Extending models of arithmetic

....... c'< .!'%-10 e

St., }'{h IEr<,~' i

R e c e i v e d 2*.; 2an,.~a:y 1 7 7 7

t h e ~ let Lt (ke~¢( b e t h e ~" ,a.,a~.e ~c . . . . o~ . . . . a k ".:1¢~a ca¢~.~_aa~>t

t

f

~,i} K ~D. ,<~ ~s a q.'#tc~ di.,;tribu~h,a, txt~ice. ~¢~en ti~ere is ,a d e f i n a b l e i:y~t ~-(v~ ~i'mt e ...... .. , / such (2) K (LD, < )

i:s a ~ q l e .

,

di~;~*ibt:!t{te h~ttice w~;h a m'~iqt<, a t o m , ¢ h e n ~,2~,'e {s . -

~ ~I E 1 D , v~-~©

(a~

A*~other p r o b l e m O~ G. .a. .l .f.m. .a. .I.¢. S iS soWeA b y at~'~ ~°,c'. ",, ~ e-. .-. -. ."." ~ cba-" ~, .,, o f mii-~iffia} i y o e ; , . =~,a.~-+~,4-,v,.~,ca~<,~

Fix some cemntabie fie'.t-..order: language e~:, ~,,~. ~s .m~.:.~t,e ~,or .x~c '~ -" ~ ~,'"~ of arithme;dc° Let 17 be any comp!ete ~heory in the Ia~.auage -, • 5:? which v".,:~ L ,I,.~J,~: , ~ .:- i:hesv: axiom~_.~ If ,=:¢iis a model of T a~td .+t J { Such ti~at Lt (.,~"/.fO~ (/& < ) ? A m o n g the t h i n ~ which will be proved here is i:hat if .;~, < ) ~s a finite distrikufive lattice, fhen the anSwer is affirnmtive for a~?.y :mode1 ;z~t. One imp0rlant instance Of :this question: was already a~Erma~{veI~ ~-~:~'~~:-"'~ ~ ~'~ k:,mtman" m" [4 °1. t l e proved tliat if (L, < ) is tl~e 2--element lattice, t;aex* *,'.-~,.,a-~,.o ~ ,~< throe is a;*: 2 ' > ; g sltd! that l[t (.#;.g_) ~ (1_. <:). We say, i~ tiffs case~ d~at ~,v :a a ...... ~,.m extension Of ,g.: : Another at~rmative answ,~r to a n instance of ',h~s qti~asti0n was 0btah~a'd t y Paris i8~.. For any C0tmtable. ~g and any complete,: distributive, comoac~v o4;.gen erat:ed la t rice (.g, < ) there is a co, mtabte ~# >-~g such t h a t Lt (,#/:4t) #~ ~L; < ). This ~:esult is: 0Pdmal f o r distribWdvel (L, < ) a n d Countable .,~t arid .#,: stoic<: Peallo:s

": * Rei~earch s u p p d r t e d ~ a p a r i 5 y N S F Grm,~[ M C S 7 , 5 , . 0 7 2 5 8

:

,S~ i17

:

:

i :

=

90

) ;:L & L<*cr

w h e n e v e r A .o-~,~ arid the:v are tx~th e o m m d f l e , thin? [A { ~'~ {~ {s ~ eomp~ete, c o m p a c t l y mo-gcmerated !a~ffce. Pafi<~'s a r g u m e n t ti~tke~; esSe)~tia~: t:~se Of ~:he c o u n t a b i l i t y ef .,~. O a i f i n a n realized ttmt evep~ fox u n c o u m a b l e 3 4 at~y' cor~sm~c~ ~ o n o f .,.xtens~on. ~ f .,¢g wotlld appare~itlv ~ave Io ~ e s s e n t l a l l v c o t m t a b l e (a-: ,a..4.: x t h e c o n s t r u c t i o n is c0m~table)~ Ti,is . . r .e s t.a t e. d . {n **~" . c n o t i o-n <:[" a dg m u s t ~an,e d e f i n a b l e t y p e a n d the e x t e n s i o n s such types i n d n c e . T h i s c u r 0 s din,~ ,<*'[))i-v~] {s ahvays well-defined, A s u s ' u a , at~ :,:-type is a m a x i m a l co,~sisIeat Set of formtflas &(v ...... v.~_~" which extend~ 7L A b ' p e is a !--tyix~ W e will use fl~e r:otat~on t:m".;o........ of ' ":~ .,¢'" , v o , . . . , %,_:)" for n - t y p e s . 'The ~,,,t~owa,~ ~" * -: ,'_. is Galhna,"gs de, ., ao defir~abie r~-type p r e s e n t e d as Definitio~ I . l o f [51].

~ ( } ( x ) - ~"4 ~ ( x ) ' L' '

' b ( < e.. ~ ( m ve

. ,5 t h e r e ~s a fovmula ~.,,,(u~ ~..~.,~'~t!,~e for aw... . . . "' 0 a ffvo, ,, x iff T P . < d x ) °

~:*'~,,

W e wi!1 use the script Ietters .~6 a n d .V, s o m e t i m e s en~bellished with subscripts, to d e n o t e m o d e i s of T. 71m m e e r ) y i ~ g sets of ~he mode~s ,.dr a n d Y will b e d e n o t e d by M a n d N, respectively, Ill ao . . . . . . a.~;.., are e l e m e n t s of s o m e (v,sua!iy ur~specified) e x t e n s i o n o f dr. t h e r ( J f f a o . . . . , a . ~ .) is ~l~e smallest st.~bstrv, cTure of G a i f m a n uses deS~?ab!e ¢ypes as a m e i h o d for mfiformly ,~,.-~mm~3,~;'"': ......... .~.c*'=~> ¢k~, "~ exteBsior)s. Defi~,itiou 0.2. If ~(th~,.,., G-~) is a d e f i n a b l e tw:>e,~ ~ t h e n .*~(ao . . . . . a.~_,z'r is a *(e,~ . . . . . v,,_~)-ex~ma~io~ cd .~,~f if for a n y f o r m u l a &(u, % . . . . . v,,_~} a n d a~ay b ~ M.. d't(a o . . . .

>- ""

"

v,~_;)-ex';c!>ioc~ :ff d < a n d this e x t e n s i o n is u n i q u e ill tile o b v i o u s sense° See S e c t i o n i of [.5i for a t h m m ~ g h discussion of this m a t t e r . W e r e m a r k here that G a i t * n a i l s rainimai emi:er~aio~s, referred to atxwe, were constructe, it in this m a a o net. 'He s h o w e d that th< r..:: i~ a d e f ~ a b | e type ,.tb,, ': "~ called a mm~iml~ " "~ type, SUCh that w h e n e v e r ¢~(a) is a t~)-e.~:~e~~:iori oi d< t h e n Ltl..a(o"');~:V~)' i has two e l e m c ~ t s S u p p o s e d{ <,,S'o if for :my b e N ,b,~:--:.:~iv :?:i: M S u c h that A@b < a the~i ,:V is a ....u~,~, e x t c n s k m of ~,(t, If whcnex~zr o~ ~: N" 3q :'::',~ ~~< M t h e n o),:~':a< a the~ ;,~-'is a n e n d . e x w , ~ s i o , . , o f .a. When.ever .~(a,> . . . . . :~..... "~ ):'s a e¢"~:,,, ~ . , t:.._, ) - e x t e n s i o n 0f ~., w h e r e t(Vo. . . . . G - i ) : i s a d e f i n a b ; e ~ t y p e , ;hen . g ( a > ~ o . , a..,.,/) is all end-extension of ,K :i

> °e a:m# p**i::~ ~a&{a-a~ arqmn>p .

.

.

v s~ (~:-"a

"7 ' s°a ]~9 arm:

.

4}soddllS *f?[ Ula"l~lt,~ A > - A ! ! . -~v.,,~,:~

7; ~:~,';~.~ec~a< ..... ~x: L:>i~g aq {|::£ sniazoa:. [} asm:l ~o sjooJd aqdi. "~:c:+.,:~r,c-, - ~&u........ I []o:~:p.:t%t5.:::_:o;:qsu:a:::OaClt:o,~vl v?:l:. ,. al~7s mxt tio::aas.,slq:. t:I

smaat~ql, a!~t-~ a ~ t " t

'::o::.::o,':£:s:::v::Lm:d ~ p:~,::.,X{am:~g:t:pkq'u:Aeq .:;~ip2aaded s:::.!?,:::u.:a:u:::.,a .,~. ........... ~.~......:~':y>SP,.~)~:UOS :~,peix~ O:I,~X 't:lTO;:J{~30 rule H N:letIl o~ 8~l I p[no:~<,. ~&

[,> 4(C<: ..... 0,,u,..,; x ~ m ) > : ">, ....0a.>>.:({"~ .... o::..~) sogs:~s clo:::,:', pue "s~tdm--(: + ~.;)s'~po;~u~ q<:tV~ ('~a ..... °a)

8:::] :':a~ ....O a A ~

"mmU:::O jo ::o::s~sb e s:o~s,m~- qa:q:v: s~G<: :sm+:u:m 1o ::::::::::::::::::::::.'.?:!~.~ ?,',_~ 9 a<:2/'aS' t:{ ~" ::: p o a o M ~x?~ lib',sa~:o~,q: o,u~ os~:LL °£i:'.,A::sods?.~:: pc:-~ ~ ,~:¢o~,tm~ •: pu:-~ V" saaa~;:fuoL-) X:d:: t[o~tIA~s:{-':~:o~:{:om$ a:::s :{b,~ ~-.:'~ :: m?:~::g uI: °g': oou!so<:o: d sss ~p.:s£?.:s::::?:: m:U ::n~::s:~ ~::?::: ~, :,.<.:£"[g] jo/A'~ ~:em-:~i :q X.:t:ssosa::sq o: us, o:qs s::::o::~ ~:,,~oS:~:;ex~:smi ( > y7+) :sq:~ ~ ~.mmc-.fuoi) :~: s}s?ti:odXq e.q&

?"

