Generalized erdoös cardinals and O4

Ammls of Mathema~.ica[1.o,_4i::15 {!97g)289-313 © qorkh_-HoilandPttblishing (_2ompa~y

J a m e s E. B A U M G A I I / T N E R * a n d Fred G A L V ! N Received 13 May 1977

t h e J.,.~. strfldng applications of t h e t h e o r y of i:arge -~r.~i~_.,q,. l-s~, be¢~ Silver's w o r k with E r d g s c,:~_~ .r..~:. ~na~s, , ]2n t~u], r~ ~ h e p, _ roves tha~ if tiaere exis~-s ~< s~d-~ thai. K-->.(oh) < ' , t h e n Ihe u n c o u n t a b l e card_inais in t h e u m v e r s e of set t h e o r y are b M b c e r a i b ! e o v e r L ; in fact, in t h e t e r m i n o l o g y of Solovay [121~ x # exls~s '<~r e v e r y x ~ ~o. Silver also o b t a i n s ~-'pper a n d Iower b o e n d s in iem-~s of LL.t r .~*" &~, N cardinals for t h e H a n f numoe_ ' r for universal L<,,,.a.,-sentences. T h e ' o u ~ o s e of this p a p e r is to ge~aeraLze t h e ;~otio'a of ,'a~,.E r d g s catalina1 aw.] ~:hereby o b t a i n m u c h s h a r p e r versions of Silver's restfits, arpee~Itahy= we w-;!l f!ad ;} condition m u c h weakez t h a n ~<--+(~)<°> w i J c h wii! suffice to prove t h e ex[ste~~ce of :~:~ for e v e r y x ~ ~, a n d we w t i s h o w t h a t ia a s e n s e t N s is t h e weakes.t ;:~ossib!e s u c h large cardinal assunaption. W e will also locate ~.,',~%e~y.,..~, ~ the: H a n ! n a m b e r for .~.==~mc~ o~ =_.~.>= a n d ~,=.,.~ languages,. IY~,,=,.=, ,.=,~.. =o.,_=,... cardhla{ p!ay a role in b o t h resntts. T h e p a p e r is o r g a m z e d as foilows, a~c~.~u=~ i cor~iains a re.view of l h e " ....... Z) E r d i s cardinals. No~?e of t h e s e rest;its are s e w ; at! a r s e s s e a t i a l y ccmeained ~ ither in Silver [11] or D r a k e [13]. O u r p r o p o s e h~_h-~.c!uding tliem is to lay a fdu~!la,'_ion for Sectior~ 9 in which .q~e definition of }2;rd6s r.,~:~a ........ c . ~ h t = n i S g e= : n e m,'; , ~Ii,l,l :z = ~ =--+b c u m b m a ~ o r m d y a'_~d m e t a m a t h e m a t i c a ! v a n d sor_~e re<{~,fi%m~d ~)rob_~ems are gi,/e~ eo~ace:rabg t h e o r d e r i n g a n d slz== "~" . . of . these . .cardinals . ann" r u. s. .~. h. s c o n c e r n i n g t5 e sets ::e~: are in Section 3,
+

.

:

Sup~ose, .._,~ha+.9S is a s t f u c t a r e apprOnriate G _ _ for, firstLordir !igJ with m=~ e=:-,.:-,;" =#.... ,~.. If < ~s a f n e a r o r d e r i n g of a s~,bset of A a i d ; is ,C0~taiv.ed in tb.e field o f <'-: t h e n * "FI{e pre{;m'at~onof th~s pa-oer was ~mrtialb swoported ~>y Natio~mi::Sck~n~Fde/da~,aa gvmm~ GP-38026 and MC576-08231. i i:,i 11": :' : i i

,LE. Ba~mvgartne*.'oF. Oa!vh~

290

is said "e be a set qf <.-i~.~.discenfb[es for ~( iff ~ is infi[fite a n d for a n y formv.la ¢'(~h . . . . . ~J=) of t h e l a n g u a g e of ?'.[ -'
'~. > qo( i ......

i,D<--> ~(/ . . . . . . /,,).

Ite;a~ai&, (I) T h e ._,elation < :~ee'.d n o t be part: of tb.e 'similarity type of 7{. (2) No~:e that sets of indiscernibles m u s t be infinite. (3) W e f r e q u e m i y s u p p r e s s m e n d o n of t h e relation < a~d s p e a k only of sets of ladiscernib!es; this s h o u l d cause no c o n f u s i o n since t h e a p p r o p r i a t e reiat,.%n w N be, ObViOltS, \~4herea,s E r d e s cardw.als are usua]~ , defined combi[mtorially in t e r m s o{ partitio~n reiat!ons, t h e r e is a very natural definition u s i n g indiseernib!es: Suppose ~¢ is a cardinal and a is an in.finite ordinal. T h e n o t a t i o n ~<-=>(a)<"' m e a n s that for a~y structure ?~ with c o u n t a b l e similarity type, iX the.. u n i v e r s e of ~!1; has cardi~mlity ~< and is w e l = o r d e r e d by < , t h e n t h e r e exists a set 2g of < i~.discernibies for ?( st~eh ~ha~ I ):as o r d e r type a ~ n d e r < . A n equ{vaient defh~.itior~ is o b t a i n e d if the r e q u i r e m e n t that 5.~ h a v e cardiaality ~¢ arid be well-ordered by < #; replaced by ff~.e r e q u i r e m e n t that < be a welb-ordering of a :.~ubse~ of t h e -~.r:dverse of ~);[ with cardinalky <. T h e neg~,tio~ of a<-->(e.:).... is written ;<-~->(c~.)<~. Th,- iirst cardff~al ~< su." t ~?f has order ty~,e . ,. ex a n d J ;~.: c.m~e::ant e.~ i X } ' for each p. T h e set X is said to be ho~aogeneov:s fi~,,la}< W e write .<-o~(a) ...... as ;:t~ abbre-,iatior~ for a - + ( e ) 2 ='. T h e equivale~;ce o~ this delini~4o<~ with the previo,4s (me ~s d s e to F
:;--.->(a.)}~"% .~he.~ :~,' ......tc~{-~}:{:"L I~s#:,{. We. make use ef an idea essentb~!iy due to ~ax£b. -and R a d o (see C~~a~g sr, d i(.-_;is]~;r ~?.'~;the argmnent was £-.~s~formu;aled in iogieaA terms by SL~_~pso~a). S~;ppose ?{ and !g~ ale struetn~e~ a~d ~ { ~ o ~!t ~s sa~J to be a re.!~tivety s.'-sar~_~,~'(~adsubst~+u¢.'~ure of ~5 provided t h a t for ar~" £ in~sIsded in t h e ,anb'erse of '[q, if 1[- h a s p o w e r ~,, ihsn every type over X ;\,hich is realized io ,~13 is a.iready ","aH=
C, ++ecq¢ized ErdSs :ardina!s

2:* 1

N o w let f : [ u + ] < + ' " + A L e t 2{ be t h e ~',:~ct:~.re w k h universe ,, ~ a : d r e [ a t k m s < (the u s a a I o r d e r % g o n v +) a n d R,,~ f o r ead~ ~! 5a~ a n d i: < k , w h e r e .7~,,~ =_ ( g ~ , . . . , g~,;..~ ,.-~.q,,..., g,}) U s i n g t h e defi~.~_i~io~ <~; ~, (m~d tl~e weli~k~;ow:~ fact t h a t ~ m u s t b e ~mcountable), ~t is e a s y to s e e !bat 52!( h a s a retative!y ~ - s a t u r a t e d s n b s t r u e t a r e ~ of cardiaaIRy >. More~-,ver. w e m a y as~:ume timt ~8 is a ~ initial s e g m e n t of ~1 relative to < . H e n c e tb.ere exists a a ~ suc}~ t h a t a~R}, a n d w e m a y define a s e q n e ~ : , ::~ {a,, : ri < g} t~at a does. D e f i n e g o n A = {ae: ~ < ~-} by letting g ( : v ) = f i x U{a}) % r all x. Since ~¢-+(ae)2:°', t h e r e exisr.s X g A stash @at N has o r d e r t y p e c-:.' a n d X is h o m o g e n e o u s for b a t h [ arid g. (.TNs is ~-~ot c o m p l e t e l y trivial if A < ~ . DeK~e it : .[ A ] <~' . o g ~ a- e"~ m s f o r both f . -->.k as. follows. . . If x e [ A I s " iet h ( x ) = 0 iff x ~s .~. .e. m +w,+)= ~(Y), w h e r e y conchs'is of t h e first .~ e l e m e n t s u~ . . . . x+ . a n d g. If x ~ [ A ] ~'~-'~ !et +' "~'~ A n d ff :,: ~ [ A ~ let h ( x ) = g(y), w h e r e g is the first ~,~ elemcetg, ~A" x. ~ v ¢.:.; infinite a n d h o m o g e n e o n s f o r h, t?-ten X is h o m o g e n e m m for b~,~th f a n d g.) B u t n o w X m{a} has o r d e r type a + i a n d is h o m o g e n e m ~ s for f+ K +c is a cardinal, let 2; = ~ a n d 2~ ~~== 2 (=:) for a : ~i ~ ~. T~eo~.er~'~ ~_o2o ~C ~ is ~.he [eas~ cardb~ag s u c h thai .< ~ ( c ~ ) / ' ~ a n d c~ i.~ a ~imi~ o r d ~ a L thev~ f o r ~l~ ~ e o.~, (2~) + -~,(a + ~, + 1);[%

P~'o,s,fo By T h e o r e m 1.i, using ti~e w e i t - k n o w u fact that a: m u s t be str~nrgb/ inaccessible a~d u ~ e o u n t a b t e .

~°e.r.¢L L e t ':2 d e n o t e t h e set of alI ftmctions m a p l i n g ~,: into 2. F o r x, y < " 2 w~th x ¢ y, let 8(x~ y) (the d f s c ~ p a ; o t c y of x a n d y) be t h e ieast o r d i n a l i f < ~< s u c h t~at x ( ~. .). .~a v(~). L e t ~... .... be a w e l l - o r d e r i n g of ~2 in o r d e r type ~'~.. (_Jose 1: A is a a i n f i i t e cardinal. S u p p o s e f : [ ~ d < ' " - ~ X is a counteraxa~m~.~ to

~ <-- (ce''<~)a• D e f i n e g :['~2]°-+' --> k as follows. S ~ o p o s e .,o. . . . . . . x~ < - x 2 < " • • <" x~+~. ff 8 ( x ~ , x2} < 8(x:;, x 0 < ' • ° < 8(x~., x~ ,. ~), let g({x~, • . . , x ~ < } ) = ~ + f ( { S ( x , , .~<,), . ~ . , O"( X k l N k"+ ~ ] l . ) , '+"

y+~ ~.~. . . . .~ a ~ u

o t h e r w i s e ,~o* g({x~,.. ., x~.+~X)= . S u p p o s e ~,~ = ~: h~s o r d e r r<~e a + ,~ with. r e s p e c t to ~' a n d X ~s h o m o g e n e o u s f o r g. W e ,;vili oh:cain a contrad!et~o~a. ~.. e n n e ,~.~: t~.] - + 3 as foUows. K x, v,.. z e ~\ m~d x < ~ < ' z,

....

~(x, y ) < ~ ( y , z),

•, v > , y , - ~ r , } - - ~

~

,%%vl--,S(y,z),

k=2

if

~(x,y)>~(y,z).

