Menzel OnTheIterativeExplanationOfTheParadoxes

7/28/2019 Menzel OnTheIterativeExplanationOfTheParadoxes

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C H R I S T O P H E R M E N Z E L

O N T H E I T E R A T I V E E X P L A N A T I O N O F T H E P A R A D O X E S

(Received 20 March, 1985)

I N T R O D U C T I O NT he r e i s a p i e c e o f c u r r e n t m a t he m a t i c a l l o r e I w o u l d l i ke t o c ha l le nge . A s t hes t o r y goe s , t he s ou r c e o f t he pa r a d oxe s o f na i ve s e t t he o r y l ie s i n a c on f l a t i ono f t w o d i s ti nc t c on c e p t i o ns o f s e t : t he s o -c aU e d i t e r a ti ve , o r m a t he m a t i c a l ,c onc e p t i on , a nd t he F r e ge a n , o r l og i c a l , c onc e p t i on . W hi l e t he l a t t e r c onc e p -t i on i s i nc ons i s t e n t , t he f o r m e r , a s G 6de l no t e s , " ha s ne ve r l e d t o a ny a n t i n -o m y w h a t s o e v e r " . 1 M o r e i m p o r t a n t ( f o r p h i l o s o p h e r s a t l e as t ), th e i te r a ti v ec o n c e p t i o n explains t h e p a r a d o x e s b y s h o w i n g p r e c is e ly w h e r e t h e F r e g e a nc onc e p t i on goe s w r ong , i n pa r t i c u l a r by p r ov i d i ng t he g r ounds f o r a d i s t i nc -t i on una va i l a b le t o t he F r e ge a n be t w e e n s e t s on t he one ha nd a nd c o l l e c t ions ,k n o w n a s p r o p e r c l as se s, w h i c h a r e " t o o b i g " t o b e s e ts o n t h e o t h e r . F u r t h e r ,s inc e t he i t e r a ti ve c on c e p t i on i s i n a p r e c i se s e nse i m p l i c i t in t he a x i om s o fZFC, t he s e f a c t s j u s t i f y t he o t he r w i s e a d ho c r e s t r i c t i ons Z e r m e l o p l a c e d ont he i nc o ns i s t e n t p r i nc i p l e s o f nai ve s et t he o r y .

I f u ll y a g re e w i t h o u r s t o r y ' s d i st i n c ti o n b e t w e e n t w o c o n c e p t i o n s o f s e t,bu t m y a g r e e m e n t e nds t he r e . T o t he c on t r a r y , I w ou l d l i ke t o ar gue f i rs tt ha t , m od u l o c e r t a i n h i gh l y p l a us ib l e a s s um pt i on s , t he i t e ra t i ve c on c e p t i o no f s e t d o e s n o t p r o v i d e a n e x p l a n a t i o n o f a l l t h e s e t t h e o r e t i c p a r a d o x e s ,a n d c o n s e q u e n t l y t h a t Z e r m e l o ' s r e s tr i c ti o n s o n t h e p r in c i p le s o f n a iv e s e tt he o r y l o se m uc h o f t he i r j u s t i f i c a ti on . I w i l l t he n c l a im i n li gh t o f the s e a r gu -m e n t s t h a t w e n e e d t o r e c o n s i d e r t h e d i s t i n c t i o n b e t w e e n s e t s a n d p r o p e rc la s se s r a t he r m or e c a r e f u l l y . T he r e s u l t o f t h is r e c ons i de r a t i on w i ll be t ha tZ F C doe s no t c a p t u r e t he i t e r a t i ve c onc e p t i on i n i t s f u l l ge ne r a l i t y . I w i l lc o n c l u d e b y o f f e r in g a m o r e g e n e ra l t h e o r y o f s e ts th a t d o e s .

P AR AD OXES AND THE I TER A TI VE C ONC EP TI ON OF SETT he c ha r a c t e r i s ti c p r i nc i p le o f the F r e ge a n c on c e p t i on o f s e t , t he s o -c a l le dPhilosophical Studies49 (1986) 37-61.9 1986 by D. Reidel Publishing Company


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3 8 C H R I S T O P H E R M E N Z E L

n a iv e c o m p r e h e n s i o n a x i o m , is t h a t( P ) E v e r y p r o p e r t y d e t e r m i n e s a s et .

T h o u g h e s s e n ti a l t o m u c h e a r l y s e t t h e o r e t i c r e a s o n i n g , P , a s R u s s e ll di s-c o v e r e d , i s i n c o n s i s t e n t . T h e t w o e a r l i e s t p a r a d o x e s w e r e t h e m o s t s i g n i f i c a n t .T h e f ir s t, t h e B u r a l i - F o r t i ( B F ) p a r a d o x , r u n s a s f o l lo w s . I s o m o r p h i c o r d e r e ds e ts s h ar e w h a t C a n t o r c a ll ed a n " o r d e r t y p e " . T h e o r d e r t y p e s o f w e l l -o r d e r e ds e t s h e c a l l e d o r d i n a l n u m b e r s . N o w , i t i s q u i t e e a s y t o s h o w t h a t

( 1 ) T h e o r d e r t y p e o f t h e s e t o f a ll o r d in a l s ( o r d e r e d a c c o r d i n g t om a g n i t u d e ) l e ss t h a n a n y g i v e n o r d i n a l a is a i ts e l f,

a n d t h a t(2 ) A w e l l - o r d e r e d s e t is n o t i s o m o r p h i c t o a n y o f i ts p r o p e r i n i ti a l

s e g m e n t s .C o n s i d e r t h e n t h e s e t O n o f a l l o r d i n a ls ( w h i c h e x i s ts b y P ) . S in c e O n i s w e l l-o r d e r e d b y t h e le ss th a n r e l a ti o n < ( a s C a n t o r s h o w e d ) , it t o o m u s t h a v e a no r d e r t y p e / 3 . B y ( 1 ) / 3 is a ls o t h e o r d e r t y p e o f t h e s e t A o f o r d i n a l s l es s t h a n/3. S i n c e A a n d O n h a v e t h e s a m e o r d e r t y p e , i t f o l l o w s b y d e f i n i t i o n t h a t t h e ya r e i s o m o r p h i c . T h u s , s i n c e A is a n i n it ia l s e g m e n t o f O n , b y ( 2 ) i t m u s t b et h a t A = O n . S i n c e O n is t h e s e t of all o r d i n a l s , t3 E O n . T h u s , ~ E A . B u t t h e nb y d e f i n i ti o n o f A w e h a v e /3 < / 3 , a m a n i f e s t r e p u g n a n c y t o b e s u r e.

T h e s e c o n d p a r a d o x is o f c o u r s e R u s s e l l 's . S o m e s e ts , l ik e th e s e t o f a lla b s t r a c t e n t i ti e s , a p p e a r t o b e s e l f m e m b e r e d , a n d o t h e r s , l ik e t h e s e t o f a llp r i m e n u m b e r s , d o n o t . W h a t t h e n o f th e s e t o f all n o n s e l f m e m b e r e d s e ts( w h i c h , a g a i n , e x is t s b y P ) ? I s i t s e l f m e m b e r e d ? I t is e a s y t o s e e t h a t i t is i fa n d o n l y i f i t i s n ' t , a c o n t r a d i c t i o n .

R e s p o n s e s t o t h e p a r a d o x e s r a n g e d f r o m i n d i ff e r e n c e t o d e s p a ir . O f t h o s ew h i c h f el l s o m e w h e r e i n b e t w e e n t h e se t w o e x t r e m e s , Z e r m e l o ' s h a s h a d t h em o s t p r o f o u n d i m p a c t . T o a ll a p p e a r a n c es , Z e r m e l o ' s r e s p o n se t o th e p ar a -d o x e s w a s u n a b a s h e d l y p r a g m a t i c . W h i le o t h e r s , n o t a b l y R u s se ll , m i g h t h a v et r i e d t o solve t h e p a r a d o x e s , Z e r m e l o , i t s e em s , s o u g h t m e r e l y t o avoid t h e mw i t h a m i n i m u m o f in c o n v e n i e n c e . I n h i s w o r d s , 2There is at this po int noth ing for us to do bu t . . . , s tart ing from set theory as i t is histori-cal ly given , to see k out th e principles required for establishing the fou nda tions o f thismathem atical discipline. In solving the problem we must, on the on e hand, restrict theprinciples sufficiently to exclude all contradiction and, on the oth er, take them sufficient-ly w ide to retain al l tha t is valuable in this theo ry.T h e " p r i n c i p l e s " Z e r m e l o s p e a k s o f h e re a r e o f c o u r s e h is se t t h e o r e t i c


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IT E R A T IV E E X P L A N A T IO N O F T H E P A R A D O X E S 3 9ax ioms ; i n pa r t i c u l a r, e i gh t o f t he t e n a x i om s w e kno w c o l l e c ti ve l y t od a y a s( s e c o n d - o r d e r) Z F C . 3

Z e r m e l o w a s u n a b l e t o p r o v e t h e c o n s i s te n c y o f h is a x i o m s , f o r g o o dr e a s ons G 6d e l w a s t o p r ov i de s o m e t w e n t y ye a r s la t e r. H ow e ve r , he w a s a b l et o s h o w t h a t t h e k n o w n p a r a d o x e s o f t h e d a y , t h e t w o a b o v e i n p a r t i c u l a r ,w e r e b l o c k e d . H o w w a s t h i s d o n e ? A s w e s a w a b o v e , t h e B F p a r a d o x a n dRus s e l l ' s pa r a dox i nvo l ve p r i nc i p l e P . T hus , i t i s a s s um e d i n t he BF pa r a doxtha t a ll t he ord ina l s ( i . e . , a l l t he th ings wi th the pro pe r ty being an ordinal)c on s t i t u t e a s e t , a nd s i m i la r l y i n Rus s e l l' s pa r a do x t ha t a l l no ns e l f m e m be r e ds e t s c ons t i t u t e a s e t . Z e r m e l o ' s i de a w a s t o r e s t r i c t P s uc h t ha t one c ou l d no ts i m p l y ge ne r a t e s e t s v i a p r ope r t i e s e x n i h i l o , a s i t w e r e , bu t on l y f r om pre-viously given sets. A c c or d i ng l y , he r e p l a c e d P w i t h h is f a m o us Aussonderungs-a x i o m , o r a x i o m o f s e p a r a ti o n :

( 3 ) V P V z 3y V x ( x E y - x E z & P x ) . 4S i nc e ( 3 ) on l y gua r a n t e e d t he e x i s t e nc e o f s ubs e t s o f p r e v i ous l y g i ve n s e ts ,and s ince i t appeared imposs ib le to genera te s e t s which inc luded a l l t heo r d in a ls o r a ll t h e n o n s e l f m e m b e r e d s e t s f r o m t h e r e m a i n in g a x i o m s , p r e m i s e se s se n ti al to t h e p a r a d o x e s w e r e b l o c k e d , a n d t h e p a r a d o x e s t h e m s e lv e s w e r et hus e f f e c t i ve l y a vo i de d .

N o w , f r o m a p h i l o s o p h i c a l p o i n t o f v ie w , Z e r m e l o ' s r e s p o n se s o f a r isde e p l y uns a t i s f y i ng . F o r g i ve n t he i n t u i t i ve f o r c e o f p r i nc i p l e P , i t s i nc on -s i s te nc y is ge nu i ne l y pa r a d ox i c a l , a ge nu i ne a n t i no m y o f r e a s on . Z e r m e l o ' sa p p r o a c h , h o w e v e r , o f f e r s n o ]ustification o f t h e r e s t ri c ti o n s i m p o s e d u p o n Po t he r t ha n t he f a c t t ha t t he pa r a dox e s a re a vo i de d . Bu t t ha t i s a d hoc . W ha tw e w o u l d l ike i s s om e s o r t o f e xp l a na t i on o f w h y t he re i s no Russe l l s e t o r nos e t o f a l l o r d i na l s , o r w hy , a t l e a s t , w e s hou l dn ' t be a b l e t o p r ove t he r e a r es u c h s e ts f r o m o u r a x i o m s . F r o m a p r a g m a t i st li ke Z e r m e l o s u ch a c o m p l a i n tw o u l d r e c e i ve l i tt l e s y m pa t hy ; t he pu r p os e o f h i s a x i om s , as is e v i de n t f r o mt he q uo t e a bove , w a s t o e na b l e m a t he m a t i c i a ns t o ge t on w i t h the i r bus i nes sw i t h ou t ha v i ng t o bo t he r w i t h s uc h que s t i ons . I r on i c a l l y , i t w a s Z e r m e l oh i m s e l f w h o e n d e d u p p r o v i d i n g w h a t m a n y p h i l o so p h e r s a n d p h i l o s o p h ic a l -l y i n cl in e d m a t h e m a t i c i a n s t a k e t o b e j u s t t h e e x p l a n a t i o n t h a t w a s n e e d e dt o j u s t i f y h i s a x i om s .

