CONSTRUCTION OF MODELS - Logicshelah.logic.at/class/Sh_c_VII-VIII.pdfCONSTRUCTION OF MODELS VII.0. Introduction The character of this chapter is different from that of II-V as extend-

CHAPTER VII

CONSTRUCTION OF MODELS

VII.0. Introduction

The character of this chapter is different from that of II-V as extending the theory does not change much; hence it is natural to use theories with Skolem functions, that is, when we want to construct models of T, we choose T,, T z T,, ITl! = ITI, and T, hrte Skolem functions; we construct models of T, and go back to their L( T)-reducts. The importance of T, having Skolem functions is that: (1) we can find such T, for every T; (2) every model of T can be expanded to a model of T,; (3) if M1 is a model of T,, every submodel of M1 is an elementary submodel. (This is done in 1.1-1.4).

Saturation is a very useful device, but unfortunately if A < A<", T does not necessarily have a K-saturated model of cardinality A. We have constructed in I, 1.7 a saturated model by an elementary chain Mf of models of T of cardinality A such that in Mf + , we realize rte many types over Mi as we can (remembering the cardinality restriction). We generalize this process (in Definition 1.6) and prove (in 1.7) that for many x's we get some weakening of saturation, e.g., of the form: if p is an m-type over M , 1p1 = A , and every q E p, IqI < p is realized in M then some r z p , Irl = x is realized in M. Notice that when A'" = A we get a K-compact model (when the length of the chain has cofinality Z K , of course); if (Vp < K,)(p<" < K,), 8,'" > K,, and h = our results become weaker as p increases, and we get none when ,9 2 y = K, > K,. However for the theory of dense linear order we can get something (see Exercise 1.7) and a generalization of this is utilized in VIII, 3.2. (There are also other applications of Section 1 to VIII, Section 2, and 3, but we do not need them in the main presen- tation.)

From one point of view a saturated model realizes many types, but from a deeper point of view it realizes few. Let TP(S, M) be the set of

397

398 CONSTRUCTION OF MODELS [OH. VII, f 0

types realized in M over ii, TP,(M) = {TP(ii, M ) : I = (a‘: i < K ) ,

a, E [MI} . Then clearly for a A-saturated model M of a countable theory ( K < A), TP,(M) has cardinality I ; ~ ” + I * I , but for a not necessarily A-saturated model, it is possible that ITP,(M)l = 2a“. We deal with these problems in 1.10-1.12 and prove, e.g.,

THEOREM 0.1: If p < 8, =- pK < X,, t9 .c K + , T countuble, 11M11 s Na+B then M hua an elementary extension N of cardinality Xcr+B for which ITP,(N)I = 2”.

In Sections 2 and 3 we deal with Ehrenfeucht-Mostowski models, generalizations in two directions, and various applications; which are preparatory for VIII.

For a theory T , with Skolem functions, every set A is a set of generators of a model, and it is natural to look for “nice”, “uniform”, such sets. The clmsical example is a free algebra in which every permu- tation of the set of (the free) generators can be extended to an auto- morphism of the model. But for any extension of the theory of dense linear order we cannot have such sets. However, by Ramsey’s theorem and compactness we can find models of Tl generated by indiscernible sequences, so the order of the generators is important. Clearly the elementary type @ of the sequence and the order type I determine the model up to isomorphism, so we denote it by EN1(I , @), and its L( 2’)-reduct by EM(I , 0).

Its importance is that we can easily find orders I with specific properties, and often the model inherits them; e.g., stability (of the model) and some of the generalizations of saturativity from Section 1. An application is

THEOREM 0.2: B”0r stable T with the f.c.p., lTll = No, I ( X a y T, , T ) 2 214.

Unfortunately this is true only for cardinalities > I T, I, so when we try to construct models of cardinality lT,I (e.g., in VIII) we have to generalize this construction. We first show we can assume w.1.o.g. that in L(Tl) there are only countably many non-logical symbols which are not individual constants, then we form a model N from individual aonstants, and define EM1(I, N 1 ) for any order I, and elementary extension Nf of N. Now we can determine properties of E M 1 ( I , P ) by those of I and W .

OH. VII, $01 INTRODUCTION 399

At first sight, we cannot expect anything better than an indiscernible sequence of generators. However, we sometimes know of a set of elements satisfying some conditions, and we want to find such a set as homogeneous as possible, which still satisfies those conditions. For proving this we always need the proper generalization of Ramsey's Theorem. In particular if T is not superstable there are v1 E L(T), and sin (for q E such that for q E Ow, v E O'w, vl&&,, a,] iff v Q q. For dealing with such situations we d e h e the indiscernibility of {G8: 8 E I> for a model I : the type of a8,co, -. , . , us(n-l) - depends only on the atomic type of (s(O), . . . , s(n - 1)) in I . The general treatment appears in Section 2 and the specific cases in Section 3. We deal there with theories with the f.c.p., and indiscernible sets indexed by trees. An important case, which does not completely fit with the general case, is uncountable D(T) for countable T,. What we get is {an: q E "2) so that the an's realize distinct types from D( T), and for every cp(Zo, . . . , Z,,,-l) E

L(T,) for every n 2 n(v) , if q1 1 n(l < m) are distinct, the satisfaction of v(an0,. . .) depends only on (ql 1 n: 1 < m) and a. For uncountable T's see a generalization VIII, 1.10.

The reader can relax while reading Section &almost no background is needed. Our main aim is

THEOREM 0.3: If h 2 2", p 2 2N0, T, countable (for simplicity), then T, has a p-universal model stable in p, of cardinality A.

If I is a well-ordered set of cardinality A, M = EM1(I , 0) is stable in every cardinality 2 ITl] (see 2.9). So if D is a good ultraflter over K,

M"/ D is K + -saturated, hence K + -universal, but it still is stable in every cardinality h = A". So assuming G.C.H., we can prove the theorem for a successor p. However, without it, we need h e r methods.

Remember that (by 111, 4.10) a type p does not fork over IN1 iff p is finitely satisfiable in !MI (though p is not necessarily over IMI). For not necessarily stable theories the second condition is still meaningful, and we can prove for it the extension property, and that it is satisfied by any type over IMI. So if M , (i < A) is a list of models of TI, con- taining, up to isomorphism, every one of cardinality s p , we can find Mi z M , such that for each E ]Mi l , tp@, U,<, M;) is finitely satisfiable in No. Let llMoII = ITJ, if T1 has Skolem functions utw generates a model of TI. For ,u 2 221T11 the model is ,u-stable ; but we work more and get it for ,u 2 2IT1l. By such a construction we prove our theorem. Notice that by such a construction we can get a sequence of indiscernibles.

400 CONSTBUCTION OF MODELB [CH. WI, $ 1

In the last section we deal with Morley numbers, i.e., Hanf numbers for omitting types.

PROBLEM 0.1: It will be interesting to find new kinds of indiscernibility. This may require proving new partition theorems or trying to find an application of a known partition theorem, e.g., Laver's generalization of Galvin's theorem 7 4 [7]& (i.e., 9 + [7]R(,,,)).

VII.1. Skolem functions and generalizations of saturativity

DEFINITION 1.1: (1) Lo has Skolem functions in T , ( = T , has Skolem functions for Lo) if for every formula p(z; 9) E Lo there is in T , a sentence of the form

(Vi7)[(34&; i7) + 'p(P(!i); @)I (P a function symbol.) Such a sentence is called a Skolem sentence of 'p = 'p(z; g) and P is called a Skolem function of 'p(z; 0).

( 2 ) T , has Skolem functions if it has Skolem functions for its language L, = L(T,).

THEOREM 1.1: (1) For every lamgwe L there is a lamgwge Lc1) I> L and a theory T(l) = T(l)(L) in L(l) such that:

(A) T(l) has flkokm functions for L. (B) Every L-model can be expanded to an L(l)-model of T").

(2) For every language L there is a language Lag z L and t h y TsK = TsK(L) in LsK 8ucJL thut:

(A) TsK has rSkolem functions. (B) Every L-model can be expanded to an LSK-model of Tag. (C) ITSKI = ILsKI = ILI.

(C) IL(1'I = p 1 q = ILI.

Condition (A) is satisfied by the dehition of 2'"). If M is an L- model we expand it to an L(l)-model as follows: we define $(a) as any element b of M srttisfying M C 'p[b, a] if M C (~s)~J(s, a); and we define $(a) as an arbitrary element of M otherwise. Clearly we succeed in

OH. VII, 8 13 SKOLNM FUNCTIONS 40 1

expanding M to an L(l)-model of T(l), so condition (B) holds. Con- dition (C) is immediate.

(2) We define, by induction on n, Ln and Tn: Lo = L, To = 0; Tn+l = T(l)(Ln), Ln+l = (Ln)(l). We let LsK = UneopLn, TsK =

Un < Tn. Clearly Ln is an increasing sequence hence any formula 'p in Lax is in Ln for some n, hence in Tn + we can find a Skolem sentence of 'p, so in TsK there is a Skolem sentence of 'p; hence condition (A) holds. As for condition (B), if M is an L-model we define by induction on n, the Ln-model Mn: Mo = M, Mn+l is an expansion of Mn to an Ln+l- model of Tn+l (by (1) this is possible). In the limit we get an LsK- model of TsE = Un Tn. condition (C) is immediate.

THEOREM 1.2: (1) Sugqme M is a model of T,, N a subnzodel of M, ( M , N are L,mOdels) L, = L(T,) and Lo E L,, and T , hae Skolem function% for Lo. Then the Lo-reduct of N ie an elementary subnzodel of the L,-reduct of M .

(2) Suppose M is a model of T,, T , hae Skolem functim, and N is a mbmodel of M . Then N is an elementary submodel of M.

Proof. (1) By the Tarski-Vaught test I, 1.2 it suffices to prove that if a E IN1 'p(s, g) E Lo and M C (3s)'p(s, a) then for some b E IN1 M C 'p[b, a]. But by assumption there is in L, a Skolem function P of 'p

(in T,), hence M C (3s)'p(s; a) implies M C 'p[P(a), a]. As N is a submodel of M , P(a) E INI, so we finish.

(2) A particular caw of (1) where Lo = L,.

CONCLUSION 1.3: Suppxe Th(M) hag Bkolem fundim, A c 1M1. Then the Skolem closure (=Bkolem Hull) of A, { ~ ( a ) : a E A, T a term} = do1 A = acl A is (the universe of) an elementary submodel N of M . So for every 6 = (bo, . . . , b,) E IN1 there are a E A, and t e r m T ~ ( E ) , . . . , T,(@

80 that bi = .;(a). we Wite it in 8hort 6 = ?(a), ? = ( T ~ , . . . , 7%). If A is (totally) ordered, we can aemme is an increaeing sequence. In any m e we can aasume there are no repetitions in a.