'
......e,,,:"" %,,;:. :,.::>. ::o:s:.:<:~?.qsu? ::x::,..::::, (;,9}7~? u:::'~, '-:"~..~4: -:a. x~: S::~,A.::..:o:.~+:::.?::~q pu:; {.:0~ s:~ a~:c~8.~<

,.~:;,..s:,. %.?~ jo :::::::::::::::::::::::. , , ,. ~- ~

"(> c[} ::: .q,-M s -~: ( @ 7 £ .~:?~::?m?@:::mH,: :~:n~s ~=",~ .~ roT& ;n~?:m:/~?v

A:~s~q~p~i~k:~~ ~:~ ~
(2) whenever ..,R<~:V
(4) w h e n e v e r .:ii(a~,.

.-

"~ " # ""~

a=_~) = X;

(5) whe~.wver ~-~"<.,ff(ac, . . . . .

a~.._~), t e e n 3~.(a~,. . . . . . <,,..~):<,,V(a,~). i

Le{ us see why T h e o r e m 1.1 implies C o ~ e c m r e A. W e assume theft the reader is famiiiar wi~h ~:he basic structure tlmorem fer finice dislvibutlve lactic :~>. ( ?o~st~L for example, T h e o r e m 7.9 of {6] or T h e o r e m liLT of [ i]. Assumi~ag T h e o r e m !. {. we will sho_w somethir~g ostensibly stro~ger iha,~ Conjecture A : Suppose that (D. < ) is a finite distrib:.~tive lattice, and let d~>.. o, ~{.~ be .~.he jom-~r~=dtkm~ elem~,nts of D, where we have m~ranged ttae da's so that if d~ < d i, men i < ~]. Then fhere is a definable (~ + l~-tvpe,, ~, ~t~o,. . . ~ ~.) such lhat whei:tever t~ A¢.(a~,,, . . , .a. .'} is ,, vo~)-extensicm . .a .~(~.'o~ . ,• . . . . . o"~...if, the~: Lt C*~:(ao,.. ., a,y~,¢Oa>: (D; < ) . tn {ac~, there is at~ isomorphism h sud~ that h(~gt(a~)) = 4 for i ~-; ~. We wiII verify the above statement bF induction on n. Let . D o = { d ~ D : ~ o t d.. ,~ d}. The~x (Do. < ) is a finite distributive lattice wi~. joindrreduciNe eiements do . . . . . ~:~_> A s an inductive h)~pothesis suppose there is a definaNe ~-a:?pe such that whe~.c~:er -¢g(ac. . . . . a,~_,) is a r~(wa . . . . . . v._.~)-exter~sion of ~.i*, then there is a~'~ isome~phism h ~ : L t (o,ff(ao. . . . . a,,_.t)!dt)-~E) o such that ho¢i~(a:~)}:~:4. N o w !et i~ < ~ . . < i~._~ < n be suel_~ tIsat d: < eL ill: "~=~o, ~' ; Use T h e o r e m L I to obtai~ an (~z + !}-ty)e t(vo . . . . . v~). That ~(Vc. . . . . v.) works is rather easily seen. We ~:;ext stai:e t}:{e ot!>:sr theeve:m~ frown which fol!aw~; Co~;jcct~.'rc B h: c:-;se>datiy the same maimer as Co~jectu:re A follows from t}hcorcm t.i.. •

~"

T h e o r e m 1,2, Suppose ~hat to(vo

.....

v~) is a

,

definable

.

(n+l)-~ype, a m l that

of Jl, ~hen .-ff(a~. . . . . . a~) is a cofinat extension of .;i~(a~. . . . . . , a~). 7 h e n th,cre is a definable (n+2)-13~pe t ( t ~ o , . °. , v"i, ~ +~~ t o ( t ~ , . .~ ,,v.) . s u c h that "whetwt~er (~) ......... ~i~(~ >. ~ . , a., ~~) is a Woger cofinato, e.xwnsio~, o f ~ ( a o , . ~ a,,); (2) wke~:<~er ..4~< . g <.,f~(ao . . . . . a,,.~.:L) a n d a . ~ , ~ N , then W'
(3) wheneoer ..... .~:~' ~,-* ~" .~,%
a , . . , ) a n d .a{ao,°.

(4.) whe~w~,er .~t( a~....... ~ ,% ) < ¢4 .<.,.ff( ao . . . . . (S'} wne~e{.~er ' .~V-<~..t({a'o,.

a,_). g~-e:'~ ,,tt,¢~:~e , , .

a.); a,,)~N~

a~). then ~,'~'(~<..o.~) ¢"~.¢ff a o. . . . . . a~ ) ,.~.)¢°(a : ,}o

C o n d m o n s (1)o--(5) o* Fheorem l.~. a r e the s>'a~g'afforward :a~}alog:ues of the corresponding'.-- conditions of T h e o r e m 11i: i : :' :

cesu.lt whL:b e~,~;e.~]s .~,od~ . . . Co:~{ecI'l~re.s A fOHOW]n':Zde~Tkieu~.

. . !c~ . . d. o. .t;bii~7~ . ~" a~~d. ;v,. aev

~ mab.~ q.>

v~;

;

9%:

=

j t : < , , C , ,<..v' -zt>q . . . .

With

'

*"

~s

this ~ o t a . t i o n ,

distribmive

~." & . . . . .

:2

p~t?;~vr

Con}ect:UPe [ }

'"'~ . . . . . . . . . .

of'

e'~ad-exS'.Hsic.,s

J

~75!. U

~

asserts 'that ,,x.q.~,~}ev;>~ ,~-,,"~' :f,,..j ~s a

', ,

"

~>,;'~.c:,

°

that I.A* k,.!o.a~,,.,;~:~~<,sJ-. < , .~min(O)~), w[~ere &!(~;~ ~s a ~(@-;,;.;.t.e;:~sb:9 o~: ,<,:~ someth{u:~ m.o;e. If (..'."A < ~ is u finhe disbfibud"ee iatt~ce a n d >s :~. . ,-. . . . ,,~'-"";s a r',~'>'-'~,~<.,x~:,~,:~, :linearly o r d e r e d . . . . . . . ~. ,~..,~ ,,......... ~s ~.,.,..a,.~a.~,:~t½~e t(v) such that Lt* (.g~(a)/.4~ ~-~(D, < , .E') wheneve:r .,i.~(r~):is a R~2).-ex.~ezisv,~.r~of ,Z.

s,~q~';~ev..,., .. ee~ch of. the ]bt.k,-tv,,~,,.~g: ( I ) E is [i~:;?orgy o,,zV.ered;~ (2) rain ( D ) e E ; (3) (i# x ~cI£. gher~ there is ez uniae.ee, y ~-." ............, ..~,,.,~*. *~',,,,~,.~. ..... (i} ~herc & ~H> z such #~at x < z < y; (H) ~ y x < z a n d z ~ E , then y ~ z . '~%en 6~ere is a ~2@nabhe type V v ) sacg¢ t.ha:g ~wk.,e * " ' . . . . . . ..... ~...... e~(~e "'"~ is o: t(v.)..ex~:e~d(,'~ o f ¢f *-- * ¢ *~T' ~ X ~ ~ X ' - - ' / t ) r'--'" ta~n gt* t.,~tka~e.a,4-=t ...... < , E). Pr~mf (~ketch% L,et doo • , d,i b e the i o i n - i r r e d u c i b t e e!ementS oil D a~ran~ed in s~Ich a w a y ti~at if ~¢~ -" ~~ .4 t h e n i < ]. Furthermore,. becm]se • • . -o ~,~, of coad~tlon ,(~*s,a b o v e , we c a n arrs~.ge t h a t : t h e r e ~s ~ set :[5; t~+:t such that x ~ E i-fi; x = s u p (.(d~>..., d ~ _ . , } ) for s o m e i ~ L N o t i c e t h a t (2~ implies t h a t 0 , - L C ~ ~'~'" :.~:. . . .u~. ~ , a d h . "?" tO~] k-)) !~IVh~,:-' ~ a ~ * {{ i ~ L t h e n d~ is ~miquely d e t e r m i n e d . N o w p r o c e e d i n d u c t i v e l y as b.~ t h e prcofs of the Conjecture,,,. H e w e v e r , m: stage i-".Lr~, e m p l o y Th.eore~u !.1 if i~.~; a n d T h e o r e m 1.2 i:f i~.L T h e r e is n o p r o b l e m w h e n i ~ l ; so ~et us check ~hat d~e h y p o t h e s e s of T h e o r e m 1.2 a~e m e t w h e n ~esL "-' Suppo~., ........ "- ~ that i:~I (so t h" a t ;.-,-,,~° a n d that w e a l r e a d y h a v e ~%(:%,., . , v~.<). L e t i(~< . • - < i~,_< i ~.:e Such'~,~.~:~c ~. ~'z0,.. o, ik} = {]: 4 < 4}* C o n d i t i o n (3) i m p l i e s t h a t i = m a x ( i n i) c {io, ~ ~ Tlms~, ;f ~kY .

c 0 f i n a l e x t e n s i o n of d f f a : , , , . . . , a},);

.

.

.

.

:

-

-

.

.

.

.

.

.

[;?

W e e~Ki this s e c t i o n b y sbowi~G -*lea> ~........'-~'~a°~, p r e v i o u s Co~-oIlary, : : : : ; : :7 : :

.

[

,

' ..~-..... 4.=..... :. . . . .

, ..

., ....