A f t e r c o m p l e t in g h i s sy s t e m o f a x i o m s w i t h t h e a d d i ti o n o f t h e a x io m so f r e p l a c e m e n t a n d f o u n d a t i o n , Z e r m e l o b e g a n l o o k i n g in t o it s m o d e l t h e o r y , sW h a t h e f o u n d w a s t h a t t h e c la ss o f m o d e l s o f h i s ( s e c o n d - o r d e r) t h e o r y


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4 0 C H R IS T O P H E R M E N Z E Le m b o d y w h a t h a s c o m e t o b e k n o w n a s t h e " i te r a ti v e c o n c e p t i o n o f s e t " .T he i n t u i t i on be h i nd t he i t e ra t i ve c o nc e p t i on is t ha t t he s e t s a r e j u s t t hos ec o l l e c t i o n s w h i c h c a n b e " b u i l t u p " i n s t a g e s b y t h e r e p e a t e d a p p l i c a t i o n o fc e r t a i n " s e t - bu i l d i ng" o pe r a t i on s f r o m a n i n it ia l c o l l e c t ion o f urelements,i . e . , t h ings which a re no t themse lves s e t s l i ke t ab les , p l ane t s , p roper t i e s , e t c .S o m e w h a t m o r e p r e c i se l y , w e b e g in a t t h e f ir s t le v e l w i t h o u r ( p o s s ib l y e m p t y )c o l l e c t i on H o o f u r e l e m e n t s . T he n e x t l e ve l H 1 c ons is t s o f t he u r e l e m e n t st o g e t h e r w i t h a ll t h e c o l le c t io n s th a t c a n b e f o r m e d o u t o f th e m , i. e. , H I =H o u ~ ( H o ) . L i ke w i s e t he t h i r d l e vel H 2 c ons i s t s o f t he ob j e c t s i n Hx t o -g e t h e r w i t h a ll t h e c o l l e c t io n s t h a t c a n b e f o r m e d o u t o f t hem , i .e . , H2 = H~ u

( H 1 ) , a nd s o on f o r t he s uc c e e d ing le ve ls . T he p i c t u r e t h a t e m e r ge s is t huso f a h i e r a r c hy o f l eve ls w h i c h a r e cumulative ; each l eve l con ta ins a l l t he ob jec t si n t he p r e v i ous l eve ls p l u s a ll t he ne w c o l l e c t ions w h i c h a r e c om po s e d o f t hos eob j e c t s ; i n Z e r m e l o ' s t e r m s , e a c h l e ve l ( s ave t he f i r s t) is " r o o t e d " i n t heprev ious l eve l s .6 A s C a n t o r d i sc o v e r ed , th e o r d i n a l n u m b e r s y s t e m e x t e n d si n t o t he t r a ns f i n i te a nd he nc e s o al s o , a l ong w i t h i t , doe s t he c u m u l a t i veh i e r a r c h y . L e t t i n g H x = u { H a I a < X} fo r l imi t o rd ina l s X , the h ie ra rchyc on t i nue s on i n t o t h e t r a ns f i n i t e , w i t h a le ve l H c o r r e s po nd i ng t o e a c h o r d i na la .T T h i s m or e p r e c i s e p i c t u r e , t he n , e na b l e s u s t o f o r m u l a t e t he a bove i n t u i -t i on c onc e r n i ng w h i c h c o l l e c t i ons a r e s e t s r a t he r m or e r i go r ous l y : a c o l l e c -t i on i s a s e t i f a nd o n l y i f it a ppe a r s a t s om e l eve l o f s om e c um u l a t i ve h ie r -a rch y .a

N o w , h o w d o e s t h i s c o n c e p t i o n o f th e s et t h e o r e t i c u n iv e r se ex p l a in t h ep a r a d o x e s a n d j u s t i f y Z e r m e l o ' s a x i o m s , e s p e c i a l l y s e p a r a t i o n ? T h e a n s w e ris : by expla in ing ho w i t i s t h a t p r inc ip le P i s fa l se . To see w hy c lea r ly , l e t usi n t r o d u c e t h e n o t i o n o f rank. F or m a l l y , t he r a nk o f a n ob j e c t i s de f i ne dr e c u r s i ve l y t o be t he l e a s t o r d i na l g r e a t e r t ha n t he r a nks o f a l l i t s m e m be r s .T h u s , t h e r a n k o f a n u r e l e m e n t i s 0 ; t h e r a n k o f a co l l ec t io n o f u re l e m e n t si s 1 ; t he r a nk o f a c o l l e c t i on c on t a i n i ng , s a y , a n u r e l e m e n t a nd a c o l l e c t iono f r a n k 1 is 2 ; t h e r a n k o f a c o ll e c ti o n c o n t a in i n g m e m b e r s o f e v er y f i n it er a nk i s 6o; a nd s o on . N o w , it i s e a s y t o s how t ha t t he r a nk p ( x ) o f a c o l le c -t ion x i s 0t i f and on ly i f x f i r s t appe ars in the (0 t + 1 ) th l eve l (~ th l eve l, fo r

< 6o) o f a n y h i e r a r c hy w h i c h c on t a i n s x . T hus , t he de f i n i t i on o f 's e t ' a bovei s equiva len t to the fo l lowing: a co l l ec t ion x is a s e t i f and on ly i f i t has ar a nk . T he r e a s on w hy P i s f a l s e i n ge ne r a l c a n now be s t a t e d s uc c i nc t l y :s o m e p r o p e r t i e s d e t e r m i n e c o l le c t io n s w h i c h h a v e n o r a n k , a n d h e n c e c an -no t be se t s . 9


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I T E R A T I V E E X P L A N A T I O N O F T H E P A R A D O X E S 4 1L e t ' s f le s h th i s o u t a b i t . F r o m t h e i t e r a ti v e c o n c e p t i o n w e ' v e d e r i v e d t h e

i d e a t h a t a c o l l e c t i o n (i .e . , a n o n u r e l e m e n t ) is a s e t i f a n d o n l y i f i t h a s a ra n k .W h a t w o u l d i t m e a n f o r a c o l l e c t i o n n o t t o h a v e a r a n k , o r a s I s h a l l p u t i t ,t o b e u n b o u n d e d ? T h is w o u l d b e t h e c a se j u s t i f a c o ll e c t io n C c o n t a i n e dm e m b e r s o f a r b i t ra r il y h i g h r a n k , i . e. , i f t h e r e w e r e n o o r d i n a l n u m b e r a s u c ht h a t f o r a ll x E C , a > O ( x ) . I f w e a m e n d p r i n c i p l e P t o s a y t h a t e v e r y p r o p e r -t y d e t e r m i n e s a c o l le c t io n , 1 ~ a n d le av e it a n o p e n q u e s t i o n w h e t h e r o r n o t t h ec o l l e c t i o n it d e t e r m i n e s i s a s e t, t h e n i t i s e a s y t o s e e t h a t t h e r e a r e s u c h c o l l e c -t io n s . F u r t h e r m o r e , s u c h c o ll e c ti o n s ar e i n t i m a t e l y c o n n e c t e d w i t h t h ep a r a d o x e s . C o n s i d e r f o r e x a m p l e t h e c o l le c t i o n d e t e r m i n e d b y t h e p r o p e r t yo f b e i n g a n o n s e l f m e m b e r e d s e t. O n t h e i te r a t iv e p i c t u r e , i t is c l e a r t h a t a l ls e ts a r e n o n s e l f m e m b e r e d , a n d h e n c e t h a t R u s s e ll 's p r o p e r t y i s c o e x t e n s iv ew i t h t h e p r o p e r t y o f b e i n g a s et . S i n ce , o b v i o u s l y e n o u g h , t h e r e a re s e ts o fe v e r y r a n k , i t f o l l o w s t h a t t h e c o l l e c t i o n o f a ll n o n s e l f m e m b e r e d s e ts is n o ti t s e l f a se t . T h e s o u r c e o f R u s s e ll 's p a r a d o x , t h e n , l ie s i n a n u n c l a r i t y a b o u tt h e c o n s t i t u t i o n o f th e s e t t h e o r e t i c u n i v e r s e . O n c e t h i s i s o v e r c o m e , i t isa p p a r e n t t h a t t h e r e c a n b e n o s u c h t h i n g a s t h e R u s s el l s e t a n d t h e p a r a d o xd i sa p p e a rs . C o n s i d e r n e x t t h e p r o p e r t y o f b e in g a n o r d in a l , w h e r e w e t a k et h e o r d i n a l s to b e s e ts g la v o n N e u m a n n . S i n c e p ( a ) = a , f o r a l l o r d i n a l s a ,i t i s i m m e d i a t e t h a t t h e c o l l e c t i o n o f all o r d in a l s is u n b o u n d e d ; i t t o o i s n o t as e t. S o t h e s o u r c e o f t h e B u r a l i - F o r t i p a r a d o x a p p e a r s t o b e id e n t ic a l w i t ht h a t o f R u s s e l l 's . G i v e n t h e it e r a ti v e c o n c e p t i o n t h e n , a n d t h e r e s u lt i n gd i s t i n c t i o n b e t w e e n s e t s a n d u n b o u n d e d c o l l e c t i o n s , t h e p a r a d o x e s a r e e x -p l a in e d a n d Z e r m e l o ' s t h e o r y , t a k e n f o r w h a t i t is, v i z . , a t h e o r y o f s e t s ,a p p e a r s t h e r e b y t o b e j u s ti f ie d .

REAL ORDINALS AND THE BURALI--FORTI PARADOX

B u t is it ? W e n e e d t o l o o k a l it tl e m o r e c l o s e l y . T h e e x p l a n a t i o n o f th e B Fp a r a d o x s m u g g l e d in t h e a s s u m p t i o n t h a t t h e o r d i n a l s a r e s e ts . F o r a ll m a t h e -m a t i c a l p u r p o s e s , o f c o u r s e , th i s a s s u m p t i o n m a k e s f o r a c o n v e n i e n t , in d e e de l e g a n t , t h e o r y . B u t a r e t h e o r d i n a l s r e a ll y se t s? A n u m b e r o f c o n s i d e r a t i o n sw o u l d i n d i c a te t h a t t h e y a r e n o t . T o b e g i n w i t h , as N i c o l a s G o o d m a n h a sr e c e n t l y a r g u e d , i n t u i t i v e l y , i t j u s t s e e m s s t r a i g h t f o r w a r d l y f a l se t o s a y t h a tt h e o r d i n a l s , a n d i n g e n e r a l m o s t m a t h e m a t i c a l o b j e c t s , a r e s e ts ; a n o r d i n a ln u m b e r is s im p l y " n o t a tr a n si ti v e s e t l in e a r ly o r d e r e d b y t h e m e m b e r s h i pr e l a t i o n " ) 1 T h e s e i n t u i ti o n s a re s t r e n g h t e n e d w h e n w e r e f le c t u p o n t h e w e l l


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42 CHRI STO PHER MENZEL

k n o w n p r o b l e m s o f i d e n t if y i n g o r d in a ls w i t h s et s ra is e d b y B e n a c e r r a f i n h i snow c la s si c pa pe r . 12 A s Be na c e r r a f f i r s t po i n t e d ou t , t he r e a r e i n f i n i te l ym a n y w e l l -o r d e r ed c o l le c t io n s o f s e ts w h i c h w e c o u l d i d e n t i f y w i t h t h e o r d i n a ln u m b e r s e q u e n c e , b u t n o a r g u m e n t s t o s h o w t h a t a n y p a rt i cu l a r c o l le c t io nr a t he r t ha n a ny o t h e r is t he ge nu i ne a r ti c le . H ow e ve r , Be n a c e r r a f c l a im s , i ft he nu m be r s e que nc e r e a l ly is som e pa r t i c u l a r c o l l e c t i on o f s e t s , t he r e ough tt o b e s u c h a r g u m e n t s . B u t t h e r e a r e n ' t , s o i t i s n ' t . N o w o n e m i g h t m a i n t a i ns t e a d f a s t l y t ha t none t he l e s s s o m e p a r t i c u l a r c o l l e c t ion j u s t is t h e n u m b e rs e q u e n c e , a r g u m e n t s o r n o t . T h i s o p t i o n , h o w e v e r , w h i c h f o r B e n a c e r r a f" bo r de r s o n t he a bs u r d" , 13 on l y h i gh li gh ts t he ( o r a t l e a s t, a m a j o r ) po i n to f th e a r g u m e n t , v i z . , t h e t e n u o u s n e s s o f a n y p h i l o s o p h i c a l c o m m i t m e n t t ot he s e t / num be r i de n t i f i c a t i on : s i nc e t he r e a r e no a r gum e n t s a va i l a b l e f o rs ing li ng ou t o ne pa r t i c u l a r c o l l e c t i on ove r a ny o t h e r , t he t r ue be l i e ve r 's c ho i c eis no m or e t ha n a n a c t o f f a it h , a g r ound l e s s l e a p ; h i s c o m m i t m e n t is e s s en t ia l -ly un jus t i f i ab le f 4