:. . . .

B y Ra:rnsey's tiaeore:m there is a a infiiite m,mog~n~.e~s ~ ~, ~ e ~ set Y % r fi Let
292

.1.P2.Baumgarmc~,F. GabAn

then 6(yo, y,}>8(y~, y_~)>8(y> yz)> . . , contradic!ion, f h is constar,,i;!y z o~ [Y]:4, then 81yo, y~)= .6(y,, Y2)= 8(Y2, y j = 8(yo; y j , a cx:,~?tradiction. H e n c e H is coastm~tb 0 o~ [g]~ arid since ¥c:_-X it fellows that g is n o t constantly 0 on X~ Suppose g is constantly i + ~ on X. T.hen ;t is easy to see ~hat if {x~: . ~ < a + I) em~.merates X in b:creasing order, t h e n {~I(x~, x,,, ~): ¢<,cv} g h o m o g e n e o u s for corm-adicdom ~"~ ~mse _.)" A2., as fo]!ows. Suppose x~ < ' . . . <'xt:~ ,. Let g({x, . . . . . x~.+,}) : 0 if[ 8(aq, xe).<- " " < 3(x> x ~ :) and {,5(x~,:cz) . . . . . 8(x~ox~,+O} is h o m o g e n e o u s for f. Now s u p p o s e X ~'~ 2 has order type ce + 1 and X is h o m o g e n e o u s for g. A s in the ])roof of Case i, we can fi~d i~.,i-lnite ~'~c__~-.X. . .s,acb that w h e n e v e r x. vo z ~ Y and x < F.-:< ~.~ z, fhen $(x, y ) < 6 ( y , z).. L e t \Yk. ~ • k e: co) e,:mmerate the first e-~ m e m b e r s of Y, and Iet ~:=8(Yk, Yk4.0 for all k~{o. F i x p~_o~. By Ramsey's t h e o r e m there is Z ~ {4:~: .'~' ~ tel. ,~;uch that f is constant on [ Z ]~. for aii i ~~'e,v~~,,4L Suppose ~ ~ h' ,<'", ' where ~'~ is {~ Iimi~ ardi;-~_aL The~ ~br each ~ e ~ , (2~) ~ is ~he ieast carc~ina! ~, s~eh #~a~ ~,-+ (e~ + ~ -'-1)2"< ~:;:'~.<.,Q ~ y '-['heo}ems ! . i s~ ~ ,~.:>. us{rig the w e l l - k n o w n

f>~ci that ~-~", c.~ -~- i `<°.

the proof of whk:i~ is esse: daby coi,J:a~ned in ~he proof of "Fiaeorem 2.6 below. ~:Le~.e;-a~'~::."¢'he idea o[ using the d{screpar:cy function to prod~oce counterexamp]es to p~i~:tJdo~ r e l a t i o n s ]s d u e to EgdSs e~ ~!l. [4].

.';~,~ch i.~c~,' ~ . - s . ( o : + ¢ 0 ~ °'. ~u~d !.:_~t v be m.i~imcd such that v - - > ( ~ ) ( ' " .

T h e n ~-.-->

P~'~{:. Silver. [I 0] has shom~ ~:hat ~,--,~ (~)~:" f o r all A < v. T h e resugt fo]k~ws n o w t,y "Fheo~
]2f:~.{~,L 7 h i s ~:m~s oI~13' the fact t h a t if v is mb~h~a] such that v--->(,~0)~ "~, t h e n U" << ~.~ w h i c h is w e ] L k ~ o w u .

Ge~teralize,'.i i fr2ds c,~,~dhaals

2~3

Pzee.et~. Suppose e<-=~(G)<% and let f:[ee]'<"'-a~2, ii~efln: ~'e!atil.-':sfur tl-e structure 0<, < , R,~)..... £ d e a r i y h o m o g e n e o u s for ~: For the converse, s~ppose ~{ is a strtxcture witk countabk,, smvtarit~, t t p e and: :suppose ~{ has cmdinaiity ~. Suppose also that gt is welt-ordered by a rc~aqo~ ::: which is part o f its sim~arffy typel L e t the universe of ?,{.be A• i:)efine [ o~x [A~ .... by letting f({x~ . . . . . x~}) be the type realized by x~ . . . . :% Since <--~ ({~}7.'.~,there is a set X of type ~e such that any two increasi,,~g n-tuples from X real:ize the same type over 9.L But then X is clearly a set of '<-imdiscer~ibles for N. !t should be remarked that t h e r e is a simple direct proof ;>f Corollary t.6 which is unlike R o w b o t t o m ' s p~oof in that ~ ~s not req, uh-ed to be a limit ordinal. ! .ct "'2 d e n o t e the set of at] functions from o~ into 2 and suppose f:.[~¢]':'"--.+*'2. For each n, define f,::[,~] . . . . *2 by f , , ( x ) = f ( x ) ( ~ e ) . Now define g:[~r]<~'-~.2 as folk)ws: Suppose x~[~¢]'L Then let g ( x ) = 0 iff j~ is constant on [x]' for a~i ~ , i < ~ . By ~¢-~*f~)['', there is X of type ce whicl~ is h o m o g e n e o u s :fox ~.~ _,,~ ~ will s~.-~filceto show that g ( x ) = 0 for aii xe[,~]<% >'ix ~-~~Y Ramsey's t h e o r e m to obtain an infinite set y c X such that R)r aH i a~nd j <: ~, .f~ i:~ constant o n [ g ] ( But th.en if x ~ [Y]~ we must aave g ( x ) = 0, s o by homogeneity of X g is 0 on [X]". H e n c e ; < - ~ ( a l ~ ' .

tn Silver's proof that the existence of ~<(oh) 9nplies ~d~.eexi,~te~>,.'eof (K~, ~hc: ::ac~ that uncountably many indiscernibles can be fom~<[ is really ne~ reflected; il suffices to k n o w only that sets of indiscemibles of every o r d e r {y[;e <,~-~.h ca~ be found, all of which satisfy the same (linguistic) type ox.er the given atructu.~'c, To make this m o r e precixe, suppose that f is a set of <-gndiscer~,~ibles over a structure '9.. Then the 7~discer~ffbiS~y ~ype t of 1' is de:gned as the set of a]! formulas (p(v> . . . , t;,) such that for some i~, . . . , i,, ~ L i ~ < . , . < i , and ?[t:¢o(i~ . . . . . i,,). Let fJ~ be tl~e least cardinal such that ff 92( is a we~l--o:rdered structure cf ~.>ov,er ~,. with conntab!e simi.larity type. then there exists ..f such that b.~:r eve:~:y ~ < <,h ti;ere . . . ~ . has order t~.q~e ~ (wh:b rcspec~ t', . [' ) C is a set f~ of ind~scernibies for. 2I such thgt f~ well-ordering) and indiscernibNty type t. V~&at S.iN;effs proof real!y shows is that the existence of ~J, Lmplies the existence of 0% ActuaI!y, it turns out that a hypothesis m .omc~ " ~ weaker than the existence Of # will suNce. A t this point, lnowever, it is probably not obvious even i:inat p. < < (~e:), ailhough it is d e a r that ,~ ~< ~(~,~). i n a~aeking this probiem~ we are led rmturcdiy ~.o the followir~g definitions. Si,~ce we ahvays work with countable sh~ai!arffy type ~;, it wili be use~'~l to fix a u~iversai countable language ~ which has infin!tely m a n y ~-ary ~tuiatiot~ and operatio~ symbols for every ~ e < < W e m a y . a s s u m e that the set ¢~f (G;Sdei mn~nbers of) fomlutas o f ~ i s recursive. L e t T b e the set of N ! md~s:::ermbfi~tS, t.sq~es m recnrs~ve sublanguages c¢ ~ . '

294

,f.E. Baumgarmer, F. Galvin

Now s u p p o s e a m a p s 7 into the infi~tite ordinals. T h e n o t a t i o n .~<-->a m e a n s that for a n y recursive s u b l a n g u a g e ;~' of ~g' and a n y c ~ ' - s ~ u c t u r e ~, if ~i h a s p o w e r ~< a n d is well-ordered by a relation < , t h e n there is a set f of < - i n d i s e e r n ibies for ~ such that I h a s o r d e r type a (.0, w h e r e t is the indiscernibflity type o f L N o t e thai if a ( i ) = n : for all ~, t h e n ;<-~>a is ibe s a m e as K-->(c~)<°L T h e !east cardinal ~< such that v,-.-.a is d e n o t e d by ~<(a). In order to ailew for different ways of coding t h e s a m e l a n g u a g e , we will usually restrict o u r a t t e n t k m to functions a which are recursivdy invadam; m e a n i n g ~hat if t~ is T a r i n g eq~fiva!ent to ~2, then a(£~) = a(@. ,~, function a is order-.preserving if t~'~--r¢2 implies a(h):~a(t?.), w h e r e ~-~T iS Turiz~g reducibility. N o t e that all o r d e r - p r e s e r v i n g functions a r e recursively invaria~t. W e say ~z i~, Iimi~ if o~(~) is a limit ordinal for e v e r y t ~ )2 E x a m p l e s of f u n c t i o n s d which are o r d e r - p r e s e r v i n g a n d ]imit are: o~U) = t h e least ordinal ~ot recursive in ~, a n d a:~U) = the least o r d h m l n o t k~ in .a T h e results in t h e rest of this section wilt s h o w tbat for every ,::~<a we m e a n that for any ¢ : [~,~11':~" --> 2 t h e r e exists .V ~. < ar;d s c-~'~2 s~ch that for all ~ ar<~ ~H!. :c ~ [X]"~ f(:<)= s(~'~) and the o r d e r type of X [s a(s}. Of course X is homogeneo~.s for f; t h e function s is a eornbiimtoriai aTmk)gae of the indiscernibility type of X, W e refer to s as the trace of X (~ela~,ive ro fl. T h e definition,s of recursiveiy i~_variant, order-preservfng, arid limit functions are easily e>:te.,ldcd to include functions o~ ~'2. W e also use ;~(a) to de,ante the least ~< sud~ tha,: a:--> c~. For arbi:~rary f~..mct[on:~ a on °'2, t h e defini~ion of ~;-~-a is surprisir~gly [ntractao hie. For exampie, v d t h o u t f u r t h e r asse.m~tioas on a we are u n a b l e to prove t h a t ff ~ is the least o~z-ifi~ai such that .~-->a in t h e obvior, s sense, t h e n ~ m u s t be a ce for worry, since in the rest of t h e p a p e r we wilt. be i~:terested priniarily in t.~se cardmais ~{(a) w h e r e a is c,1 _le>-pre,serw~g. T h e results i'n th,~ l~st g~r~: ~.>fthis secfior~ s[ro~gly s a g g e s t tha~: the best generatizatio~ ~f E~'d6s car(finals is give~ by t}-e c,rder-pre-servie.g functions. See especially T h e o r e m 2.16. it is not essentkg to use t h e 'tuft,as partial ordering ~.~- in t h e definition of mder-.preserving functions. T h e proofs s h o w that a w e a k e r o r d e r l g g w~li work, b a t there e;.~:~ernsto be:: n:o point in trying to define this minima1 o r d e r i n g explicitly. A n y ordering s t r o n g e r t h a n the m i n i m a l o n e witi w o r k too, as long as t h e bet ,0f predecessors of each e l e m e p t is countable mad every countable set of e l e m e n t s has

Ge~*era~izedErdiis caedh~uL~

2 )5

a n u p p e r bc und. F o r e x a m p l e , d?e ( ~,'iering -~, v. 7! work, w h e r e x ~ , y iff x ~s A ~, in y. h s h o u i d b e r e m a r k e d , h o w e , er, t:ha~ the res.ults in Sections 3 a~d 4 r e q u i r e t h e o r d e r i n g to be definable in a r e a s o n a b l y ~:~imp]e way. in t h e rest of this section, e a c h time a cardina! ~¢(a) is. me¢itioned ir~ a t h e o r e m t h e taci~ a s s u m p t i o n is m a d e that ~¢{a) exis%, i.e,, ~hai d~m'~ exists ~< such i:hal. g< -'-> a .

