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
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