L e t ' s m a ke t h i s c l e a re r s ti ll . A s no t e d a bove , t he no t i on o f a n o r d i na l i s as p e c i f i c i n s t a n c e o f t h e m o r e g e n e r a l c o n c e p t o f a n o r d e r t y p e . N o w , f o r a l lc u r r e n t m a t h e m a t i c a l p u r p o s e s t h e r e is b u t o n e a x i o m w h i c h g o v e r n s t h ee x i s te n c e a n d p r o p e r t i e s o f o r d e r t y p e s , v i z . ,

( O T ) V x V y ( O T ( x ) = O T ( y ) - O R D E R E D ( x ) & x ~- y )i .e . , t w o o r de r e d s e ts ha ve t he s a m e o r de r t yp e i f a nd on l y i f t he y a r e i so -m or p h i c . I n t he s pe c i a l c a se o f w e l l -o r de r e d s e t s , t h i s be c om e s

( O R D ) V x V y ( O R D ( x ) = O R D ( y ) -= W O S E T ( x ) & x ~ y ) ,w he r e O R D ( x) i s t he w e l l - o r de r t ype , o r o r d ina l , o f t he w e l l -o r de r e d s et x .F r o m a m o n g t h e m u l t i tu d e o f c o l le c t io n s o f se ts w h i c h s a t is f y O R D , it h a sp r ov e d u s e f u l s i m p l y t o single ou t a pa r t i c u l a r c o l l e c t i on o f s e ts w h i c h a r et he m s e l ve s w e l l - o r de r e d t o s e r ve a s t he o r d i na l num be r s : f o r e a c h w e l l - o r de rt y p e a w e c a n s i m p l y c h o o s e a s et o f t h a t o r d e r t y p e t o b e t h e o rd i n a l a ;s uc h a re t he vo n N e u m a nn o r d i na ls . Bu t de s p i t e the c o nve n i e nc e o f t hea p p r o a c h , t h e r e i s s i m p l y n o a r g u m e n t , n o me t a p h y s i c a l j us t i f i ca t ion , ava i l -a b l e f o r ta k i ng t h i s c o l l e c ti on , r a t he r t h a n a ny o t he r w h i c h s a t is f ie s O R D ,a c t ua l l y t o b e t he ord ina l s . Thi s i s Benacer ra f ' s cha l l enge .

S t r i c t l y s pe a k i ng , o f c ou r s e , Be na c e r r a f ' s c ha l l e nge doe s n ' t r e f u t e t h et he s is t ha t t he o r d i na ls a r e s e t s; a s no t e d , one c a n c ons i s t e n t l y m a i n t a i nthe thes i s desp i t e i t s un jus t i f i ab i l i ty . But the cha l l enge ra i s es s e r ious doubt s ;


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IT E R A T IV E E X P L A N A T IO N O F T H E P A R A D O X E S 4 3

i ts u n a n s w e r a b i l i t y s e e m s t o c o n f i r m , a s o u r i n t u i t io n s f ir s t w a r n e d , t h a t t h et h e s i s i s s e r i o u s l y m i s g u i d e d .

W h a t t h e n a r e th e o r d i n a ls ? A l t h o u g h f o r p u r p o s e s h e re w e n e e d n ' t a d d r es st h is q u e s t i o n ( w e n e e d o n l y c a st d o u b t u p o n t h e t h e si s t h a t t h e y a re se ts ),t h e r e is a n a n s w e r s u g g e s t e d i n t h e w r i ti n g s o f C a n t o r a n d R u s s e ll w h i c hs e e m s t o m e t o b e d e a d r ig h t . W h a t , e x a c t l y d o e s O R D s a y ? I n es se n c e, th a ta n o r d i n a l is s o m e t h i n g i s o m o r p h i c w e l l - o r d e r e d s e ts s h a re i n c o m m o n . I n -t u it iv e l y , t h o u g h , w h a t t h e y h a v e i n c o m m o n c a n n o t b e a s e t, f o r w h a t t h e ys h a r e i s a c o m m o n s t r u c t u r e , i .e . , a n a b s t r a c t u n i v e r s a l , a p r o p e r t y ; i s a se t ,b e i n g a p a r ti c u la r , i s j u s t n o t t h e s o r t o f t h i n g w h i c h c a n b e h a d i n c o m m o n i nt h i s s e n s e. W h a t t h e n a r e th e o r d i n a ls ? T h e y a re p r o p e r t i e s o f w e l l - o r d e r e dsets. 16

N o w , t h e i n t u i t i v e d o u b t s , B e n a c e r r a f ' s c h a l l e n g e , a n d t h e p l a u s i b i l i t y o ft h e C a n t o r - R u s s e l l v i ew a ll m a k e f o r a p o w e r f u l c a s e a g a in s t t h e t h e si s t h a tt h e o r d i n a l s a r e s e t s. S i n c e , a s w e s a w , th i s t h e s i s is e s s e n ti a l t o t h e e x p l a n a -t i o n o f t h e B F p a r a d o x , t h a t e x p l a n a t i o n i s n o w i t s e lf d u b i o u s . T o s e e t h ism o r e c l e a rl y , r e c al l t h a t t h e i n t u i t i o n b e h i n d t h e i t e r a ti v e p i c t u r e is t h a t w es t ar t w i t h a n a r b i t ra r y c o l l e c t io n o f o b j e c ts a n d " b u i l d u p " b y m e a n s o fc e r t a in s e t- b u il d in g o p e r a t i o n s . T h e p a r a d o x e s w e r e a v o i d e d , a n d Z e r m e l o ' sa x i o m s j u s ti f ie d , b y n o t i n g t h a t s u c h o p e r a t i o n s c o u l d n e v e r y i e l d t h e R u s s e lls e t n o r a s e t o f a l l o r d i n a l s, w h e r e t h e o r d i n a l s w e r e c o n s t r u e d a s s e ts . B u t i fw e r e j e c t t h is c o n s t r u a l , a s w e s e e m c o m p e l l e d t o d o , t h e s t r u c t u r e o f t h eu n i v e r se o f s e ts , w h i le s ti ll e x p l a i n i n g R u s s e l l' s p a r a d o x , n o l o n g e r p r o v i d e su s w i t h a n e x p l a n a t i o n o f t h e B F p a r a d o x ; f o r o n th e it e ra t iv e c o n c e p t i o ns i m p l ic i te r , t h e r e is n o o b v i o u s r e a s o n w h y t h e r e c o u l d b e n o s e t o f a ll o r d i n a l s :s in c e t h e o r d i n a l s a re n o t c o l l e c t i o n s o f a n y s o r t t h e y h a v e a r a n k o f 0 , a n d s ot h e c o l l e c ti o n o f t h e m h a s a ra n k o f 1 a n d is t h e r e f o r e , b y t h e d e f i n i ti o na b o v e , a s e t .

T h e c o n s e q u e n c e f o r Z e r m e l o , o f c o u r s e , is t h a t t h e a x i o m o f s e p a r a ti o no n c e a g ai n a p p e a rs a d h o c ; t h e B F p a r a d o x is a v o id e d s i m p l y b y e li m i n a ti n gt h e p o s s i b i l it y o f p r o v i n g t h e r e is a se t o f all o r d i n a l s in s p i te o f th e f a c t t h a t ,f a r f r o m j u s ti f y i n g s u c h a m o v e , t h e i t e ra t iv e c o n c e p t i o n p r o v i d e s p o s it iv er e a s o n f o r t h i n k i n g t h e r e is s u c h a s e t . W e a r e o n c e a g a i n l e ft w i t h a p h i l o s o p h i -c a l l y u n s a t i s f y i n g s i t u a t io n .


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4 4 C H R IS T O P H E R M E N Z E L

T H E C A N T O R IA N W A Y O U T

L e t ' s t a k e a s te p b a c k a n d m a k e s o m e m o r e g e n e r a l o b s e r v a t io n s . F ir s t o f all,t h e a b o v e s e e m s t o c a l l f o r a m o r e c a r e f u l c h a r a c t e r i z a t i o n o f t h e d i s t i n c t i o nb e t w e e n s e ts a n d p r o p e r c l as se s. I n Z F C ( i. e. , Z F C s u p p l e m e n t e d w i t h c la s st e r m s ) a ll a n d o n l y p r o p e r c la ss es a re u n b o u n d e d . G i v e n t h e a b o v e , t h is is a si t s h o u l d b e ; u n b o u n d e d n e s s i s t h e e s s e n t ia l d is t in g u i s h i n g f e a t u r e o f a p r o p e rc la s s. B u t p r o p e r c la s se s a l so h a v e a f u r t h e r p r o p e r t y i n Z F C : t h e y a r e i n ap r e c i se s en s e " t o o b i g " t o b e s e ts . T o s p e ll t h is s e n s e o u t c l e a r ly , l e t u s a s s u m e ,a s w e s h a ll p o s t u l a t e e x p l i c i t l y b e l o w , t h a t t h e r e a r e j u s t a s m a n y r e a l o r d i n a l sa s y o n N e u m a n n o r d in a l s, a n d f o r c o n v e n i e n c e t h a t c a rd i n a l n u m b e r s a r e as p e c ie s o f o r d i n a l n u m b e r . 1 7 I n l i g h t o f t h i s , l e t u s s a y t h a t a c o l l e c t i o n has adef ini te cardinal i ty f f a n d o n l y i f i t is e q u in u m e r o u s t o a v o n N e u m a n no r d i n al . N o w , i n Z F C ( w i t h o r w i t h o u t u r e l e m e n t s ) , e v e r y s et h as a d e f in i tec a r d i n a l it y , a n d i t is e a sy t o p r o v e ( t h o u g h , o f co u r s e , n o t in Z F C p r o p e r )t h a t n o p r o p e r c la ss d o e s; n o v o n N e u m a n n o r d i n a l c a n b e m a p p e d o n t o ap r o p e r c la ss . W h e r e Z F C is o n e ' s t h e o r y o f s e ts , th e n , a ll a n d o n l y p r o p e rc la s se s a re t o o b i g t o h a v e a d e f i n i te c a r d i n a l i t y a n d h e n c e p r o p e r c la s se s a r et o o b i g t o b e s e t s. T h e r e a r e t h u s t w o e q u i v a l e n t w a y s o f d i s ti n g u i s h in gp r o p e r c la s se s f r o m s e ts - C is a p r o p e r c l as s i f a n d o n l y i f C is u n b o u n d e di f a n d o n l y i f C is t o o b i g t o h a v e a d e f i n i te c a r d i n a l i t y .

F r o m o u r n e w p e r s p e c ti v e , h o w e v e r , t h e se t w o c o n d i t i o n s ar e i n g e n e r alb y n o m e a n s e q u i v a l e n t . F o r a ll t h e i t e ra t iv e c o n c e p t i o n s a y s , s iz e h a s n o t h i n gw h a t e v e r t o d o w i t h w h e t h e r o r n o t a g i v e n c o l l e c t i o n is a s e t; th e o n l yr e le v a n t f a c t o r is u n b o u n d e d n e s s . I n li g h t o f o u r r e a s o n i n g a b o u t t h e o r d in a l s,t h e i d e a t h a t t h e s e t w o c o n d i t i o n s d i v er g e is n o m e r e f l i g h t o f s p e c u l a ti v ef a n c y . T o t h e c o n t r a r y , t h a t r e a s o n in g , to g e t h e r w i t h t h e i t er a ti v e c o n c e p t i o n ,p o i n t s t o a u n i v e rs e w h o s e s e ts d i v id e i n t o t h r e e , r a t h e r t h a n t w o , si ze s :f i n it e , i n f in i t e , a n d i n C a n t o r ' s t e r m s , a b s o l u t e l y in f i n i te , o r " m a t h e m a t i c a l l yi n d e t e r m i n a b l e " , i .e . , t h e s iz e o f t h o s e s e ts w h i c h a r e t o o b i g t o h a v e a d e f i n i tec a r d i n a l i t y . I n p a r t i c u l a r, t h e u n i v e r s e c o n t a i n s a se t O n o f a ll o r d i n a l n u m -b e r s . I w i l l c a l l t h i s t h e " C a n t o r i a n u n i v e r s e " . 1 8

N o w , t a k i n g t h e C a n t o r i a n u n i ve r se t o b e th e o b j e c t o f o u r s e t t h e o r e t i cr e a s o n i n g , I c la i m , e n a b l e s u s t o r e m e d y t h e u n s a t i s f a c t o r y s i t u a t i o n w e w e r el e f t w i t h a t t h e e n d o f th e l a s t s e c t i o n . T h e r e w e r e t w o p r o b l e m s : t h e lo s s o fa n e x p l a n a t i o n o f t h e B F p a r a d o x in t e r m s o f t h e s t r u c tu r e o f t h e s e t t h e o r e t i c


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IT E R A T IV E E X P L A N A T IO N O F T H E P A R A D O X E S 4 5u n i ve r se , a n d t h e r e a p p e a r a n c e o f a n a d h o c c h a r a c t e r t o o u r ( a rg u a b l y ) b e s tt h e o r y o f s e ts . L e t ' s b eg i n w i t h t h e B F p a r a d o x .