'ghe~a~ex,~ 2 3 ° Suf, f~ose a is a f~,~c~ioa:~ ml "'2 a n d ~ ~ r a n g e (a)° 7 h e ~ ,: % >-'-*((~)<% P r e e L S u p p o s e ~ = a (s) a n d [ : [~ ( a ) ] <''' --> 2. D e f i n e g : [~¢(a )]<~' --~ 2 as fotiows, if x ~[~:(a)3 '*, let g ( x ) = s ( . ~ 0 iff x is h o m o g e n e o u s for f. By ~ ( a ) - ~ - a d~el:'e exis<,~ X c: ,,< a n d s'~'*'2 s u c h ~hat g ( x ) = s ' ( n ) f o r atl x ~ [ X ] '~ a~d ali n, a n d X has type: a(s'). W e assert t h a t s" = s . Lct ~ ~bto~ : . . . . . . . .oi . ~p,, o c c u r a m o n g vl, • . . , o,,. , ,_. .x,o w ¢lehnt: ~ : ]: : [~]<~ --> 2 as follows. K x e [~]2,, for s o m e ,% t o t / ( x ) = 0 if[ f o r aii i < ~ any t w o i n c r e a s i n g s e q u e n c e s f r o m x (of the a p p r o p r m g e ~.<°.,o, e~se b o t h satisfy t h e p,e g a d o n of 9~ If x e [~<]>~+~ for s o m e ~,~,let f(x} = 0 {ff s o m e i n c r e a s i n g se(iuence f r o m x ~,a~sue,~ :?,. N o w c h o o s e X a n d s satisfying "~he defi~itioi~ of ,,<---~.~?. By a r~ow i'm~-li~ia: argui~nent, w e n t a y use ~ a n 3 s e v ' s ~ h e o r e n i t o o,htai1~, f o r e a c h ~, ;~ ,5¢2~ x r= ~:-~

~ ' t ~ ~ ÷~,~.a~ ~- f(x). = O. H e n c e s ( 2 n ) = 0 for #.! n a n d it ~ii~,.!ows- tilat /< ~s a se~- c,f --pa'eservhag arid equiva1-::.~t. H e n c e X has t y p e ~>a(s) so ~--.--+a : '

296

1.2£ Baumga !c~er,F; GaIoi~a

In view of T h e o r e m 2.2 we m a y regard o r d e r - p r e s e r v i n g fu_ncdons as being defined o n both " 2 and 2-: S::ppose c and b are h m c t i o n s f r o m " 2 into t h e ordkmls. )is t h e r e a n y simple way to tell w h e n an(a)< ~,:(b)? M o r e p r e d s e l y , if a a n d b h a v e rar~ge included :n el::, say, is there a relation definable in t e r m s of sets of tow r a n k which, is equivaler~t to ~:(~,~)<:<(b)? W e he.re ,~een ab!e to find several sufficient condO< o n s , b u t n o n e of t h e m s e e m s to be necessary al.~.o. L e t us s a y that a < b iff t h e r e is s such that fo-, a~ s' >-,.s, a(s')< b(s'). T!~teer'an~ Zo3o < is a we,.;L..fou'~dcd partial ordering of ~II fu.ncfio:ts mapping ~"2 into

zhe ordb~aM Tbeerepa 2°4. ~%e Axiom of Det,:rm.inateu~ss impgies that < is a pre-~e~Iordering

of the ~rcu~siuely incariaut f,dncfions. Pa'e~fL T h e o r e m 2.4 follows f r o m a result of Mar~,::~ w h i c h asserts that if the A x i o m of D e t e r m i n a t e n e s s holds, ther~ every set of T ~rh~g d e g r e e s eRher contab.s or is disjoint fror_,: a set of t h e form {s': ~,~-:.s'}. w h e ' e s is the degree of s. 'Fhee',re>'.~ %5° .{g:~ < b a~sd a in.: orde:'-£~reser~)ir~g, d:e~a ~<(~)<~<(b). Pu~,e;!o S u p p o s e ~<(a) -~ .,-:(b). For each ordbml ~ < ~(b), let ~) he a com',terexample to <~-~.a. Deii;~e f : £ a ( b ) ] ......... 2 so that ff ~n < ~ < " " < ~.,, then f({6, . . . . . g,}) = Now s:;pp~se s a ~ 2 :is such that a(s')',~eneous for J;:.. If ss is t}:e trace of {'~?e X: .1 < ~} @en ss~.rs::~.rs> H e n c e ,'.~(sO~-..a(s ). :0"* {.r~e X : ~,? <~} has o r d e r type >~e(s~), eomradfcting t h e fact ~. ~a~ Ji w:,s a cou::terexample to 4-+~0 :~:&'.
Ge~u~.~,dh:ed />J,;5~ caMb~mls

297

%~me~'e_<~,~2o~° &repose a(s)--~a(r(s)) for ati s<'<~,, ~f a , ( s ) < 5 ( s ) S a t at~ s, ae.~*

P~,'ee,.~, Suppose .<(a)>~<(b). Lee ¢~ and f be as in the proo{i of F} o r e m 2.5 and specify ~ha-~f(0) = 0. Let :c be h o m o g e n e o u s % r .f with trace s, a > : ~,~q-~(~<::ase{~~ X has at b a s t a(s) predecessors in X. The~ {re ~ X : ~ < g} is homogeneo~,'~, 5:,~ {5 ~**~.d if ks trace is s', then ~ ( s ' ) = s. Bat since a(s'):s; a(r(sg")= a(s), this contradici::, '.b'fact that re. was a coanterexample to ~-->m Ete~X~o (I) T h e o r e m 2.6 remains true ff ¢ Js any function s~ch that r(s)iaQ d e p e n d s onty o n s(0), s(!) . . . . . s ( n - t ) . However, a simpk: diagonai argmx~-~:nt produces a function c~ for which there ~s ~_~o such r satisfTing the ame~ded hypothesis of T h e o r e m 2.6. (2) Anod,.er shm.pb diagonal a r g u m e m wiii produce order-preservk~g a b. c s~'ch that a < c but a < b < c . Thee. ,<(e)<~(c} so either a'(a)<.<(b) o r a ( b ) < ;~(c). E i t h e r case gives a~ example of order. 2reservh~g a', b' such that ~d-'-~')< x (b') but a' < b'.

s e~"2, and he~ce a ( a ) ~ , < ( b ) . Mo~'ee~:er, d' a :°'2-->*h, ~hen b may be ch<~ee .~o ~hat b :~2-~w~ also. A similar" ~'e'm.dt holds )Cot o and b defined ¢ ~'~ T. .~aree~o Simpiy Jet b(s) = sup {a{s'): s' "-%-s}. The~m~oe~ .....~o ]f C is a set o[ f~nc~io~s such rk!:'t a .z.-->~ .for ,.J~.a ~ C aF~d Chm; ca~'dinality ~-T", then there is mder-presen~ing b :"2-+,:0~ s~ch that a! < b f:or eve~7 aeC. ~r,~e,~ W e may assume <2 has card~a!itv 2"L Let C=-PA: ~ ~ ' 2 ~ ~~-~'~.~. a nmps 7: inte ah}. 7'o see the rest, note that i:[ a and b are defi~ed o n T a~d ~2 respe¢Ibe12, t[w.~ by Theorem_ 2.7 there, are order-preservhi~:a ; , ~ ~h' ,s0 t h a t a;~'.t)>~ ~(t);f.or ~,-"~i~,~.. ,_ ~and b ' ( Q ~ b ( s ) for aii s~'~2. H e ~ c e a ( a ) ~ - ~ ( a ' } and }((b)~.~(b'). Bz~t ~0~e by

T h e o r e m 2.8 t h e r e is o r d e r - p r e s e r v i n g c such t h a t d , b ' < c . H e n c e by T h e o r e m 2.6~ ~(a'), ~ ( b ' ) < ~(c). T h e r e f o r e s u p { ~ ( a ) : a m a p s T into ~ } = supl~'<(a):a m ~ p s ~'~2 i n t o ~e~}.

P~:eof,, By T h e o r e m s 2.6--2.9. i,?1,~-~.z'i~° W e n o w have the i n f o r m a t i o n advertised earlier. L e t a~(s) be ~he least ordinal n o t recursive in s, a n d let a~(s) b e the least ordinal n o t A~ in s. T h e n b y Ti-~eorem 2.5, ;¢(~)<~(a~)<~,:(a~) for at! c ~ < ( ~ . By T h e o r e m s 2.8 a n d 2.9, -~(~2)< ~_:~.Since ~((;~) is stromgty inaccessible, we h a v e ~, < ~(.e~b) by "[)]eorem 2.10. It follows f r o m u ~ p u b l i s b e d results of L a v e r a n d t h e first a u t h o r t h a t T h e o r e m :2.10 c a n n o t be, improved. Fc.r L}st;!mce, givea a c o u n t a b l e tra~sXive m o d e i of set theory in which it is true that :;(a,~) exists, h is possible to p r o d u c e mod~/s J,f~, i ::: 2, 3, 4~ such ::hat in .,~f~, T% = H~, 2 '~, = ~'~4, a n d cf ~ = X, T h e rest o!: this sectio~ is cono:~rned with ~he sizes of £he cardinals ~(a), pari~cular!y w h o a a is order-preserving. If a is a fu~;ct:iop, or~ " 2 o r T, let a ' d e ~ o t e (telnporarily) t h e f u a c t i o n .;uch that ~'(s) = c~.(x) ff ~(s) is a limit ordi}?al, and c~'(s') : o.(s).k 1 otberv~ise.