THEOREM 1.4: Suppoee [MI = {ai: i < A}, L = L(M), h > ILI, h ie regular. Then the set C of a < h which satisfy the following, is a closed unbounded 8ub8d Of h:

A , = {ai: i < a} is (tlte universe of) an elementccry idmodel of M .

Proof. By I, 1.3(2) C is closed. Let us prove it is unbounded. By 1.1(2) we can assume w.1.o.g. that T = Th(M) has Skolem functions. So

402 OONSTRUUTION OF MODEL8 [OH. V'II, 9 1

a E C iE A, is olosed under the functions of M. Clearly Idd A,1 < A hence for some /3 = /3(a), do1 Aal G AB. 'SO for m y a. c A, if a,,,, = /3(an), a = Un a,,, then do1 A, = A,, so we finish.

DEFINITION 1.2: (1) &(A) = { B : B c A, IBI < A}. (2) %(A) = { B : B c A, IBI 1 A}. (3) S A ( A ) = { B : B G A , IA - BI < A}. (4) #$(A) = { B : B G A, IA - BI 2 A}.

DEFINITION 1.3: ( 1 ) H is a set function if it is defined on sequenoes {ti: i < A} (ti any entities),

H({t,: i < A}) c ( 8 : s c {ti: i < A}} and

H({ti: i < A}) = {{ti: i ~ q : 8 E H(A)}.

(2) We sometimes use aa a& of indioee sets other than A, mainly orders. H is a pure set function if any permubtion of A maps H(A) onto itself. H , 5 Ha if H,(I) E Ha(I) whenever H l ( I ) is defined.

Renucrk. Clearly SA, S f , SA, S$ are pure set functions.

DEFINITION 1.4: Suppose H,, Ha a m set functions, M a model, A a cardinality. Then

(1) M is (A , H I , Ha)9aturated when for every set A s 1 M 1 of cardinality 5 A, ordered in some way in order type A, and p c S m ( A ) ; if for every B E H,(A), p 1 B is realizes in M , then for some B E H,(A) p tBisre&edinM.

(2) M is (A , H,, Ha)-compact when for every sequence r = {qt(2; a,) : i c A} (ai E 1611, q, E L ( M ) ) if every p E H,(r) is realized in M , then some p E H a ( r ) is realized in M.

LEMMA 1.6: M i s A--act iff it i s (p , SN,, S1)-wm~pact for my p < A. M is A--eneoua iff it k (p , rSn,, 8')-eaturated and No---, for every p < A.

(2) If H1 5 H1, Ha 5 H' , M k (A , H I , Ha)-saturated [-comw] then M is (A, H1, H " - s a t u r a [-compact].

(3) (A, H I , Ha)-Baturatim and mpa&ne%8 are p e r v e d by &fin;- tional expansion. (See I11 6.14.)

(4) (A, H,, H 2 ) - ~ p a & n e % 8 is preserved by adding individual constants and by taking reducts.

Proof. Immediate.

CR. VII, 5 11 SKOLEM FUNCTIONS 403

DEFINITION 1.5: Let M be a model of T, (L = L(T) = L(M)) 11M11 = A 2 K + ILI , K is regular. Then M is called (A , K)-wturated if the following holds: (R(I?), E) is a model of ZFC, consisting of the seta of hereditary power < I?. Some N E R(I?) is a (2”) + -saturated model of T and as a set N is of hereditary power 5 llNll, 94, = SA,, (a 5 K ) is an increaeing and continuous sequence of elementary submodels of 93 = (&I?), E, N , A ) (N, A serve as individual constants) such that:

(1) 93, has cardinality A. (2) If a E l93,l, and a is a set (in R(R)) of cardinality 5 A then b E a -

b E ISal. (3) (IS41: p 5 a) ~ 9 3 , + ~ (hence, e.g., there is an ordinal in 93a+l

bigger than any ordinal €BE, hence the set of ordinals €!BK has cofinality K).

(4) Let N , be N as interpreted in 93,. Then M is isomorphic to N,. (Notice N , E by (3), and that N, is an elementary chain as 93, is, and L E S,.)

Remark. It almost always suffices to msume N is A- or at most A + - saturated. We can replace “(R(I?), E) is a model of ZFC” by weaker conditions to ensure the existence of such a’s, but our theorems will not be changed.

THEOREM 1.6: (1) If A 2 K + ITI, K regular, M a nzodel of T of cardinality 5 A, then M has an elementary extension N which is (A , K ) -

eaturated. (2) If Lo c L, the saturation of DeJinition 1.5 holde, and Lo E 23,

then the reduct of M g N , to Lo is ako (A , %)-saturated. ( 3 ) Let M be (A , ~)-saturated. If 2IL1 I A, M is K,-saturated. If Ax = A,

K > x , M is X+-compact and X+-honzogeneowr. If 21LI 5 A = Ax, M is

Proof. (1) and (2) are immediate. For (3) see the proof of Theorem 1.7( 1 ) below.

THEOREM 1.7: Suppose M is a ( A , K)-saturated madel, p a cardinal s A. If A@ > A > 2, let N, = mink: f 2 A} (hence 2, < N,, cf(H,) 5 p; p < a - N$ < 8,). Let A = K,. Then:

X+-BdUV‘d&. M is No-hO?TlQgent?W.

(1) If A = A<#, K 2 p then M is p-compact and p-homogeneous. (2) If 2” < A < A”, y - a < cfp, cf K, # cfp, and K # cfp then

(3) If 2’ < A < A,, y - a < cfp, cf N, # cfp, p < K , then M is M is (p, sn,, s : ) - c o m p ~ .

(p, s,, S1)-compact.

404 O O N S T R U ~ O N O F MODEL8 [CH. VII, g 1

(4) If 2Y < A < AY; and for any 8, a < /3 5 y =+ /3 - a < Ks; p < x 5 A < K a + x y x # K then M ie (x, SHo, S;)-~rn~)a~t.

(5) (i) If 2fl < A < AN; and for every 8, a < 8 5 y =- 8 - a < NE and p < x < K 5 A < t h a M is (x, S:, S1)-compact.

(ii) If A = AMo and x < K 5 A, then M is (x, Si, r91)-com~t. (6) The dde7nentS (1)-(6) b l d with 8d~t'dion of c0rn'p&12e88;

but for (1) we s b l d aesume 2ILI s A, in addition.

Remark. In parte (4) and (6)(i) we can replace ''8 - a" by "/3".

Proof. W.1.o.g. we shall use the notation of Definition 1.5 and assume M = N,.

(1) Suppose p is a sequence of m-formulas over M of length p, ) which is finitely satisfiable in N. Aa K is regular and >pl, p is over Na for some a < K. Now L, Na E IL( 5 IINaII = A, hence the set @ of m-formulae over N, belongs to b,+,, hence SN(@) E Sa+ ,, but [ & ( @ ) I = A'" = A, so p E BY(@) == p 6 (by (2) in the definition). So p is realized in Na+, (as Sa+, is an elementary submodel of (R(K), E, N, A) and N is (2"+-saturated; we use such reasoning without saying). Hence p is realized in M = N,.

The p-homogeneity is proved similarly.

CLAIM 1.8: Suppose u E IS^,^^, u c 1SASEl, S c u, IS1 < Kt 5 IuI. (1) If IuI < then for some VES~,, , S G v, lvl 5 Kc and if

cf Kc > JSl even I v I < Kc. (2) If I u I s Kc+,, cf 181 > j, then for sow v ESA,~ IS n v( = (81,

I v I 5 Kc and if cf Kc # cf (Sl even lvl < Kc. (3) V c f K c > 181; and x < KI;=xlSl < A, then in (1 ) K p , < A+K5+,

so S E %A,B; and if cf Kc # cf 181, x < Kc == xlSl < A , then in ( 2 ) we can aesume v c 8.

Proof of 1.8. (1) Let lul = NC+,, and we prove it by induction on n. Clearly Kc+, 5 A, and there is in a one-to-one function g from Kc+, onto u. For I < Kc +, let u, = {g(i ) : i < e}, so clearly u, E SAPE and if 181 < cf Kc+,, then for some I , S E u,; as )u,J < Kc+, the conclusion follows by the induction hypothesis. If (81 2 cf Kc+,, n = 0, u = v ie sufficient.

(2) Let lul = Kc+ ,, and we prove it by induction on I.

OH. VII, 5 11. SKOLEM FUNCTIONS 405

If ( = 0 or ( is a successor the proof is as in (1). If ( is a limit ordinal cf I u I < cf l#l;hencedefiningu,,i < Nc+easin(l) ,there~((i) , i < i, < cf 181 so that u = U ,,,, u ,,,,, hence for some i, 18 n ue,,,l = 181, and of course lue,,,l < IuI, so using the induction hypotheeis on ueco, 8 n ueco we get the desired conclusion.

(3) Immediate, by Definition 1.6, condition (2).

Continuation of the proof of 1.7. (2) Let p be a set of m-formulas over N,, lpl = p , p finitely satisfiable. We should find some p' G p, Ip'l = p which is realized in N,; so we can replace p by any subset of cardinality p. As of p # K for some B < K, the set of formulas in p which am over N,, ply has cardinality p. So p1 is a subset of @ = the set of m formulas over Np, which E % ~ , @ + ~ . By Claim 1.8(2), (3) (taking p, = S , @ = u , 5 = a) and the assumptions, some p, E ply !pal = p, is a member of S,,, + ,, hence realized in N, + hence in N,.

(3) Let p be a set of m-formulas over N, of cardinality p sucih that every p' E 8Jp) is realized in N,. We should prove p is realized in N. Let p = {p,(Z, ail: i < Cl) and choose 6j EN, which realizes pj = {p,(Z, at): i < j}.

As K > p there is 8, a 5 8 < K such that a,, bj E N,. Using 1.8(2), (3) with {6f: j < tc) = 8, the set of finite sequenoes from N, = u and a = 5 ; some AS" r {Ef: j < p}, 18'1 = p, belongs to 8,. As 93 satisfies "there is a sequence 6 E N which satisfies any formula p(Z, 7i) (a E N6) which is satisfied by 6i for every big enough ~ E S " also SF+, satisfies it, so this 6 necessarily realizes p (in N,). So we have proved (3).

Now we need

CLAIM 1.9 : Let Sc be as in Definition 1.5, N, = min{N, : Nz > N,}, 2 ~ < N,< N t ; a < / 3 ~ ~ * / 3 < K B ; p < ~ 5 K r < N a + x . I f ~ ~ 1 8 t I , S E u E ISc(, lul < K,+,, IS1 = x then there is S' E S , JS'I = p, S' E lScl.

Proof of 1.9. We prove i t by induction on lul and for fixed u by induction on X. Always there is in 93, a one-to-one function g from lul onto u, and we let ue = {g(i): i < 8 for I < lul.

