We have y = Lla = 6a ' a = 6U (1 + P ) f o r a c e r t a i n P , s i n c e u and Y + K . Thus, from t h e d e f i n i t i o n o f I&, we g e t ,
= e l 0 = 6,' 0 = T g,'rZP 5.01 2 e , f o r e v e r y ~ E i n g U ~ E . Thus, 6 e x e m p l i f i e s
(i) p
-
L e t E = u o h - ' ; s i n c e K E P and p i s r e g u l a r , e Ep. cp and suppose t h a t t h e r e i s a u ' P w i t h & l P = L I P . L e t
K
5.2.1.6
E p
i s a bounded subset o f p.
For u E S choose g, E T ( u ) u ,
ul.
D e f i n e f o r u E S,
L'a = y l ; t h e n h
PI
A du'a =
P and t a k e y
Assume Lla =
Then t h e r e i s no u exemplifies p 6 E wci(p).
PIuCal.
-n
t h e s e t Cu : u E p
EP
373
4
E
17 : y
-+
A
E
u C S : ~a ~ y 1 1E w c i ( p ) .
P AWa(a E
A
+
6'a
C
a) *
YEP
A
WCj(P))).
Thus, t h e weakly compact i d e a l i s normal. ( i i ) i s a v e r s i o n o f Fod o r ' s Pressing-Down Lemma. Assume t h e h y p o t h e s i s and l e t d e x e m p l i f y S E w c i k ) . PROOF OF (i). c; t L e t D = Cy:yEUISa:aEy}}. F o r y E D l e t g ' y = n { a : y E S 1. Then
a
g ' y C y. D e f i n e f o r u E A, hu'O = 9'0, h u ( l +a) = dglo,ula. Let L E ' p be such t h a t f o r every a E P t h e r e i s a u E A, a 5 u such t h a t h u l a = L l a . L e t L'O =
P.
L e t a E P , t h e n k u l a = L l a f o r some u E A.
j'y = L ' ( l f y ) we have t h a t 6 P , u J a = j l a ,c o n t r a d i c t i n g
wci(p).
4.3.2.3.
( i i ) i s o b t a i n e d from ( i ) as Fodor's lemma 4.3.2.4
6P
But,then d e f i n i n g e x e m p l i f i e s SP E
i s o b t a i n e d from
9
Thus, e v e r y weakly compact c a r d i n a l i s Mahlo.
PROOF,
subset o f cc.
Assume P i s weakly compact and l e t S be c l o s e d unbounded Then S E d u w c i ( p ) , P ) , t h e dual f i l t e r o f t h e weakly com-
374
ROLAND0 C H U A Q U I
p a c t i d e a l . Also, S L O I n p E d u ( w c i ( p ) , p ) , because S L O I n p i s c l o s e d Thus, C I A n p E d u ( w i b ) , unbounded, and R G E d u ( w c i [ p ) , p ) , by 4.4.3.4. Thus C I A i s p ) and, hence S n C I A E d u ( w c i ( p ) , p]. Hence S n C I A # 0. s t a t i o n a r y i n p.
PROOF, Suppose t h e h y p o t h e s i s o f t h e lemma and assume t h a t f o r eve r y i n a c c e s s i b l e K E t h~e r e i s P E K such t h a t SP nK i s n o t s t a t i o n a r y i n K . Let
g'K =
n {P : P
EK
A 1 S t a t (SP n
For each
K,K)}.
CIA let
K E
normal f u n c t i o n t h a t enumerates a c l o s e d unbounded subset o f
8,
be t h e
disjoint to
K
'g'K.
Since g i s r e g r e s s i v e and C I A n p E d u ( w c i ( p ) , p ) ( b y 5 . 2 . 1 . 7 ) , such t h a t g ' K t h e r e i s an A C C I A n p , A E d u ( w c i ( p ) , p ) ( b y 5.2.1.6) f o r every K E A i s c l e a r l y unbounded.
=P
x.
Then, f o r e v e r y disjoint with S t h a t f o r every
P
c1 E
5.2.1.9
(I
E
4,
enumerates a c l o s e d unbounded subset o f E
e
wc c M~(CIA)
wc.