P~o~!L T h e p,:oof is a variatio~~, on the proof of ' ] ~ e o r e m !.3 so we oaly outline h. . . . . . . --'~ ~ ['~]':~---2 as follows: if [-: ~. . . . . . ',Z'. } ~: [" 2]':'" a n d z ~ < " ' • <::' x z . ( w h e r e < ' is w e l ! - o r d e r i ~ g o f ~ 2 [n t y p e 2'). i~:t ~,tix~ . . . . . . . x : , , } l = O iff 8 ( x ~ , x ~ ) < , 3 ( x ~ , x 3 ) < . ' ' < a ( x z ...... x?..) a n d { S i x . :q, ~): l '-':&i ' < 2~'} is homo~m.neous f o r ]: I f [x~ . . . . . . x ~ , . ~} c- ['<2]:" '~: a n d x~ < ' . . :::' ~t,,. ~, le~ ;',d{::: . . . . . x~,, ~ ~}) = 0 ifi for some y

[]/~(x. xl, 0: t :~a ~--~;2~"~}]f', f(y) = (k L e t X be hor;~ogeneot~s for g w~th trace s and ,~rder type :;/(sL L.,e~0,'. : ~:~< a'(s)) enume~.ate X. As ir~ ~ h e p r o o f of 1'heorcrr., t, it turns ou~ ~hat 8(x., x~,., ~) <: 8(x;.~, .x~ ~,) whenever a' < ~, and :* ~ .on,.,c.e,.. . . . . :~ for Lf. H e n c e the latter set b_,ab trace ~' where s'(i~)=x(2,,zd i), and ft h a s ordur type a'(s). B u t now s' is Turi~?g equivale;tt to ~. '.:n (~'(,~) = a'(s'), a'~d ~his co~tradic~s the !~ypothesis about: j:,

0.re pairw~se disioi;~t, e a c h S~, has o r d e r type ;r~ for s o m e ~,~.< ~<(c~), a n d if e~ < ~,

G e~dralized Erd,i;scaa'd#a
2}>?

then every e l e m e n t of S,= pre'cede:~ every ebmeaxt of ."b For each o:.2 be; a eounterexampte to ~c,:,--:< and i e i~ :~A] <'~'-~'2 be a corn>. tetexampte to a - + a . Define g:[~(a)]<~°-->2 as R.~{!ovs: Suppose x~i)~(a)] :>. Then let g ( x ) = 0 ~g e ~ h e r (I) all e!emen~:s of x lie in the same S~ an:~ x is hemage=~eoas far i~, or (2) eacl~ e l e m e n t of x hes in a different S,~ and [~3: x N S~::/(" ,~ iioraogeneous for i . Suppose x ~ [ ~ ( a ) ] ~ + k Then iet g ( x ) = I] iff either (1) h."*!ds a~d %i ~ :.omy e[x]% £~(y)=0; or else (2) holds and for ~orae y~[{(?: x ns,~#0}]% i ~ },i: 9. S~tppose X is h o m o g e n e o u s for g with ~raee a and order type a(sL it is :~mdne ~:o check that either X is homogeneo~is for some j~ with ~zac,~ %.~ring equivMe~t to .% o r eise [X C] S{.,i~ I for aii ¢3 and then {fi: X im,S~.~# 0} is h o m a g e n e o u s for ~k wi~.h trace Turing eqtfivIaent ~o ,v. Either case yie!ds a :ongradictiom

s*ro~ggy i~mecess~bge. ~e~o By T h e o r e m s 2 . t l a~d 2 . t ? L*.~ . this. c ~se . the . a ' oi~. 7'hearern ~'~.~.l coinckh;:.~ with a. G{ver~ a ~function a on ~"2 and . . . . . %-~- . . . . . a(s)+n.

° ~

by '

.r,. . 234° sup {2~': ,,
' n}(x)=

::

T h e proof _~s o m h t e d , si~'~ce it is only a . m g n coinpheat, on of the proof of Theorem i.i. ....

g=

L,

.

•

~(~ + ~) = (2;0 + <-. Thee~'em 2.35° I f a is a~ order-presevoi**g jhnc~.iem fl~e~, ahe,~e is e.~ ,,>0N~:>prese~'vb'~g Hmi~ .f~mctiml b s',~ch *~hag fo~' some ~, a(a) = ~(b + r,.). ~_~;a{:[. Let b(s) be ti~e largest 1Lmi~ ordinal ~ s,~eil that ~<<-a(s), W e t b: ~md the a,,~ are all order-preserving; and ~<(b)~-.g(¢,,) v{ g 0..'],)z-~i-~(f~.) whenever }~ ~ e:o~. h re!lows than there exist m and ~ such th~i~ ~<(~b.)= ~< for alI ~ m m O1' c,~i*~ie g<(b) ~ ~, %\% d a b n ~<(b) ~<. Suppose .~(b) < ,<° For ead~ i ~ ~;%let../~ : ~{b}] ::"~~-:0.;}: be a coumterexample m : ~;(b)-~>a~. For ~ : m a n d ]'
t

300

]2L Bau ngdrtner F. GMvin

set of indi,s'.;ern~bles for 2~ w h h ~ndiseernibi~ity type t and o r d e r t y p e b(t). N o w f o r s o m e 2 ~ m, a~(t) = b(~). (Reem~ 0?at Mnee a~ i.s o r d e r - p r e s e t , r i n g w e m a y a s s u m e a~ i,s defined or~ bo~h ~'2 ar_~d T.) Clearly X is h o m o g e n e o u s % r j;, a n d h s trace s relative to .~ is r e c u r s i v e in t. B u t X h a s o r d e r t y p e e~ ( 0 ~ a~ (x)~ contradJctiag t h e c.q(4ce Of

fi-

~ •

W e cond~Me with a proog by h ' d u c t i o n o n i ~ n m t h a t ~4(a,,,_~)= ~¢(b+]) f o r s o m e j. W e are a k e a d y d o ~ e in case i = 0 . L e t m - i . = n a n d s u p p o s e ~(a~)=,,~(b + ] ) . If ,~-(a, _,,) = ~,:(a,,) we m e d o n e so s u p p o s e ~{a,, .~) > ~¢(a,~). If j = 0, then by T h e o r e m 2.14 (and C o r o l l a r y 2.13) ~do~,)+--.'.%-i-1 so ~(a,,)+--~a~_~. H e n c e ;<(¢-,~_~) := g,:(a,,) -~ = ~:(b e t). N o w s u p p o s e .i ---~>I. T h e n e<(a,,) = e<(b + D = A + f o r s o m e )~ by C o r o l l a r y 2 . ! 5 . N o w ;q-/:,a,~, so s'x~C>af, w h e r e a~ ~s as in T h e o r e ~ 2 . 1 i . Since a~',(s)~a,~_~(s) for ail s, 2%¢a,~_~. H e n c e ,:(~,.~.~)>2 a, O n t h e o t h e r hand, by T h e o r e m 2 . t 4 , (2"~)-"-~¢~ + I, so (2a)~-~.a,,_,. H e ~ c e ~<(a,,_0 = (2x) ~ = ~ ' d b + ~ + !). T h e folLowi~_~: t h e o r e m ae~era!izes $ ~ v e r ' s p r o o f [10] t h a t .(~(~)-- t~)x A < ~(a:) w h e n c~ is a !h~ait o]:dina!.

P;_'(~eL ~'b; c~ a~d c h o o s e f so t h a t for a!~ -~'~c,., a(~ , ~ e .

v,,~/~ , . ,, ,~),,): g ( f ' n . . . . . .

*~,~}) = : O f .

Let

for all

L e t f:D-~(a)] ...... ~A, a u d

~?~:(~<(a),<,R,.S.)

..... . C h o o s e

a set ) (

of ~indisce~.bies ~or ~i wi~i~ ~dJscerMbJlity ~ype t' a n d o r d e r type .~( ~ ~, By m e a n s Of o ~ e of l~. .i:¢ t_r~cx~s :.a., i!sed eavlier h~ t~i!s - section, w e m a y ensm*e h a t ., ~ r ~, ; s o W e m~s!- shov. ~.]~at ?,2 is homoae~,eot~s for fi L~e t ( ,,~ . se < a d" ' )/, \ ~, . r . u m ~~," ,_. . a t . ~ -. . Y in i:~cr~ £&~g order. Ca,~e 1: For a~t n <(o, .t({x~ . . . . . , x,,}) =.{({a;, ~ . . . . . %,})~ ~ this case we are . e ~71,, do~ae,. f o .r i~. ~:,. y .~ i / ~ a m mbit~aDz. a n d z -c [ X i " :is ci~osen so ~hat ~:al ",~ .~.a i"~ eiemeP~ts are i~::ger t h a n "~;~e~e!emer~ts of x U y (whiei" ~s possible s ~ c e a ( # ) is a t 5 ~ t o dh~!~[i, t h o u by h v J i s c e ~ ' M b i i i t y f(.~)= f ( z ) a n d f(y} = ¢'(z), so ~ ( x ) = f ( y ) . _. o r 2: F

C,.;~s~z

some

. . . . - . ,. . . -

•

~
"

•

-

~

~'~ •

j'({x~ ..... ,-2t~

f/

.

r x ~ , } ) . -7": f~(-.~,..~ .......

.

~" B ~ : ' f ( f ,

.

.

'

., _ r . ...:~m}}.

F

,~ 2 ~ A .

fir~-t

s-appos~

-, & u + ~)}. T h e ~ i~or e a c h ~ < : a ( ~ " ) ~

~et

. . . . _¢(r,. ,)4 By ~nd~scernibii~ty o~ J(, I / = i v , : ~ < a ( # ) } ] s a set eli ==~. }{%'ri()e }"" is [~o~s~r)i~eiieoas f o r Ig a~;d leas o r d e r '> ;'*~;~~...,.,,;;~ . . . . . . ~.~, :: ' ub " a_~sO. ' @ ~ eai,zt )r sluice ~(~').... is iirMt. S~.~- the. ~race s el i ' rc~_adve ~(,,~ .>~ is clearty ~ecursive in t', s o Y h a s o~:der 1~.rpe ?~.q(s), co~?trad~ctn~g ~:~e choice of g.

~ '. <,,.rd~.ze,~ [~

&,s v a r a hat~

30 t

T h u s , i o r e x a m p l e , ff a~(~) is " ' . ~..... " .... a,~ ;¢(a:~)-->(#e)7.]'" f o r all ~\ < ~ ( ( a 0 and all e: <.~,~. A s in S e c t i o n 1, we m a y o b t a i n ~he f'o[iowh~f5 ::xt<:as,ou: Thee~'e~¢~ 2o~8o Suvpose a is orde>prese:.-°vir~g a~d timig. . . . . . . . ~ ( a + . n ) ~ °' ¢"~r agg a <~'~(a) and agt a: c=r a n g e (a). T h e r e are still m a n y o p e n p r o b l e m s . T h e fettow[m~,~ in pardc~fiar, ~'~,,.~ ,~, :~,.:-, mentioned. P~,re~b~e~ i.o Is it p o s s i b i e to c h a r a c t e r i z e {(a, b): a, b : " 2 - e to, a a d e; ( a ) < ~;(b)} in t e r m s of sets of low r a n k 2 D o t h e r e exist m o d e l s ~ff~ a n d ;~{:, of s e t t h e o r y suci~ t h a t ./d.~ a n d J~:~ah a v e t h e s a m e real n u m b e r s a n d t h e r e are e, b e_.,fit g~.% s u c h t h a t in ,~fft, e < ( a ) < ~ ( 5 ) while in d42, ~<(b)<~e(a)? P~zeble~e 2o L e t a be al~. a r b i t r a r y fgnctio,~ o n ~2. Is e a a cardinal?

the least

P ~ : o ~ e ~ 3o Is T h e o r e m 2 . 1 6 t r u e f o r a r b i t r a r y f u n c t i o n s ?