P r e N s i l i o n l.,S, .Let D be a n y finiw lattice. ~et E si~;D,r ,.,~,.{,A_,.r ,e,,:?7' ~ be a n er~d.. ex~ensio~ o .f!~. . . . such . . tl,.at .L t * (.~g./iEt ~ ( . D i < ; E ) : T**e:,~ . .,atid'~es~ .. ~.... .;4-->~;g':'~ 0f. Comtlarv 1 . 4 : :

9.1

3'.[i

&:,;i ,,~.d

Pr(~d. (t) and (2) a~'e obvim~s. For (3)!e~ x ~LT, a~_d suppose L'(.;%)=X where H : L t * (,.¥'/.,R'~,, ~.'0._.~ <:, E). is an isomorphism. Then, l e t z b< fl~e. immediate - x = m a x (.:,~, g-, !et. z .:= m a x w,,.a'~ successor of x in E ~ or if - : . L.e t . h.( # ' A ~• z, and.]c~ g ={.;% :..,%~.,%
2~ ~ e

proof of'"I h e o r e m L L

i~ T h e o r e m L I , if n =0~ then the ~heorem~ius[ asserts ~he exis~e~ce of -YV "0 "

"3~

"

~,.~..I~.~.,.,., O f l!11111llla~.tvlxeS.,

(:iuite

IS

a

some

detai~ in Sect~on 3 of [51}, We wiiI .~:~eeda rcthaed version of his constructiom which we present iu T h e o r e m 2.5, Before we give this construction, some properties of mi~Ymai t3~;es w{I[ be discussed. In the following ! e m m a we cotiect toged~er several of GMfman's characterizations of mk~imal types. See T h e o r e m s 3.6 and 3.i3 of [5]i. L e m m a 2 . L FB:" an ~m~_:oar,ded ~ g

~(~ "~ each o f

~s e~g&>aIe.,~g .to tgoH,

being a m_:g~abna/ type:

((Vv ".--.xXq,(~.0-~ g(~, .f(u, e0~ = v))] is a :consequence of T; ',"~),.,:. for ereo; term ?'(~. v) there is a .fbrm:da 6(>'~. ~ t:,~,,,,°*.~,~.~ J.,. J.~' ""'~ o w h:,r:~:~ :.p.u'~ "', ".,~--:~.-r(~,~

iV,, "- x)(O(v).,-,,..~f ( . . v),:,, ~ '~?\: z W

........ ., ...-:..<:

"

. .......

~,)}---~

CV ~ : ,'q ""0 4 : ~P".~) = v))]

is a co,~equence o f T : (3) .,ebb.even,,

term .f(~c v) q,,~.,':: is

Vu:~[(~v,(Vv:>

.q e'....... '-*~,,ma~,, ~}(>)~? K t ' )

")(~b(;-)-"'f(.

:~1.7:~~" ~h~,:?t ~'hi, s,c.~n,:~i~ e

:,:):' !~,'}'~,,,'

( ( V :>: > x ) ~ V v - , > x K. < s, ( v ~ ) . - , . . . . . . ,

•

........

.

~

. "" .....

.,~

" .~

~

.,¢'2IJ¢v.v,{

.-*.

is a consequerce o f 7:

The condition (3) above Ca~.~be expressed informNty i~ a more dcsc.rii:atiXe= w a y by., saying that for each f(u, a) there is a foi:mtfla 6 0 0 c; ¢:v) such that ': ,':.~q-~" cacti ~¢ :

,

i

:

.

r%.¢#t 2;~ ++++>+~+d+:@Of#+r+,+'h++v+~ff,::

95

o;!~zle~ c~Ji~+t8~:.++.+.~.+~£>x++o~'~e<;-+13<¢;~('b}."*(,~Te~,-u~ . . . . . . . 4.~° v +,-~+~,='a'.,~+,,>+,R D£°Ob]et}:?.

m a k e use o f this ++~ q m ' i b r

~,~ G a i f m a a (see +page, 266 of [51]) by ~+ep!acisg (i) av+,~d(23 by a ~d+++,leccmd~tk;,~ x

,14%q,

+

+aI+~g

+'I,

:*+~

W e ca~t Stiet~tb' st'ce.n£then (3) of Len~+s+~a 2. i. i~ Q~e :tbilowk+£ w~-,w • • t]emm~a 2 2 b~>r cat+) m+++++++a+ +~v+e +(+;} +++ ,+ +++4+ t e m t , >F ,+, , -a} +, +~+,r °

+ + + \ ....

+

' V ~ : : ; + x [ ( B ~ v ( V e + ' > .......v . ~' +:+"' s°'~ = + ~ ' ) ) v ~ t "~ ; ; ~',:-~ ]b:+%

(V~++V,:)+,(x<~,+<>,~, '"+ ~ ........ ,;, , + . , + >,s+++,+',~, ,.-.~,+ +:,'~+.':~,++st++ +.~,; < f ( <

,a:+'+;Ul.+.+. ~)~i i s e v e n K . a i ] v

Pro~ff.

"L e t

eventtmilv+

O++o ,+" . c q'"- =++ ~ ( v ) constant

or

err:her b e s-u c h

corlst++~+.~ o r that

eventuaH~

Hlcreasm~

+)Y))?

o i ~ 9+tab ~ < 1

~~o :+ +:+........ + u t.c+e ~ ,~,,+,j - ~ ' / +t,d,, v } ] +,~'~" +~ £+a~c~to++ ~: " " .... ,~

is e~!: +"b.@r

o n e o - o c < e o~t~ @0V:+' +`',/+~ ~

W e wM define two. t e r m s g(~+, tO al°~d h(~, i,Q w ~ o s e v~:fiues are of ~t¢~:est oo,]y for 0~ose ~+ for w h i c h Xvi~f(',¢ v ) ] is eve:ama!ly one+.or~e, On @o(v)+ Co~,~+der a n y

.

. .

.

.

.

.

.

j

.

)-"

] \+.%

z/]j

Let ..... "+,.~,, w) > Q+

N o t i c e that each of the foltowh~g is a cort~-eque, nce of T:

:

Vv(h(u, ~.n~-h(u, " v + i)), Vx~v(h(u, v ) = x), W.u,Vv:~(h(u, ~.t) + 2 < h(u, v~) A 4'o(V0 A #'o',~h)+%&+.~',':-h;< A~', V:,))+ s u c h t h a t 11-Vv(6(v)-÷&o(v)), and s~tch d~at tB.e fu.~acdor~ Av[h(u, v)] is evm~tuMly one+orie On (b(z,) and also i s eitl~.er eveL,+-. ,,m, eve~ Or t.k~,ra~ay e v e n t u fll'~ o d d on ~b(t?}: F. .r. o. .m . the a b o v e c0~~seque~ices of T it e:+:,s~+.y'•-':.tolbws" ' titsi: /~,,~.~\H,+ c.v, ....+,.5} is eve~m~.ailv . . . . b~creasin+: . or~ ~6(i}), +=,+"+'t

Now

let O(t~)et(v)tx~

L e m m a 2+3. Fbr each m i n i m o l LvPe t(v'~. . a~.s . . . . . .e:aca . . . . een';l f(~, t~) the~e as a fc:~'m.+eb:e

8++(r~)+~: t(t/) such that the senwetce w+,vt+:a~[((v,.,> x)(44~,)7-/(u,, .,,)= fCt++,+.+~,))) i v((v<

:

> x'~CVv..,> . . . . x')(e~(r, ~.,,, d,C,u.,'~=++f ( u . v,)'+ +:+~vu v + ~

i + t l +:O~+,.~eql+gepl(:¢ Op+ T+

: :

: ::

+ ....

i

hfformath<., tb:c f o r m a l sente,~ce i~ the l e m m a expresses ~.he[fact t h a t fo~' v.~.v,, ~ ~~" ee m~d ~> vhe functio,~s X;.~[t(~,.,. v ) ] a.*ad ,\o[f(:~'~..*.07 a,'e e i t h e r e v e m u a l l y idemica! o n ¢(t') or else h a v e d i s i o i n t r a n v e s o ~ d~(.~,).

.....

~ ~h"e~ m L.e~,.:n;as ? ~-":'' alld "~;.,.° Primf. T h e characteriza~o,~s of mmm~z~, " * ~ :vpes.. will b e utilized several t k n e s i,~ this ~>roof.

F o r clarity we p r e s e n t the proo~ ia~fo~;mal!v, Let ,:; ~#,,~ <¢...~e,.)be s~ch t h a t for ,.v.m.,,,.J'v {a>:~ e,.~ca' m mc'~ ......unctiot~ ,.h.'[f{m.. . .~,'~]. .{S .c........... . ~ ' e i t h e r Co "c.~,{t~,,RtOf SD"iC{Iw i~{rt'{':lq N" (¢ V , i r

"~ {a k ¢@ I

or:,
[0 iff(u> o) '=[(u> e3 g(,.,> ~',:. "] = ] [ ; ff [[(m. ' ...... ~"" i t]

.?\$I.I .....

z#"

.t k~2,

~'~

.