Case I : lul s 24,. Then, of Ha s p < x, for some < Ha lu, n 81 2 p, hence (by Definition 1.6(2)) 8' = ue n 8 E SC, so we get the conclusion.

Case 11: lul > Ha, x a limit cardinal. Then for some ~ ( 1 ) < X, p < ~ ( 1 ) and I u I < Na+xcl,, so choose 8, E 8, l8,l = ~ ( 1 ) and use the induotion hypothesis on u, 8,.

406 OONSTRUCTION OF MODELS [OH. VII, 4 1

Cme 111: IuI > Xu, x is a successor, 1.1 > x, IuI singular. As X u < 1.1 < Xu+x , x regular, necessarily IuI has confinality <x, hence for some 6, Iu, n Sl = x so use the induction hypothesis on u,, u, n S .

Cme IV: 1.1 > Xu, x is a succesaor, 1.1 > x, 1.1 regular. Then for some 6, u, z 8, so use the induction hypothesis-n u,, S.

Cme V: 1.1 > Ha, x is a successor, IuI = x. Let 1.1 = A:, then for some 8, luel = Iue n Sl = Al, so uee the induction hypothesis on uey u, n 8.

So we finish the proof of 1.9.

Continuation of theprmf of 1.7. (4) Let p be an m-type over N, of cardinality x (which is finitely satisfiable). If cf x # K then there are /I < K ,

p* G p , Ip*l = x, p* is over N,; and then we proceed as in the proof of 1.7(2) using Claim 1.9 instead of 1.8. If cf x = K , x # K, x is singular (so a limit cardinal) hence there is x(1) < x such that p < x(1) s A < Na+xcl,, x(1) is regular and # K . So replace p by p* c p , 1p*1 = x( 1) and proceed as before.

( 5 ) (i) Let p be a set of m-formulas over N of cardinality x; such that every q E S; (p ) is realized in N,. Let p = uf pr, pi has cardinality x, the pi's are pairwise disjoint. Let zf E IN,] realize p - pf, and aa K > x for some < K , 6f E lN,l and p is over NB. The rest is like the proof of 1.7(3) Using Claim 1.9 instead of 1.8.

(ii) Now the proof is trivial. (6) The proofs for eaturativity are similar.

DEFINITION 1.6: (1) Let i be an infinite sequence from N, ii = TP(i i ,N) = {tp,(in6,0,N):6E~bI((6afinitesequence)}.

(2) TPA(N) = { T P ( i , N): i is a sequence from N of length A}.

DEFINITION 1.7: For X = S, C

TPX"(a', N) = {E r is a set of formulas p(3; qCl,, . . .) such that for some 6 E IN[, N t p[6; afo,, . . .] for every p E and X = C * Irl s K , X = S == only S K

variables appear in r} TPX;C,(M) = {TPXK(iiy N): 5 a sequence of elements of N of length A}.

Remark, Clearly from TPSi(M) we can compute TPA(N) and vice voi%a. .

OH. VII, 8 11 SKOLEM FUNCTIONS 407

LEMMA 1.10: Let L(M) = L(N) = L, and add new variablm q, i < A, to L.

(1) 2A s ITPA(M)I s min[2aA+'L'y llMllA]. ( 2 ) TPX$(M) is a monotonic function in K , A ; but if K 2 A , [X = C *

K 2 11.13, it is a constant function in K ; and if K 2 ILI:ITPC;(M)I = ITP&(M)I (in fact, one is computable from the other); and ITPSi(M)I = I TPA(M)I (in fact, m e is computable from the other). A180 1 TPQ(M)I I I T P m w I *

(3) 2A s ITPfl(M)l s min[2(A+ILI)x, I I Jf 11"- (4) 2A s ITP8;(M)l s min[2(Ax+a'L' )Y 1 1 ~ 1 1 " . (6 ) If M i s K+-compact, ITPCz(M)I s 2A+ILI, and if N is elementarily

equiwalent to M and is K+-compct, TPCZ(M) = T P a ( N ) . (6) If M is K+--eMOUB (e.g., K+-sdUrded) t h ITP&(M)I S

2A+ILI.

( 7 ) If M is stable in A (i.e., m < o, A E lMl, IA1 s A implim ISm(A, M)1 s A) t h ITP,(M)l s 2A+lLI.

Proof. Immediate.

The next theorem will be used in Ch. VII I in computing the number of models for an unsuperstable theory.

THEOREM 1.11: Let 2" < A; K < A =- K" < A; Man L-model, llMll s h, ILI s p, then there are x I p, a hngmge L, = L u {Q,: i < x s p} (&,--one place predidm) and an L , - d l M, w h tirat

(1) M , b p+-lumLOgeneow,; weover if Lo c L,, &, E Lo, then the

(2) The reduct of M, to L is an elementary extenAm of M . ( 3 ) M , ?ma cardinality A.

L,-reduct O f MI i8 p+-homogeneOU8.

(4) p l l = Uf<X&f(Ml)Y m d i < j =+ QdMd = Q,(Ml).

Proof. W.1.o.g. llMll = A. If A' = A, take x = 1 and extend M to a p+-saturated model M , of cardinality A. So assume A p > A, and let x = cf A s p. Choose an increasing continuous sequence Mf, i s x so that L(Mf) = Lf = L u {&,: j < i}, Mf < (My &,),<, where &, = 1Mj1, and IM*l = 1611, /[Mi 11 < h for i < x. Expand M to an L,-model M, by defining Q,(M,) = lMf I . Let D be a good N,-incomplete ultraflter over p, (exists by VI, 3.1) M, = MY,/D, and let M , be the submodel of M , with universe U,<,&,(M,) . By VI, 2.3 M , is p+-saturated. As Q,(M,) is (the universe of) an Lf-elementary submodel of M,, &,(M3) is (the

408 OONSTRUCmON OF MODELS [m. MI, 0 1

universe of) an LWementary submodel of M,, hence M, is an L,-elementq submodel of Ma hence an L,-elementary extension of Ma. As IQr(Ma)l < h a h IQWdI = IQWdI’/D < &hence 11JfiII = A Now if A c ]Mil, IAI 5 p, p E P ( A ) , thenp is realized in M, iff it is realized in Ma in some Qi(M,) iff it is finitely satisfiable in some Q,(M,) (by the p+-saturativity of Ma); and this depends on the type of A and on p only. Hence Ml is p + -homogeneous.

Remark. If the other oonditions are satisfied but p < ILI 5 h we can get M, satisfying (2), such that for any Lo c L,, lLol 5 p, Q, E Lo the redud of M, to Lo is p + -homogeneous.

(By 1.10(2), we can wume p 5 x) .

Remark. If h 2 2 X + I L I , we oan find suihble K, or hX+ILI = X and the conclusion holds by 1.10(6) aa a (A , A)-saturated model is X+-s&turated by 1.6(3). But we cannot always satisfy “ y - a < x + ” .

Prmf. We use the notation of Definition 1.6. Note that aa 2ILl < K 5 A, M is Ko-saturated by 1.6(3). By 1.10(2), (3) it suEces to prove ]TP.8;(M)l 4 2X+ILI. Suppose not; then there me sequences it from M of length x for i < (2X+lLI)+ such that for i #j TPSfi(i , ,M) # TP.8’(iijS M ) . As ( 2 X + l L I ) + < K , and TPS”(i , ,M) ha9 cmdinality s 2x+IL1 < K, for some C < K, i, is from N , and TPS’(iif, M) = TP.8’(iii, Nc). W.1.o.g. N, = min{NB: Nj 2 A}; so we can assume of Nu 5 p (otherwise N: = N, hence h = N,, hence M is p+-saturated, so our conclusion follows from 1.10(6).), and ( 2 X + l L I ) + < Nu. Let D be a regular ultrafilter over p(1) = p + ILI (exists by VI, 1.3), which belongs to So; and its power is 2’+ILI s 2x+lLI s K s h so it is ~ 8 , . As Sc+, C “ N c haa cardinality A”, in SC+, there is an elementary embedding of Nf(l)/D (as interpreted in 9, + l ) into Nc + , = PC + 1, which is an inverse to the natural embedding of Nc into Nf( l ) /D .

Now define TPS’(S, B, M) as we defined TPSN(i, M), but restricting ourselves to 6 E B, end let TPS;(C, B, N) = {TPSM(’a, B, N ) : ii a

CH. VII, 0 11 SKOLEY BUNOTIONS 409

sequence of length x of elements from C). Let us prove by indudion

(*I on PI

IfC, BE^,, C, B E N, then there is B ' E ~ , , B c B' E N,, 123'1 < IBI+ + N, and TPB;(C, B , N,) haa adin- ality s 2X+ILI. Moreover there is a formula B E L(93), with parameters from 93,+1 such that for suitable B, C, 93, ( x q ( 2 , B, c), and 9, c e[E, B, q.

Proof of (*). For the definition of B use the construction below and a choice function (in 93,) for the family of sets of hereditary cardinality I llNll (we essumed N is in it).

Case I: I BI < K,. B will be the image of Bfi(l)/D (ultrapower com- puted in 93,) by an elementary embedding f of N$(l)/D (ultrapower in 93,) into N , where B u C c IN*] N , < N , N , E b,, N , is KO-saturated, '29, C " llN*II = A"; where f is an inverse to the natural embedding of N , into N$l)/D. As I BI < K,, all functions from p( 1) into B with range of cardinality s p are included, aa IBI" < N,, 2fi(l) s A. As D waa regular, clearly any m-type over A E C, IAI c p, is realized in B' iff it waa finitely satisfiable in B. It is eaay to check that B' satisfies our demands.

CaseII: 1B1 2 K,, IBIisregular.SoIBI > (2x+ILI)+.In93,thereisa well ordering of B, so let B = {bi: i < IBI}, Bj = {bf: i < j}, and let BF = (B, u Uf<j Bt)'; and B' = Ul<lBl 23: satisfies our demands; otherwise, aa cf I BI > (2X+ I L I ) +, some Bj' fail our demands. (Notioe that )B, u Uf< BTl < I BI, so B; is well defined by the induction hypothesis.) [The primes are from (*).]

Case 111: 1B1 2 Nu, IBI is singular. Define B' aa in Case 11: by wumptions cf IBI x, so let a(i) , i < 8 = cf IBI, be an increaaing unbounded sequence of ordinals < I BI, then B' = Ute B&, and B,*) c B& for i < j. Hence I TP(C, B', N,)I s I[rP(C, B&, Ne)l s (2x+ILI)x = 2x+ILI. So we have proved (*).

= f [ + 1, then, we get a contradiction by B' and the properties of the &'a.

Let us finish the proof of 1.12. Choose B = C = IN,l,

DEFINITION 1.8: H is a bi-set function if H({ul: i < A}) is a family of pairs of subsets of {al: i < A}, and H({ui: i < A}) = {(Al, A2): for some (w l , wa) E H(A), A , = {af: i E q}}. We can use index sets other than A. A set function H is used also aa the bi-set function 8,1?({af: i < A}) = { ( A , 0): A E H({a,: i < A})} .