~n this s e c t i o n and t h e mo×i we ~--:~'~ ~vm .isolate., a. c e r t a i n d!ass of f a n c t i o n s a ~ d c o n s i d e r a p p l i c a t i o n s of the a s s o c i a t e d gene~a~aze.~ . . . . ~':-" c~ F,'dA< ._........ e a r d m" a ! s . i , , ; o . ~ , a.... vy [61.~. w e say t h a t a f o r m u l a ~ of a set t h e o r y {s 2",7. ff e; is of {i:~e .F ~o~.o;\.a~ form

W ~ e r e e a c h Q ~ is a b!ock of e×bstenfiat q~a~tifiers, e a c h O2e÷~ ~s a ~.:o~.. of u n i v e r s a l quantifiers~ a n d t h e o n l y qum~tifieps f.n ~ are b o u t < l e d quantifiers of the f o n n ( g x s . y) a n d (:2x~ y). A f o m ~ u l a i s D ~ >: '~' ~.~s-'÷~>:~a:adon~ is .~,,, ~' L e t H C d e n o t e the coiiectiol~ of h e r e d i t a r i l y c o a n t a b i e set?,, A ~ -~s<_ (resp. ~<.,). iff X is d e f i n a b l e o v e r H C by a ~,,,v (resp. f;~). fom~.~fla ,a,.th o:a r m L , . t ~ , f r o m H C . A set is &~ [:f it is b o t h Z,, a n d 17,,. A r e l a t i o n £', c = ' 2 x ~ is f~,~netionag iff for all s a ° ' 2 tlaere exists G (m,. T h e m a i n r e s u l t of this s e c t i o a is t h e f o l i o w i a g . . ~ , ~ e : a ~ oox, a:appo.~e tha~ for eeerv E~ fi{~l¢~ona~ R x ~ to, x v exist,%

}{CAR) exisa. The~ ~o~."ad

302

J.kT. B,.aumgam~ee;I;: G_alviii

Before provi~g Theorea~ 3o I, we wiil n e e d s o m e facts a b o u t t h e f u n c t i o n s a.~. A functional re.,a
2~ and c~u.=as. P~:.e~£ T r i v i a l q2he~'ela 3.3,~ Le~ C be a coungab[e co[tectima o[ N~ flmefional m!atio~s. Then ehe~e is )-~ )%mctfe~al S such t~at a s is orderopreserving and limi~ and for all R ~ C gin.s! al.I s, aMs) < as(s). Pr@~~f. By T h e o r e m 3 2 we m a y a s s u m e that a]t t h e relations in C are closed u p w a r d . It is w e i l - k n o w n (and easy to see) t h a t v sets are ciosed u a d e r co~ntabh; . ~ e . . . . . . ~om,. L e t = I~C. The~i S' :is Z{~.arid functional, a n d a , a ( s ) ~ a s , ( s ) for all R ~ C a n d all s, L e t (s, ce)~ S iff l'¢s' < ' ; , r s ) ( ~ ~- a ) ( s ' , B ) ~ S') a n d ~ is ] h n i t Note that b o t h q u a n d f i e r s are b o u n d e d so !; is N,. M o r e o v e r tts is c!eady erda)rpreserving, a n d ~s.(S); {~ o r T x {o2Let ~*~ = s u p {,'<(a~ }: R is 2 t a~?d h,'-,:c{ional}. By T h e o r e m s 3.3 a n d 2.6, we have

b: f©iiows that ]<. < g. ~n fact, as v,,e shail s h o w later., if a:,(s) is t h e ieasi ordir~al ~-~oi: A~, in s., .~hen t.c~< g < ( a J . Q ~ d~,~; o i h e r harM, if a~(s) is t h e leas~ ordinN n o t recursh.e in < ti,e~ it is easy to see that t h e r e is a 2:~ [mmtiona! R w k h a~-: or,,. For ~e¢: at d'.,<-set r / defined by SoIovay in [12] is essentially an Mdfscermbi~ky '4,pc o v e r ~he structure {L[x], {, x). T h e feiiowing s t a t e m e n t s are e q u i v a ! e m (see [!0! a n d [12]): (1) .C" exists. (2) T h e ,m?eomatab~e eardiaals in '~<, the universe of set theory° f o r m a p r o p e r class of < .bdiscernibles for (L.[x], c:, x), w h e r e < is t h e u s u a i o , d e r i ~ g o n 1lee ordb.a:s. (3} For s o m e ordinal/3, th<~re is an u a c o u a t a b t e set of <-indiseerreiNes for the s~.ruclure (L~[x], e, x), Where < is tt~_e cai~onk:N w d [ - . o r d e d u g of Lr.,[x]. N o w we begin tfm proof of 'Theorem 3.~. W e witl act~m!!y p r o v e (3) for x = a iea-,ing it i:o i h e r e a d e r to r e b t i v i z e the proo£ for arbitrary Xo S'.::p~-~o?e ,' satisfies

Ge~eralized ~rdSs cardinals

3{}3

(4) t is an ~ndi-:;cen~ib~itytype in the iang~age of se4 theory w h b h eon~ins all the axioms of v n blow if V = L~ then 5 k o [ c ~ '~ :<,"q-',~s f o r all formtflas are de!~mMe. W e ~ a ~ assnme, t h e r e f o r e , that if ~ satisfies (4}: } m ~ there a,. ~yrnbo.ts ava~!abb to denote all terms o b t a i n e d b y com~osing Skoh~an functioss. O f course; a.%, f o r m u l a bd w h i c h s u c h t e r m s occn:r is equbvalent to a for~r~nia iri ,otving orfiy <. K t satisfies (4), t h e n fer every ordinal ce >~-w, ti~ere is a canonical structure C(t, ~} defined as follows. I e t & - ~ < ~} be a set of n e w co~!~mt:-:, ~:.,~,.~t,~-le~' ~J;.~"be d~e coi~ectlon of all t,erms o b t a i n e d by applying" the defhmb!e n'"":.~,<~-,~=~ ~~c~,o~,~* '~ to t h e ce. Define a n equivalence relation ~:~ on U2~'-"by s~.~.ag'~;~i¢ { c ; . . . . . . . . :. ] ~ %(c b . . . . . . % ) if ~ , < . . . .
belongs to L ]Tt is readily, c~ecked that the structure C(t, a ) is weI!-defi~ed, h4o~eover {[c~]: ~ < e.~} are a set of <-i,~discernibies for C(t, ~) with ~ndiscernibi]ity type L if C(t, o),,) is w e l l - f o e n d e d wRh respec! to E.. t h e n by s t a n d a r d ar~,~,enm~ ..............'~",, ~,,~," ~ is i s o m o r p h i c to s o m e L m w h i c h mus¢ t h e n h a v e an u,~ceuntable set of <-i~adi..<:e~~> ibies. Titus if .d for s o m e ~ sat:~sfvmg (4), <7(4 ~o,) is wel]-fov.nded. we wi!] be do~e. Since any d e s c e n d i n g ~s-seqne~ee ~:~ _.(,~ ~h) i~avo!ve.~ o~Jy c o n n m b l y m a n y terms, h e n c e
,

.

.

see that (5) holds aft (6) t h e r e exists t satisfying (4) s u c h that for aU o: < e~, C U a

'" °,. . . . . . . .

S u p p o s e (6) is fa[se. W e wii]. o b t a i n a contradiction. L e t R :~-TX,.o~. be ti>~ s.-~t of at! ( t c~) such that e h h e r (7) t does n o t satisfy (4) and e~ > r,), or (8) t satisfies (d), ~ ~ e , a n d C(~, e,:) is n o t wei!-founded. o_n~., (6) is false, R is fenctio~aI. W e assert: t h a t N {s a~. D: is d e a r ti~at (-/) {s arithm.etieak berate c a n b e e x p r e s s e d with boup_ded q u a n t m e r s (V}.~~ c~ ....... arid so is all ot (8) except ,-or t h e a s s e r t m n ...... C(< ~ ) b n o t well-bu~,ded: B ~ C(t, ~ ) is n o t weil.-fom:~ded iff _q~ ([ m a e s o) into G~ a n d V~fff0~ + i ) ] ~ ~(~)]): !t ~s e a s y to s e e that [f0~ + i)a[f(~,)] is a boende~,~,~{u~ma~-~e~ sta!:ement abo~at % <~:.,~', a n d ~, so R is 2;> U s i n g Theorem. 3.3 c h o o s e N{ fm~cti~na~ S so that ,~S ~ o r d e r - ~ r e s e r v b ? , a~d Ifrni~ a n d ~
30~

Bau~gartt~er.~< G~h)in

J.H

inaccessible so (L,~, c-) is a mode; of Z / : + V = L. I e t X be a set of <-b.discen-dbles for (L~, ~s) with h-zdiscarnibilhy type Z a n d order
s

F.

Let aa(s) be the least ordinal not A~ in s. The~ az is ~ rder-preserving and limit so u(ai) is stz~ngly inaccessible. The@!~:em~ 3°5. Sappose R is Zt and ]#nctionat. 71~en u ( a ~ ) < ~<(ai). PzO@fo Xt will suffice to consider only the case when a~ is order-preserving. W e wiit sEow that a~ < a ? . Let us suppose that some standard system of coding countable ordh~als by elements of "~2 has be~:n settled upon. Let R ' = {(s~, s~): sa codes an ordim~I ~ such timt (s~, re) ~ £}. As s weil-Ar,own, since R is Z~, R ' is N~. Say ~;~' is N:! in the p a r a m e t e r ,s' e ~'2. Now for each s~, {s2 : (s~, si) e R'} is E~ iu s~ a~d s', horace, by the well-known t h e o r e m of N o v i k o v - K o n d o - A d d i s o n (see Sh,)enfieM [S]), must contain an e l e m e n t which is k~ in s~ and s'. It follows that for any s~ ;~TS', ae,(s~) is 4~ in S~. H e n c e an(s~),rs'. 2~t foKews that tz~ < ~:(a~). H e n c e the existence of ~<(a~) implies the existence of x # for a~l x ~ ~. Moreover, ~(a2) seetas to be the least such cardinal of the f o ~ a ,<(el) for "reasonably" defiaable a. J. Steei has recently shown the foi~owing: Assume the Acdom of Projective Determinater~ess. Suppose that f~ is projective]y definable and functkma!, and that a~a is recursively invariant, if ~ ( o a ) < ~ ( a ~ ) , K k{ZR) "~. ~C<~S}.

OF {:,)u~rse h at:;o follows from T]heorem 3.5 that a ; is not of the form a e for i~'t

hmctior~al ~ !~ is not diNcult i:o see that there is a ~]~ functionN R st~.ch that C72 =:

{i

[t s natural. [o ask w h e t h e r the existence of £-t is in s o m e se~se the weakest ~argc cardinal axiom which implies the existence of x ~ for a!l x ~ e~. T h e next theorem is intended as an answer to that question. TEe¢,~e~:_~ 3X£ S@pose x ~ e,~ c,.~.~d;R is f~{nctr:o.~;at ,',n-~,dv gef-in{4@ wi:h pmameters ];~n:~ D~C
r<-~aa, ~hen, in L[x], ~<-~,a'.