~ ,~ cb~O.'.~<=-u.u .>e such *,kat Vt..'.{<~,(~)-.~d:o('.0). a.ud for each u~. ~:q, d~e ~,- - ~ ac.[gC,.,~,~ . ,,~..,.:,u)}. is eve~tuailv., c o ~ s t a n t o n ¢ ~'<.~e.\Ve .r~o,.~,...d e l i ~ e a tev-m ,....,>hi",. ....... g~. ,., so dmt the whw,,~*~,%~p r o p e r t i e s hoid: ......~:.p~....... m . ~ i . are such t h a t {or s o m e x , whenever x<-e~< ¢", ,i -,~ \ o-,.of, O.'W-,' )A <~*~'2,;, then ,f(u~, ~'>,t , :-'o ~i ~ ,~#" " ~ ,.,. . " *'"~,.,> <'~. ~'~-.f ( >. , ~. . ~.;.~ . . .1 - .h e. n . h. O. , . , ~ . . .',~ ~s t h e iarS:est .< s ,,~'ta that there: is a s e q u e n c e go . . . . , z~ w h e r e z o == o aa~., for ; < it. ~..~+..: = max

({ z: ~: < q ,,. <,% (z } ,,. f(~e~, z '~ = f( m. ;:~)}?.

such tha~ Vv.(tT~0.-,~d~.-~(-'.0) a n d for each ~t> t~> the h m c t i o n M..'[h0,~., ~<-,u)j is e k h e r e v e m u a l t y eve~ or eve~mmily o d d otl g~(*0, w e claim that this ~ ( e ) satisfies the coo.dkio.u of the l e m m a . T o :~ee this c o n s i d e r a r b i t r a r y u~ a n d ~ o K, for i = ! or i = 2 ,

OF

LZ4V:,~> ,~,'~(<~&.'.b.-- f ( u ~ v) = f(,'<> v)), the~:~ . there is n o ....... *q-,0r~ So w i t h o u t '~ " of "~.tl,,i,~ . . . . . . ~'* th~...... x:~ call ass!line Ih~'t t h e r e is

some x such that

so tlm~ (1) a b o v e applies, Let tts lit a d d i t i o n a~%t~:xo t~!a[ V~hVt 2[
.

.

.

.

.

.

"cA f~u~' ~:d ?~;ri 'b:~ ;~-sOOqa '(L):;~o.,t % .}0 s~au.,~nbasuoa mv:::I9)pue:(g)saaualu:as ~V+Va aea~:> Sl ::~ (:~)~L; 7[0 S~IU~adi:,:~d attl ptie (,q 'n'lf Io u01i!uiaP aql tuoi R

= -~)z K ',,,qmui% atl$ <" <'<'~{' ¢,~g t~v d g aai ;aoN T Z

i

~q: (o. *i)q~ ~-~Ifl";lleU[R "{iaQq) [:~} it> au;~ul~-~la oql-¢i aql i~mm~rl }e (¢j~ satisfies 'lOl:itppl u~
s:~S~X';: ~.<~~£)~.1 [g,a@l a R £ q+.~t Y()(? :aC!- IO ".Iu~tual'a ql-d otl:i a[l ((~ 'd)q I !0"l *l~a~t

"i.t~]oIsIp/uemua,~t !i~.i cX u o [(a %=)t.J/y p u i ~:X" u o [(a ,*n)glIa¥ . . . . . . a=;b',--, r g ' q ! 1~:I11 't~',X {IO IY~IISUO9 K l l e ! l l i i o a a

loll

$[ [ ( a { i ~ l ) i ] / ,

g io i X {io I/ilgUOa

+q[~n:[l'~*.-'~ ~ou ~,~*[(o.qr~g]o; V }[ letll sl,I~,~SV (L)'(IImL~.>-:t "tX u o soa~ur~:~ lu[o[sw, $Alt[

L,~F>~+{.,*,>"..... ()I}a ..IO

UO

L(c~ ./,i,..~y ,..'~c,:.~:~un:~ z,ql +~:t~ pue ~n UOe,a .~;)} lmU sX+s (9) pro3 '~X uo a u o - a u o .vo ~ueasuoa dlim~:u, aA~ st ffa 'n'~g]ay uo~lamlj gtll °n qm?,a ~o} l*:~ql s/,t~s b ) u~-~U, "(a. ~K)4~ & l pau{}ep uoutate.d o*ta jo SosseIa oaualean~ba a:)u!is}p aa~ ~'X pu~ ~X ....... :+,J,,,-,.~.k~'~'~" "paa~:moc~un . . . . . . s~ qaN,w }o gst~p oauolv,unbo. W e e suouuJed. + gu!ug[op s~ ~{.) v,',%o~., ,.=....... ~.. a q pinoq,i',. (a "/,)# pu~ (~ v()0 sqntu,m}, aq~. (1:0-(!i "~,svaum'.'.as },,:) ~sne:~aI1

({{".~. ,,;,.. +-

, ..................

~. . . . . . . . . . .

~(c~ "i)4V, X < aE),
{l,1 :~ i /

.1,

+i~'~,q

""

.......

"

,"~I~) 1((:,~i> !~$ ..>, y. 'i~iV~ V3,='~ l,l,i:i Z~ $,4

',,~.:

i]~Ib ~,~

iitli

~K.)Yli?ilt/,!i[l(30 1©

£?I~ ,,[q 5sOddnll A'~oN

,'~ t :~ ~ w ~ ~i:~;~O,,'~ ~ ] ~ : ~ i ~ , ~ ¸

w h e r e y:,-:+"~y+~ a n d k~ X:++:':{o: ~,{y,+" + i,<'},so ~hat X; i S at~ e q u i v a l e n c e class o f the p+artition d e f i n e d b y 0(~,5, v'< S t m e o s e t h a t g ( u . v) is;e~,eniuat:v oae+oa:{ o.+: >2. f o r i = I . 2. N o w . bY way of coilt.radiCtiov., s u p p o s e g( +++_ + L.',)=: +{'++~ ,+.L'+f+L,'~,,:i:d+itra.;tV large v i + ~ : , so t h a t g(++;, h:(}'++i th+jh(Y> z , ) ) ) ) = +(u+ h+(v4+ he(he(V,+ +z):))) f o r a r b i t t a r i l y large <,, z : such t h a t 4{~+:(h+{y:~, z:ib+ a a d ~?.z:)~+ Thtks+ ] ~ {+t , )+ +) , -+

++'>+,liSt ~ + " V'I +

+':[ ~)~ ' : =,' ,

1:+¢+ ~1X+' +::, + .X'+ . . . .}. +. lJt+'{ }+'. £<~ `"m+I , _:.,,,+,', ,?,+ S i n c e

v++ ~4 .... V,,~ i t f o l l o w s

ttiat

+

h2(h+(>.., z J ) ¢ h.(lh(v+, Z2)), so ttmt the hmcti0-.s k t>IrfCO,+.+... + x>+~),. +, ~:)] .... rex! ;'t++ktU, ~.,)]_, are ever:really+ com~tant m~c:+ {demfc:: ++m +a,{;.>}+ t ~ ~" rI~ r . .".' ,. .u..+, v-,).+., " (..,. . . , , c a : ' :,~.+ <

k++[g~;+> cmmo++ " +¢+ +>P] " co=tradicfion+ i-j]

evet+tuai{y

~e

onc-ot'~e

o}:

X+,

fher+b-,¢

v~c.id+++< ,+ ' ++

W e now c o m e to tl~"~ ...... c o n s t r u c t i o n of. the type . .,~d~+~.~ . . "~ W~S ~rot];I+~sed .... a t the begkmimg of this sectio:+_. T h e o r e m 2+S. L e t q>. . . . .

ok.+ ~ be Jisti+++c'+c(>+,:sm++t:+y+,+:bo/sn+~+,. +is'++~<+a++d ],e+ "Fu :~ T be a c o m F M ' e ~h~'oo' i:.++ ~hc ;;a+++:4>':~g,~ • :~k{co . . . . . c~° ~})+ 77,+~+n de:are is a type

i+x. z+ z_

+,i,~

\¢7\+.:0+

+ o + +

x+_+ "++ f\CO+ ~" .+

C k. + _ + ~ + .

UJ~+

.

..

+

%:_~, Z+.) ++,

,

+

-

e~Ct~ CO~+SgC+pf fe+f'f?{ :'(" +

+++\<,{>

,

+ .

+

c+,
(2) ~hen:" are +Sq.+wm+s ho(v,) ....... h:~
c;++_> +:0

Sot each ] < k ; (3) for eac~n +°erm f(co, i . . ,

c~:+.~, .'+, v) the~+ #: a ):~rmu~a (c+:+. . . . . ca-:. v) a++,d a ~enn g(+;') s u c h Na~ g w sct++++ence

~(++,o. . . . .

V u E i : ¢ f ( ~ w t V ~ > .v)(,+++,c.+,. . . . . v ((V~+ > x ) ( 4 ( c + . . . . .

c+~. ~. v}-.. + / ( c o . . . . . . <~>

q

c:~_., v)(=

~+ u, v} :,'+"+v))

C)--L>' g(f(c+ . . . . .

c+_+, u, +,)) = .++))]

i t is d e a r f r o m c iauses (1) and (3) and L e m m a 2<1 that fl~e a b o v e : y p e t(co . . . . . <::~:<, v) must Oe nmm+..a~+ T h e k e y f e a t u r e of the t h e o r e m is ,,:ha: li~e term g t w ) h~ (3) is a ~erm of ~+ so that it in n o w a y d e p e m t s upcm ~he co~st.mts c~ . . . . . henever t, +,~+0~ the conclusion_ (2) is v a e u o u s l y ~rue, a n d (3) is true for all minL,"m+, types b y T h e o r e m 4. I+ Proof+

utiti+d~:g :~ ~-/~+ ++~ ~ w e . can . . assmne, . . . without . . . . ioss cd' ,
By

that k = i.

Let L~(c..+ + %+ u, aI:d c'-: W e wi|t -~ ++ ,,++:,<}: ~ +.o,~ + e : i t m < <+~:< a(: i:¢'ra'~s irp¢oiv~m <+ de(++:e, ckg..'C>,+2), for e~:ch /<:++o so +m, ..... ,+~ +~+.~+c:':+,+,+" t1:e f o ! J o w i n g ser~tences ~s a \st:\

,

+ O I ~ S C q ~ t @ l t C C OP T 2

VyYx(++

>

x)4,~(y, v),

VyW,(,/,.:+d>

:,)~ 6,(y,

r)).