T he Ca n t o r i a n un i ve r se g i ve s u s a n i l l um i na t ing , a nd t ong ne g l e c t e d ,pe r s pe c t i ve on t he B F pa r a do x w h i c h w a s i m p l i c i t i n the w r i ti ngs o f Ca n t o ra nd f i rs t e xp l i c it i n s o m e e a r l y pa pe r s o f J ou r d a i n . 19 Ra t he r t ha n de n y t hee x i st e n c e o f O n , w e d e n y t h a t O n i s m a t h e m a t i c a l l y d e t e r m i n a b l e a n d h e n c ede n y t h a t ( in i t s na t u r a l o r de r i ng < ) i t ha s a n o r de r t yp e , a nd a c r uc ia lp r e m i s e o f t h e a r g u m e n t i s b l o c k e d .

B u t is t h e r e r e a l l y m u c h p h i l o s o p h ic a l g a in h e r e ? T h e B F p a r a d o x a f f o r d st w o w a y s o u t . B o t h i n vo l ve d e n y i n g t h e e x i s te n c e o f c e r ta i n o b j e ct s w h o s ee x i s t e nc e o ne i s i n i ti a ll y i nc l i ne d t o a c c e p t . A r e n ' t w e l e f t i n p r e t t y m uc h t hes a m e pos i t i on no m a t t e r w h i c h w e o p t f o r ? I t h i nk i t is c l e a r w e ar e no t . T os ee w h y , l e t ' s f i r s t r e c ons t r uc t ou r p r o b l e m . O ur re a s on i ng a bove le d u s t ot h e c o n c l u s i o n t h a t

( 4 ) T he o r d i na l s a r e no t s e t s,f r o m w h i c h i t f o ll o w e d b y

( 5 ) T he i t e r a ti ve c on c e p t i ont ha t t he o r d i na l s c ons t i t u t e a se t O n . A t the s a m e t i m e , how e ve r , i t s e e m s t ha t

( 6 ) E v e r y w e l l - o r de r e d s e t ha s a n o r de r t yp e ,w h i c h i s i n c o n s i st e n t w i t h t h e c o n j u n c t i o n o f ( 4 ) a n d ( 5 ) , w h i c h a re b o t he s s e n ti a l t o t he Ca n t o r i a n un i ve r se . I w i l l t a ke ( 4 ) f o r g r a n t e d . T ha t l ea ve s u sw i t h ( 5 ) a nd ( 6 ) , a nd t he que s t i on t he n i s w h a t r e a s on t he r e is t o c hoos e oneove r t he o t he r . Bu t t he c ho i c e i s p l a i n . T he i t e r a t i ve c onc e p t i on i s t he c on -c e p t u a l b e d r o c k o n w h i c h c o n t e m p o r a r y s e t t h e o r y i s b u i l t . H e n c e , t o d e n yi ts ge ne ra l va l id i t y w ou l d be a d r a s ti c m ov e inde e d , one t o be un de r t a ke non l y f o r t he m os t p r e s si ng a nd c oge n t r e a s ons . D oe s t he p l a us i b il i ty o f ( 6 )p r ov i de s uc h r e a s ons ? I t doe s no t ; t o t he c on t r a r y , i n de ny i ng ( 6 ) w e g i ve upno t h i ng t ha t i s e s s e n ti a l t o t he t he o r y o f s e ts . L e t m e e l a b o r a t e .

O ne c a nn o t de n y t ha t t he f a l s it y o f ( 6 ) g i ve n ( 4 ) a nd ( 5 ) i s puz z l i ng ; it i s.Bu t w hy , e xa c t l y , i s i t ? O r d i na ls , a s w e ' ve s ee n , ar e ( o r a t le a s t a r e i n t i m a t e l yc on ne c t e d w i t h ) t he o r de r p r ope r t i e s , t he o r de r t ype s , o f w e l l - o r de r e d s e ts .W h a t t h e i n c o n s i s te n c y o f (6 ) w i t h r e s p e c t t o ( 4 ) a n d ( 5 ) s h o w s , th o u g h , i st ha t s om e s e t s , de s p i t e t he f a c t t ha t t he y a r e w e l l - o r de r e d by a pe r f e c t l y


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4 6 C H R IS T O P H E R M E N Z E Ld e f m i t e r e l a ti o n , n o n e t h e l e ss d o n ' t e x e m p l i f y a n o r d e r p r o p e r t y ; t h e re j u s tis n o s u c h p r o p e r t y . T o m a k e th e p o i n t m o r e e x p l ic i tl y , l e t r b e th e u s u alyo n N e um a n n o r d i na l , a nd c ons i de r t he s e t w ~ = 6o - { 9 ) - T h e w e l l - o r de r e dsets < w, E ~ 6o> a nd a r e i s o m o r ph i c und e r t he m a pp i ng w h i c ht a ke s e a c h e l e m e n t o f 6o t o i t s s uc c e s s o r i n ~ . T he s e t w o w e l l - o r de r e d se t se x e m p l i fy th e s am e o r d e r p r o p e r t y , viz . , t h e real or d i na l 6o . Cons i de r nowthe se t s a nd . These se t s to o a re i som orp hicunde r t he m a pp i ng w h i c h t a ke s e a c h f i n i t e o r d i na l i n O n t o i t s s uc c e s s o ra nd i s t he i de n t i t y m a p o t he r w i s e . N one t he l e s s , de s p i t e the s i m i l a ri ti e s t o t hep r e v i o u s c a s e , t h e r e e x i s t s n o o r d e r p r o p e r t y w h i c h t h e y b o t h e x e m p l i f y ;a nd t h a t i s puz z l i ng i nde e d .

Bu t i t i s a ls o i l lum i na t i ng . F o r ou r a pp r oa c h y i e ld s a n in s i gh t i n t o t het r u e n a t u r e o f t h e B F p a r a d o x : i t is n o t a p a r a d o x o f se t t h e o r y a t a ll , b u t o fp r o p e r t y t h e o r y . T h e r e a l p u z z l e d o e s n ' t h a v e a n y t h i n g t o d o w i t h t h e e xi st -e n c e o f se t s, b u t w i t h t h e e x i s t e n c e o f p r o p e r ti e s . C o m p a r e t h e p r o p e r t y f o r mo f R u s s e l l ' s p a r a d o x . L e t E X E M P b e t h e r e l a t i o n t h a t o b t a i n s b e t w e e n t w oob jec t s x and y jus t i n case x ex em pl i f i e s y , i .e . , j us t i n case y(x ) , and l e t Sb e t h e p r o p e r t y t h a t h o l d s o f x ju s t i n c a se E X E M P ( x , x ) . I t is r e a so n a b l e t ot h i n k t h a t e v e r y p r o p e r t y P h a s a c o m p l e m e n t 7 P , i .e ., t h a t p r o p e r t y a t h in gha s j u s t i n c a s e i t l a c ks P , a nd a l s o t ha t f o r a ny p r ope r t y P a nd ob j e c t xe i t h e r P x o r 7 P x . C o n s i d e r t h e n t h e c o m p l e m e n t o f S, a n d l e t u s a s k w h e t h e ro r n o t S ( 7 S ) . N e i t h e r a n s w e r w i ll so , o f c o u rs e , si nc e S ( 7 S ) = E X E M P ( T S ,7 S ) = 7 S ( 7 S ) . W e a r e t h u s f o rc e d t o d e n y t h e e x is te n c e o f a p r o p e r t y w h o s ee x i s te n c e s e e m s to f o l l o w f r o m p e r f e c t l y r e a so n a b l e a s s u m p t i o n s w h i c h , w h e nm a de i n o t he r c a s es , a re w ho l l y unp r ob l e m a t i c . A nd t h i s , i n es s e nc e, is j u s tt he puz z l e w e a r e f a c e d w i t h i n t he BF pa r a dox . O nc e a ga i n , t hough , i t i s ap u z z l e f o r p r o p e r t y t h e o r y . H e n c e , w e s h o u l d n ' t e x p e c t t he s e t t he o r e t i cun i ve r s e t o be a b l e t o p r ov i de a n e xp l a na t i on . A c c o r d i ng l y , t he Ca n t o r i a nun i ve r se j u s t a c c e p t s t h i s puz z l e a bo u t p r op e r t i e s a s a b r u t e f a c t : i t a l low s a llt he o r d i na l s t he r e a r e as i ts u r e l e m e n t s ; w h y t he r e a r e j u s t t hos e o r d i na lsi s non e o f i t s c onc e r n .

W h a t n o w o f o u r s e c o n d p r o b l e m , t h e a d h o c c h a r a c te r o f Z F C ? L e t ' sun pa c k t h i s a l it tl e m or e c a r e f u l l y . I n l igh t o f t he a bov e , t he p r o b l e m i s n ' ts o m u c h t h e a d h o c c h a r a c t e r o f t h e a x i o m s it se lf , b u t t h e way in w h i c ht he y a r e a d hoc . A ny s e t t he o r y i s go i ng t o m a ke c e r t a i n c ho i c e s i n o r de r t os k i r t t h e p a r a d o x e s t h a t a r e a r g u a b l y a d h o c i n s o m e r e s p e c t . T h e p r o b l e mw i t h Z F C i s t he pa r t i c u l a r c ho i c e i t m a ke s , a c ho i c e w h i c h pu t s i t i n d i r e c t


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IT E R A T IV E E X P L A N A T IO N O F T H E P A R A D O X E S 4 7c o n f l ic t w i t h i t s c o n c e p t u a l f o u n d a t i o n s . A s n o t e d , t h e a x i o m s in a c e r t ai ns en se e m b o d y t h e i te r a ti v e c o n c e p t i o n o f s et : th e n a t u r a l m o d e l s o f Z F C( w i t h o r w i t h ou t u r e l e m e n t s ) a r e c um u l a t i ve h i e r a r ch i e s . Y e t i n o r de r t oa vo i d t he BF pa r a do x t he a x i om s r u l e ou t t he pos s i b il i ty o f a s e t w ho s ee x i s t e nc e s e e m s t o follow f r o m t he i t e r a t ive c onc e p t i on . T hu s , de sp i t e t hef a c t t ha t i t e m bod i e s t he i t e r a t i ve c onc e p t i on , i t c a nno t c a p t u r e t he i t e r a t i vec onc e p t i on i n i t s f u l l ge ne r a l i t y . I n pa r t i c u l a r ( a s w e ' l l s e e e xp l i c i t l y be l ow ) ,Z F C i s no t t r ue i n t he Ca n t o r i a n un i ve r s e .

B u t o u r n e w p e r s p e c t i v e o n t h e B F p a r a d o x s u g g e s t s a w a y o u t w h i c hd o e s n ' t d o v i o l en c e t o s e t t h e o r y ' s c o n c e p t u a l u n d e r p i n n i n g s: b y i n c o r p o r a t in gt h e d i s ti n c ti o n b e t w e e n d e t e r m i n a b l e a n d i n d e t e rm i n a b l e s e t s, w e c a n r e m a i nf a i th f u l t o t h e i t e ra t iv e c o n c e p t i o n b y f o r m u l a t i n g a t h e o r y o f t h e C a n t o r i anun i ve r s e w h i c h r e t a i n s t he s p i r i t o f Z F C bu t i nc r e a s e s i t s pow e r . S o t ha t i sw h a t w e ' l l d o .