;~hc{xsf Firs~ ft shguM be remarked that ~:~,~ ~s simply the restrictior~ of a~ ~o "~2 f~ L[x]° Suppose timt ~ ={(,% ~),¢~(~,~, y) ho~,ds ~ ]{¢C}, wb_ere y .~HC ~'~! and <2 b, Z~. T h e re~r{dvization o f Lc~,,w% v

Thus

if w e

.,~ ~-~....... g

!et R':={(s,~):!,[x]!="o{s,e~,,y) holds

[~ HC'},

f,~_ R ' :

Gene:'Mized Er&'Jsc~;~rdi~tais ,~ -.,.,

,,, ~

305

c~ ~ G4

M o r e o v e r , i{' is func{ional i~ f.[x] since foc eacB s {~: t h a t .~c~ (s, ~ ] c P.. is J.~J~ir~ p a r a m e t e r s f r o m ,,....~4~L'::~l, h e m : e Is absolute from ; .. o o ~ . H e n c e letting a ' be ch~, as defined ip. L [ x t , we ~-)e d~at a ~ s t h e ret:iri:::ik;,~ ;ff ar~ to °'2 N L[x]. ']'he rest. of t h e proo[ is a c o m b i n a t i o n o:[ t h e a p p r o a c h o~. B,,~ .,,~>; and f,~sher [ 1 ] to N t a b s o l u t e n e s s and Sib,,e:-:, (~)<~ in L. T h e Skot%m n o r m a l f o r m t[~morem i.:<:~ Cimrch [3]) says that for a n y first-order s e n t e n c e ~ (which, for simplicky, we a s s u m e does n o t invo[ve operation symb~)Is b u t m a y h a v e relation or cons~an~ symbols) ;;hers ~s a~ effectively ~ "'°'~";~ '~ s e n t e n c e o-(e) of a larger similarity type, again, with <<~ ~i~<;r~tio~; %a~.~b:~ls, st ch tha~ (9) o'(~) h a s t h e form Vx~ . • " Vx,, ~y~ " . - ~y,,~ ~ , w h e r e ~{~is q! a~Vifler-ofrce~ (10) a n y m o d e l of ~ m a y be e x p a n d e d to a m o d e l o2 o-(~)), and ( I I ) if ~[ is a m o d e l of (r(,.~), t h e n the reduct o~L !~i to t h e sim;2arity 'L'pe of d; is a m o d e l of q~. H e n c e ~ h a s a m o d e l iff u~(4~,) ~as a model. S u p p o s e tb as a b o v e is fixed. G i v e n %-o fi,aite structures R}~, fS~ in t!ne q m i h u i v / type of "...., c.,,). Now, .given . . auv . . se~me~:~ce :"2f ~._.~i .~i. . . .~.[~c~ .. statement

.

~2

)

a~"

,:

•,

....

.

s~a

nc=o~, s u c h t h a t 2 3 . , ~ < ~ , ,

f o r "-,li ,.,, w e h a v e

tO,,,.:,,, '' e,, :¢%, ..~ r , ~,'}

Also.

am,

cotmtabie or fimte m o d e l ot o ( ~ ' is of the for_.'n U ~;:, for s u c h a sem~et:ce. H e n c e ~p h a s a m o d e l i f f < i s no~ ,J~,,:-found~d. S u p p o s e )~ is a binary relation syrnbo! occ~:rring in ~#. C o n s i d e r the .'let of aii pairs (93, h), w h e r e >~3is a finite structure at; above a n d h m a p s 93 Mto the ordinaie~; in such a way that if ~ E a b , t h e n h(a)<.h(b). Let (~-~ },~)<(f:,3, ha) ~:ff '% ~ a n d h~ = ,'h. N o w it is easy to see that 4~ ha~ a w e l l - f o u n d e d mode1 (with resp~:~ct to E~) iff the n e w :eIation .< is welt-founded. [f ~h in:plies t h e axiom of exter~s:onalRy with respect to t.~e s y m b o l 7E, then we me} also couciude i:ha¢ ek ~m~.; a it~ms[~L:e m o d e l iff < is well-founded. For a m o r e c o m p i e t e discussion.. ~ee Ba::v, ise a n d Fisl~er [1]. W e wiii p r o v e T h e o r e m 3.6 for the case in which R is Z,-defimff~le wit['c:u: p a r a m e t e r s a n d x = 0. T h e m o r e general s r g u m e n ~ is left to the r e a & m S~ppose ~ - ~ e a . L e t r : [ ~ ' < ] < ~ - 2 au,.;*~sup~.~ose~f e e W e will show that there exists a set X h o m o g e n e o u s for f which satisfies the definition of x ~£.-,. iff a ce~t:ain partial ordering, beio~gh~-g to L, is weI!.-fomaded. Sh~ce weiVfo,.mdedrr:e~-;s is absolute, t h e existence of such a n X in t h e real w o r m wH! in-;ply the existe.~x o f such a~ X in k ;Let ~ be a ~N,. f o r m u l a s u c h that f£ ={(s, a ) : (HC, ~:5)~s,_. , , , a41,,,. C h e o s e ~aew c o n s t a n t s s a n d r~ a n d let ~ be tSe c o n j u n c t i o n of q¢(s, ~) a,nd th.e a x i o m of ,de a s s u m e t h a t E is t h e bh~ary relation ~ v m b ~ occ~Th~£ iu ~L ex,~-e,nsiop, aiity. ,~. . . . P be t h e set of nil q u a d r u p l e s (Z, g. ~'~ ~L h.) S~mn" t h"a t i,et ( 1 2 ) ~ ~s a fi~ite structure ~ ~," ~--4~ ~*, t~,~ ~ *~ ~. ~qq~)~ w ~ ~ ,~,,,e~°s~~

Z E . Naut*~garmer, K Gab,in

306

i~duded (i3} h (i4) Z (t5} g

in o) a n d such that Vk, n c - ~ , ~ t £ ( k , 2 n ) iff a m < n k = 2m. m a p s ~a into ~ t so @ a t fh~Eab implies h ( a ) < h ( b ) . is a finite set h o m o g e n e o u s for f. m a p s 22 o n t o tt~e E - p r e d e c e s s o r s of e~''~ in :'8 so {hat s S < {

~ff

(15) V2+~6-~, 9,~ts(2~O==O iff f is constantIy 0 or~ [Z]'L L e t P be part~a!ty o r d e r e d by ~iettklg (Z~, g~, 93 t, hO < (Za, g> 93> h:z) iff (~3~, h ~ ) < (~2.~>h:z) as above, Z~ ;2 2-7~ g~ ~ g~ a n d ~ n .~ a~ n G 9d2 b~t n q ~ i N o w s u p p o s e P is ~ot well-founded. L e t (Zo, go, g~o, h o ) > ( z + , g~, '~-:, h : ) > + " ', a n d let ~ = U ...... 55+,. T k e n ~ O arid ~ is a we!l-fovmded m o d e l of t h e axit_~-~ of e.cmx?skmaiity. L e t t : ( e , E ) = (D, ~), w h e r e D is a transitive set. N o t e 'that by the dedir~ition of <% @e universe of ';~3 mus~ be ~.~. By (i2), X ( 2 n ) = ~+ for nil ~+e (e. L e t 2 = [ j ..... 2.%. T h e ~ Z is h o m o g e n e o u s l o t f a~d by (15), X(~'a% = c% the o r d e r type of Z. By (16) .t's ,-.s+~ ~ ~s - the trace s of Z reIative to f. B y t h e definition of G ( ? 3 t e ) b g , ( s , a ) , a n d since D ~ H C a n d 2~:~ f o r m u l a s are absob~te ~apwards, (HC, ~)t:p(s, c,_'). H e n c e Z satishes t h e definition of -,<-->.~'~. Now s u p p o s e Z is a h e . m o g e n e o u s set for f which sa isfies t h e definition of +'~-~ ~;~. W e claim P is n o t well-founded. L e t s be ~he trace o:~ Z m~.d Iet ~;{~(s) = rx. T h e n (HC, ~ ) t q ~ ( s , ~ ) . H e n c e t h e r e is a c o n n t a N e transitive set D v,t~ch that s , a ~ s © a n d (D, s ) P p ( s , e). Let @=(D,c_-,N, . . . . . R.:) b e t h e expaRsior~ of ( r .~, ~) to t}te s~nffar{~:y type of cy(,.)) such that f.:~t.,r(@). L e t ;~,:o_,-.->D be a bijeetio~ s u c h ~:hat: t ( 2 n ) = .+~ for all ~.+e on. Say X : (m, E, S ~ , . . +, Sa) ~ @. ?Let h m a p ~e into a)~ so {~a¢ E(n+, n) implies h ( m ) < h@} for a!l ~+~,~.t. L e t g:Z-~,o~ be so that g applied to the ~th m e m b e r of Z is t +'{~). Now it is easy to cL'oose a dccreasi~++~ ++:%seqae+ace ((Z_, g .. ~,,,.h,0:. n ~ ~o) so that U ..... Z,~ = .Z, ,+..~...... g,, = g. U ...... >zh. = (a:, E, S{ . . . . . Sv) a;M [j ...... h,, = h. D'ence P is no~- v.e~t-fouJxded, a n d

tee proof is eomple~e+
i%+eeL S r p p o s e ~:.~ i.s 25~-defirtaNe f r o m a p a r a m e t e r z <~HC. Let : g c ~ be such ~ha~ z ~ H C r ~ j ~f ~'~'--+~;'R, ~-he~ by T h e o r e m 3.6 it w s u t d be ~rue in L[x] ti,.at ~<.--,+a;~ B~.:~tsi~_ce x #' does not b e l o n g +/a £. [~, 71a n d in fact £ [ x ] e x # does r~ot ex-?st, it foIlo,;vs float tL~.t exis~e,~ce of ~<(@.,) d o e s n a t imp!y the existe~ce of x # f~r every X.

Of course. {:he asst!mptio+~ thai x :~'+exis~+s for aH x _q ~0 d o e s ~;,ot even ~mply t h e exis.te~~ce ~.,,e a stroagIy irmceessibie cardi~aL "I%e ceason is t~mt if/;{ is a model of Z F C ph~s tire asser~.ior; tha~ %% .g e., x 'v exists, a+~d if ~¢ is t h e smal!est s.:rongly i~aaccessible in .R.. @e~ il~e collection of atI se% o;[ ..£-rank ieas t h a n u: fs a m o d e l o{ 2;£C ~ogether wi.th t h e assert%n tI-~at Vx g +~ x*' ex~s+.~san'd t h e r e are no s~aongiy

h~accessibie cardh~a~s,

Generafized F~diJ::cardinals

3fi7

On the other hand~ ~", ::: ,:~:ristence of ()x~~m,. :., p,:~e.-, ~ the con:£%e.~cy o f the exis.~?~c< of some quite large cardiua!:;. T~ea~re~a 3o8,, S t ' p p o s e R is flenc*io:~a ~,e*m:/ .,~.~ d e f i n a b l e ]:}'om parame~er,~ ie~ ..... f o r s o m e x c= ~ . ~ f x ~ exists, .;hen it ~.~ te'~:e i ,. L[x~ chat e:(ar~) e:d,'¢s.