T h e type ¢(cc,, tO witi +x' any type which includes {aS+(<~: :.) : i < ,go}:

L:src,~:::Lb~g

-

~<,..{v. ~:~ {~e {.!:e f o r m u a a t:sserh~':g t ' v ~s ~v.(:v~}:{~ : h e {e..'m v,V~tbe: " . . . . . . ~:

¸¸9.9

°

r~owe:r ,..~ the y-~:~: -"'..~-, " : ~ ~ y v Jv3P('&~

s-'y--=

. . . . .::) . . . b y h@o!.v), ~:, v~ L e t ,#~(......... ~ b e she for'me;ia ~,P,,:~:~ ar:d the r o l e of ~(~. °¢-; . Iem:ma., . . . and '~ d , , , - ~'' 4~(y, v:i, e x i s t e ~ c e is gummt~teed. by. ~hat T o see : h a t th{s Construct::o:: work~,;.
>w,, a:av ,* ~: A::tj:vc¢:, ae, :02 :s ao{ e v e v>~1~v cons~a~t oi?

e~,:+:l,co, ~ , the:~: ~heI'e- [s ~: ,.~uch ~hat ~

fH.? \ _.- ? ~: v¢o~ ~ "z W.,-~ e ,_.:v'%.¢,vJ2W',) t, ~*:z "- 4,co, ;,'OA ~i~,:-:.-_(Y, e:)/~ & + ~(c.,~, v2)

/\ x < ::-~A X < re) --~ ~ (Co, :{,. -....'~'~,~, ~,~-,.,,.&-,,:.~,,:, . . . U:)).

i:: a p r e v i o u s v,:.rs:on of this p a p e r , the fo!h>wi:ag 1emma was p r o v e d u n d e r the ....4 4 " , ' . ~ ~ ~v,~(,v~,,,~o t h a t Y ( a ) is a ~:(~)),-exte:~sio:; o¢" JV fc.r s o m e mm~,na: type 0~,(e,~..,,F E e prese:,t, more. ~e:~era!-.. stateme:':t ,,e.~,v,~:,._,._L, : .........., e,~ .... we~.: ~.,o ...... ~i,,:.. ~-4-,-~:;fiedop r o o f , is d u e to H , Ga.ifma::. L e m m a 2.6= S~,~x,ose e:a;a: N'(c~) is an e~d-ex~e~:si¢?n m[ ~; :~:d supFos:e ~h~ ac

Pr~mf. S u p p o s e b ~¢N~> T h e n t h e r e is a t e r m f(x, y) a a d c e+.N such ~hat b = f ( a , c). Then' define d = ~ y { f ( a , v) = b~. S i e c e ,-i:--~ c e :~ ÷ it foIlows ~ha~ d ~ ~ A.lso d is defi~mb[e f r o m a and b, s o t h a t d ~ ~\~> T h u s d es.iN)~C~~'.~ so that d ~ ~'4~ a~d h "=f(a, d) ~ N:. Thus ,~¢o.g N~ a n d by s y m m e t r y .Nr~~2 ?~> ~[

Definition 2.7. A {g + 1)-tb2ae ~(v0,.. v~) is " "' definai~e ff the ~_~,),emg holds:: W h e n e v e r c a , . . ° , c~_.: are distinct co"~s*aat sytnbo:s not h: :~g, a u d a d e f i n a b l e t y p e f o r T;~° Th:. previous, defiMtiop, c a n b e f o r m u l a t e d in tbe~, ~o~owh~<~_ .... ~ equiw~Aeut way... T h e vk) iS relativelY d e f i n a b l e 2ff for a,~v formula 4G~, vo ........ V(~: t : e r e is a f o r m u l a ,:r(u. v~,,~:, v>._;? ::mh t h a t for a~w ~e:m f~".t, ., ~

(k 4,1)..type t(~o.,

t ( v { : . . , , r~)~ D e f i n i t i o n 2.8. Suptx~se t h a t t<,(vo, . . . . v,,o..0 is a d e f i n a b l e v 4 v p e ; a:~d ~h(v_.> ;o°, t~a) ~s; a relatively d e f i n a b l e :i:k:4= J ?-.tvi~e. Vui@~ermore: sixpp0se i6 < ° : ' <: ;'~,,..::< ?~,

100 and

.i'o~::[ &L!~:,::~

that ¢(t~o .....

r.{..~)~{t%,

e::)

: ,

~ff : o r e s >

"

~

t~,

~". . . .

~

"*

Then define

i n t h e f o l l o w i n g w a y : Supp<). c k + t = t~ a n d t h a t .¢0--" • ° < I > .~< n . w h e r e i < . , . -.

i~--,_}~ tie . . . . .

h_O = O. T h e > f o r a,n~ f o r m u l a ,~(uo: i : ?~4, ..:~ ,~,,,,. : o.

f o r m u i a < ~ (uo, •.

*'

"¢D

u > > <:,

>~:.. ~) s u c h t h a t ~ '~ --, ....

.*.'.,~) "*~', ~,1,~,~.~,.'~

e # ~.

2[.

iff

i t is e a s y t o s e e t h a t t(t'o . . . . . F~,:thermore.

v~,~, v,)~2-~ t(v~. . . . . . . h)r,

cop, s i d e r a f o r m u l a

j~.-~ ,vo,

v~...~)~,oo . . . .

. ........

Defiaifion

,.},(~. vi, ' . . . . . .

,

2 . 8 is a ~ (a~ + l)--;i~*pe.

: ; . . ~ ) ~ I(vo . . . . . .

~;,)o It t n r a s o u t that t ( ~ . . . . .

m u l a ~:r(z~, u o . . . . . . ~.r><, ~.)o. . . . . t e r m s /'o(~:c,. . . . . . u>..~) . . . . . k--.~(vo . . . . .

~;~) d e f i n e d h

i~ is e a s y m s e e t h a t t,-(vo . . . . .

e~. o <,-... . . . .

:,,0) a n d t~(v~, . . . . . .

v , ) is e v e n a d e f i n a b I e type. < ....... t~,,), ' t h e u

t.berc is a ~e.>o

t~k--~) s u c h t h a t t-or a n y c o n s t a n t t e r m ~ a n d a n y .f;~(vo . . . . . v,...~), 4(~,.fo(vo . . . . . vk-.~,.). . . . .

v~)et,(ve ......

, ~ _ . ~ ~ t:~.(ve, ~ ; . ,

t h a t % r a n y c o n s t a n t t e r m % cr(.% % , . . . ,

c'a)

iff

(r(w,f~(vo .....

v~_~) . . . . ,

e:~:)..Also, t i m r e is a [ o r m u l a r t ~ ) s:ad~ t~,_., t:~,,,. • . , ~h~. ,?< ~,~(v~,...... , *)...~) iff

;(vr) e T. T h u s °

iff

l { e n c e , i(v~. . . . . . .

~t.) is d e f i n a b l e .

Definition

.g{(ao . . . . . .~'({co; . . . .

h (co,.

2~8 c a n 'Ix: v i e w e d in m o r e m c ~ d c L t b e o r e f i c t e r m s . ~Hu" ~ose a,._.~) is a t , ; ( v o . . , v~i:~)-extension o f ,.U.. T h e n w e can: c o n s i d e r t h e

c£_l}).-structurc ( ~ ( a o . . . . ; a,, &.), a~,, :. : . , a~2_~). T h i s ~;t~-uctu~ h a s ;' . . . . . . "~

....

a

!

.2

a,~) :is a ~kt:>~a,

.

v.;%ex:tension 0'.~ J~ ,

, >-.......~......~, b'<. ,:ts ~n ti~e }'ivpot~e.~is ~',~:',.!l.:..'dt~m~w(:.r> {Asi C.~:.<,,. ,'~} b.,e k>~-'I- '=~ ......~¢'. iI:i i l l ']{'.<'.Ori~;
i~(Cc~,

,,,~,v~,o s(it. t{ i ; m -''I ' ~

~

:

£% "i ':= a'..~'~

~ ( i , ) O > ,- , u :, {'*~,~ I S

of ..!L (1) .id.q~ ..... ..... .. ,

, ~

,,~_~

......... ,i.*K1.]v.r.~2~ 2

.

" - ~ ~',.~

,

~iI]l~;?aie

,

*

~.,) is ar~. ep.d-.,exfe~!do:,~ Oe " ~: # g g l U . , , , . ~" . . cat_,) is ~ t * " = * n *~"

:,.

R{-[¢- o ....

; ,,

*

......