T H E F O R M A L T H E O R Y

O u r f ir s t t a s k is to g e t c le a r a b o u t h o w t o r e p r e s e n t t h e n o t i o n o f m a t h e -m a t i c a l de t e r m i na b i l i t y , i . e ., t he no t i on o f ha v i ng a de f i n i t e c a r d i na l it y a ndhe nc e ( w he n w e l l - o r de r e d ) a c l e f ' m i t e o r de r - t ype . F o r w e ne e d t o knowp r e c i se l y w h i c h s et s ar e d e t e r m i n a b l e i n o r d e r t o g e t a n i d e a o f h o w m a n yord ina l s the re a re , s ince i t i s on ly determinable w e l l - o r de r e d s e ts w h i c h ha veo r d e r t y p e s . 2 ~ F r o m t h e a b o v e w e h a v e t h a t a se t i s d e t e r m i n a b l e i f a n d o n l yi f i t i s e qu i num e r ou s t o s om e von N e um a nn o r d i na l. Bu t t h is is no t t oo r e ve al -i ng . W ha t ' s s o s pe c i a l , one m i gh t a s k , a bou t t he von N e um a nn o r d i na l s ? W hys h o u l d t h e y d e l im i t d e t e r m i n a b i l i ty ? W h y d o e s n ' t i n d e t e r m i n a b f l i ty b e g in at ,s a y , t h e f ir s t u n c o u n t a b l e v o n N e u m a n n o r d i n a l, s o t h a t t h e o n l y s e ts t h a te xe m pl i f y de f i n i t e c a r d i na l i t i e s a nd de f i n i t e o r de r t ype s a r e c oun t a b l e ? W e l l ,f o r g e t a b o u t t h e v o n N e u m a n n o r d in a ls . L e t ' s a p p r o a c h d e t e rm i n a b i l i t yf r o m a d i f f e r e n t , m o r e i n t u i t ive a ng le . H e r e ' s the i de a . W e be g in w i t h one o rm or e s e ts t ha t w e i n t u i t ive l y r e c ogn i z e t o be de t e r m i na b l e . T he n w e c on s t r uc t" b o t t o m u p " f r o m t h o s e s e ts b y m e a n s o f o p e r a t i o n s t h a t ( i n tu i t iv e l y ) l ea df r o m de t e r m i na b l e s e t s t o de t e r m i na b l e s e t s . T he r e s u l ti ng c o l l e c t i on C ~ c a nt he n be u s e d t o c i r c um s c r i be j u s t t h e de t e r m i na b l e s e t s: a se t w i ll be de t e r -m i n a b l e j u s t i n c a s e i t i s e q u i n u m e r o u s t o s o m e m e m b e r o f C Cr.

W i t h w ha t s e t s s ha l l w e s t a r t , a nd w i t h w ha t ope r a t i ons ? T o be g i n w i t h ,w e d r a w on t he i n t u i t i on , r e f l e c t e d in the a x i om s o f a r i t hm e t i c , t ha t a l l f i n i te


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4 8 C H R IS T O P H E R M E N Z E Ls e t s a re de t e r m i na b l e . 21 F o r pu r p os e s h e r e , g i ve n the ope r a t i ons w e w i ll u s e ,i t s u f fi c e s j u s t t o pos i t t ha t t he e m p t y s e t i s de t e r m i na b l e .

W e c ou l d l e t t h i ngs go a t t ha t a nd a l l ow on l y t ha t f i n i t e o r d i na l s ( i . e . ,o r de r t ype s o f f i n i t e w e l l - o r de r e d s e t s ) a r e a l l t he o r d i na l s t he r e a r e . F r omC a n t o r ' s g r ea t w o r k , t h o u g h , w e k n o w t h a t t h e i n fi n it e i s n o t t h e c h a o ti c ,m a t h e m a t i c a l l y i l l - b e h a v e d r e a l m i t w a s f o r s o l o n g t h o u g h t t o b e . R a t h e r ,t he s a l ie n t f e a t u r e s o f i n f i n i te s e t s a r e c ha r a c t e r i z a b l e in t he s a m e m a t he m a t i -c a l t e r m s a s f i n it e s e ts ( e s pe c i al l y i f w e a s s um e , al ong w i t h Ca n t o r , t ha t a l ls e t s c a n be w e l l - o r de r e d ) . I n pa r t i c u l a r , w e c a n a s c r i be w e l l - de f i ne d , de t e r -m i na t e nu m be r p r ope r t i e s t o i n f i n i te s e t s w h i c h a r e r i go r ous l y c ha r a c t e r iz e dby a ge ne r a l i ze d t r a ns f i n i t e a r i t hm e t i c . 22 W e w i l l f o l l ow Ca n t o r he r e . F o r o u rpu r p os e s i t s u f f ic e s t o pos i t on l y t he e x i s t e nc e o f a s ingle i n f i n i te s e t f r omw h i c h w e w i l l be a b l e t o p r ov e the e x i s t e nc e o f a de t e r m i na b l e i n f i n it e s e t .

W ha t t he n o f ou r de t e r m i na b i l i t y - p r e s e r v i ng ope r a t i ons ? T he r e i s l a t i t udehe r e ; t he c ho i c e s c a n va r y . Bu t g i ve n t he p r o j e c t a t ha nd , m y c ho i c e i s s i m p l yt he s e t - bu i l d i ng a x i om s o f Z F C, i . e . , p a i r i ng , un i on , pow e r s e t , s e pa r a t i on ,r e p l a c e m e n t , a nd c ho i c e . 23

A c l e a r p i c t u r e s hou l d n ow be e m e r g i ng : ou r pa r a d i gm a t i c de t e r m i na b l es e ts a r e j u s t t he pu r e s e ts o f Z F C ; o u r c o l l e c t i on Cr a bove i s j u s t t he pu r ec um ul a t i ve h i e r a r c hy V . 24 T hus , a s e t i n ge ne r a l w i l l be de t e r m i na b l e t ha t a llp u r e s e ts a re e q u i n u m e r 0 u s t o s o m e y o n N e u m a n n o r d i n a l, it s h o u l d b e c l ea rt ha t t h i s de f ' m i t i on i s e qu i va l e n t t o ou r o r i g ina l e xp l i c a t i on o f de t e r m i na b i l i t y .

N o w l e t ' s m a ke a ll th i s r i go r ous . W e w i l l w o r k i n the s t a nd a r d l a ngua ge o fs e t t h e o r y a u g m e n t e d w i t h a o n e - p l a c e f u n c t i o n s y m b o l O R D . I a s s u m et h e s t a n d a r d s e t t h e o r e t i c d e f i n i ti o n s o f ( x , y ) ( o r d e r e d p a i r) , x ~ y ( x ise q u i n u m e r o u s t o y ) , a n d x W O y ( x is a w e n - o r d e r in g o f y ) . I ta k e w e l l- o r d e re ds e ts t o b e o r d e r e d p a i rs ( x , y ) s u c h th a t y W O x , a n d I d e fi n e I z l t o be t he" d om a i n " x o f t he w e l l - o r de r e d s e t z = ( x , y ) . A s l i m i ti ng ca s es , I c ons i de r~b a nd ( x ) , f o r ob j e c t s x , t o be w e l l- o r de ri ngs o f t he m s e l ve s , a nd he n c e ( ~b , ra n d o r d e r e d p a i rs o f t h e f o r m ( ( x ) , ( x ) ) t o b e w e l l -o r d e re d s e ts . 2 s R e l a ti v et o t h i s de f i n i t i on o f a w e l l - o r de r e d s e t , I t a k e x ~ - y ( x is i s om or ph i c to y ) t obe d e f i ne d i n t he u s ua l m a nne r . F i na l l y , the no t i o n o f t he t r a ns i t ive c l o s u r e o fx T C ( x ) i s e s s e n ti a l to ou r d e f i n i t i on o f de t e r m i na b i l i t y . F o r va r ious r e a s ons ,h o w e v e r , t h i s n o t i o n c a n n o t b e q u i t e d e f i n e d i n a n y o f t h e s t a n d a r d w a y sf o u n d i n m o s t s e t t h e o r y t e x t s . ( C f . f n . 2 7 .) I n th e a p p e n d i x i t is p r o v e d t h a ti t c a n b e d e f in e d i n a n o t h e r w a y w h i c h c a u s e s n o p r o b l e m s .

W e i n t r oduc e t h r e e ne w de f i n i t i ons :


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IT E R A T IV E E X P L A N A T IO N O F T H E P A R A D O X E S 4 9( D 1 )( B 2 )(D3)

S E T ( x ) - d f -7 3 y ( x = O R B ( y ) )P U R E ( x ) =- d f S E T ( x ) & V y ( y E T C ( x ) D S E T ( y ) )B E T ( x ) = d f 3 y ( P U R E ( y ) & x "" y )

T he i n t u i t i ve m e a n i ng o f t he de f i n i t i ons s hou l d be c l e ar . S e ts a r e t he ob j e c t sw h i c h a r e no t o r d i na l s ; pu r e s e ts ha ve on l y s e ts i n the i r t ra n s i ti ve c l o su r e s ;a nd , a s m e n t i o ne d a bove , a de t e r m i na b l e s e t i s a s e t t ha t is e qu i num e r ou s t os om e pu r e s e t .

N o w f o r t h e p r o p e r a x i o m s o f t h e t h e o r y . W e b e g in w i t h w h a t I w il l c al lt h e C a n t o r i a n a x i o m s .

( C 1 ) V x V y ( O R D ( x ) = O R B ( y ) - W O S E T ( x ) & D E T ( I x I ) & x ~ - y )(C 2 ) 3 x V y ( y E x - - 3 z ( y = O R B ( z ))( C 3 ) V x V y ( x = O R B ( y ) D V z -7 (z E x ) )

C1 s a ys t ha t t he o r d i na ls o f t w o w e l l - o r de r e d s et s x a nd y a r e i de n t i c a l j u s t i nc a s e x a nd y a r e i s om or ph i c a nd t he i r dom a i ns a r e de t e r m i na b l e . C2 s a yst he r e i s a se t o f a ll o r d ina l s . C3 m a ke s e xp l i c it w ha t w a s a e gue d a t l e ng t ha b o v e , viz., t ha t o r d i na ls a r e no t c o l l e c t ions o f a ny k i nd .

N e x t w e i n t r oduc e t he s e t t he o r e t i c a x i om s p r o pe r . A s s ugge s te d , w e ta keo v e r Z e r m e l o ' s a x i o m s ( i . e . , Z C ) s t r a i g h t a w a y , m o d i f i e d a p p r o p r i a t e l y f o ru r e l e m e n t s . I i nc l ude e x t e n s i ona l i t y a nd f o und a t i on , w h i c h a r e o f c ou r s ee s s e n ti a l t o t he i t e r a t i ve c onc e p t i on .

( Z 1 ) V x V y [ S E T ( x ) & S E T ( y ) D V z ( z E x - - z E y ) D x = y ](Z 2) V x [ 3 y ( y E x ) D 3 y ( y E x & V z T ( z E x & z E y ) ) ]( Z 3 ) 3 x ( S E T ( x ) & V y - 7 ( y E x ) )( Z4 ) 3 x ( ~ bE x & V y ( y E x D 3 z ( z E x & V w (w E z = w E y V w = y ) ) ))( Z 5 ) V x V y 3z V w ( w E z = w = x v w = y )( Z 6 ) V x 3y 'v ' z ( z E y = 3w ( w @ x & z E w ) )( Z 7 ) V x [ S E T ( x ) D 3 y V z ( S E T ( z ) D z E y --= V w ( w E z D w E x ) ) ]( Z 8 ) V x [ V y ( y E x D S E T ( y ) & V z ( z E x D y n z = ~ V y = z ) ) D

3 y V z ( z E x D 3 w ( w E z & z E y & V u (u E z & u E y D u = w )) ) ]( Z 9 ) V x [ S E T ( x ) D 3 y V z ( z E y - z ~ x & ~ 0(z ))],

w h e r e ~ ( z ) i s a f o r m u l a o f ou r l a ngua ge w i t h z f r e e no t i nvo l v ing y .A l l t h a t r e m a i n s in F r a e n k e l ' s a x i o m o f r e p l a c e m e n t F , a n d h e r e w e m u s t

b e c a r ef u l. F o r i t i s F w h i c h m a k e s Z F C i n c o n s i s t e n t w i t h th e e x i s te n c e o fO n . T o s ee t h i s, c on s i de r t he f o r m u l a ~0 (x , y ) t h a t ho l d s i f a nd on l y i f y is a


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50 CHRISTOPHER MENZEL

y o n N e u m a n n o r d i n a l a n d x = O R D ( ( y , E ~ y ) ) . ~0 is f u n c t io n a l , s o b y r e p la c e -m e n t o n O n i t f o l lo w s t h a t t h e y o n N e u m a n n o r d i n a ls c o n s t i t u te a s e t S . I t ise a s y t o s how , ho w e v e r , t ha t S is t r a n s it i ve a n d c on ne c t e d , he nc e S a l so i s byd e f i n i ti o n a y o n N e u m a n n o r d i n al . B u t t h e n S E S , a c o n t r a d i c t i o n , s i n ce n ov o n N e u m a n n o r d i n a l is s e l f- m e m b e r e d .