- "

~'~'¢e2o A s usual we assn.,me x = ( ? and leave the c'd: {:c> the reade?~ As we r e m a ~ m d above, tim existence, of 0"* implies ':he existep, ce ,£ a :}c>ed ur_~bounded class C of indiscernib~es for L. In fac~, C is even a Class o; b~v~=scernibles for (L, ~, z), w h e r e z is the p a r a m e t e r occarring in a £ , defieitio~ of iq'L Let :< be %e ahst n~ember of C and let C ' = CO,,<, If ~,~> a;.~ i.~ L, the~ ~et f be t h e minimal (i~a ~he canonicN weii-order*:r,~3 of L) counterexample t a ~<~ a~v Z);-:~ f g definab.~e f r o m ~ and z. But now by indiscernibiiity C' ~s hom~:~ge~eo~s for ]. tt follows that the partial orderir~g (P, < ) [~ ~he proof of T h e o r e m 3.6 i:~ sot weI!-fou,~aed, hence is r.~ot weIl-fou~ded in L. But this comradicts the choice of J: H e n c e ~-->a,,a in f.. [t .~'oliows that for any ,~ in the cano:..-ecai class of indiscer~i~ bles, 2, --~ a~. tt is now possible to fiad a very elegant proof of the p;o~e×ister~ce of t~e cardbml ,Lq, as follows. If ~ exists, then 0/"" exists. Hence if ~ is any f~.mctionai re]ation which is N, definable fro:m p a r a m e t e r s in H C ~', the~_q ~(~<) exisha so by T h e o r e m 3.8, ~<(a~) exists in L. H e n c e ~.q exkts in L H e n c e 0 "" exists i~ L, contradiction t The en~or h~ the argumem: iies ia trying io conclude from T h e o r e m 3.8 that i~ f ,J va~.l, exists ~o:_ ~ " e,,e~"~v ~-,t ftmctionaf }2. . .-~ . !tnrns ov-t that there ,~.~:~"-:~~ ' .................... ~q'~ ~'i,~,~ c~ which are kT~ and ftmc1.k~nal if~ L, but for v,hid~, there are no .S~ :[unct:io~m{ relations S in the rea~ world such that a~ = as -i L. H e n c e T h e o r e m 3.8 dnes not apply. Such an ~,~ is {(,,. ~.e): o: m
In this sec~on we c o m p u t e I{la~[ n u m b e r s for several ta~s_guages -~n terms of ge~eralized ErdSs cardinals. Recall that the formation_ r~ales for [ormutas ,of ~he infinitary language L ...... are tke same as for fi~st-order }ogle except for the additioa of the ~,.oLowlng: (1) if @ is a countable set of L , -formulas then the c o r d u n e t i o n / \ @ a~d the disiunction V ~) are L ...... -fonmdas. ~2) If ¢ is an L -~orma!a and x,, Jc,, x > . . x,, are va~abIes then (:~xi) x ( E x , . ) • " " ( ~ x , , ) " ° • ~9 a*~d ( V x ~ ) - • ( g : 2 ) ' " " q~ a r e L ....... - f o r u m ! a s " u.siversag it it ~ A n L ,~-sentence ~s ' as the form (Vx~)Wx2)" o a~. , where 4' ~s quantifier-free .

. .~g .

.

.

.

d.E. Bamngenner, f. Galvin

3(18

L,et h~ be t h e H a n f m ~ b e r of universal L ...... -sentepces, Le., h~ is @e least cardip.al v such t;hat +or a n y un>"versa~ ~ - + L.,,,~ ,- s e n t e n c e 9- if ~ h a s a mode! of eardhnality >~, t h e n p h a s m o d e l s o f mbitrarily large cardin.agty, if -~ is " : ~'- ' ' a n d ~ is a m o d e l of ~p, t h e n a n y s u b s t r u c t u r e of~ 9~ is a m o d e l of ~9 also. H e n c e h~ is atsn ~,f~eleast cardinal v s u c h that for a n y universal L~.~ - s e n t e n c e ~, if @ has a m o d e l of cardinality v t h e n 9 has m o d e l s of aH infinke cardinalities. L e t h:~ be t h e least cardinal ~ s u c h that for a n y c o u n t a b l e set ..v of first-order sentencv.s ii>'eh,ing a binary relation symbol < , if N h a s a m o d e l ~f of p o w e r v* s a c h th>t
~,~i~

.~oao S a p p o . s e t h a i t'or e v e r y .Yd, ]:~,mctional r e l a t i o n f~, ,. t a R ) e x i s t s . T h e ~

b,~. --. .a~ . . -- ~>." . . . .bc~

s u p {,<(a~): R is Z-, a n d f u n c t i o n a g } .

.> r p . wi&n',.d . . para,',~eters, . . . . ~hen J 'w- . R ,~ exists, T h e u

~" =

h~ =~sup i~c t' a a . .". N is .t % n c -

T h e proof cf Thd0>'.em 4,2 is sp, easy variant of the proof of T h e o r e m 4 . i , to which the rest of this s e c t k m is devoted.

}i-~ee[~ h2 ~- i~3: S u p p o s e Z is a c o u n t a b l e set of fu-sv.order s e n t e n c e s i n v o l v h G < . Suppose also that ~ ]-,'as a wei[-orde~ed ~aodel ?~ of' p o w e r ~-%° V/e m a y assm'r~e that the s e n t e n c e asserting that < is a linsar ordering b e l o n g s to 2 1 B y definition ci' h~, i~,. every infiniee cardina!ky Z h a s m o d e l s i~l which < de~lotes a weil£ou~ded reiati©n. B u t a weii-founded tiuear ordering is a well-ordering. [~s < [h: Let Z be a c<>untab!e set of s e r g e n c e s iwJolvh?g E~ a n d s u p p o s e that N ha.- a weliofourlded m o d e l 9{ of cardh~adity h > Let < be a symbol not c o n t a i n e d in Z a n d let T ' consist of Z t o g e t h e r with a s e n t e n c e saying that < is a @lear orderin 8 a'~d the s e n t e n c e bet

.

,

Si~ic~: E ~ is weii-fou~ded, t h e r e is a welf-ordering <':' s u c h that @{,
Gem~ralO:ed ~:~r~~" c~rdi~.a]s first-order

se~'~te~tce o J

"P*" ~ : ....... :~ c

509

~

(4) e v e w ;s~, }de1 c'i o~' is a m{~c:l.:5 ,_~i' ~>~.a u d (5) every mode1 of G can be expm~d-d ~:~; a mode{ of G'. rv£oreover~ the c o r r e s p o n d e n c e betweer~ ~- ~ ~:' ?s sufficiently m~i[nrm so that aJt Z ~s a set of f~rst-order s e n t e n c e s , r h e a (41 ~ J u {5) r c ~ d n i:rue w!"~ei~o is replaced

b' X and o~' by ~'={G': G c £ } . N o w s u p p o s e ~ is c o u n t a b l e arid V has a we~I~ordered m,~,-,"_ of {x~rdL~aiitv ~.

}.et ~ b e tt~e c e n j u n c t i o s of AN' a~d ( V v { ~ ) ( V v d " " ' ( V v , , ) . • • A ( - ~ v , . ~ < ~'~), • a •we.,r-oxen.areS. " ~" ,~ 'i~her~ ~p is e q u i v a i e n ~ t o a u n ~ v e r s a i w h i c h asserts t h a t < ~s L,,,~ - s e n t e ~ c e a~d, by (5) for Z, p h a s a mode~ of ! ; o w e r }h. H e n c e p !~as m o d e i s in all infinite cardina]ities so, by (4), .~ ~' ~as welL,~rdered, . ~ moi~eh; ~ all i~di~fi.e

cardinalities. Le~,~ea~s 4.4, S,~@Bose @at for ,ve~,~,. : 2~ ~:m~.ctio.~al re:~tio~ ~,r# ~(a.e,. . e~.:is:s. 7'hcu h t ~J. f5

,

~:reoL S u p p o s e t h a t @ is a m~iversai t o,,~.--sentence a~d @ h a s mode~s of ai! . )a cardinalities
x,~ y ) - , .,O(x> . . . ~ x,, g~tx~ . . . . .

,,~.,

w h e r e p is an off -to~muta. If ~ is a S k o l e m ~:ype, t h e n j u s t as it.,. g,ae p~oo~ . . . . . . o~ T h e o r e m 3.1, for eaeia irtinite ordina~ G there ~s a cax~_on~cal structt~re CO, ~) associated w R h L [f ~2(~,c~.)~4, f o r all ~, t:he~ we a r e d o n e . s o s u p p o s e C'(.~,a:)~-n@. Let ~ @ =~vo~v~ " • " ~~v,,' ' ' . d/, w h e r e O is quantifier free. rl~,en there are te~n~s "to, ~ . . . . . "r. . . . . . ~c ~(~, ~x) s u c h that C ( g ~')bd~('r~;, ~r~. . . . ) ~'~t ti~e {e.rms r,, ca~ invoK, e (rely c o u ~ t a b l y m a r G indiscer~ibles, so j u s t as i_~ t h e p r o o f of T h e o r e m -L i ~here m ~ s t b e a co~tab~e o r d i e a [ ~ sv.ch that: C(.,I; ~).b~e;. H e ~ c e w e m a y a s s u m e t h a t f o r eve G S k o l e m ~ ' - t y p e t t h e r e is an ordhsai G. <:oh s u c h that C(t, c~)b----~@. D e f i n e £ ~; T> e~ a~d ~s (s is an ass~gm~eat mappir,g {v~, : ~ ~ ~} into C(¢, c~) suct~ t h a t C..t e:/bg~(~,@~,, s(,,~/,..)). U s i n g t h e defi~itio~ of C(~, e ) and t h e stm~dard [email protected]~-~ of i~atisfact£m, o ~ e s e e s easiIy t h a t t h e a s s e r t i o n "C(~, e) ~ @IS(So), s(v~),. ,..)" !~# A. ~ e~ .~. aP,'d @.

3t 0

].E. Baumga~-~.eer,F. Gal~in

cardbmlity r<(a,~.). W i t h o u t ~n.m of gemerality, we rF~ay a s s u m e that Sko[em fun_cfions have been a d d e d t:o e x p a n d 917 to a n ~ ' - s t r u c t u r e . L e t X be a set of h~discer~ibles for 2{ with type t a n d o r d e r type c~ = ae(t). T h e n t is a S k o l e m ~gf-type so by h y p o t h e s i s C(t, ce)b-qe}. B u t C(t, a ) is obvio~ws!y i s o m o r p h i c to t h e Skolem huli of X in ~t so, since @ is universal, C(< c~)~:4~, co~madiction. This e s t a N i s h e s [he i e m m a . Le~r~m 4o5. Suppose that for every Z~ functio>:at ~vlatioa R, e<(a~) exis*:~. 7;hen

~,~ ~ h ~ . Kc~,e.>{L h wi!t aufiqce to show that :;~' R is Zj a~d fun.etional a u d *<(c!a) is strr_mglv inaccessible, the.~ t h e r e is a co~mtab!e set ~: of first-order s e n t e n c e s c:ontaiMeg t h e b k m r y relation symbol E s u c h flint Z h a s a weil-fo~,.nded m o d e l of cardinMky ~(a,a), b a t rlone of cardinality >K(a~,,). S u p p o s e ~-~ is ~ac~,eanable "~ ~ " f r o m a p a r a m e t e r a e r i C . By a simple coding argumerJt we m a y a s s u m e a _%.{o. N o w let 2~ coasts{ of (9] the axion, s %r Z F C , (10) the assertion (writter~ in ~he l a n g u a g e of set: t h e o r y with E instead of mad a special co~starat a to d e n o t e a) that :<(a~z) does n o t exist, t1 i) the assertion tiaat a g ~ a~d (] 2) for each n c an, t h e assertion % E a " if n <: a, m~d t h e assertion "-~n E a " if