We must prove by i n d u c t i o n t h a t
BY 5.2.1.7,
with
4
A n M~(CIA)
be- weakly compact.
f o r every a E p.
c1 E p.
1.
p E M ( ' ) (CIA
We have t o prove t h a t
L e t C be any c l o s e d unbounded subset o f p.
")(CIA).
It
i s stationary
P
@
such
A such t h a t u > a and d u l a = L l a .
= 0, c o n t r a d i c t i n g t h e assumption t h a t S
P
Assume t h a t cc E M ( " ) ( C I A ) M
E
K
i s normal and hence D L - l i s c l o s e d unbounded i n p.
THEOREM,
p E M (")(CIA)
E 'p
d u ( w c i ( p ) , p ) , we o b t a i n an
p there i s a u
S
PROOF, L e t
A,
Since A
n K .
i s easy t o see t h a t Clearly D L - l inp.
K
-
Then
CI E
S
P
=
By 5.2.1.8. M ("(CIA) n C np CI i s s t a t i o n a r y i n p f o r e v e r y P E a. t h e r e i s an i n a c c e s s i b l e K E C (such t h a t S nrc i s s t a t i o n a r y i n K f o r a l l ~
E
n K a.
But
a r y f o r every
P
K
= u
E a.
(C n (p Thus,
K
- a) n E
K).
P
Hence K
E
C and S nrc i s s t a t i o n -
P
M ( " ) ( C I A ) n C, i.e. M b ) ( C I A ) i s s t a -
t i o n a r y i n p ; thus, p E M (at ' ) ( C I A ). I f a = u a +O, r e s u l t i s clear.
Therefore, p E
t h e n M ( a ) ( C I A ) = n {M ' P ) ( C I A )
we have proved t h a t
@ E
:
PECX};
M ( " ) ( C I A ) f o r every
M ' ( C I A ). L e t C be c l o s e d unbounded i n p.
a r y i n ~t f o r e v e r y a
E @.
By 5.2.1.8,
Then T, = C n M ( " ) ( C I A ) there i s a
K
E
thus t h e
c1 E
p. Hence
i s station-
such ~ t h a t T, n
K
is
A X I O M A T I C SET THEORY
375
A s t a t i o n a r y i n K f o r e v e r y a E K . Thus, K = u ( C n K ) E C and K E M ( C I A ) , A i.e. M ( C I A ) n p i s s t a t i o n a r y i n p. Hence P E M ( M * ( C f A ) ) and thus, p
+nM ~ ( C I A ) .
S i m i l a r l y we can prove t h a t W C c M A ( M A ( C I A ) ) , M and so on. I n general, we can prove,
A
A A ( M (M ( C I A ) )
PROBLEMS
1.
Prove t h e i m p l i c a t i o n from r i g h t t o l e f t i n 5 . 2 . 1 . 3 .
2.
Prove 5.2.1.10.
5.2.2
LARGER
CARDINALS,
Other l a r g e r c a r d i n a l s have been e x t e n s i v e l y discussed i n t h e l i t e r a t u r e . F o r a thorough d i s c u s s i o n and a c h a r t see Kanamcri-Magidor 1978. I n general g i v e n an i n t e r e s t i n g p r o p e r t y P of c a r d i n a l numbers K , we i n v e s t i g a t e whether o r n o t P ( K ) holds d o r some c a r d i n a l s K . We w i l l be concerned m a i n l y w i t h c o m b i n a t o r i a l ( n o t metamathematical) p r o p e r t i e s P . None o f t h e c o r n b i n a t o r i a l l y d e f i n e l a r g e c a r d i n a l s , appearing i n t h e c h a r t on pages 265-266 o f Kanamori-Magidor 1978, t h a t a r e l a r g e r t h a n weakly compact can be shown t o e x i s t i n B C . For i n s t a n c e , i n e f f a b l e c a r d i n a l s can be defined ine66abLe i f i s o n l y i f p i s uncountable i n a c c e s s i b l e and as f o l l o w s : p f o r e v e r y d E IIc"a : u € p 1 t h e r e i s a s t a t i o n a r y X ~p such t h a t d l X i s coherent. O b v i o u s l y (see 5.2.1.3) p b e i n g i n e f f a b l e i m p l i e s t h a t i s weakly compact; b u t i t a l s o i m p l i e s t h a t t h e s e t o f weakly compact c a r d i n a l s bel o w i t i s a s t a t i o n a r y subset o f p . Now, f r o m t h e e x i s t e n c e o f an i n e f f a b l e c a r d i n a l p ( a s w e l l as f r o m t h e s m a l l e r s u b t l e c a r d i n a l s ) i t f o l l o w s t h a t t h e r e a r e II:
-
indescribable cardinals f o r every
K ,
v Ew.