,.%~..~)-" t;~ltC;e I" a!g;,r~-,, "

'

,'~ c

"

'

,:~ '¢,

"

/k,L]})o

+l+{~eiL~ there ex{st : < ,M(am + ~, ++=_.) and a t e r m f6% .'+'+ s~tch that .+4¢1(ao. . . . . (% ~r' , . . . . . . . +l+(a+c>,++., a,..;)>a._+= v(d). Tlms, a~, e N i (3) I_et W'=.,g(+;,, . . . . . . a,,---,3 a='~d a = ~n; a n d t h e n appIy Lem:ma 2.6, ;,~s "eb;+ is ,,,,~n~,~a.=~a: ...... ~.,,a'.~.,- ..e'o-,.,,..('I~ ~-~-~._Tbe~>e.m ,_ '~ fi i has

k_ l* ~tle,(~.av~:. f r o m

[z,)

7}~

T h i s c o m p l e t e s ~he p,roo~ ,.;:~fT h e o r e m 1. L

3, ~ t e pn~ot o f 11tcmrem 1..2, T h e p r o o f of T h e o r e m I : 2 will g e n e r a l y fol!ow the iines o f q~,.... ~,~; !)K)of -<', .... T h e o r e m 1.1; tt¢~wever; t h e r e is a p r o b l e m wihich immed~ate>J s~rface.s: T h e ;:'~':;e Constructed in m e p r o 0 f : o f T!~corem L t (in De.fimition 2.8? w a s cof~st:.~ . . . . .u~. . . ,.~.,~,u~x~ ~..0,2 +'amalgating *~ tw:o t31,~es-2~. One was the n-tN3e ~ ~ , . . . . . hypoi!msis, aild dt~ e t l l e r was t h e m i n i m a l t y p e w!~ose ..... e~is~e~c{.' " w a s p:¢oved i1~ Thco~.em 2-5. N o w w e k n o w that: a~l! m i n l m a i t y p e s are e~d,ext~sr.~sk~,~'~a~,typ,:~s, b~.:ri: wI,at we n e e d in t h i s case a r e cofinal e x t e n s i 0 ! s, t.t is easily: seer~i {Pr0pos!tio~ 2,,2 0f ~ 5 ] ) t h a t if . ~ / < Y , w h e r e N is a m i m m a i e x t e n s k m Of ,:,i.¢, ' v~ tacit t<:.o ~}.D toM-,eXtension Or a cofinat e x t e n s i o n o f w.° Wh&t iS ,~eeded ,he~, {s to fi..,~d ..............~-, m i n i m a ! cofin~t e x t e n s i o n s ; £ ' I l i e s e . of course, c a n n o t b e Co~:~sta~ci,ed b Y mear~s o f minimal t ~ s : neverthelesS, as shown w Blass [3], for countabk" .,~:./.t?~ere ¢:d:4S a minimal cofina[ eXienSiOno T h e conSm~eti0n w e w(?{1 use i a i-:~e:i-;:~-o0~,of Thc0:renl~ ! . 2 i s very similar: to tim ole, u s e d b Y BlaSs, c o m b i ~ e d with the t e c h n { q u e o : f .~,,(?Clt¢~t ,~

:::

i:

:

?

I02

.i~.?:{, Sc,h~r:~'#f

Definition 3:~i.. W e s a y t h a t a v.~ o, ~eo~',~, ti,::<~, . . . . . a v~:)-definabge if <.ach of:[i~e :(o!lowin~z ._ h o l d s -

~,. :

(2) b(vo, ., c',~ { s a germ s u c h t h a t .......... ~ " " .... each j ~ k and m < a~; (3~ f o r e v e r y f o m m t a ~}(~, vo,-o =~a~-")t h e r e ia a f o r m u l a vk)) e ~v' "~*<,. . . . , O~,,, \e k>O', . . . . . .

:>;~),

the~

,::,

:'

x

,5V/'tv0,..'' . ,, ~,:.'>, >, .....

~re,(~¢ v~>

,,~~

>:: ~} e.i ;{~:><. . . . . . . . ?', ~) {ff

t~k ~+ ~{,~2o . . . . . . . .

N o t i c e t h a t (2) a~x~ve i m p l i e s O~at if ,~r < ,~,~ a~c~ io . . . . . .

i,,~ ~ k. t h e n ((c¢ ........

It is e a s y m s e e t h a t a n y r e l a t i v e l y b(~..'0. . . . . v a ) . d e f i n a b t ¢ t•ypc :(~:.~. . . . , v,~ , ~ ~;s d e f i n a b l e . T h i s ]ns~ veqz~h'es tt~e: obscr~ a d o n that (a' < i~(v~ . . . . . ~:~¢))¢ : (vo . . . . . . v~..~) f o r ead:~, c<:mstan't t e r m rr, S o m e t h i n g m o r e ear~ b e show,a, f o r w h i c h w e ~ e e d t h e foHowi~,g defieitio~?, w h i c h is a n a l o g o u s t o D e f i n i t i o n 2.8. •

D~fi~ilion 3.2. S u p p o s e t h a t to(t:o, . . . . h ( v o . . . . . v k ~ ) is a r e t a t i v d y b(vo . . . . .

*

x

•

~.,~","~ ~s a d e f i n a b l e {n. , 1: t y ~ . , an<~ tha{: t;~)~defir~bte (k +2)-~Wpe. F u : r t h e r m o ~ c ,

~.:<)e&~(>.,, . . . . . G) Also suppose that (o'}~b(tg eac,:h ] ~ e~ a n d ~7
....

, <,))¢:h~.(¢o, ~. ~, G~) f o r

ha t h e fo!~owb~g w a y Si~pposc ~ -+. g = n a n d t h a t j o ' < " " ' < . k - ~ ~; ~, w h e r e {re . . . . . i~;}(?{] o .....

}.~.}=~.

('iNe % r n m I a % Zf ~v(~.'.o,...,

For

ae, y f o r m u l a

,p(:< '.h~. . . . .

~)de~

4((el,, ....

~%~.,},

c o m e s f r o m D e f i n i t i o n 3, t.)

G,+,,) is as in ~ , e previo~,s d c ~ b l i t i o m thc*-~ it c a n I~: v e r i f i e d t h a t

t~ (~. o. ~ ~ ,~ . . . . . ,~.. , ~ c:: t(t;o, ~. ., <, . , h T h i s cm~ b e d o n e i n a m a n n e r S k n i k ~ to t h e d i s c u s s i o n folk>,vb>~ .Oefinitio:~ 2°8.

if the f o l h ) w i n v h o l d ;

a.~,_~) is~a t(~;o, ~ e x t e n s i o n of , , ~ ( a e ,

G:,~)-extension a,..).

o f ,~, t h e ~

d¢,(ac,~

. , :,

i

a.i~t't is a m i n i m a l : ?

103

v: :] f o r

Wl~c~

extension

.;d(ao, ....

of .{{(a;,.

-

~ "

end-,.~xte~s~on

~ a;~).

fc~:maib:.abIe

mcdized

~ ~-" " ' Qeh~11t{Or{$,

in Peap~o ',"*~-",'~.< ..........

p~opert.y

in Pea~o

w{~ich

we

, '-~,~

:

W e ar¢" n o w ,.l o. i.n .~ I;0 m a k x ? s o m .e

rot'lOwing

,;f =,,~>:m°.

.a.X.A. 5. . D?~:r!.,OS<)o f -V~b>CK! - a : , ; - .' W p:~J

°--.<-

descnge

!-, ,, e , ~ ~ a " -

,.v>.~c~,

~.,.~o;:'~.~ady, b u l ; "',"<":"°>, ~,,x~ ca~. e..';~s-gv

bc~

ik>;.-

a'dtlametic.

S ~ @ p o s e X is a set: s u c h '4"a,,,.~. . .~,,.~,:~~ ..... iX)" ---.>.c,,.z," 3;), ~md sap)".os¢~ t h a t h . / : X-.+{(}, I.}

m'~d gi~ o"',..X~.~r-.~.It~. ~ , L} a r e hn)ct~<.~ns W h e n e v e r

i ,~-~z. T ~ e ~ t t h e r e is a )....... : " c i;< ~:.;
t h a t ,..,",,-,~,. ,. ( Y ) : = y ,~-,d~, ~,'~ o{s ho~:;~o.~,-e,aeo~,~s~ ~, . . . . . for each "I"[le~1 w e

q~

'~''~*

.-,,

> < < y + !i} =

" "

\

_is ~74 " --uQ:}~r.:Oe~: ' " ~.., t h e r e

K~" !:{ ( W

~.~ z _

,~-:{< ~,~, y ) + z

is s o m e

" ~" /

~o s ~ c h t h a t d:~e ,sen,~e~4ce t~l-*

is a co.~,se~ 'de rice

he

y) ~,

<

v
m<

~,:e'" a;:~d e a c h

.---..fi,~-a <~ ~ ....... e. . . . :

* °.,>

'2'

Ir~

1,-~ '

~.

l,,~h<.~vs.~.g Ic~:-l~va is ar~, a n a l o g i z e

<.,~.-'~L e m m a

.4.4.

Lemmu 3,4. ~ugj;ose that O(z, u) is accepmbi~ and '~ ~ ' J]ere is ~e~ acceptable 4~(z, v) s~mh that the sentences

(1) VzVv(g,(z, v ) ~ o(z, v)), ,-, v~ # v::-~, g(u, v~)# g(u, v j))], (3~ . Vz(Vu~ . . ~ z. ) ( V u. a ~ .z)[NV,Vv:~(~)(z, . . . vt) . A @(z, %)-~ gh.,_~,,"'u,,~ ...... ~,-.~,_O~ ~.:~, va))) v (Vvo)(::, v ) - + g(u~, v) = g(u> v)))],

:

2

[

NotiCe. that ~entm',ees (2)-(4)correspond to sentences ~;-'~" , (-n,~ :m°//ex,:rm~a 2o4o ....

e~.|,

w e ~,iil proceed quite informallY.., For eacK ::Iet ,X? =.{-o: t;~,, ......... u~r,~ ,~,~',4'

1041

3.H. ~<:hmed

,

then define g~}: [A~ ]~:=-~_{J_?,!} so that flmfc~k,,.~~4~mhcflds: w h e n e ¥ c r ~,. c,, ~a.'X~im~d •u~ < v > then = "-: ] : if g(i, v~)= g(; ~t-.'.) if g[i, vi) ~ g((, v,)

e~(v,~, t;;)= ........