:~-,e,,.,.cousse. "7i E a " is an abbrevia'Jon i'oi a se.~*enr:e . . . . Lqvalvbtg . . . . . o~qy . . E and a.) Si~ice ;<(.'-~e<)is s[rong~y inaccessibIe, > e have (R0<(a~,0), ~, a),~-Z, w h e r e R(~-(a~}) is ¢he coliecdon of aIi sets of rm~k <~e(a,~j. S c~ppose 9t = (A, E, b)b Z, w h e r e 8g is we11-founded a n d o£ carc!iuaihy >,<(a~v). Si~ce 9! in, a weli~founded mode:, of t h e axio "-~of e x t e n s i o n a t k y , 9~2is i s o m o r p h i c to a *~ax~si{ive set. H e n c e we m a y as weli a s s m ~ e that A is eransitive, E = % a a d (by ( i i ) arm ([2)) b = a . Sir~ce "!{ has; cardinaiity >~¢(0~),, it m~isi (z .... which is; a c o u n t e r e × a m p l e in ;;¢[ " ¢4[0. B u t ~*o,.e. m g u i n g as in the proof of T h e o r e m 3.6, ff~e exNter~ce of an a p p r o p r i a t e g.ona.~gepeo-~s set for ~" ~s equivaler~t to t h e non--welJ-foundedp, ess of a certain partiai, ordering belo~agir~g to 9I. B~t if the partiai orderh~g is weH-fo~,mded in ~t, it mt~st really be weli-founded (st,ace ~;?.f is). Note timt wc are using here ~he absoh:te~mss of {he Z t defi~fitioe of R a n d the fact ~bat 5 =-a, so the partia! orae~mg >; the s a m e w h e t h e r k is defined b~ 9[ or t h e real worM. ~->'p '- ~* bc a ge~n~in~ co~.nrierexample ~o ~<(c~z~)-+at<, contradictiom T h u s Z has no raodeis of power >~,:(a,~.~). "

"

"

"

*"'-

"

*

[O

~" ~......... t l ) in modifying ~ae a b o v e o ---~ . . . . . . p o s s i b k to h a v e Z (wifich ,:nm.~{ be fi~aite) eontai~'~ Mi o f ZFCo ]t i::. easy t:> see, howeve:, thai- o~>}y fmi~eiy mat~y a x i o m s of Z};C are .<~sed in carrying o u t the p~-oof.

<-"

;~;i'i::'d Erdi s cm'di~mls

3ii

(2) It fs also pos~.ible to s h o w that ;:h~ !f h exists, ~s the klanf n u m b e r for well-ordered m o d e l s of L .......-sentences. Tt~e L,..,.,-~,~. ~?tences are g e n e r a t e d by the s a m e f o r m a t i o n n~les as first-order togic except tha~ c o u n t a b l e e o n j u n c t k m s a n d d~sjnnctions are allowed. Thee; the m e t h o d s fn t h e proof~: ahoy:: show that if c, fs an L ....... s e n t e n c e a n d cy h a s m o d e l s in every ce.rdina!ity < ~ , th<~:~ ,- has m o d e l s in a!l infinite cardinaiitics,

i n this section we sketch a generalization of T h e o r e m 4. ] for t.,~ .,~--scnte~~c,:s~ w h e r e ~ is an fiafinite cardinal. T h e f o r m a r i o n ru!es for L,~ ,,~.,Jormtfias are t!m s a m e as for first o r d e r !ogic except that conjunctions, disjunctions, and quantifier blocks of length ~e~ are allowed. L e t h,~ be ~he H a n f n u m b e r fo~ universal L~, , ~ - s e n m n c e s . in o r d e r to characterize h,~ k ~s necessary to genera}~ze ;he notion of an Erda;is cardinM stitl further. W e give the combinatorial definition h e r e a n d leave h to ~he r e a d e r ;.~ generalAze the definition using fi~discerMbiiity types. S u p p o s e e m a p s ........2 (t~e set of a!1 fm~et[ons f r o m eox ~ into 2) into .<+. W e write ?~-~-a if t~e foi!ewh~g i~oids: For each ~ ~ a~ a n d c~ ~ ~, let j],,~ :[X]"--~2. T h e n the:re exists X =c-~.~ and s ,5 ...... 2 s u c h that for all ( n , e , ) ~ a ~ x ~ a n d aI1 .~:~[X] '~, f i ~ ( x ) = s ( m c ~ ) , a n d 2< l~as o r d e r - t y p e a ( s ) . O n c e af, ain we cN! s the '~t'~ce of x, a n d we let ,fie) d e n o ; e ii~e least X such that A ~ . a. if R c ...... 2 x ~{+ is 2~nctiona!, then ~ : ...... 2-÷~<~ m a y be defined as before. a . ~ a e ~ 5o~o Suppose that fi)r every R ~ ...... 2 × ~c ~ which is ]%mcffoaa~ c ~ d 2~, ou;~*

1-f,~, (&e co~ection of sets heredi;(~ily of p~wer ~ ) ,

~c(aa) e::i:¢,< Le~ ~,,~ be 8~.e

supremv.m o f a~ such ~(rha). %hen h~ = b~,~. ] ~ m ~ i T h e p r o o f that h,,~_,.,~ ~s o m i t t e d , since Pl {s j~st a o~.~m,.,..~.~,;a~, generalization of rile ear]iier proof. © n t h e o t h e r hand, a new ~vno ~ ~~~'q - ~.~h,, is necessary, since t h e old o~ae does n o t ge'-eralize. S u p p o s e R..=c. . . . 2 × .e~"- is functional a n d ~,'~ o¥'er H,~+ in a p a r a m e t e r ~.~~_.~.4~,~. W e w~il p r o d u c e a u m v e r s a t L~.,~,-sentence. ~, wL,~ m o d e l s of a~i carcfma.~"~es~{C <~:(aa)~ but ~mne in cardkmtities m ~ ( % ) . F o r simplidt-¢ We a s s u m e e~-:(s)~ {," for ali s. T h e s e n t e n c e q~ wil! b e written in a language conta_~h~g ~.~-ary r eiat~op, syrubo!s .~,,,, r~ for all ~ < o~ a n d a < ~, t o g e t h e r vAth a binary reiaiion symbol ' <. W e abbreviate s~rings of quantifiers like Nx~):~}x~o. ~ B ~ . . . (~ < < ) as ?~:~. T h e s e n t e n c e ~ wili be t h e c o n j u n c t i o n of a s e n t e n c e q-'0 asserti1~g t h a ~ < ~s a w e i b o r d e ~ n g a n d a ,~entence V~ of t h e f o r m

where each % i<8,

is q u a m~"-" mer4ree

~" ~ee s e n t e n c e qh w~l a s s e r t t]%{f ~ e

p a r t / d o n s j-.... defined by J,~, ~,~ . . . . . a , , ) = 0 iff N,,~<(.~,.. a,,~ h o l d s ( w h e r e a t < " • " < a,,) e o n s t f m t e a c o u ~ t e r e x a m p l e to k--> an, w h e r e k is tlae cardinaiJty of t h e m o d e l of cp. L e t p : ~: x ~<.-* :< be a Mjection. T h e f o r m u l a % ~s a [ t .D" }' ,, (..... (:I ,~ {I ~ y ,){..,,~ )(-i ~: )) ~} ~.[. ~-~, I

'

'

I

~',.,'.~¢~ 'Z 1 ~S

T h u s , if we define a relation S o~ {;%: c~ < ~} by letting :,~ E x# if[ Y.,~,.m < ~, t h e n ~0 asserts tltat E g e x t e n s i o n a l and ?~ t o g e t h e r w i t h Pu asserts that E is weil.-%unded. CYhis pect.cliar de~:nition o f E is used s o l e l y to a v o i d i e t r o d u c i n g a n y c o n s t a n t s o r o p e r a t i o n s y m b o l s . ) H e n c e there is a transitive set T a~,.d an i s o m o r p h i s m wifich carries ({x~ : c~ < ~<},)E) o n t o (T, e ). Frmx; ~ o w o n w e u s e a~ E x a as m~ a b b r e v i a t i o ~ f o r ypu,.~}
\

,',i

v:%=:,d].

!qe.x{ we w a n t to be sure t h a t z(~ is ca, ried i n t o ~he para~-.~.eter .2. L e t e~' be a t~:aDsiti',<'e set such t h a t !,:(I = ,.c arm a ~_a". L e t ] : q,'-~.~< be a b i j c c t i o n such ~hat

i ( ~ ) = 0 . ~'{c.w !;st ?s b e

i

A

A[(z;~;,;=.-~<,;,ziu:>=-r~3-->-nx,., E:S~].

b.c c ?,' ~,f4 bcc

lqow let ~)(s~ c~ d.} be a Z<~osnm3a which despots R i~l ~,<,. Let ~(~, % zo) be ti~e q~aafitifier--;:ree formula obi:ained from d~ by ~ep!acing s by ~', a: is}, v, a b~, z0, every q~aantifier of the forn, (V$')~(2) by /\,~ ;~(&) and every quantifier of the iTorm (~Y)k'(Y)

by

V~x(x<~}.

V ~ = *~ ,"

Vv

Let

=-J,~ a

},'im..s ' G asserg~: thr~t ~,~ a ~ d

% be

'-i;('-~, th,.

u are carried

z,.:;,. ir~to

objects s~andTug in

the relation

.[;£ I;o

e?x:~i o t ~~e';.

"qe;,:~ ~,:e warn ~o say t~at ti?e predecessors of v coincide witi~ {&~: ~z < ~<]-.Let ~rs be [(s~ =

•" " "

/'\ A [ i l p ~

" AX

T7 ~,

.

".......

T h e f o r m ~ d a ?~; s a y s ~hat {~<~: ~ <~<} ;:~rr.,, a s e t ~'~ <-ir~diseerMh~es for the r e l a t i o n s ~,,~, a n d t h a t at is c a r r i e d i n t o ~}~e t r a c r'~f {~,, : c~ < a}~ Lee ,(, b e

Ix /5 ? ]

:=: ~.,

w h e r e X(u, b, c, d ) is a No f o r m u l a in H,,, w h i c h a s s e r t s t h a t t~(b, c) = d, a ~ d 2 is o b t a i n e d f r o m ~ as ,'} w a s o b t a i n e d f r o m 0 in ~rs. Finally.. w e a s s e r t ,..~a~"~t h e o r d e r t y p e of {t,.~: ee < u'} {s ~vrec{se!v t h e o r d i ~ a i i n t o wL,i c h v is c a r r i e d . Let. v7 be r

-;

/\ L ~° < ~'~"-" A ((~:, :r,, ,,,,s,~ = xa)---> x,, ~z :';a ) [. I t is e v ] d e ~ t , l.herefore, "~5'xt if ~ h a s c a r d i n a l i t y A ;~ ~< aiad <'~: weii~or&~rs ~?~, t h e n ~t~-q2 it]: t h e R~',~ g i v e ~ c o u n t e r e x a : m p l e ~o A - ~ - a ~ ~it f o l l o w s t h a t Vi i~as a n~odei i n p o w e r A ~ ~ iff k-~-.a.:. 535s c o m p l e t e s t h e p r o o f .

[I] J. Ba~'~,qseand E. Fisher, Tlr~ 5 haenfiei~ absoluteness ie~a~.ma,~srae! .L Ma~h. 8 (1970) 329-339. [2] C.C. Chang ansi H.~. Keisler, Model Theory (North-Holla.ad, /kmsterdam, t973). [3! A. Church, ~atrodoction to Mathema~:k:al ?_.~gicVeL ! (Pd.~ee~.ou Uaiversity Press, ,~ri~c~:~, N.J. 1956}. ~4] P. Fra6s, A. Hajnal and R. R d o , Pa~ ~_ifionreiatiop.s for eardhiN numbers. A,.4~ ~