IIt- i n -
d e s c r i b a b i l i t y i s a metamathematical p r o p e r t y o f c a r d i n a l which I w i l l n o t d e f i n e here ( s e Drake 1974 o r Kanamori-Magidor 1978). Weakly compact c a r dinals are IIi- indescribable
I t has been shown i n Tharp 1967 t h a t a l -
though f o r each n E w we can prove i n B C t h a t t h e r e i s a IIi- i n d e s c r i b able cardinal there i s a
,
1nV
t i s n o t poss b l e t o show ( i n B C ) t h a t f o r every
indescribable cardinal.
VEW
,
Thus, t h e e x i s t e n c e o f i n e f f a b l e
376
ROLAND0 CHUAQUI
(and subtle) cardinals cannot be proved i n B C .
The larger cardinals: Ramsey, measurable, strongly compact, super compact, etc. are inconsistent with G'ddel axiom o f constructibility, which i s consistent with B C ( i f B C i t s e l f i s consistent). T h u s , their existence i s n o t a consequence o f B C .
REFERENCES
P. Bernays 1976
On t h e phu&!em ad 6chemcLta 0 6 i n d i n i t q i n ax.LomuLLc b e t theohy, in Sets and Classes, G.H. MUller (editor), North-Holland Pub. Co.,
Amsterdam, pp. 121-172. R.Bradford 1971
R.
C a h d i n d udddition and ,the ax&m V O ~ .3 pp. 111-196.
06
c h o k e , Annals o f Math. Logic;
Chuaqui
1978
BaMqA' h b t h w k y , in Mathematical Logic: Proceedings of the First Brazilian Conference, Arruda, da Costa and Chuaqui
(editors), Marcel Dekker, New York.
1980
I n t u ~ r u Cand 6uhcing mod&
doh t h e h p k e d u a t i v e theahq
Dissertationes Mathematicae, vol. 176.
06
clla66e6,
N.da Costa 1980
A madeL-theahuXd apphaach v&Me bhd.ing tm apmXotclh6, in Mathematical Logic in Latin America, Arruda, Chuaqui, and da
Costa (editors) North-Hol land Pub. Co., Amsterdam, pp. 133-162.
F.R. Drake 1974
Set Theory: An Introduction to Large Cardinals. North-
Holland Pub. Co., Amsterdam.
H.B.Enderton 1972
A Mathematical Introduction to Logic, Academic
1977
Elements of Set Theory, Academic Press, New York.
.
York.
Press, New
H Ga ifman 1967
A genUra&zation
06
Makeo'6 methud
YUU~ Israel ~~U J. ,of
60"
oMaivLing b g e caA.dhd
Math. vol 5, pp. 188-201.
K.G8del 1940
The Consistency of the Continuum hypothesis, Annals of
Math. Studies, Princeton.
377
ROLAND0 C H U A Q U I
378
P. R.Halmos 1965
Naive Set Theory, van Nostrand.
T.Jech 1978
Set Theory, Academic Press, New York
A.Kanamori and M.Magidor 1978
06
6c-t theohy, i n Higher Set Theand D.S.Scott ( e d i t o r s ) . L e c t u r e Notes i n Mathemat i c s , 669, S p r i n g e r Verlag, B e r l i n , pp. 99-275.
The evo&ilu.tion
ory, G.H.Miiller
U g e ax&tnA i n
J . L. K e l l ey 1955
General Topology, van Nostrand Pub.Co. P r i n c e t o n .
K.Kuratowski
and A.Mostowski
Set Theory, Second E d i t i o n , N o r t h - H o l l a n d Pub.Co.,
1978
Amsterdam.
A.L~VY 1960
A x i o m A c h m & 0 6 A a o n g i v l @ ~ L X yi n axLomcLtic A & of Math. v o l . 10, pp. 223-238.