Also, define )~'~' ' X2 -.~.{0, t} so timt

h.~(v~ = {0 ~ "

if g:J, ~.') :=; e(i. ,.~)

~vl. i f g(:?, v) ~' g(], u)

From *h~' . ~ d,.tLut~cm ~ °,," °~ s of e(<, y) and .o(2,.. '" -v5 it foHo-~vs. that for, each e .-'.,m + 1. th~,~'e is a s.~, "v',~ .X\ such that ~,a .... e, .~

X"

=,¢

h~, vaAea~.4 :~-#.. '~; ~,i 4 z. It .fOllows_ .... .[ .a.u. .k. . !. . . V / ~ 'C] ; a* 1 ~, eas~y from this h o m o g e n e i t y that for each. u ~-: z the, function hvtg~,~*; >~ ~s e.m~e~ co)~stmU or o n e - o n e o n Y;, and that for each z~, u.,-'~: z, the ftmctions M>[g(u~, ,~YI and A.v[~;i"~.~. v)] are either identica! o~ Y; or else have disjoint ranges o ~ Y;. This handle.., .... sev.tences ( 2 ) and t,~> ~';' of t h e ' ~emma. N o w we wilt inductively dedne Y~ g;Y~ so that if g > ~ , u + l theu card (Y~)~: .*:?{:-° : r - m - I ' L For z ~;:m 'let }% ~={k N o w consider s o m e < .... m, l e t and which

is h o m o g e n e o u s h'r each _g~.~ a n d

card ( R } ~

Y~ ( w + 1 )

card (Y..,.]~z V ~(uv z - m - I )

W<2;

W"<=

2"}en set -v_.~-,,~. _ >_~.~ y~.:. if ~ ~::iz aud )~v[g0,, . .u}] is. o n e .< m e . on ~ ,~:. I:mu g(u, e~a R L so that i

...... ---car(, (R b>/3[~, ~ - m - 1 ) . C h o o s e any g~ g,: Y Sud~ th:~:i: c:,~.d ( 31, ) = f~(Z, ::,-. m ,~ i) required.

'

[3

3 , 5 , Supg~se h~(t:o. . . . . Irk) iS a defirJl{¢i~: {~"~Vpi~: a};h:l b(/-~o; , , ; -l!v) ~'.~-(;~ t~r~tt such that (v'~"~ b(vt~. . . . . v~,)) e to(V, . . . . v~) Nr each i ~-~k ond m < a> Tlwn t~iere is' a relativety b(:vo, 2 . . , v~)-mirdmal type t(vo, : i v~.:+:,)g- t~i(~.',:,.. . , V<:)such that;

Theorem

is a COfinal :extensi° n ` of .~-(ae ...... , a~);: :

:

: ::

i[ ;::

::',---m-/.*~,,: q<~-?~,? .{~*ff-~)~+:0 i~! .!(,a-~z)~ g|nm:O} ? ~o a:~ua:~s~xa aqt slla.%~: etnmal

o:9. :<~u:~ "(c¢"rW[ &: (o,~P35'1o $1ol aq~ pue. (a.f2)~O Kq p~Ke,ld ~I (~a¢z)8 ~o alOl : ('(>~a)h,/S':{U.~[01 oqt }0 ~X~]OtS~XO ag.i *.© v.>~ea~q ...........~ ? s a q t aI::,o .topis:~oa Oi Saagp:~s ~1) ~:'~ pue ~a ~al¢l~!!.:ga .-.-~:,:i:~:,f; ~ ~:.vo,,~:: su]:~al , : e ~ae.~;~mmo {o~-> i :(~+:.a ~n){g) :~a:] "(a ~d):~o paugap /{~3>:©.:~e ct,¢}3U ov ]i?et ~i~oo~[ o(o> ?, ti317~:),~o]: ((a 'g)~(~ <-R((f 'a)~:t'~0} DA~'a',A~_L llgI[I qoiqg s~muJ:~0~, a:0e:id~o.o-e ].0 {~a~> i ' (a ~'W0x. a:~anbas ~ augap 0t gu}o~ ~.ou oa~: %k\ :

'L j o a a u o n b : a s u o a e, s~

'(.,t~ ~; a >. : ~.,(a 'tz)°O)aA,~EZA

"C,::'~ = a v ( z "z',,d > ,~q,~'~E

:(-:., , .... oo:)o: .;: :4 [(:'*>>~ = H ~"*~

- • ,

~o,.

::., ~ ::," %~

'

~ - , "

°

e

%'F',<#

,,. ,,,

,-e..,t

'~:-a.~1~/; ,gO f g' £~

a,t::'~,, .: f

~'q~,

~ ? ':" "~'}" v ..,, "

>~:w~;~ ,E:f

{,~)'"

,

bL

"

c'~)) ~ ~(vo . . . . . ,'~) for c~:~ch m < ~, tha~ t(t:~, ~. , , v:~/:~ ~s ~)~sistent. T o finish tt~e p r o o f that tOt.~, ; . . , vk +..:) is a ( k ~2),-b~c:e, i"~ :suffices t<~ show Camptete~ess. St? [~:t 4@0, - • ; .. vk+~l~,b e anv~._fovmala. . . . . . . The.~:_ . . t!.,.erc, . . . . ~s . .s o m e i < e ~ for w~v:'ch ./:(u. ~...-.~..,~' ~ ~ ~s the t e r m ~x[&(ho(c,-+~'L ~, ¢,, , " ~- " ~ v =~'-? F~'-,',- the d:eii:~.,,itio~ of ~+~(vo . . . . . vk÷~), it easily follows ,qmt fl~e Sentence

4(vo,.

va ~

We

~tow s h c w

o ..... T h u s eiIi:e:" w ,

; .:

o~

&(,%,

is h~ t " "

that

~(v; . . . . . . .

:',.~

.-

.5..>J,(~,a . . . . "°

~ Thi.s ......., " ~

~; that

o~: .t) is re~afiveiv b ( % ....... ~ va"-definabi~. Coi.adi-

"'a,k-~.a~<->, ~ ~,.""= 0~., F r o m (2). of L e m m a 3,:; it !\A[ows tt~at

~s a c o n s e q u e n c e of 12 T h e r e f o r e , -~ c, tvo, • o . , v~.Lh_v'uk ~t~A. ,t % . . . . .

l',k+~)""~ ~ ( ~, :~'o. . . . . . .

..

easnv ~ c . v,~) is th.e d e s i r e d , o., ~adta. H e n c e , '~' ' . . . . . ~::~1) is relatlve~y b(tk, . . . . t k ) - d e t i n a b l e . ,~vo N{::-;:t we !<~tow tlKlt ~*,U~o, , . . , ~" - ~ h + l / ~ is relatively b @ o . . . . . ' " " " W e ba';e ' r~)-mmm~a~. ,.~.,~.~d~ show~ c o n d i t i o n {t"~of DetiMtion 3.3 Fk)r <2), s u p p o s e that ,t( (ao . . . . . . % , ~ ' ~v . ~',~.~.,)-extensio:~ of ~l( C o n s i d e r s o m e term ~ (u, ~:~.: 1), {rod !or c 4: M, t~o ti~at we are c o n s i d e : i n g the, et:e.,nc.~ ~(c, a~.,.~) of .ft.(at,, . . . . . ~:~, ~), S u p p o s e S~(c, ~:h,. ~):~:.~4(a,~. . . . a,.). F r o m ( 2 ) o f LenlDla 3.4~ it follows that e.

,, ~.:; ~.:: ,- ~ / ~ V u ~ t q h . _ , , t t e

.....

A4~+dao,.

..ft(a~ . . . . .

a~ ,) {'~¢-(,:t, ~ (c., c.'~+ ~)) ~-::~~ ~, ( k : a r } y ~:xtension . . . . o f. .,,~(ao, . . a~:)..:xo t h a i t('v~,,: ..... ..~

minimal.

(Ik~ Vl )

, a~,

v d).~,IAc,

vj:#" t;(c~ v~'!}.

!!!erl : . ~ ( a o . , . ° , - ~ - , , ) is a m i t i i m a I ' "~ ;~ r e l a t i v e l y / ' ( ~ 0 1 . . . t, ~,

. . . .

T h e c o n d i t i o n s ( 1 ) ' ( 3 ) Of the t h e o r e m wilt c o w b e t;,~::~:'~:e¢l~ or (1)it is e~tough to o b s e ~ e that the f o r m u l a

. . . . . . . . . . .

is i:: ;Q v e~' ~G:

.

¢}~,+it "j:°

.

.

:

"

:

e2¢. ),. F o r ~,2) :>'<:~ ~...... a b e a d y ~ee~: tha~: ~'~, e,

I = v ~:,'(,~

.

"

:

q'7

~

:

[

: ::: proof oi (4} iS sh:a.i:m ~o t::e 9:'oof of (~) o: ":1:ao::.sm 2,5° C0::s:c~e:c :-:om:,c ~'f:, :

~ I.,~t :(w} :.~e the i:@i:m

"~Vr'd;~(~

-(h,('Q

b.(a£L " "

.~t (:t.gu~

~ z," z O

:'trope: <,, :x:'.quiyCd by (3) oI:'~he ~m~x;~rem, TlJs com:.~let<'~s ~a~: :~,:~,o,_.of .:t;e<;:~-~:>

W e are ::ow r e a d y for the p r o o f <,,f "~i:~~ ; ...... t.2. ~ Q" vO a a d as in the h)~)othesis~ of t:hat *:',~---~°~m°.:,,.~ ..... Then there ~s s i:e:rm

L, .... <--..<<~'~:~-:~.~.... noe

b tvo ..... t - : ) si:ch th::t ~,v~~:' ~ -: :.u~o. . . . . ." ~k": ),'.:=.to(Vo..... "~" ~, :':''

":4,\?

whenever

,J"<<;n

and

.~n.+< a~. I.et t-,tv:~,..'. :, 'G)~ bc the (k + i).4yp~; st:ch it:at ~:,~<4:,~" r~.. o., %), ~..... ,~o{vo,'"..,, v. ?, W e ::o: apply T h e o r e m q._~,<:(wl.~h :he role ~,: ''~ to(t'~o,.• + , v:], behtg p~ayed by -''i}0 :~. ~,,,:~,~:,ly b('::o-... ,v~,:)-mi::~mal tyTx: ~:tv:>.. v):.+::) :--,,t ,, J~) to .:eta . . . . . . . . . .