W h a t a r e w e t o s a y a b o u t t h i s ? O n e m i g h t a r g u e t h a t t h e i n c o m p a t i b i l i t yo f r e p l a c e m e n t w i t h t h e p i c t u r e h e r e s i m p l y s h o w s t h a t t h e p i c tu r e is b o g u s .B u t t h a t w o u l d b e t o o h a s t y . T h e i n t u i t io n o n w h i c h r e p l a c e m e n t is f o u n d e dis t h a t i f y o u ' v e g o t a se t M , t h e n i f y o u r e p l a c e th e m e m b e r s o f M o n e f o rone w i t h c e r t a i n o t he r ob j e c t s , t he r e s u l t w i l l a l s o be a s e t . F r om a na i vep o i n t o f v i e w , t h e i n t u i t io n a p p e a r s s o u n d ; i t is b o l s t e r e d f u r t h e r b y t h e f a c tt ha t i t i s t r ue i n t he un i ve rs e V o f pu r e s e t s, a nd i nde e d i n a ny h i e r a r c hy t ha tbe g i n s w i t h a de t e r m i na b l e s e t o f u r e l e m e n t s . S uc h h ie r a r c h ie s , t houg h , I ' v ea r gue d , don ' t r e p r e s e n t t he i t e r a t i ve c onc e p t i on i n i t s f u l l ge ne r a l i t y . O n t hem os t ge ne r a l c on c e p t i o n , t he r e i s no l i m i t a t i on o n t he s i z e o f t he c o l l e c t i onw e be g i n w i t h ; t h i s is w h a t c a us es p r ob l e m s f o r r e p l a c e m e n t . S i nc e t he it e r a -r iv e p i c t u r e is t h e b a si s o f o u r c o n c e p t i o n o f s e t , a x i o m s m u s t b e ju d g e d i nt e r m s o f t h a t c o n c e p t i o n , n o t t h e o t h e r w a y a r o u n d . A n d w h a t t h e i te r a ti v ec on c e p t i on t e ll s u s i n t h i s c a s e i s t ha t F r a e nke l ' s a x i o m is f a ls e i n ge ne r a l ;j u s t a s s o m e p r o p e r t i e s c a n d e t e r m i n e u n b o u n d e d c o l l e c t i o n s , s o s o m e f u n c -t i o n s o n s o m e s e ts h av e ra n g e s t h a t a r e u n b o u n d e d . A n d j u s t a s w e m u s tm od i f y p r i nc i p le P i n l igh t o f ou r de e pe r in s i gh t i n t o t he s t r uc t u r e o f t he s e tt h e o r e t i c u n i v e rs e , so a ls o w e m u s t m o d i f y r e p l a c e m e n t .

H o w i s t h i s t o b e d o n e ? S i m p l y e n o u g h : w e m u s t r e s t ri c t t h e a x i o m s u c ht ha t i t o n l y a pp l i e s t o de t e r m i na b l e s e ts . T h i s , a s w e ' l l s e e , w i ll e n s u r e t ha ta n y s e t w e c a n p r o v e t o e x i st in Z F C b y m e a n s o f r e p l a c e m e n t c a n als o b ep r ove d t o e x i s t i n ou r p r e s e n t t he o r y w i t hou t g i v i ng r i s e t o t he p r ob l e ma b o v e 3 6 W e t h u s a d d t h e f in a l a x i o m o f o u r t h e o r y :

( F ~ ) V x V y V z ( ~ 0( x , y ) & ~0 (x , z ) D y = z ) D V x [ S E T ( x ) & D E T ( x ) D~ w V u ( u ~ w - ~ z ( z ~ x & ~ ( z , y ) ) ) ] ,

w h e r e ~O (x , y ) i s a f o r m u l a o f ou r l a nguage w i t h x a nd y f r e e , no t i nvo lv i ngW.27

I w i l l c a ll t h i s t he o r y C Z F ~ . I t s i n t e nde d m ode l , o f c ou r s e , is t he Ca n t o r i a nun i ve r s e , i. e ., m or e p r e c i s e l y , t he c u m u l a t i ve h i e r a r c hy V ~ t ha t r e s u l t s w h e nw e b e g i n w i t h O n a s o u r s et o f u r e l e m e n t s , v i z . ,


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I T E R A T I V E E X P L A N A T I O N O F T H E P A R A D O X E S 5 1

V g n = O nv O n g O n v O n

O n .V ~ . a < X ) , f o r l i m i t X ,w h e r e o f c o u r se V ~ = w g V ~_ ~ , l o ~ O n ) . = s A s I s h o w i n th e a p p e n d i x ,w e c a n n o t h o p e t o p r o v e t h a t C F Z ~ i s c o n s i s te n t re la ti v e t o Z F C , s in c eC Z F ~ r b - c o n ( Z F C ) . I t is , h o w e v e r , c o n s i s t e n t r e la t iv e t o Z F C w i t h a n i n -a c c e s si b le . F u r t h e r , w e l o s e n o f a c t s a b o u t t h e p u r e s e ts p r o v a b l e i n Z F C ,s i n ce , a s w e a l so s h o w , t h e r e is a n a t u r a l i n t e r p r e t a t i o n o f Z F C i n s id e C Z F ~r .M o r e p r e c i s e l y , w e s h o w t h a t t h e r e is a f u n c t i o n rr f r o m t h e f o r m u l a s o f t h el a ng u a g e o f Z F C i n t o t h e f o r m u l a s o f th e l a n g u a g e o f C Z F O s u c h t h a t f o r a n yf o r m u l a ~o o f t h e f o r m e r l a n g u a g e , i f Z F C } -~ o, t h e n C Z F ~ r k ~ r r, w h e r e ~0 re s s e n t i a l l y " s a y s t h e s a m e t h i n g " a s ~0.

C ZF Cr t h e n is j u s t w h a t w e w a n t : a Z F C - s t y l e t h e o r y o f th e C a n t o r i a nu n i v e r se . S p e c i f i c a ll y , C Z F ~ i ) is a s s t r o n g a s Z F C , i i) p re s e rv e s th e v i e w t h a tt h e o r d i n a l s a r e n o t s e t s , a n d i i i ) r e m a i n s t r u e t o t h e i t e r a t i v e c o n c e p t i o n o fs e t i n it s f u l l g e n e r a l i t y b y a l l o w i n g t h e r e t o b e a s e t o f a ll o r d i n a l s . S o a ll i sw e l l; R u s s e l l 's p a r a d o x s ti ll r e ce i v es i ts i t e ra t iv e e x p l a n a t i o n , t h e B F p a r a d o xis r e l e g a te d t o i ts p r o p e r d o m a i n , a n d o u r s e t t h e o r e t i c a x i o m s a r e n o lo n g e ra d h o c in a c o n c e p t u a l l y o b j e c t i o n a b l e w a y .

A P P E N D I X

Defining TCT h e f ir s t o r d e r o f b u s i n e ss is t o d e f i n e T C ( x ) i n s u c h a w a y t h a t w e c a n p r o v ei n C Z F ~ w i t h o u t c i r c u l a r i ty t h a t e v e r y s e t h a s a t r a n s it iv e c l o s u r e . ( C f . f n .2 7 . ) T o d o s o w e w i ll n e e d t o d e f i n e it so as t o g e t a s u f f i c i e n t ly f i r m g r ip o no u r b o o t s t r a p s .( D E F 1 ) T R A N S ( x ) - d f S E T ( x ) & V y ( y E x D y C_ x ).( D E F 2 ) T C ( x ) = d f ( t y ) [ T R A N S ( y ) & x C_ y & V z ( T R A N S ( z ) & x C_ z D

y C z ) ] , i f s u c h a y e x i s ts , ~ , o t h e r w i s e .B y D E F 2 , o f c o u r s e , i t i s tr iv i al t h a t V x 3 y ( y = T C ( x ) ) . W h a t w e w a n t t os h o w is t h a t t h e a d d e d c l au s e o f t h e d e f i n i t i o n is s u p e r f l u o u s , i .e ., t h a t


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T H E O RE M 1 . V x [ T C ( x ) = 0Y ) [ T R A N S ( y ) & x C y & V z ( T RA N S ( z ) & x C_C_z D y C _ z ) ] .

T he t r i c k is to ge t j u s t e no ugh r e p l a c e m e n t t o do t he j ob w i t h ou t c i rc u l a r-i t y . T he f i r s t s t e p is t o s how t ha t ( t he y on N e um a nn o r d i na l ) c o i s pu r e , a ndhe nc e de t e r m i na b l e , w i t ho u t u s ing F ~r . T h i s i s e a s il y a c c om pl i s he d w i t h a f e ws im p l e l e m m a s t h a t f o ll o w f r o m t h e e m p t y s e t a x i o m Z 3 , t h e a x i o m o fi n f i n i t y Z 4 , a n d s e pa r a t i on Z 9 . W e t he n p r ove t he f o l l ow i ng ve r s i on o f the c o-r e c u r s i o n t h e o r e m .L E M M A 1 ( c o - r e c u r s i on ) . L e t a be a ny ob j e c t a nd l e t r ( x ) be a t e r m . T he nt he r e e x i s t s a f unc t i on G s uc h t ha t

G ( a , 0 ) = aG ( a , n + 1 ) = r ( G ( a , n ) ) .

T h e p r o o f p r o c e e d s i n t h e s t an d a r d m a n n e r i n t e r m s o f f u n c t io n s f a + 1 o ni n it i al s e g m e n t s ( a } x n + 1 o f { a } x co t ha t a g r e e w i t h t he de f i n i t i on o f G asf a r as t he y go . W e c a n t he n u s e Fr on co t o ob t a i n t he se t f = af n + l l n E c o } .W e the n ha ve G = U f .29C O R O L L A R Y . F o r a n y o b j e c t x , t h e r e is a f u n c t i o n H s u c h t h a t

H ( x , 0 ) = xn ( x , n + 1) = H ( x , n ) u U H ( x , n ) .

( D E F 3 ) T C ~ ( x ) = d f U { H ( x , n ) [ n E co },w h e r e H i s t h e f u n c t i o n g u a r a n t e e d b y t h e a b o v e c o r o n a r y g iv e n x . 3 ~L E M M A 2 . F o r a ny ob j e c t x , T CC~( x) e x i st s .

P r o o f B y 6 o- re cu rs io n, F ~ o n w , a n d u n io n Z 6 .W e c an n o w p r o v e T h e o r e m 1 b y m e a n s o f th e f o ll ow i ng :

L E M M A 3 . V x [ T C ~ ( x ) = ( t y ) ( T R A N S ( y ) & x C_ y & V z ( T R A N S ( z ) & x C_ z Dy C z ) ) ] .

P r o o f . T h e p r o o f b r e a k s d o w n i n t o th r e e p a r t s . L e t x b e a n y o b j e c t .( i) x ~ T C ~ ( x )


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ITE RATI VE EXPLANATION OF THE PARADOXES 53

T h i s i s o b v i o u s s i n c e x = H ( x , 0 ) C T C C r ( x ) .( i i) T C ~ r ( x ) i s t r a n s i t i v e

S u p p o s e a ~ T C ~ ( x ) . T h e n f o r s o m e n E ~ , a E H ( x , n ) . I f b E a , t h e n b EH ( x , n + 1 ) = H ( x , n ) u U H ( x , n ) C T C ~ ( x ) .

( ii i) T C ~ ( x ) i s t h e s m a l l e s t s u c h s e t .L e t z b e a s e t t h a t i s t r a n s i ti v e a n d s u c h t h a t x C__ z . W e s h o w t h a t H ( x , n ) C z ,f o r a l l n E c o. O b v i o u s l y H ( x , 0 ) = x C z . S o s u p p o s e H ( x , n ) C z , a n d l e ty E H ( x , n + 1 ) . T h e n y C H ( x , n ) u U H ( x , n ) . I f y E H ( x , n ) , y E z b y o u ri n d u c t i v e h y p o t h e s i s . S o s u p p o s e y E U H ( x , n ). T h e n 3 w E H ( x , n ) s u c h t h a ty E w . S i n c e , b y o u r i n d u c t i v e h y p o t h e s i s , H ( x , n ) C z , w E z . B y t ra n s i t i v i tyo f z, y E z . S o H ( x , n + 1 ) C z . H e n c e , f o r a l l n , H ( x , n ) C_ z . S o U ( H ( x , n ) ln E r } = T C ~ ( x ) C z , a s r e q u i r e d .

B y L e m m a 3 t h e r e i s a l w a y s a s e t s a t is f y i n g t h e f i r st c o n d i t i o n i n t h ed e f i n i t i o n o f T C , a n d h e n c e , a s w e w a n t e d t o s h o w , t h e s e c o n d c o n d i t i o n i nt h e d e f i n i t io n i s s u p e r f l u o u s , i . e. , T h e o r e m 1 is p r o v e d .