410 CONSTRUCTION OF MODELS [CH. VII, 5 1

DEFINITION 1.9: M is (A, H,, Ha)-comp&, provided that for m y set F = {vf(z, 4): i < A} of m-formulas over M , if for every (F,, Fa) E

H,(F) some sequence in M realizes (I',, Fa) (i.e., realizes F,, and fails to satisfy any formula in Fa), then some sequence from M realizes some pair from H2(I').

DEFINITION 1.10: For an ordered set I: (A) DC(I) = {Ia: a E I), 1, = ({b E I : b < a}, {b E I : b > a}). (B) The pair (A, B) is a ( A , p)-Dedekind cut of I if A < B (h.,

a E A , b E B =- a < b) and A v B = I, A is the cofinality of A, p is the lower cohality of B (i.o., the cohality of B with inverse order).

DC,(I) = {(A, B): (A, B) is a (A, p)-Dedekind cut of I , (A, p) E W), DC*,(I) = ( (A, B): A , B E I , A < B, and for no c E I A < c )) E W } .

DIPINITION 1.11 : (1) A (A , +family is a family of sets of cardinality A such that the intersection of any two distinct members has cardinality

(2) AD(x, A, p, K) holds if there is a (p, K)-family of x subsets of A; < K.

where x > A 2 p 2 K.

Remark. Baumgartner proved the consistency of AD(2n0, N,, K,, No) -t 2*0 > N,.

EXERCISE 1.1: (1) If AD(x, X,p, K) holds, x' 5 x, A' 2 A, p' 5 p, K' 2 K , and x' > A' 2 p' 2 K' then AD(x', A', p', K') holds.

(2) If AD(x, A, p, K ) holds then A" 2 x; if x = 2A then A" = 2A. (3) Prove that A D ( A + , A, p, p) holds when of A = p. (4) Use (3) to show 1.8 cannot be improved in this case.

CONJECTURE 1.2: Claims 1.8 and 1.9 cannot be improved (at least in ZFC).

EXERCISE 1.3: Find the order between the set functions S,,, S:, S', 8: (dso Werent p's).

QUESTION 1.4: Investigate the logical connection between the various notions of saturativity (implications, and independence).

QUESTION 1.5: Investigate the various kinds of compactness for dense linear order.

OH. VII, 8 21 GENERALIZED EHRBNFEUCFI“-MOSTOWSKI MODELS 411

QUESTION 1.6: Let I = z,,, I,. Investigate the connection between the compactness (and saturativity) (with various notions) of I ; J, I,.

EXERCISE 1.7: For every order J, IJI I; K there is an order I 2 J, IIl = K such that DC((A,fl)l(I) = 0 if A, p + KO, A # p.

QUESTION 1.8: If we use the construction of Definition 1.6 but without assuming that N is saturated, does the big model still bequeath its properties (such as (p, Sh, S,*)-compactness, -saturation, etc.) to its submodela.

EXERCISE 1.9: Show that in 1.6(3), 1.7(1) and 1.7(6) we can replace (6 X-homogeneous” by “strongly X-homogeneous”, if K s x or N is strongly X-saturated.

VII.2. Generalized Ehrenfeucht-Mostowski models

Here we shall deal with the pseudo-elementary class PC(T, , T) where

DEFINITION 2.1: (1) PC(T,, T ) is the class of L-reducts of models of T,, of cardinality 2 IT,!, L = L(T), L, = L(T,), T is aa usual complete, T G T,, T , has infinite models.

(2) T , is a conservative extension of T , if any L,-model of T1 of cardinality 2 I T , I can be expanded to a model of Ta.

Clearly if T , is a conservative extension of T , and I Tal = I T , I , then PC(T2, T ) = PC(T,, T ) ; hence we can replace T , by T , without loss of generality. Also if we want to prove the existence of some models in PC( T,, T ) we can replace T , by any extension of the same cardinality. Hence if T , has an extension of power I T , I which hae some additional properties we can assume T , has them. We list below some such assumptions on T, . Note that if Ti + is a conservative extension of T,, T , = U, <, T,, I Ti I = I T,I, then T , is a conservative extension of To.

Assumption I: T , has Skolem functions. (See Theorem 1.1(2).)

Assumpth 11: T , is complete.

Assumption 111: There is a countable LC, E L, such that for every formula ~(3) and term 7(2) in L,, there are a formula R(3,y) and

412 OONSTRUOTION OF MODEL8 [CH. VII, 8 2

function symbol P(Z, z) &om LE and individual constants c,, c1 in L, such that

(1) (W[Cp(Z) = m, c1)3 E T1, (2) (W)[.(Z) = P(Z, C')] E Ti. For this it suffices to prove:

CLAIM 2.1: 80me coneervative ezteneiOn T , Of TI, = 8&b$eS

Aem?n#on m. Proof. It suffices to define by induction on n theories T", and countable languages L* s L,, = L(Tn) such that To = T,, T"+l is a conservative extension of T", IT"I = lLnl = lTll and for every p(Z), ~ ( 2 ) in L, there are R(Z, y), P(Z, y) in L,*+, and individual constants cl, c1 in

such that (1) and (2) in assumption I11 holds for Tn+l. (Then T , = P, L; = U b* satisfies the claim.) If T" is defined let &*+ , consist of (m + l)-place predicates RE(Z, y) and function symbols P:@, y) for m < w , and

Tn+' = T" U {(W)[q(Z) e ( Z , c:)]: (p(Z) EL,, Z(Z) = m}

u {(vZ)[T(Z) = 4)]: T(Z) E L,, d(5) = m}.

Clea,rly d the demands a m satisfied, so we have proved the olaim.

I denotes an index-model, i.e., a model whose elements serve as indexes; usually I is just an order.

DEFINITION 2.2: If J c I , 8 E I , then the atomic type of 3 over J (in I ) is

atp(Z, J, I ) = {tp(Z; i): Z(Z) = Z@), Z = ( ~ 0 , . . .), i E J , I C ~ [ 3 ; Z] and cp(Z;jj) is an atomic, or negation of atomic, formula in L(I) }

We omit I when its identity is clear. Clearly atp(3, J, I ) = tpd(3, J , I ) where A = the set of quantifier free formulas.

DEFINITION 2.3: If 8, t E I , J c I , 3 - t(mod J) (in I) if atp(3, J) =

atp(t, J ) . In both definitions, if J is empty we omit it.

EXERCIBE 2.1: Check the meanings for I an order.

NoWh. If {&: 8 E I} is an indexed set, 3 E I, 3 = (8(O), ~ ( l ) , . . .) then sj = ti#(o)-6r(l)n....

CH. 5 21 QENERALIZED EHRENFEUCHT-MOSTOWSIU MODELS 413

DEFINITION 2.4: (1) The indexed set {&: 8 E I) (& E 6) is called A-n- indiscernible over A :

(A) If 8 - t then I ( & ) = l (6J. (B) If 3 - t , Z(8) = l ( t ) = n, ~(3; g) ~ d , a E A then Q C &; a] =

Q[k a]. (2) If 8 # t implies 6‘ # & we d our indexed set non-trivial (we

use (2) rarely-mainly in estimations of cardinalities). (3) We adopt the same conventions for shortening aa in Definitions

I, 2.3 and I, 2.4. (Note that this definition and Dehitions I, 2.3 and I, 2.4 am Consistent.)

Assumption IV: When I is an index-model we wume for every 3 E I there is a quantifier-free formula d(Z) in L(I) such that for 8 E I, B - 8 iff I b 0[8].

DEFINITION 2.5: K ( I ) is the c l w of L(I)-index models I, such that for any I E I, there is 8 E I for which atp(3,0, I , ) = atp(t, 0, I). So for every I there is I, E K(I ) , 1111 s IL(I)I, such that &I,) = K ( I ) (without assumption IV, we would have only /Ill s 21uOl). The c l w of orders is Kord = K((w, c)) = K(w).

LEMMA 2.2 : If I , E K ( I ) , {a, : s ~ l ) is an indiscernible indexed set in a ntodel M of T, , then there is an indhcernible indexed 8et {a8: 8 E I,} in a d M‘ of T, , ~h tirat if I E I , 8 E I,, atp(8,0, I) = atp(t, 0, I , ) t h tp(&, 0, bl) = tp(Zj, 0, MI). A8 T , h Bk&m f u n d h a we mn a88um lM’1 = c1{a8: B E I , } and then M’ is uniquely determined (wp to ieo- w p h h pmerving the 4).

Proof. Immediate, by the compactness theorem and the dehition of W).

DEFINITION 2.6: If {68: 8 E I) is non-trivial and indiscernible in M, then the # of {g8: 8 E I) which we define as {&): M k &I} and the index- model I, E K ( I ) determine M’ so we denote M’ by N1 = EM1(Il, #), and its reduct to L by N = EM(I, , a). Such # is called proper for (I, T,). We call {a8: 8 ~ 1 , ) the skeleton of N1 (and N) (of course, really we cannot reconstruct the skeleton from the model, but we behave a8 if we can).

LEMMA 2.3: T h e is #proper for (w, T,) .

414 OONSTRUOTION OF MODEL8 [=. 8 2

Pruof. It suffices to prove in some model of T, there is a non-trivial indiscernible sequence {a,: i < w}. Let M, be a model of T,, & distinct sequenoee &om M,. It suffioee to prove

I' = Ti u {v(%) ~ ( e ) : v(E) E L,, I N 8; I , f E W }

U {zs # Zt:8 # t < w }

is consistent, for this it suffices to prove the consistency of a finite

where I(Z) - f(Z). Let A = {pl: Z < n}, m = maxk<,, Z(b(k)), a d let {& k < w } be a A- S m-indiscernible sequence (exists by I, 2.4 and I, 2.3(3)) and interpret Ek by 6Nk); 80 in M,, r' is satisfied.

Asmmpth V: If T is unstable, for some formula 3 < g in L and some 6f, i < w , (I(!$ = I ( @ = Z(q)),6" < 6k iff n < k. Then (Z < ~ ) E L P . If T has the f.0.p. then T n L; has the f.0.p.

subset I" c T1 u {vl(Zal)) ql(Zi(1)): Z < n} u {iE, # 3: I # t < w}

LEMMA 2.4: If T W unstable, thre W 9 proper for (0, T,) whkh W ordered by < , i.e., (3,, < ?Em) E Q iff n < m.

Pruof. AS of 2.3, only to I' we tldd {(Z,, < Em)u(n<m): 7~ < m < w}, and use the &,'a from assumption V.

Similarly we can prove

LEMMA 2.6: If 9, i8 proper for (w, T,) and T , is complete, T, 2 T,, t h there W 91 2 whkh is proper for (w, T,).