1979
Basic Set Theory, Springer-Verlag, B e r l i n , H e i d e l berg-New York.
t h w a y , Pac. J .
J. D.Mon k Introduction t o Set Theory, Mc Graw-Hill, New York.
1969 A.Morse 1965
A Theory of Sets, Academic Press, New York.
A.Mostows k i 1950
Some -bnpnphediccLtcve dedinitiuru i n t h e uxiomuXic Math. pp. 111-124.
Set
theohy, Fund.
.
S She1 ah 1979
WuLktiq compact c a h d i d : i c , v o l . 44, pp. 559-562.
J .R.Shoenf iel d 1967
A cambinatmid ph006, J. o f Symb. Log-
Mathematical Logic, Addison-Wesl ey Reading.
AXIOMATIC S E T THEORY
379
A.Tars k i 1949 A.Tarski 1965
L
.Th a r p
1967
Cardinal Algebras, O x f o r d U. Press, New York. and J.Doner An extended CULithmuXc 95-127.
oh o / r d i U numbem, Fund. Math. v o l . 65 pp.
On a s e t t h e o t r y o h Bexnuyb, cl. o f Sym. L o g i c vol. 32, pp. 319-
321.
INDEX OF SYMBOLS
Symbol
Informal explanation
d:
P 1
Page
P r i m i t i v e Language
5
Negation
5
Imp1 i c a t i o n
5
v
Disjunction
5
A
Conjunct i o n
5
Equivalence
5
Universal q u a n t i f i e r
5
W 3 3!
Existential quantifier 'There e x i s t s e x a c t l y one
5 . . . I
5
--
Identity
5
E
Membership
6
@x[Y 1
Substitution
6
Ax Class
Axiom of c l a s s s p e c i f i c a t i o n
6
Ax E x t
Axiom o f e x t e n s i o n a l i t y
Ax Sub
Axiom o f subsets
7 7
4JU
Re1a t i v i z a t i o n
Ax Ref
Axiom o f r e f l e c t i o n
7, 131 7
B Ax GC
Bernays c l a s s t h e o r y
8
Axiom o f g l o b a l c h o i c e
8
Ax C
8
BC
Bernays c l a s s t h e o r y w i t h c h o i c e
9
Ax Reg G
Axiom of r e g u l a r i t y
9
General Class Theory
13
Ax Em
Axiom o f t h e empty s e t
13
Ax Num
Axiom of numbers
13
Extended language
13
Description operation
13
Classifier Substitution
14 15
e
U
{:I Yx 0"'
Xn-l
[rO...r n-1 1
381
382
ROLAND0 CHUAQUI
Ax Def
Axioms of d e f i n i t i o n s F and G a r e t h e same operation A and B a r e t h e same notion Universal c l a s s Empty c l a s s Subclass Proper subclass 21, Union 21, Intersection Difference, complement Sing1 eton Pair Ordered p a i r Cartesian product Ordered p a i r of c l a s s e s Power s e t Identity relation Membership r e l a t i o n Inclusion r e l a t i o n Diversity r e l a t i o n Re1 a t i ve compl ement Converse r e l a t i o n Composition of r e l a t i o n s I d e n t i t y r e s t r i c t e d t o the f i e l d of R Domain of R R r e s t r i c t e d in i t s range t o A R r e s t r i c t e d in i t s domain t o B Image of A by R Superclass of t h e 7 ' s such t h a t @
F - G
A- B V 0 C C
U
n
CAI {A, B l (a,b)
Ax8
[&Bl P ID EL
IN
Dv
- R R-1 RoS RO
16 18 18 20
20 21 21 26, 27 26, 27 21 22 22 24 24 25 26 33 33 33 33 33 34 34 34 36 36 36 36 40
Class o f d i s j o i n t non empty sets X i s an equivalence c l a s s o f R R i s a p a r t i a l ordering t h e R-least upper bound
44
t h e R-greatest lower bound
46
44 46 46
AXIOMATIC S E T T H E O R Y
383
L u b R [2, Al
z i s a l e a s t upper bound o f A
46
G1 b R ( z,Al
z i s a g r e a t e s t lower bound of A
46
ULO
Upper s e m i l a t t i c e ordering Lower s e m i l l a t i c e ordering L a t t i c e ordering
46
47
CLO
Complete upper s e m i l a t t i c e ordering Complete lower s e m i l a t t i c e ordering Complete l a t t i c e ordering
ORM
Connected Simple ordering The i n i t i a l segment o f R determined
LLO LO CULO CLLO
co so WF
wo R'x
( 7 :@)
X
' A ( FI
BA
"4
! A(F1 ! A
Mo W
S t
P -v
Rv, R-'
x;
exp, p V RW
TA
W == -
?