Interpreting ZFC in CZF~W e m o v e o n n o w t o o u r m o r e s u b s t a n ti v e t h e o r e m s . T h e i r p r o o f s ar e q u i ter o u t i n e a n d w i l l o n l y b e s k e t c h e d . W e b e g i n b y d e f i n i n g a f u n c t i o n 7r f r o mt h e f o r m u l a s i n t h e l a n gu a g e o f Z F C t o th e f o r m u l a s o f t h e la n g u ag e o fC Z F ~ r a s f o l l o w s .( D E F 4 ) ( x @ y )~ r = ( x E y )

( x = y ) ~ r = ( x = y )( 7 ~ ) '~ = 7 ~ ~-(~ ~ ~)~ = ~ ~ ~( V x ~ 0 )" = V x ( P U R E ( x ) n ~0~ )

l r, t h e n , s e n d s f o r m u l a s o f Z F C t o f o r m u l a s o f C Z Fr w h o s e q u a n t i f i e r s a r er e s t r i c t e d t o t h e p u r e s e t s. I n p a r t i c u l a r , i t s h o u l d b e c l e a r t h a t a s e n t e n c eo f Z F C a n d i ts c o r r e sp o n d i n g C Z F C r t r a n s la t i o n " s a y " t h e s a m e t h in g .L E M M A 4 . I f ~o i s a l o g ic a l a x i o m i n t h e l a n g u a g e o f Z F C , t h e n ~0~r is a logica lt h e o r e m i n t h e l a n g ua g e o f C ZF r

Proof. I u se e s s e n t i a ll y t h e p r e d i c a t e c a lc u l u s w i t h i d e n t i t y f o u n d i n


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5 4 C H R I S T O P H E R M E N Z E LE n d e r t o n ' s A M athematical In troduc t ion to Logic w h o s e a x i o m s a re p r o p o si -t i o n a l t a u t o l o g i e s a n d g e n e r a l i z a ti o n s o f i n s t a n c e s o f fi v e a x i o m s c h e m a t a .M o d u s p o n e n s is t h e o n l y r u le o f i n f e re n c e . 31 T h e l e m m a is p r o v e d f i rs tb y s h o w i n g h o w t o d e d u c e t h e t ra n s l a t io n o f a n y a x i o m i n th e l an g u ag e o fZ F C f r o m t h e a x i o m s i n t h e l a ng u a ge o f C Z F ~ r . T h e n o n e c o m p l e t e s th ep r o o f b y s h o w i n g b y i n d u c t i o n o n t h e n u m b e r o f i n it ia l u n iv e rs a l q u a n t i -t ie r s t h a t i f ~ i s a g e n e r a l iz a t i o n o f a n i n s t a n c e o f a n a x i o m i n t h e l a n g u ag eo f Z F C , t h e n C Z F ~ r b - ~o r 9( D E F 5 ) Z F C ~r = d f {~o~r [~o i s a n a x i o m o f Z F C } .L E M M A 5 , I f Z F C }-- % t h e n Z F C ~r } - ~0~r.

Proof. A n e a s y in d u c t i o n o n t h e l en g t h o f p r o o f .L E M M A 6 . I f ~o E Z F C 'r , t h e n C Z F ~ b - r

P r o o f . I w i ll r e s t ri c t m y a t t e n t i o n t o r e p l a c e m e n t F , w h i c h i s t h e l e a sto b v i o u s c a se . T h e m e t h o d o f p r o o f i s t h e s a m e i n al l c a se s . W e l e t 'V Px a b b r e -v ia te ' V x ( P U R E ( x ) D ' a n d ' 3 ~ ' ' 3 x ( P U R E ( x ) & ' . S u p p o s e t h e n t h a t V Px, y ,

P P Pz (~ 0 rr (x , y ) & ~ 0 rr(x , z ) D y = z ) . W e w a n t t o s h o w t h a t V x 3 y V z ( z E y - 3Vw Ex (~ o ~ r( w , z ) ) ) . S o l e t x b e a p u r e s e t . B y F r t h e r e i s a s e t y s u c h t h a t V z ( z Ey = 3 w E x ( ~ r ( w , z ) ) . L e t yC r = { u E y [ P U R E ( u ) } . N o t e t h a t y ~ r i s i t s e l fp u r e . C l a i m : V z P ( z E y ~ - - 3 P E x@ ~ r (w , x ) ) . T o s e e t h i s , l e t z b e a p u r e s e ta n d s u p p o s e z E y ~ . T h e n 3 w E x (~ o~ r(w , z ) ) . B u t s i n c e x is p u r e , w i s a s w e l l ,s o 3 P ( ~ o rr (w , z ) ) . S u p p o s e t h e n 3~ w E x (~ o n( w , z ) ) . T h e n b y F % z E y . B u t zi s p u r e , s o z E y C r, a n d o u r c l a i m i s p r o v e d . H e n c e , C Z F ~ t-- F ~r.T H E O R E M 2 . I f Z F C ~ % t h e n C Z F ~ r 1 - ~o r.

Proof . I m m e d i a t e f r o m L e m m a s 5 a n d 6 .S o w h a t e v e r Z F C s a ys , C Z F ~ s a ys a b o u t t h e p u r e s e ts . A s c la i m e d , th e n ,C Z F ~ i s a t l e as t a s s t r o n g a s Z F C . T h e o r e m 2 a ls o y ie l d s t h e f o l l o w i n gC O R O L L A R Y . I f C Z F ~ i s c o n s i s t e n t , s o is Z F C .

P r o o f S u p p o s e Z F C i s i n c o n s i s t e n t . T h e n f o r s o m e ~0, Z F C 1 --~ o& 7 ~o .B u t t h e n CZFr } -- (~o & 7 ~ ) 7 r, s o b y d e f ' m i t i o n o f rr , CZFr 1 - ~07r & 7 ~ r r.

I n f a c t , w e c a n s a y m o r e , viz.,


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I T E R AT I VE E XPL ANAT I ON OF T HE PAR ADOX E S 5 5

T H E O R E M 3 . C ZF r 1-- c o n ( Z F C ) .I w il l o n l y s k e t c h t h e i d e a o f t h e p r o o f . T h e c o n s t r u c t i b l e u n i v e r s e L i s

d e f i n a b l e i n C Z F ~ . A s is w e l l k n o w n , t h e r e i s a d e f in a b l e w e l l -o r d e r i n g< L o f L . T h i s w e l l- o rd e r in g , o f c o u r s e , a s so c ia te s e a c h e l e m e n t o f L w i t h ano r d i n a l . S i n c e t h e r e is a s e t O n o f a l l o r d i n a l s i n C Z F r t h i s m e a n s t h a t w ec a n m a p L o n e t o o n e o n t o O n , a n d w e c a n u s e t h is m a p p i n g t o d e f in e a ni n t e r p r e t a t io n o f E a m o n g t h e o r d in a l s. S o m e w h a t m o r e p re c i se l y , l e t ~ b eo u r m a p p i n g f r o m L in t o O n . T h e n w e d e f i n e a R /3 t o h o l d i f a n d o n l y ~ ( x , o0a n d ~ o (y , /3 ) a n d x E y , f o r x , y E L . U s i n g t h e ( r e s t r i c t e d ) d e f i n a b i l i t y o fs a ti sf ac t i o n, 32 w e c a n sh o w t h a t C Z F ~ [ - -( O n , R ) b Z F C . A c o r o l la r y , o fc o u r s e , i s t h a t C Z Fr is s t r i c tl y s t ro n g e r t h a n Z F C , b y G 6 d e l ' s s e c o n d in -c o m p l e t e n e ss t h e o r e m .

The Rela tive Consistency o f CZFrW e t u r n n o w t o o u r f in a l t h e o r e m . L e t Z F C b e t h e t h e o r y Z F C + T h e r e isa n i n a c c e s s i b l e c a r d i n a l . T h e n

T H E O R E M 4 . I f Z F C is c o n s is t e n t, s o is C Z F ~ .W e f o l lo w e s s en t ia l ly t h e s a m e s o r t o f a r g u m e n t t h a t w a s u s e d t o p r o v et h e c o r o l l a r y t o T h e o r e m 2 . T h a t is , w e w i ll d e fi n e a f u n c t i o n v f r o m t h ef o r m u l a s o f t h e l a ng u a g e o f C ZF r t o t h e f o r m u l a s o f t h e l a n g u ag e o f Z F C a n d s h o w t h a t i f C Z F ~ r k ~ 0 , t h e n Z F C [ -~ ou , f r o m w h i c h ( g i v e n t h e d e f in i -t i o n o f u ) o u r t h e o r e m f o l lo w s . W e b e g in b y d e f in i n g ( i n Z F C t h e s e t At o w h i c h q u a n t i f i e r s o f C Z F ~ f o r m u l a s w i ll b e r e l a ti v iz e d . K h e r e is a n i n-a c c e s s i b le c a r d i n a l .( D E F 6 ) A o = { ( K , a ) l a < ~ }

A ~ + I = ~ ( A ~ ) w A ~A ~ = u { A ~ I a < X } , f o r l im i t X < t ~A = u { A ~ l a < K } .

N o w w e d e f in e o u r f u n c t i o n u . A w o r d o f c l a ri fi c a ti o n f ir st . I n t u it i v e ly , O R Dd e n o t e s a p a r t ia l f u n c t i o n o n t h e C a n t o r i a n u n i v er se . F o r p u r p o s e s h e r e h o w -e v e r i t is e a s ie s t j u s t t o t h i n k o f O R D a s a 2 -p l a c e p r e d i c a t e , s i n c e t h a t w i lla l lo w u s t o d e f in e u m o r e s i m p l y . T h u s , ' x = O R D ( y ) ' c a n b e t h o u g h t o f a s a n" a b b r e v i a t i o n " f o r ' O R D ( y , x ) & V z ( O R D ( y , z) D z = x ) ' .


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5 6 C H R IS T O P H E R M E N Z E L( D E F 7 ) ( x E y ) ~' = ( x E y )

( x = y ) " = ( x = y )( x = O R D ( y ) ) v = " y is w . o . " & o r d ( y ) < K & x = ( K , o r d ( y ) )( - l ~ ) ~ = 7 ~ ~

( V x r ~ = V x e A rW e n o w n e e d t h e a n a lo g u e o f L e m m a 4:L E M M A 7 . I f ~0 i s a log i c al a x i om i n t he l a ngua ge o f C Z F % t he n 9v i s al o g ic a l t h e o r e m i n t h e l a ng u a g e o f Z F C T h e p r o o f p r o c e e d s e ss en ti al ly a s in t h e p r o o f o f L e m m a 4 .

N e x t w e w a n t t h e a n a lo g u e s o f D e f i n it io n 5 a n d L e m m a 5 .( D E F 8 ) C Z F ~ v - -- d f { ~o I ~o i s a n a x i o m o f C Z F ~} .L E M M A 8 . I f C ZF r 1 - 9 , t h e n C Z F ~ v t - 9 v .A ga i n t he p r oo f is e s s e n t ia l l y i de n t i c a l w i t h i t s a na l ogue .

I n o r d e r t o p r o v e th e a n a lo g u e o f L e m m a 6 w e n e e d t o i so l a te th e s i m p l es tc o n d i t i o n s i n Z F C t h a t c o r r e s p o n d t o t h e d e f i n e d p r e d i c a te s S E T , P U R E , a n dD E T . S E T i s e a s y e n o u g h :L E M M A 9 . Z F C I - V x E A ( S E T ( x ) v - x E A - A o ) .

P ro of . F or x E A , S E T ( x ) V - 7 3 y E A ( " y is w .o ." & o r d ( y ) ( g & x =( K , o r d ( y ) ) ). B u t b y c o n s t r u c t i o n o f A , t h is h a p p e n s i f f x ~ A o i f f x E A - A o .

T o d e a l w i t h P U R E , w e n e e d t o m o d i f y t h e s t a n d a rd Z F C d e f i n i t io n o ft r a n s it i v it y s li g ht ly t o a c c o u n t f o r th e p r e s e n c e o f " u r e l e m e n t s " ( i .e . , m e m -b e r s o f A o ) .( D E F 9 ) t r a n s ( x ) = d f ( x = V 3 y ( y E x ) ) & V y ( y E x D y C_ x )( D E F 10 ) t r c l ( x ) =d f ( t y ) [ t r a n s ( y ) & x C_ y & V z ( t r a ns ( z ) & x C_ z D y C_ z ) ]O n t h i s de f i n i t i on , i f u i s a n u r e l e m e n t , t he n t r c l ( u ) = ~ . N ow w e c a n s howL E M M A 1 0 . Z F C ~ V x ~ A ( T C ( x ) ~ = t r c l ( x ) A ) .T h e p r o o f is s tr a i g h t fo r w a r d .