For Q, proper for (w, T,), we can investigate properties of EN(1, Q,) (I E R(w)) , but for casdinals s I T,I we cannot get much. Hence we first introduce a generalization.

DE-ON 2.7: If 9 is proper for (I, T,) let N1 = EM1(I , @), N' be the submodel of N1 with universe the set of interpretations of terms (with no free variables). (As T , has Skolem functions, N' is an element- rtrg submodel of N1.) Let N* be N' expanded by adding the relations:

R ~ , ~ , ~ = {a E N': ~1 c 'p[~(q, a)], when I E I, I c em} for every atomic formula 'p(Z) E Lf, term c(3, 9) E LE(l(Z) = l ( ~ ) ) and quantifier free formula e(3) E L(I) satisfying assumption I V (we cthoose one e(Z) for each atp(B, 0, I), B E I). Let T* = Th(N,), = L(N,),

= Lz u {RB,o,s: 8, 'p, c as mentioned above}, so IQI s No + 1L(I)l and T* satisfies assumption 111 when L(I) is countable.

OH. WI, f 21 GENERALIZED EHRENFEUCE’FMOSTOWSKI MODELS 415

We behave as if T, determines 9, and say, T,, or a model of T, has a property, if 9 has it.

LEMMA 2.6: 8wppoee N i s a maEel of T, (heme an elementary extenaim of N,) where 9 is proper for (Io, T,), and I E K(Io).

Then there is a unique L,-modeZ W , w1 = cl(m U {a#: t?~I) ) m h that

( 1 ) F o r e v e y R , , , , ; ~ L , - L , , ~ ~ ~ E E I N I , I E I

(*I when I C ep], N C R,,q,t[E] o M1 C p[z(h, a)].

We denote this d by EM1(I, I?), and it8 L-red& by M = Ehf(I , N ) . We call {a8: s E I } the skeleton and N the ba& of M1 (and of M ) .

( 2 ) M’ is a &el of T,, and an elementary extension of the L,-reduct of N . The indaed set {Z8: 8 E I } is indi8cernible over IN1 in AP.

(3) For every E E IN1 I there are I E I I I, 6 E IN I and z E Lf (we k d e r them aa determined by E ) m h that i5 = T(%, 6). Also (*) holds f o r any T ,

p E L, for suitable f m d m (wring mitable individual mtant8). The elementay type of Z is determined by atp(I, 0, I ) , tp(6,0, N). Also

(4) If in I only finitely many atomic types of sequences of length m are realized f o r each m: then for every fornula V(T(%, g)) ( p , z E L,, I varied over I , $i over N ) there is a Boolean cornbination #(S, g) of formdm O,(a) E L(I), e,(g) E L, m h that M1 C p(z(q, E ) ) ifs #(& a). [Being formalistic # is not a well defined formula, but its meaning is clear.]

lP1ll = IlNll + 11I11*

Proof. There is no problem in the proof.

Remark. If M , satisfies (1) it automatically satisfies (2), (3) -and (4).

DEFINITION 2.8: I is A-atomically-stable if J E I , IJI 5 A implies 18T(J, I ) ! 5 A where A = &(L(I)) is the set of quantifier free formulas of L(I). (Note that if IlIll 5 A, I is h-atomically stable.)

LEMMA 2.7: ( 1 ) If N is proper f o r ( I , T,), N stable in A, and I h- atomically-stable then M1 = EM1(I, N ) is stable in A.

(2 ) If N is proper for (0, T,), IlNll 5 A and I is a well ordering then EM1(I, N ) is stable in A.

416 tYONSTRUCTION OF MODELS [CH. VII, 0 2

Proof. (1) Let A E IWl, IAI 5 A, then there me J E I , IJI 5 A and B E N, IBI 5 A, such that for every a E A there are %(a)EJ, ~ , E B and 7, E Lt such that a = 7,(7ii(,), 6,). Noticing that if at = ~(79,,,, h1) %(l) E I , 6I E IN1 for Z = 1, 2, and %(1) - S(2) modJ, tp(6,, By N) = tp(Eay By N), then tp(@, A, iK1) = tp(aa, A, Ml); the conclusion is clew.

(2) Immediate from (1) and part ( 5 ) of the following claim.

CLAIM 2.8: Let I E K(w), J s I .

and t > 81 0 t > 81.

(1) 81 - 81 mod J if for any t EJ, t < 81 o t < 81, t = 81 o t = 81

(2) (80, . . ., 8,) - (to, . . . , tm) modJ ifst - ti mod J for i = 0, . . ., m

(3) The number of atom&? typea of 8eqwnces of length m from I i8$nite andat < 8 , 0 t t < t, for 0 5 i , j 5 m.

(even <(2m)!).

where d = &(L(I)). (4) I i8 h-&?&dy 8tabk if 1&(J,I)I 5 IJI f or J E I , IJI 5 A,

(6 ) If I k well ordered, I k A4omictally 8tabk for euch A 2 No. (6) If A 2 2b, I k A-Ut~WnMy-8tabk iff I k A-etabk. (7) If I = zae, In; J, I , are A - a t ~ W l y - s t a b k Orders th I i8 A-

atoncieauy ekrbk.

Proof. (l), ( 2 ) and (3) are immediate. (6) will not be used hence we leave it as an exeroiae fo the reader. (Hint: use the Feferman-Vaught theorem, see, e.g., [CK 731.)

(4) Suppose I , J are a counterexample. So IJI = A, J E I , p i = atp(%{, J , I ) l (SJ = my for i < A + , pi # p ,

for i # j. As we can replace {p,: i < A+} by any subfamily of same ccudinelity we can aasume w.1.o.g.

(A) atp(Siy 0 , I ) is constant. (B) Letting Si = (a!, . . . ,Y-l>, then for each k, 0 I k < m,

atp(6, J, I ) dependa only on k (for this use m times the aasumption I(atp(8, J, I ) : 8 E r)( 5 A).

So we get a contradiction by (2). (6) Using (4) we prove lRj(J, I)I 5 A when I JI 5 A. Define a function h on I : h(8) = min{t: t E J , 8 < t } and if for no

t E J t > 8, let h(8) = a0 (h is well defined aa I is well ordered). So the range of h has cardinality I JI + 1, and if 8, t 4 J, h(8) 3: h(t) then by (1) atp(8, J, I ) = atp(t, J, I ) . Hence we finish.

(7) by (4) it is immediate.

CIE. m, 5 21 O E N E U E D E~ENFEUCET-MOSTOWSKI MOD- 417

CONCLUSION 2.9: (1) If A 2 ITl] then there ie M E P C ( T ~ , T ) of cardinality A, which W 8tubk in evey p 2 IT1]; 80 i f T W unatabk in lT1l, Af W not IT,I+-univereal.

(2) I f p 2 2”, h 2 lTll then there is Af E PC(T1, T ) of cardinality p, whhh ie A + - ~ n i ~ s a l , x-etabk for x 2 2”, h when T W tlnekrble Af is ?lot (2”) + -universal.

Remark. In ( 2 ) if 2” = A+ the mult on universality is sharp, but in general not. If pA = p we can assume M is A+-satmted x-stable for x = XA (using ultrapowers). Sharp results, using finer methods, appear in Section 4.

Pvoof. (1) Let N be proper for (w, Tl ) , and be of mrdinality I TII. Then EM@, N ) is of oardinality A by 2.6(3); it is stable in p 2 lTll = llNll by 2.7(2).

If T is unstable in I T , I, there is a model M of T which is unstable in I T l I, of mrdinality I Tl I + , hence Af cannot be elementarily embedded in EM@, N ) .

(2) Use a A+ -saturated model N of cardinality 2” whiah is proper for ( w , TI). Take EM(p,N) as the required model and use Theorem 11, 2.13(2).

LEMMA 2.10: If M1 = EM1(I , N ) , N p o p ~ f ~ ( I , T l ) ; t h ITPX;(Afl)l s ITPX;(N)I + lATPXf(I)l where X = 8, C and where ATP metam3 that in &$nition 1.6 we replace t p by atp.

Proof. Similar to that of 2.7.

LEMMA 2.11: S-e I is a h e and h+-saturated or& and N W h+-cumpact a n d p ~ for (0, Tl ) . Then Afl = EAfl(I, N) W (A, SN,, S1)- compact.

Proof. Let p be an m-type over M1 of cardinality A, and every q E

SNl(p) is realized. W.1.o.g. h 2 lLll.

(*) There is F = ~ ( g , Z) E Lt such that any q ~ & ( p ) is realized by some F(&, 6), 3 E I , 6 E N .

Otherwise for each F EL: there is a counterexample qo so q = Us qr is a countable subtype of p , and is not realized, contradiction. So for every q E SNl(p) there are S(q) E I , 6(q) E N such that T(%(~, , b(q)) realizes q. Let D be an ultrafilter over Sh(p), such that { q ~ S ~ ( p ) : ~ E ~ } E D for any v ~ p . Let J E I, IJI s A, B E INI, IBI s h be

418 00NS"RUUTION OF MODELS [OH. WI, 8 2

such that p is a type over dcl(B u {a8: 8 E J}). We can find, by the A+-sa.turation of I and N , f E I and E E IN1 such that:

(1) For every 'p(iZ, E), 5 E B, 'p E L ( N ) ,

N C v[E, 51 * {q ~ S b ( p ) : N b &q), 51) E D-

(2) For every # E L(I ) , 3 E J ,

I #[t, 31 0 {q E SNo(P): I f= #P(q), 311 E D.

It is easy to check ~(s, E) realizes p.

QUESTION 2.2: Investigate the connection between (A , H,, Ha)- compactness (or saturativity) of N , I and of EM(I , N ) ; in particular when the H a am S,,, S:, Su and S$.

QUESTION 2.3: (A) Investigate the connection between non-(A, H,, Ha)-compactnees (or saturativity) of N , I and of EM(I , N ) when N C T,, @ is ordered (see Lemma 2.4).

(B) Show, for unstable T , the existence of models which satisfy simultaneously various compactness and non-compactness conditions (also for mturativity; we mean of course for the various (A, H,, Ha)- compactness notions).

EXERCISE 2.4: Suppose A > p > lTll am regular, bl = EM(I , @) I a dense order, @ proper for (w, T,) . Show N is ( A + p*, DC, DCo,,,)- compact, when I has no (i) (A, p) or (p, A) Dedekind cut, (ii) (1, A) or (A , 1) or (0, A) or (Ay 0 ) and (1, p ) or (py 1) or (0, p) or (p, 0 ) Dedekind cut.

EXERCISE 2.5: Show that if T1 i < 8 is an increasing sequence of complete theories, @: proper for (I, TI) 4 5 @, for i < j < 8, then Ul<d @: is proper for ( I , U l < d T1).