46 47
by x
We1 1 -founded We1 1 -ordering The value o f x by R The function determined by
47 47 47 47 47 47 49 52
7
and @
F i s a function with domain 8 and range included in A €3
The c l a s s of functions 6 with A [ 6 ] Generalized product Product o f a function F i s a permutation of A The c l a s s of permutations of A Mono tone The c l a s s of natural numbers Successor Addition of natural numbers Substraction of natural numbers I t e r a t i o n of r e l a t i o n s Mu1 t i p l i c a t i o n o f natural numbers Exponentiation of natural numbers Ancestry r e l a t i o n T r a n s i t i v e c l o s u r e of A Class o f well-founded s e t s Isomorphism Isomorphic embedding
53 55 55 55 56 56 56 57 67 67 73 75 75 76 78 80
82 90
95 95
384
ROLAND0 C H U A Q U I
Rx
R r e s t r i c t e d t o OR(xJ
97
I S [ A, RJ A=8
A i s an i n i t i a l segment o f R
A i s equipollent with 8
105
AS8
A i s s m a l l e r o r equal i n c a r d i n a l i t y than 8 Cardinal a d d i t i o n
105 108
ZCx' 7 : 4 1 FN
General i r e d c a r d i n a l a d d i t i o n
108
Class o f f i n i t e s e t s
110
DO ( R l
R i s a dense o r d e r i n g
117
MKT
Morse-Kel l e y - T a r s k i Theory
123
NBG
von Neumann-Bernays-GBde1 Theory
123
Zermel o-Fraenckel Theory
123
Ax Un
Axiom o f unions
124
A tc8
ZF
97
Ax Pow
Axiom o f power s e t
124
Ax Rep
Axiom o f replacement
125
Ax I n f
Axiom o f i n f i n i t y
125
MKT'
MKT-Ax I n f
MKT R
MKT
+
+
Ax Em
125
Ax Reg
125
The r a n k o f X
143
RX
The c l a s s o f s e t s w i t h rank l e s s than p X
151
ds X
D i s t i n g u i s h e d subset o f X
154
P X
Type o f x w i t h r e s p e c t t o R
155
On
Class of o r d i n a l s
156
In(F)
F i s s t r i c t l y increasing F i s completely a d d i t i v e
163
F i s continuous
165
Cad(F1 Con(F) Nor m i l ( F )
165
F i s normal
166
F"
Iteration o f f
169
Fp(F1
F i x e d p o i n t s of F
170 172
Addition o f ordinals
173
Multiplication o f ordinals
179
Main o r d i n a l s o f a d d i t i o n
182
Main o r d i n a l s o f m u l t i p l i c a t i o n
182
Ordinal exponentiation
184
AXIOMATIC S E T T H E O R Y
Main o r d i n a l s o f exponentiation Equi pol 1ence re1 a t i o n Cardinal number o f x Class o f c a r d i n a l s a i s l e s s than o r equal t o b a i s l e s s than b Addition o f c a r d i n a l s Multiplication o f c a r d i n a l s Exponentiation o f c a r d i n a l s
385
188 192 192 192 192 192 193 193 195 197
Class o f well-ordering Enumerator o f ti Type o f h Class o f simple ordering types Inverse of type c1 Ordered sum o f t h e 7 ' s according to R Ordered sum
197 258 258 259 259 259 259 260
Ordered sum Addition o f ordered types
261
Product o f orderings
262
Product of orderings Product of orderings Multiplication o f types o f orderings An ordering isomorphic t o t h e rationals The order type of Q R i s scattered u i s c u t in R Completion o f R R i s a continuous ordering The continuous c l o s u r e of R An ordering isomorphic t o the r e a l s The type of Q c
262
26 1
263 264 265 269 269 268 268 269 269 270 271
386
R O L A N D 0 CHUAQUI
S u p ( S , [ F ( x ) :x E A ] )
S i s a l e a s t upper bound o f