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I T E R A T I V E E X P L A N A T I O N O F T H E P A R A D O X E S 5 7

L E M M A 1 1. Z F C + [ - V x E A ( P U R E ( x ) v - = x ~ V ~ ) ,W he r e V~ i s o f c ou r s e t he i n i ti a l s e gm e n t o f t he pu r e c um ul a t i ve h i e r a r c h yV d e t e r m i n e d b y u . T h e p r o o f o f L e m m a 1 1 is a s t r ai g h t f o r w a r d in d u c t i o non E , m a de e a si er , f o r t hos e w h o w i s h t o w or k i t ou t i n de ta i l, w i t h t he he l po f t he f o l l ow i ng r ou t i ne d e f i n i t ions a nd f a c t s.( D E F 1 1 ) g z E A , 0 A(Z ) = d f / ~ a ( V y E Z ( p A ( y ) < I X )) .( D E F 1 2 ) g z E V ~ , p ( z ) = d f ~ c ~ ( g y E z ( 0 ( y ) < a ) ) .( F A C T 1 ) V z E V ~ V a < K ( p ( Z ) < a m z E V a ) .( F A C T 2 ) g z E A g a < K( pA ( Z ) < a = Z E A a ) .( F A C T 3 ) V z E A V y E Z ( p A ( y ) < p A ( Z )) .( F A CT 4 ) V z E A ( z E V~ D pA ( Z ) = p ( Z ) ) .( F A C T 5 ) V ~ C _ A .L E M M A 1 2 . Z F C + ~ V x E A ( D E T ( x ) v - c a r d ( x ) < ~ ) .Proof. D E T ( x ) v - 3 y ( y E V ~ & x ~ y ) , b y L e m m a 1 1. B y e l e m e n t a r yp r o p e r t i e s o f V ~ , t h is h a p p e n s i f f c a r d ( x ) < ~ .L E M M A 13 . If ~0 E C Z F ~ v , t he n Z F C }- ~o .Proof. F o r e a c h m e m b e r o f C Z F ~ v th e l e m m a i s a m o r e o r l es s s tr a ig h t -f o r w a r d c o n s e q u e n c e o f t h e l e m m a s a n d f a c t s a b o v e . I w il l p r o v e t h e le m m ai n d e ta i l f o r th e C Z F ~ r e p l a c e m e n t s c h e m a F ~ .

L e t i f ( x , y ) be a f o r m u l a in t he l a ngua ge o f CZF r w i t h x a n d y f r e e , no ti n vo l v in g w . S u p p o s e t h e n t h a t V x , y , z E A ( f f ( x , y ) V & i f (x , z ) ~ D y = z ) .L e t x E A - A o be s uc h t ha t c a r d ( x ) < to . T he n w e c l a im t ha t 3w E A V u EA ( u E w - - 3 z E A ( z E x & ~ ( z , y ) V ) ). B y r e p la c e m e n t , 3 w V u ( u E w =3 z ( z E x & ~ ( z , y )V )). L e t w , # = { u ~ w l u ~ A } . C lea rly , f or u E A , u Cw ~ = 3 z ( z E x & f f ( z , y )V ). S in ce z E x D z E A , w e h av e u E w ~ = 3 z EA (z ~ x & i f (z , y )V) . So a l l we n eed to show no w is tha t wCr E A. Clea r ly ,w ~ C A . By A C, t he r e is a f unc t i on f s uc h tha t f : a ~ x , f o r s om e o r d i na l- - o n t o~. S ince ca r d( x) < K , cx < to . Th us , by Fa c t 5 , r @ A. N ow , l e t g : x ~ g bes u c h t h a t g ( z ) = P A ( Y ) + 1 i f r y ) V , a nd g ( z ) = 0 i f t he r e i s no s uc h y .Con s i de r now t he f unc t i on h = f og . h m a ps o~ i n t o ~ . C l e a r ly , Ra ng e ( h ) c a n -no t be co f ina l in ~ s ince cx < K and K i s inaccess ib le . Thus , su p( R an ge (h ) ) < ~ .Le t 13 = sup+(Ran ge (h) ) . T hen V y E wr 0A(Y) < /3 . Th us , by Fac t 2 ,wr C_ Aa .B u t t h e n w ~ E A f l + 1 C A .


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L E M M A 1 4 . I f C Z F r ~-~ 0, t h e n Z F C [ - ~0u.P r o o f . I m m e d i a t e f r o m L e m m a 8 a n d L e m m a 1 3.W e c a n n o w p r o v e th e s e c on d o f o u r tw o m a i n t h e o r e m s .

T H E O R E M 4 . I f Z F C is c o n s i s t e n t , s o i s C Z F ~ .P r o o f . S u p p o s e Z F C is c o n s i s t e n t , b u t t h a t C Z F r ~ - ~0 & 7 ~0. T h e n b y

L e m m a 1 4 , Z F C + k ( ~ 0 & 7~ 0) ~. S o b y d e f i n i t i o n o f v , Z F C + t--~ov & ~ 0 ~ ,c o n t r a d i c t i n g t h e c o n s i s te n c y o f Z F C

A f i n a l n o t e . I t s h o u l d b e c l e a r f r o m t h e d e f i n i t i o n o f v a n d f r o m L e m m a1 4 t h a t w h a t w e ' v e e s s e n t ia l l y s h o w n i s t h a t t h e s t r u c t u r e J / = ( A , O R D ~ ,E r A ) , w h e r e O R D J = ( ( x , ( ~ , o r d ( x ) ) ) [ x E A & " x i s w . o ." & o r d ( x ) < ~ } , i sa m o d e l o f C Z F ~ . T h u s , w i t h a l i tt l e m o r e w o r k , a n d u s i n g t h e ( r e s t r ic t e d )d e f i n a b i l i t y o f s a t i s f a c t i o n o n c e a g a i n , w e h a v eT H E O R E M 5 . Z F C k c o n ( C Z F ~ ) .

Z F C i s t h u s s t r i c t ly s t r o n g e r t h a n C Z F C r. A n o p e n q u e s t i o n : I s t h i s r e s u l tb e s t p o s s ib l e ? T h a t i s , i s t h e r e a p r o p e r e x t e n s i o n T o f Z F C w e a k e r t h a nZ F C s u c h t h a t T [ - c o n ( C Z F C r ) ? S i m i l ar l y w i t h r e s p e ct t o T h e o r e m 4 : I st h e r e a T w e a k e r t h a n Z F C s u c h t h a t i f T is c o n s i s t e n t , t h e n s o is C Z F C r? 33

N O T E Sx K . G 6 d e l , ' Wh a t is C a n t o r ' s c o n t i n u u m p r o b l e m ' , i n P . B e n a c e r ra f a n d H . P u t n a m ,Philosophy o f Mathematics, (Eng lewoo d Cl i ff s , 1964) , p . 263 .2 E . Z e r m e l o , ' I n v e s ti g a t io n s i n t h e f o u n d a t i o n s o f s e t t h e o r y I ' , i n J. v a n H e i je n o o r t(ed.) , From Frege to Gbdel (Cambr idge , 1967) , p . 200 .3 A c tua l ly , Zerm elo spec if ie s o n ly seven ax ioms . O ne ax iom however - h i s ax iom ofe l e m e n t a r y s e ts ( A x i o m I I ) - c o m b i n e s b o t h t h e n u l l s e t a x i o m a n d p a i r in g . T h e m i s si n ga x i o m s a re r e p l a c e m e n t , l a t e r a d d e d b y F r a e n k e l , a n d f o u n d a t i o n .4 Z e r m e l o d i d n ' t f o r m a l i z e h i s a x i o m t h i s w a y , b u t i t d o e s s e e m c lo s e to w h a t h e h a din m ind , e spec ia lly i t s second-order charac te r . H is ac tua l word ing ( t r an s la ted in to Eng l i sh )w a s: "Wh e n e v e r th e p r o p o s i t i o n a l f u n c t i o n (Klassenaussage) F(x) i s de f in i t e (de f in i t )fo r a l l e lem ents o f a se t M , M possesses a subse t M F con ta in in g as e lements p reci se lythose e lements x o f M fo r which F(x) i s t rue . " (Op . c i t . , p . 202) . For the con t roversys u r r o u n d i n g t h e m e a n i n g o f def ini t here , see G. H. Moo re, 'B ey on d f i rs t -order logic:T h e h i s t o ri c a l i n t e r p l a y b e t w e e n m a t h e m a t i c a l l o gi c a n d a x i o m a t i c s e t t h e o r y , ' Historyand Philosophy o f Logic , I ( 1 9 8 0 ) , p p . 9 5 - 1 2 7 . A s M o o r e s ho w s, Z e r m e l o h a d n o t h i n ga p p r o a c h i n g t h e i d e a o f a f i n i t a r y f i rs t -o r d e r f o r m u l a i n m i n d b y a defini te Klassenaus-sage.E . Z e r m e l o , ' U b e r G r e n z z a h l e n u n d M e n g e n b e r e i c h e ,' Fundamenta Mathemat icae,X I V (1 9 3 0 ) , p p . 2 9 - 4 7 .6 Cf , ibid., p p . 2 9 - 3 0 .


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ITERATIVE EXPLANATION OF THE PARADOXES 59

7 Although I assume the existence of transfinite ordinals, the assumption is not essen-tial to the cont ent of this paper.a I say some cumulative hierarchy, since, of course, distinct collections of urelementsyield distinct hierarchies; so not all sets, indeed only the pure sets, will be members ofevery hierarchy.9 This way o f distinguishing sets from o the r collections was first elaborated by Miri-manoff in his 'Les Antinomies de Russell et de Bural i-F ort i et le probl~me fondamentalde la th~orie des ensembles,'L 'Enseignement Matheman'que 19 (i 917), 37 -5 2.1o More carefully, that every proper ty o f sets or urelements determines a collection,lest we generate a "collection theoret ic" form of Russell's paradox.11 N.D. Goodman, 'The knowing mathematician ', Synthese 60 (1984), p. 22.1~ p. Benacerraf, 'What numbers could not be', Philosophical Review 74 (1965),47 -7 3. Benacerraf confines his discussion to the natural numbers. What I say here is anatural generalization.13 IbM., p. 58.14 One might point out here that we often take convenience and elegance to be sufficientindications of the truth of a physical theory; why not hold the same about a meta-physical theory of what the numbers are? There are a number of problems here. First, itis plausible to think that a false but convenient or elegant physical theory will eventuallyhave to conf ront recalcitrant data for which it simply cannot account. No correspondingclaim can be made about the me taphysical theories in question. Second, a further, Ithink indispensable, criterion a physical theory must mee t in order to be coun ted true isfruitfulness: the theory must account for phenomena other than what it was designedfor, and it must lead to new, unexpected results. Now there may indeed be somethinglike fruitfulness in the developmen t of set t heory, though even that is open to question.But even if we grant that, I don 't t hink i t is of any help here. For there is no reason at allto think that the content of set theory would be any different than it is today if mathe-maticians had chosen some other co llect ion to serve as the ordinals; any recursive collec-tion isomorphic to the collection of yon Neumann ordinals would have been just asfruitful. This is hardly surprising, since it is the structure which both isomorphic collec-tions exemplify which set theorist's are interested in, not necessarily any particularinstantiat ion of it. Third, many phi losophers (e.g., Bas van Fraassen) simply deny thatthe convenience, fruitfulness, elegance, etc. of our theories tell us anything at all aboutwhat the real world is like. In the case in ques tion, I am inclined to agree.15 Cf. G. Cantor, Contributions to the Founding of the Theory of Transfinite Numbers,tr. by P. E. B. Jourdain (New York, 1955), pp. 1 51 -2 . Cantor, something of an idealist,didn't put things quite this way. He held explicitly that an order type is the "generalconcept which results from [a simply ordered set] M if we abstract from the nature ofits elements ..." (pp. 151 -2) . But, in good psychologistic fashion (for which Fregecastigated him in an unpublished review, cf. his Posthumous Writings (Chicago, 1979),pp. 68 -7 1) , he took abstraction to be a mental process which resulted in somethinglike a menta l picture Of M somehow minus the nature of its elements (cf. pp. 86, 112).I filter out this at best marginally coherent feature of Cantor's position and take hisgeneral concepts to be full fledged abstract properties. For RusseU's view's cf. Introduc-tion to Mathematical Philosophy (London, 1919), pp. 56, 93. I should note that Russelldiverges from the account here in his full blown no-class theory, where the ordinalsare propertie s of relations, not propertie s of sets. In a more recent publication PenelopeMaddy argues persuasively for the view that cardinal numbers ar

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