DEBTNITION 2.9: I hw the extension property i f When T, is complete, Q1 proper for ( I , T,) , Ta 2 T I , then there is @a z proper for ( I , TI) .

EXERCISE 2.6: Show that if is proper for (I, T,), I has the extension property, then there are T , 2 T,, I TaI = I T,Iy d& z a, proper for I, T,, such thrtt for any J E K ( I ) , EM(J, 0,) is KO-homogeneous.

OH. VII, 8 31 ON THE f.c.p., AND UNIFORM TF~EES 419

EXERCI8E 2.7: Suppose T1 = Th(M,), M , = (w, < , R,, . . . ), T, has Skolem functions, and that we get @, proper for (w, T,) &B in 2.3 for b, = n. Then for any I E K(w), N = EM(I , @), 8 E I

I{t €1: t < 8}1 5 l{b E IN]: N b b < aa}l

5 I{t E I : t < 8}1 + min {2No, 1T11}.

Moreover for each K we can find such a T,, lTll = K , such that the second inequality becomes an equality. (Hint: See the proof of 3.1.)

EXERCI8E 2.8: Let w denote here a finite set, and say M is (A, H,, H,)*-compctCt if for any indexed set {y,,,: w c A} of formulas over jK if for every 8 E H,(A), ps = {yW: w E 8) is realized in M then for some 8 E Ha(A), ps is realized.

(1) If H,(A) is closed under countable unions, N is A+-compact, I is (A, If1, El,)*-compact (or vice versa) and N is proper for (0, T,), then EM(I , N ) is ( A , a,, Ha)*-compact.

(2) Generalize from I E K ( o ) to I E K(Io). (3) Generalize to saturativity and Definition 1.8. (4) Check the implications among these notions.

VII.8. On the f.o.p., uniform trees, and ID(T)I > IT1 = KO

We continue the conventions of Section 2, with various additional assumptions on T.

THEOREM 3.1: 8wpp8e T 7uza the f.c.p. (finite cover property). If @, i8

proper for (w, T,) then there are T , 2 T,, lTal = ITl] and @a and y(z; g) E L, O(g; Z) E La m h t7&:

(1) @2 2 0, is proper for (w, Ta). (2) If N 1 = EM1(I , @a), 8 E I , then O(N1; 4) 7uza cardinality

I{t E I : t < 8}1 + min{lTll, 2&}.

(3) {rp(z, a): a E N1, N 1 C O[E, an]} is not realized in N 1 but any proper &set of it is realized in N 1 .

Proof. Let M' = EM1(w, 0,). By assumption for some y(z, 8) E L and infinite set W E w for any n E W there are (i < n) such that

M' C A (W[ A d z , 5,,.,)] A -(W[ A t<n d z , 5,J] - f e n j-8

jrrt

420 O O N S T B U ~ O N OB MODELS [OH. VII, 9 3

We expand M1 by: P = {ti,,: n E v, sequence of functions F, P(7i,,~i0 = 6n,1andQ = {6n,,:i < n , n ~ W}andR = {6n,1n~n:i < n, n E W), and by adding Skolem and other funations so that we get a model M" such that T a = Th(Ma) satisfies all the assumptions from Seation 2. Now for any finite A E La = L ( T 7 , by Ramsey'e theorem (eee I, 2.4) we can find an infinite J A c {Zi,: n E v which is A-indiscernible. So we culn find a maximal consistent set of formulas 0, in La in the variables itn, n < o, such that for any formula $(Zo, . . . $4) E

La if for some finite A, E La for all finite A, A, E A c La, and all Z,,(o,, . . . , E Jb(n(0) < n(1) < . .), Ma C $[aM0), . . .] then $(%, . . .) E 0,. So clearly 0, 2 0,, and is proper for (o, T,).

Let B(E; g ) = R(g; E) , then (3) is easy to cheok so only ( 2 ) remains. suppose 8 ie a counterexample. Using P we see 1 e(lN1 1 ; a,)! r I{F(q, q): t < 811 = I{t: t < 811.

By d e w g Ma properly we am ensure that 1 B(lN1 I ; G,) 1 r m i 4 1 T 1 I,%). So we prove one inequality. Now suppose I B(N1 ; 4) 1 > A,, = I{t : t < 811 + rnin{l T I 1 , 2Ho). So there are distinct ri(b(,)) E B(IN1l; 4) for i < &+. As we can replace them by any subset of &ality A,,+, we can assume all the J ( i ) are similar over {t: t < 8).

Also, if A, r ITII we can assume 7, = r for any i. If A, < ITII n e d y A,, r %. Define an equivalence relation among the 7,: 7, N

r j if for any f E W, rl(&) = ~ ~ ( i 3 ~ ) , or rl(i&), r j (B f ) Q in the model Ma. Clearly the number of equivalence classes is s 2n0, so we can assume all the riys am equivalent. As q(%(,,! E Q(N1). q(7i8(,,) = r0(8,,,); we can assume a ( i ) = tn(8)"€(i), f[l] c 8, t ( i ) [ j ] > 8. Now if R@, rO(Z, 8, Z l ) ] + [rO(it, g, z l ) = r0(itY g , ~ , ) ] E A and E, m, By i E {i: a, E JA), (I(@ = I ( € ) , z ( E ) = ~ ( i ) = Z(t(i)), and ~ [ i ] < m < E l i ] , iu] then necemady

(otherwise n 1 I{Z: R(amy a))] r l{rO(a;, am, aE): suitable B}I r No). So all the rO(ZZHi)) are equal, contradiction.

THEOREM 3.2: 8 u ~ e T is stable, M1 = EM1(I, N ) , N proper for T I ) .

( 1 ) Let v(E, 8) E L, E, T E El. If tl, f a , J1, ga are i n ~ r e ~ n g sequences from I , = l ( 6 ) = m,,l(s,) = l ( 4 ) =.m2 and g1[i] = g 2 [ j ] o t ; [ i ] = fa[ j ] , ccnd 6, E N then M1 t T[T(%,, a), 6(qa, 611 = v[.r(aZl, a) , d ( ~ i ~ , , 6)]

(Remark: I7cstead of if,, aa, we could have taken if1, . . . , &.)

CH. VII, 8 31 ON TILE f.C.p., AND UNIFORM TREES 42 1

(2) #uppose A > 11 N 11 is regular, and every interval of I hm cardinality # A (including the intervds with an endpoint &a). Then M , the L-reduct of M', is ( A , # ~ , # ' ) - c r n n ~ ; and moreover (A,#A,#l)-mpact. (The aame h0ld.a for saturation.)

(3) In (2), instead of A regular, it aumes to msume I hQs ru) interval of cardinality cf A; or for some A, < A, any interval of cardinality 2 A, has cardinality > A. We can replace " A > 11 NII " by " N h+-compact ".

Prmf. (1) Here we can wume I is dense. Suppose we have a counterexample. Then, by assumptions we can find go, . . . , Ik such that:

(i) So = and glniik - 81-82. (ii) a2[i] = 8,[j] -Qi] = <[j] (0 < 1 < k, 0 < i < m,, 0 < j < m2). (iii) For each I there am i (Z) , j (I) such that for n # i(1) S"[n] = g'+'[n]

and gl[j(Z)] < a'[i(l)] if€ not gl[j(l)] < i?+'[i(I)], but for j # j ( l ) , S,[j] < d[i(I)] iff gl[j] < #+'[i(l)].

BY (i)

M' c 'p[t(%,, a), 6(ha, 6)] 0 M' c 'p[t(%,, a), 6(Q, i)],

M' c 'p[t(a~,, a), 6@&, 6)] 0 M' c 'p[t(%,, a), 6(%&, 6)] .

But as we are dealing with a counter example

M' c 'p(C(%,, a), qqa, 6)) 0 M' c 7'p(e(afl, a), 6(afp, 6)).

M' c 'p(t(%,, a), 6(%, 6)) = l'p(?(%,, a), 6(7ij+l, 6)).

So for some 1 s k

Let U, E I, U, < U,+', and U, is between #[i(l)] and $+l[i(Z)]. Let v:, vt E I for n < o be defined such that: they are of length l(gl), I(&J if j # j ( l ) , V:[j] = S,[j], if j # i(Z), v 3 j ] = 8 [ j ] and "[j(4,] = U,, vi[i(Z)] = U,. It is eaay to check that for n # k M1 I= 'p[d;, a;] = +a:, a:] where a: = ~(9"~ a), at = 6(az, 6). Hence for # = 'p or # = 7'p M1 C #[a:, a:] A d: # d; iff n < k, ao T has the order property contradiction by 11, 2.2 and 11, 2.13.

(2) Suppose every q E &(r), I' = {'pf(3, &): i < A} is realized in M1. So let ~ E W I , M ' + r p t [ ~ , b , ] for i <j. So let ci = ~ ~ ( a ~ ( , $ ) ($EM, Tt E LE). Of course, each ct is increasing. As A > llNll by renaming we can assume that for i < A cf = c,

8 = 8, (as we can replace Ef by a, when j > i). In the same way, we can assume that for each I, either all Zif[Z], i < A are equal, or for every i, j S,[l] = q z ] =$ i = j.

422 CONSTRUCTION OF MODELS [CH. WI, 5 3

Let Z(0) < - - - < Z(k) be those for which the firat w e occurs, and again we can wume that for n, m # Z(O), . . . , Z(k) i,[n] = 8,[m] iff i = j , n = m. So Z(n) + 1 < Z(n + 1) implies {e E I: eo[Z(n)] < s < 8o[Z(n + 11) hae cardinality T A hence > A (for - 1 s n s k + 1, where we stipulate I ( - 1) = - 1, Z(k + 1) = the length of a,,). Hence we can define i E I, @) = Z(io), %[Z(n)] = i#(n)], and if n # Z(O), . . . , Z(k), then a[n] does not appear in ti, where b, = ah, 6,). Then by part (1) ?(ai, a) realizes p.

(3) Proof similar to (2).

CONCLUSION 3.3: 8 p e T i8 S a k , with the f.0.p. A 2 lZ'11, and 8 a set of c a r d i d 8 A E 8. T k n there is M E PC( T, , T) such that:

p 4 A, ~ € 8 , then for s m v(z,a) E L , bl omits a T-type p of cardinality p, m h that any proper &type of p b realized in M .

(2) If m i n { * o , lT1t} s p < h,p$8,pier@arorp = min{*o, lTll}, then M ie ( A , 81) -compt .

(1) If min{lT1l, 2no}

Proof. When lTll < 2Mo this is immediate by 3.1 and 3.2, aa there is an order I, IIl = A such that for every p, lTll 4 p 4 A: p E 8 * for some s E I I{t E I: t < 8)) = p, and p $8 == I haa no interval of cardinality p. When lTll 2 2% we can use Exercise VI, 6.4, 6, Exercise 2.9 and take an elementary submodel, or 3.2(3), laet phrase.