273
The c l a s s of alephs a has c a r d i n a l i t y l e s s than b The c l a s s of beths Hartog's operation The local axiom of choice The generalized continuum hypothesis
27 6 278 280 282 288 295
AI a4b
Be H
AC CH
[F(xl : x ~ A ]
HC
OI
(r,R)
Cof
cf ( R l cf
S N(4 SN
RG ( X I RG IA
c1
(X,
C u b ( X,
r) )
5
The c l a s s of i n i t i a l o r d i n a l s i s cofinal w i t h R degree of c o f i n a l i t y of R degree of c o f i n a l i t y of r X i s singular Class of s i n g u l a r s e t s X i s regular Class of regular s e t s Class of i n a c c e s s i b l e s e t s X i s closed i n r X i s closed unbounded i n r
r
The enumerator of X
EX
Iim[Fa:
A
a E p ]
The l i m i t of Fa when
ci
tends t o
295 297 300 300 300 3 04 304 304 304 311 314 314 314 314 315
2 ( K ' X :X E A )
The diagonal 1 imit X i s stationary i n r The d e r i v a t i v e of F The ci d e r i v a t i v e of F The diagonalization of F MKT + Ax GC The i n i t i a l ordinal equipollent t o A Cardinal sum o f o r d i n a l s Cardinal mu1 t i p 1 i c a t i o n of o r d i n a l s I n f i n i t e sum of c a r d i n a l s
315 317 318 318 318 321 325 325 325 327
n
I n f i n i t e product of c a r d i n a l s
327
lim
[ F :aEr]
S t a t [X,
dF
a
I')
daF
FA MKT - C A
a tCP a eCp CX
(K'X:XEU)
c x
AXIOMATIC S E T T H E O R Y
C V K
Cardinal exponentiation o f i n i t i a l 332 ordinals Class of subsets of of c a r d i n a l i t y 334 l e s s than v Weak exponent i a t ion The exponential s c a l e of i n i t i a l ordinals
Stat ( K , x , u ) sat (x,a)
Comid
(K,x)
c o d 1 (K,X)
Pr
(x,cO
c ub ( K )
Tree(T)
ht Lx,Tl L e v r (TI H t (TI Stree ( T ' , T ) Path { B, T )
K-Aron(T] Mod WCZA
n
387
{Fi : L E I }
The cardinal successor o f K Class of l i m i t c a r d i n a l s Class of strong l i m i t c a r d i n a l s Set of proper f i l t e r s over a Set of proper i d e a l s over a the dual f i l t e r ( i d e a l ) of t h e ideal (filter) x x i s K-saturated the s a t u r a t i o n o f x t h e ideal x i s K-complete t h e f i l t e r x i s K-complete x i s a maximal ideal ( f i l t e r )
334 335 336 338 338 338 340 340 340 340 341 34 1 341 34 1
The closed unbounded f i l t e r T i s a tree The height of x in T The r - l e v e l of T The height o f T T' i s a subtree of T B i s a path through T T i s a K-Aronszajn t r e e Class of u w i t h P u ' a model of M K T C
344 34 6
Class of weakly i n a c c e s s i b l e cardinal s
356
Class of i n a c c e s s i b l e c a r d i n a l s The fixed point of EX
356 359
The sequence [ R * { d : B E Y ] i s decreasing The sequence [ R*{al, a € Y I i s cont i nuousl y decreasing I n t e r s e c t i o n of operations
359
341 347 347 347 34 7 347 354
359 359
388
ROLAND0 CHUAQUI
FOG
Composition o f o p e r a t i o n s
359
Diagonal i n t e r s e c t i o n
359
D i a g o n a l i z a t i o n o f a sequence o f operations
359 359 360 360
Class of l i m i t s o f subsets o f X X i s closed i n
r
w i t h respec t o Y
X i s c l o s e d unbounded i n respect t o Y
r
with
361 362 362
The Mahlo o p e r a t i o n
363
Class o f weakly compact c a r d i n a l s
366
The weakly compact i d e a l
372