~

*

,,<~,.~,_:~,:h~ ,,.~,~.,t,ton~ ( 1 ) - ( 3 )

"°' .... ~<,--o,

.,

G;+:J

As :~:rcady me~mone~z'

that theorem.

"-" '~= :'o(v,:..... t~v:> ...,'

D,~)::<

~

Then

5(~,, ,~

' ~ ,.A: .....

, + . :

D

' ~-" :<'~- 3 . 2 :sel-

v-. ~ . v~+.J

v:,~.:) Js a definable (n+?)-ty:~e...~ .°~. it ,:-e::,~a:i~sto._

ve'dfS' c o a d i t i m : s ( t ) - ( 5 ) o:f T h e o r e m 1.2, so s u p p o s e t h a t d,t(a.:),. ,, a,.:..,) ~:, a t(vo . . . . , :,,,+:)..extension of AL ( ! ) .,ff(ao . . . . . a,~+:) :is a Cofinal e x t e n s i o n of ..~{(ao,°.., a~,). F o r , by Tiaeo;em . . . :s . a c o f b a I extension: Of ~gv-%,G,., c~'~ '3.5[i), . d r , a(: % . , . , . , ~ k : a..) . .~. . .~-Y2c} . . . . . ..,.,g(~.o:. .¢ " ' ' a.,._:)r'< ,'.-:b(% .... as ) w h e n e v e r i ~ n . (2) S u p p o s e .,t~,'.~ .. ~ .,.,,..~,. ~ =

f(a,

a,,+:). T h e n b y (3) o f T h e o r e m

.:~.~ t h e r e is a t e r m g ( w ) s u c h t?mt .&~ ao . . . .

a ..:)k~L,+~ = g(d). T h u s , a,, ~.: ~ N (3) This proof is the anat(:gue of Lemma 2.6, S:~ppose ...Se(o,,~+.~)<..q':~, ,,t%..<

N~-M(ae..~, ., a,,), N o w using:the, n o t a t i o n i:: the ~:~roof, of ~[]~eorc~m ...... 3 : , t h e r e is term: . < ( u v] s u c h that ...~ta0;..'. , a . . ,)b:(d=~ o.,.[:¢c,a~.o~..,3"~for ,,,:---<~e,.... . . . . .c. ~: M(a:,, ~. G(L: L e t g(:;:~ G + : ) b e the i e : ~ : :

some

l*uf3'[~[~i+:qhott;,+.t), "

.

.

....

;L<(V~-< ;, t., A 5,,,t, t,~ =-' .v

. . . AVv~Vt,"(6,.i.::(h~,(tt,+:),.

G ( v ,", + ~ .

q; ' "~`

[

<

i08

~

T h e n i t is clear t h a t ,~{a~i,

c~:...N,~ T h u s c .eN;, .

,~.

' ~ .,,.~n . . ......

.s o .t h a t . d e. N , ~.

" ......~"*, t~,~it ,,-~...... ' ,... c~,N'~ so '{h~i: i h a ' t ..... ~ ~°.~ :;:.',~"2, so~ b y sVl:no~

. W e . h a. v e . s h .o w n

mere,, X , = , , ~ " , , : *% v4) This folDws easily from T h e o r e m "; v,~,:~-~ (5) This follows easilY from 2~-~eorem '~.-w',~ This completes the proof of ~ e o r e m I '~

:

:

'

'

:

4. A c.haracler~,~fion o~ mlni,mt t y ~ s t n this section we give the previ0usl 3 ptofl~ised characieri:~ation of mi~.?i:om.l types. C o m p a r e this t h e o r e m wfih L e m m a ;.~ t

" r h e ~ e m 4.t, l f

'

"

is

a

" "

~,m~ a ;'erie:~, ~( w) sz{ch t h a i #u: se~ tenc<~

ormug,~

VuVx[(~w(Vv

> :,.)(+(v) ..... :,.':,< ~,~•

=

w',~...... ,, (W,, > x)(. +(~:)-~" gff( ~- ~0) = ~:;]~ .

.

.

.

.

is a co,,~sequenca o f T ,

Proof, Le{ &(v)v5 f(,~) be such that !;<~th the seme,,~ce ~r~ t'"):,~of Lemma 2~ i m~d the sente~:ce i.~ L e m m a 2.3 are co, asequev, ces of T, t)efi~ae th.e ~e.-rm h~w) to be

T o see that this choice of d:,(v) a n d g(w) works, we p r o c e e d informal!y. F o r a n y i:~ ....,:~-':- z b e such that whe~:lever u,~.. . ~¢2~,*~ . . tlie~l t h e f u n c t i ) n ~,~'~('}',,u;. v';: ~ . - - o , , ~o,t,..)~" is eh:her o n e - o n e o r c o a s t a n t , a n d cithm ~ g~

x

o r else (V~:......~.,,,~.,,,.,?,'.4~,:-..... .,~,.~ ~,q~ v ) = f v( u) ~z ,. . . . . Let x>z be sach thai w ~ e n e v e r u > u,,.~; u a n d {(
6(v~)/'4,(va)"%f(u,,v~)~:sf(<:, (Vv,.~ iv.-.x)(4~(v. . . L( Y,.r .

,c%(> I~he.n ~-:> ~,,~.~'d,{,....,--,. .,. ~,.,,,., ..,~~,~,~'~o v ....)), ~..1::::; i:(~ v 3 5 ~

:m I.,ar{icular . . . . . . . . . . . . . . . . . . .(Vl . . ~::~ v":'('b{p) . , ...... f .I.lfl_( , ( u u)),, ~:~)':: ~'(~,~ :',~73 g(j'(u,v))=v),

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1{ desired, a slight s t r e n g t h e n i n g o f fhis t h e o r e m can: ~ : obt;~ Paed by >:tplacin~ t h e "T" in the s~atemen~: of the t h e o r e m by: "Peano's axk~mS'~: Y

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O v t . .t:h:-~t : , .... > ~ - : ~-,. 5 :: ~ : ~ O:L :.,, :. ~...., :C:?~!ee: . . . . . ~~.~:. ~.>>:b :,,voIf~:a::g [:5c~:

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Ph.IX .

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to be a .... "

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f'f 3:2 ,:: $.%*x~ \ % , ' a .Y*' ~ x *i'V " ~ }'= . . ~w' &"± " {..""" ,

1977) is ,-:.ienific:mt:~,re~a:e~:~ tO i~.~.Sp::~?;ero A~: ~[,-4YOe :S ~:[~-e:aat:~ral :,.,...~,_.~,~'.~'~,::':'~,~:,..~s,..,,.,,~.,~'~ ...... of a~: ~ °.bq)eo:h: ~v:,:cb t:~e fr::e ",:a:::al:>lesa:'e ~k,,3:::.'the:set :.,::,.~ : ~.:,>.-. ;:~.::g.~,:,.:>+:< :,-_,, ~,,::~:.~:~,~: e,~..,:. a~:d @2; ............... P' ~,o" .... ge:~e:-aliz~tio:: o:f Conlectme .A. !LeL .:) i:>~ a co:::phrte, ~,a,-...~:-,:.:.. distribut:[ve lattice ~n wi:ich eadh co:~'pact eie~ne~tt has at most cou~xtabiV ;:):arty compact p r e d ¢ : c ~ : s s o v s . "[]:e:~ ti:e;'e ::~ a:: S-type f ~.u,.~ ""~ ,:}.~>.t.~s>>,:..~,...,,.., .4:."is :-: *.oe:xte::s:or~.c( ,g, the:: Lt {~GA"} ~.D.. :f'h:s :'es:,qt:is ea....:s.,~.~.. ..,,?"~.m~,~,.~.a~ for :,.iSi:£iDva:i~e ::i1:}:t£¢.~S;

Refm'e~¢~ [[) d, Bifk}.'off~Lmttlcc TL;cor% Amer, Me~l~, Soc, OaiL P:~b:XXV, 3rd cditi0r.,, (r~rovid~:mc,4R..L, ~'2] A:'~d~c::s:?k~s~ ~I]:e ::::e:.~ectio::~o~ ~K~m:ta:~dardmodels of mqdm~etk:, J. Symb. Logic 37 (!972) !03-.I0(?, {3] Aadreas Elass; O~. ce:mi:~ Vpes m:d models for a~.tbmetieo J. S~r~b. Logic 39 (!974) I5!-!62. [4] }t2.Gaifma'a. O;~ loc£d :uitha~etic fm:ci:io:~sand the&.... a~)9,~:"-~:a~e~.oasfo~ cetas'!.ruc~i~gty2~es of Pear,o's arifi~metk:, M~.q:~o.;~a.~icN:.egic and !"om:~atloas of S~'t ':"heory~ y. Ba>Hiilei ed.] (North.. tio:ta~.d, Amste>da~.~, ~970} 105--121. i:15] I'L Gaifman. Mode:s a:~d b'p~s o~ Peano's a:i~-l~:::etic,Annals of Math. Logic 9 (1976) 223-306. {:61 O. Gr~itzm', Lattke Theory (W< H, F~a'zmaa a~d Co°, San Francisco, f9"70). [7] R. MaeL~wett, ant? R., Sl*cker, M~Jelle der Arithmetic, Iafmitistic Mett;ods (Pergamo,~ P:ec-e aed PWN, Warsaw. 196!) 2.5%G63. [g] J,B Paris, On nrodels of arithmetic, Confereacz~ h: Ma.:hema~ca? L gie, Lo:~don '70 (Spr:n:ie~Ver:ag Lecture No:es 255, 't972) 252-4~8(k

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