CONCLUSION 3.4: 8uppo8e T i s sa le with the f.c.p. If Nu 2 N, + lT1l, t48 = min{lT1l, P o } , then there are in PC(T,, T)

at Zeaat 2Ia-B1 n o n - i e ~ p h i c models of cardinality Nu.

Proof. If a - 8 2 w, this is immediate by 3.3. When a - 8 < w we look at &: for some finite A c L, n < w, there is a finite A-n-indiscernible set I c N such that dim(1, A, n, p ) = p}. For suitable T, this is not difficult, and we leave it to the reader. (Compare with VI, 3.9.)

DEFINITION 3.1: For an order J, I is called a (J, /3)-tree if I is a model with universe G JsB = (7: 7 a sequence of elements of J of length sp}, such that 7 E 111 =- q a E 111; and with the relations Pa = (7 E 111: Z(7) = a} for a 4 8, the lexicographic order < , and order 4 of being an initial eegment, and the function h, h(7, v) = the longest common initial segment of 7, v. I is the full (J, /3)-tree when 111 = JsB. We denote I by JsB. Note that up to isomorphism the claas of (J, /3)-trees for a11 J's is K(w'8). If for some J, I is a (J, /3)-tree, then I is a 8-tree.

OH. VII, f 31 ON THE f.c.p., AND UNIFORM TREES 423

Remark. If I , f E I ; I ,., f iff Z(I[i]) = l(&i]) and for any i , j ; ai,j =def

max{a: B[i] 1 a = B[j] 1 a} = max{a: t[i] 1 a = f [ j ] a}. (In another notation, l ( h ( g [ i ] , ~[ j ] ) ) = Z(h(E[[i], E[j])) a n d ~ [ i l < g[j]e-E[[i] < E[j]).

DEFINITION 3.2: (A) T has a uniform ,%tree of the form ((cpa,ma): a < 8, a successor) (cpa E L, ma s w ) if there are a,, for q E ASS such that :

(1) a,, realizes p,, = { c p a ( f , anla): a < 8, a successor} for q E B A .

(2) For any a, r ] E Aa,v E AB,7iv realizes <ma + 1 of the formulas ~ ~ + ~ ( f , Sin-<r)), i < A. (We can aasume that if 6 < /3 is not a successor q E Ad, then a,, is empty.)

(B) The tree (and the form) are called strong if for any set w s A,

LEMMA 3.6: (1) Definition 3.2 holdejor s m e A iff for every A. ( 2 ) If 8 < K ( T ) then T h a uniform ,%tree, ma = 1 for all a. (3) If T is stable, /3 < K ( T) then T h a strong uniform 8-tree, ma = 1,

(4) If T hua the strict order property then T h a strong uniform 8-tree,

( 5 ) If D ( s = 2, L, GO) = GO then T h a strong uniform Ho-tree,

(6) If K < KrOdt(T), K is regular, then T h a strong uniform K-tree,

for all a.

with ma = 1 for all a.

ma = 1, for all a.

ma = 1.

Proof. Immediate, see 111, 3.1 for (3), 11, 3.9 for (6) and 111, 7.10 for (6). Clearly, (1) and (4) are immediate.

For (2), note that if on J = As* x {0,1} we define an ordering < : (7 , i ) < <v, j> i fq 2 v, i = 0 or q = v, i = 0, j = I , 01 p Q q , v where p = i ( q , v ) # q, v and q < v (in the lexicographic order). Then assuming T is unstable (as (3) has been proved) by the order property, there are cp E L, a, (8 E J) such that tcp[Z8, at] iff 8 < t. Then let

a = ~ < , o > - ~ < , * l >

cp(J ln~ , , a,) = [cp(fl, B<,,,OJ = -lcp(fl, ~ < W , I > ) l .

and

THEOREM 3.6: (1) If T h a uniform @-tree of the form ((cp,, ma): a < 8, a a 8ucce88or), then it h such a tree which i s indisernible ( w h

424 CONSTRUCTION or MODELS [OH. VII, 8 3

we ukw the 8-tree acr a n index set). In fact, JsB h a the extendm property (see Definition 2.9).

(2) B y the assumption of ( I ) , there is a @ proper for 8-trees and T I , euch that V M = EM(hs4, @), then {a,,: q E hsB} is a uniiform 8-tree of the form ((v,, ma): a < 8: a memasor).

(3) I f we dart in ( 1 ) and (2 ) with a strong tree, we get one. (4) BY rep hi^ cp , ( f ; g ) by & ( f ; 81, gal = v a ( f ; a11 A i , ( f ; Y a ) ,

we can wmme in (2) t7& if M = E M ( I , @), q E I n ha, no seq- in M satis+ infinitely many formulae rpa+ a,,-<,>), qh(i ) E I . That is, we get a strong uniform tree of the form ((cp,, No): a < 8, a a memasor).

Proof. (1 ) Suppose {Si,: q E hsB} is a uniform 8-tree; and choose h 2 sa+,, a, r IT(. By adding dummy variables to the qaYs we can assume the a,'s ctre distinct.

X t sufXces to prove the consistency of

u {7va(gn, g v ) : a = y + 1,l(v) = a, l(q) = 8, v r Y = q r Y Y V ( Y ) Z r](Y)l

u {!I,, # !I" : r ) st v7 l(!I,,) = l(!J")). So it suffices to prove the consistency of all finite I" s r. In I" the set of a which me l (q) where appeam in I" is finite. So by renaming we can aasume all such a's me <no. (We rename in a corresponding way the anye.) By the combinatorial theorem 2.6 of the Appendix we can fhd an aeeignment showing the consistency of I".

(2) , ( 3 ) Follow easily. For (4) notice that for the @ from 2, for no 6 E &I 80 S I{i: cpa(6, a,,-<l>) 7d6; a , , - < ~ + ~ > ) } l .

THEOREM 3.7: 8 w p p ~ e IT1 = KO and let v,,(P) E L ( T ) for q E 2<@ be euch thud

(a) q 6 v =- w ' a v v ( f ) + v,(f)l, (8) W 3 ~ ) [ ( ~ n - < 0 > ( 5 3 A ~ , , - ( f ) I ,

(7) b(3x)vn(f). ThenTha;sanzodelMand7i,~Mforq~2",andafunctionh:2<"+

2<" such that: ( 1 ) q 4 p impl iu h(q) 6 h(p), and h(qh(0)), h(qn<l)) are 6-

incomprable, so for q E 2" we can define h(q) E 2" w the unique v E 2", euchthuth(q 1 n ) Q v f o r n < w.

OH. m, 5 31 ON THE f.C.p., AND UNIPORlK TREES 425

(2) a,, realizes p: = pi(,,), p t = {‘p,+,(i~): n < 0).

(3) Call i j , i E 2’ 8imiZar over my i j - 6(mod m) if (i) ij[ll r m z ij[kI t .zfor k # 4 (ii) $1 m = $3 1 m for any 1. Then for p(0) E L there i 8 m, euch t7td if i j , i are 8imilar over m 2 m,,

(4) If T E L, 7(aq) realizes p: then v = q[Z] for some 1. Moreover there ie nz, such tW if i j - +j(mod m), m > mi and v E 2m,

(6) If r, are j h i t e set8 of formulas in L, r,, E r,,,,, UneP, r, i8 the

then M C cp[a;] = cp@~].

v # go] r m, . . ., then MI= -tpncV,[~(aq)].

set of formdm of L, then in (3), if ‘p ( j j ) E r,, we can chooee m, = n.

Proof. Choose a model M , of T , and Z,, E lMol , M , C cp-,[Z,,] for r) E 2. We use 2.4 ofthe Appendix for the reasonable fm’s (for each ‘p(Zl, . . . , Z,) there is m m such that f ( i j ) = 0 if€ f ( i j ) # 1 iff M C ‘p[s]). Now letting M = Mb/D, a,, = (. . . , E,,i~,,,nly . . . ) /D , D be any non-principal ultrafilter over w, clearly (l), (2) and (3) are satisfied. Now it is trivial to get (6) by renaming, and (4) is quite easy.

EXERCI8E 3.1: We can assume in 3.7, that there are distinct b, E M such that

(i) {bn: n < w} is indiscernible over U {a,,; r) E ‘2}, (ii) (My b,, b,, . . .) satisfies (1)-(6) of 3.7. [Hint: Let N < M y b, E IMI, {b,: n c w} indiscernible over N. Apply

3.7 to T’ = Th(M, P, boy . . .) (for P = N) and cp,,(z) such that CP[a,,] (more exactly, we have to repeat the proof).]

EXERCISE 3.2: Show that in 3.2(2), we can replace “llNll < A” by “ N is (A , SAY S1)-oompact.”

EXERCIBE 3.3: Suppose #,, E L(T), T countable, p an m-type over 0, N a, model of T , omitting p , and {{I,$,(??): N C #,[6]}: 6 E INI} has cardinality 2Mo and 2Ho is real valued measurable (see [So 711).

Show that T has a model M omittingp, and there me ‘p,, E L(q E @’2) which are Boolean coqbinations of the #,’a, and a,, E 1M1 aa in 3.7. [Hint: Let D be a normal measure over 2n0, and 5, E INI, (i < 2Wo) such that p , = {#,(if): N C1,4,,[5~]} are distinct; and w.1.o.g. T has Skolem functions. Now we define inductively Ela E D (a < wl) such that S6 = nf<a8f, S,,, c 8,, 8, = 2n0, and: if i (O) , . , ., i(n - 1) are distinct, and ‘p E L, N C p[b,(,,, . . . , 6,(,- ,)I, then for some j E Su - Su+

426 CONS!l'RUCTION OF MODEL3 [m. VII, 8 4

N C cp[g,, &l), . . . , &,-l)]. NOW we shall define indu~tively tp,,(. . . , z,,, . . . such that

(i) for every a for some i(9) E Is,, N C t p ~ . . . , & ,,), . . .I; (ii) If v(7) E {f(O), ~"(1)) for 7 E 2', then

tpn+i( - . ., zD, - - . )pea-+l I- tpJ. - - 9 ~ v ( n ) , - - )"ep;

(iii) When 7 4 v(r)) E 2" for r) E P, tp,,(. . . ,z", . . . ) is an approxima- tion to the desired properties of ~n7iv(,,)n--- (i.e., we allow only finitely many terms, and note here we should decide why a term does not realize p). For further help SIX [Sh 761.1

M . 4 . Semi-definabilify

DEFINITION 4.1: (1) If D is an ultrafilter over I (I E 161") then

Av(D, A) = {tp(Z; a): si E A , (6 E I: ktp[6; a]} E 0)

(Clearly it E Ism(A), it is consistent as D is closed under intersection). (2) An m-type p is semi-definable over I if there is an ultrafilter D

over I such that p c Av(D, Dom p). (3) An m-type p is semi-definable over A if it is semi-definable over

Am. (4) tp*(C, A) is semi-definable over B c A, if for any E E C, tp(E, A)

is semi-definable over B E A. (This naturally extends to not-necessarily complete t y p e s with infinitely many variables.)

( 6 ) p E Ism(A) (B c A) is stationary over B, if p is semi-definable over B, and it has no two contradictory extensions which are semi- definable over B. (This is not necessarily consistent with the previous definition of stationary Definition 111, 1.7, but our meaning will always be clear.)

Remark. Notice that if T is stable I an indiscernible set, D a non- principal ultrafilter over I then Av(D, A) = Av(1, A).

LEMMA 4.1: (1) A type p ( p s i b l y with injnitely many variables) ia semi-dejnable over I(A) iff everyjnite subtype of p is realized in I(A).

( 2 ) Every type over a model M , i8 semi-dejnable over ]MI. ( 3 ) If p is a type over A , and is semi-dejnuble over B, then there is a

complete type over A extending p which is semidejnable over B (the 8ame hold8 for I instead of B).

427 CH. VII, 5 41 SEMI-DEFINABILITY

Remurk. Part ( l ) , implies, by 111, 4.10(1) that if T is stable, B = lUl, then p is semi-definable over B i E p does not fork over B.

Proof. ( 1) Let p be an m-type semi-definable over I. If p c Av( D, Dom p) for some ultraflter D, and q c p is finite, then

{C I: c realizes q} = {C I: b Q [ ~ ; a]} E D ,(Z;;a)sq

(as D is closed under intersection). So {C E I: 5 realizes q} # 0 80 q is realized in I.

Suppose every finite q c p is realized in I. For each Q EP let J, = {EEI: C satisfies Q}. By clssumption the intersection of any finitely many J, is non-empty, hence there is an ultrafilter D over I such that Q ~p J@ E D, by 1.1(2) and (3) of the Appendix. Clearly Q ~p * Q E Av( D, Dom p).

For types with infinitely many variables the proof is similar. (2) Every type over a model is finitely satisfiable in iK, hence the

(3) It suffices to note the following; and use (1): (i) If p,, i < 8, is an increasing sequence of types, which are finitely

satisfiable in B, then Uled p , is finitely satisfiable in B. (The proof is immediate.)

(ii) If p is finitely satisfiable in B, ~(3; 3) a formula, then p1 = p U {v(Z;a)} or p , = p U {-q(z;a)} is finitely satisfiable in B (or both are).

Because otherwise there are finite q1 G pl, qa G pa which me not satisfiable in B. So q = (ql u qa) n p 5 p and is finite, hence satisfied by some 6 E B. If b ~ [ 6 ; a] then 6 realizes ql , and otherwise it realizes q2, contradiction.

result follows from (1).

LEMMA 4.2: (1) If C is a subsequence of 6, tp(6, A ) is semi-&$nable over B thn tp(C, A) is eemi-de$nuble over B.

(2) If p k q (e.g., q c p ) and p is semi-dejnable over B, t h q is semi- dejinuble over B.

( 3 ) If B' c B, p is semi-&$nuble over B , t h p is semi-&$nuble over B.

Proof. Immediate. Note this lemma shows the consistency of the definitions of semi-definability.

428 OONSTRUCTlON OF MODELS [OH. VII, 6 4

LEMMA 4.3: (1) If p is semi-de$nable over By then it doecr not &it over B. (2) If p is a complete type over A , eemi-&$dle over By B c A, and

every qESm(B), m < w , is realized by some F E A , then p is stationary over B.

Proof. (1) Supposep c Av(D, A), and tp(6, B) = tp(E, B), 6, E E A . If v(Z; 6) E p , then 8 = {8 E B: hpp; 51) E D. But for any 8 E B bp; 61 = g(G; Z], hence 8 = {8 E B: b[8, a}, hence v(E; Z) E Av(D, A) hence +P; 5) $ p , so p does not split over B.

(2) Suppose ply pa are contradictory extensions of p which are semi- definable over B. By 4.1 (3) we can mume they are complete types over some C 2 A, SO them is iz E C q ( E , 8) €ply -Q(Z, 8) Choose 8, E A, tp(8,, B) = tp(iz, B) and w.1.o.g. p(2; a,) E P . So p2 splits over B, contradiction.

DEFINITION 4.2: q = { ( p , B): p E ~ ~ ( A ) , B E A , IBI < A, p is semi- definable over dcl B}. LEMMA 4.4: (1) satiejk9 the fobwhg a&ma (eee IVY 8ectbn 1):

and (XI, 2). Ax (I), (11, 11, (IIY2)Y (III,l), (111, 2), I V Y (VII), (VIII), (IX), (XI, 1)

(2) A(Fi) = A; p(Fi) = a. (3) If T hm 8kolem functions, then also Ax(X.1) and (X.2) hold.

Moreover if IDompI < A, p i s over B E A, IBI < A, t h there ie a complete type q over A &ending p mch tluct (q, B) E Fi. Also Ax(II.3) and (11.4) hold.

Proof. (1) For Ax(II.1) notice that if 8 E B c A , l(8) = m; then D = (J c B"': ii E a) is an ultrafilter over B"' and Av(D, A) = tp(iz, A). Ax(1V) is Lemma 4.2( 1). For Ax(VII), it suffices to prove the following: suppose dcl B = By dcl C = C, B c C; and B s A, tp(Z, A u C) is semi-definable over B u C, tp*(C, A) is semi-definable over By a E A and v(Z; a) is realized by some Z E iz u C. We should prove q(Z; a) is realized by some i? E B. We can assume Z = 8-Zl, El E C, so Cq[Z, E l ; a], hence for some izl E C C @,, El; a]. As Z,, El EC, for some a;, Z; E B, C &, 5;; a] and we finish.

The other axioms are trivial. (2) Immediate. (3) If T has Skolem functions, for any By cl B is the universe of a

model. So by 4.1 (2) every type p over cl B is semi-definable over cl B. So 4.1(3) implies our conclusion. The proof of Ax(II.3) and (11.4) is easy.

CH. VII, Q 41 SEMI-DIEFINABILITY 429

LEMMA 4.6: Let P = IN[ , and (N*, P ) be an elementary submodel of (Q, P ) m h tlrat IN1 E IN*I. Then for any sets A , B there are elementary mappings f, g which are the klentity over IN!, IN*! r q . m h that if A' = f ( A ) , B' = g(B) then:

(1) tp,(B', IN*I u A') is eemi&$nuble over IN*I, (2) tp*(A', IN*I u 8) i.9 smi-de$nuble over INI.

Prmf. By 4.1(2) and (3) we can extend tp*(A, INI) to a complete type over IN*I which is semi-definable over INI. Hence we can define an elementary mapping f, which is the identity over INI, such that tp*(A', IN*I) is semi-definable over IN], where A' = f ( A ) . Now let

I' = tp,(B, IN*[) u 0 u Y, where

@ = {-,cp(Zs; a, 5): 6 E B, E A', E E IN*I (Q E L) and

Y = {l$(Z6; B, 5): 6 E B, a E A', E E IN*I (+ EL) and Q@E; a, 2) is not realized in N*},

#(6, g, a) is not realized in N).

Clearly it suffices to prove I' is consistent (if x,, I+ b' is an assignment satisfying I', let g map c to c for c E IN*I and b to b' for b E B. Now g is elementary aa tp,(B, IN*I) E it satisfies (1) aa @ E r m d (2) &B

Let r' be a finite subset of I'. As tp,(B, IN*[), 0, Yare closed under !PG r).

conjunctions, we can mume, by adding dummy variables,

r' = {6(36, q, 7q(5, a, q, -@(s, a, q> where a E A', 6 E B, E E IN*I, CO[6, E l , +(6, gy 5 ) is not realized in N. So, as P = "1,

a, a) is not realized in N*,

(Q, P) C (W[ A P(#[iI) -+ +@, a, El] A e(6, el. t

As (A'*, P) is an elementary submodel of (a, P) and 6' E IN*I

E IN*I, there is

By the second conjunct, 6' satisfies O ( q , a); by the first one for every a' E INI, C7+[6', a', 131, but tp@, "*I) is semi-definable over IN1 hence by 4.1(1) finitely satisfiable in N, hence Cl+[6', a, a. So in order to realize r', 6' has to satisfy l Q ( f 6 , a, a), but as 6' E IN*I this follows from W s definition.

430 <IONSTRUO!l'ION OF MODELS [m. VII, 8 4

THEOREM 4.6: Por every p 2 2A1, A, 2 2IT1I, PC(T,, T) has a A,- universal model M of cardinality p which is stable in A,. (Hence if T is A,-unstable, M is A,-universal but not A:-universal.)

Remark. The construction here wi l l serve additional theorems.

Proof. We shad describe here a construction for a cmdinal A > I T,], 2<" s p such th t for A = A t we get the desired model. Of course, we can assume T, has Skolem functions.

Let N be a model of T, of cardinality lTll, P = INI, and (N*, P) an elementary submodel of (El, P) of cardinality ITII. Let M1, Ma be p-saturated models of T, extending N. By Lemma 4.5 (and 4.2) we can aasume tp*(lM1l, IM'I), tp*(1HaJ, lM1l) are semi-definable over IN*I, IN1 reap.; hence do not split over them (by 4.3(1)). By 4.3(2) tp,(Wl, PI) is stationary over N. Note that if a, 6~ wl, t p (q W*l) = tp(6, p*l) then t p ( a , w l ) = t p (6 ,wl ) (as for any CEWI and tp(z,g), t=tp(a,c) EE ~ ( 6 , c ) as tp(q PI) does not split over p*l); hence the number of ~ES"(W) , m < w , realized in 1M2 is < IUmSm (N*)l Q 2ITll. We can interchange the roles of JP and M2 (with N instead of N*) .

Let {M,: i < p} be a list of models of T, of cardinality < A, such that up to isomorphism, each model of T, of cardinality < A appears in it. As Ma is A-saturated, we can assume all the M i are elementary submodels of Ma. We define by induction on i models Mf such that:

(i) tp,(lM:l, lM1l) = tp,(lM,I, IMlI); more exactly there is an elementary mapping f,, f, is the identity over M1, fi maps M , onto M?.

(ii) The type of iKf over M' u UjCi M; is =mi-definable over IN!. Let M be the Skolem closure of ufCg M:. Clearly M has cardinality s p (as 2CA s p) and M is x-universal for every x < A. Suppose x+ 2 A so llMfll s x for each i, and we shall show M is X-stable. Let A E 1x1, IAI s x, but @(A, M) has cardinality >x. W.1.o.g. A is the Skolem closure of {M:: i E w} where IwI s x. Suppose Zi E 1611, i < x+ realize Werent types over A, so by the definition of M a, = T,&, . . . , 6,,J where n = n,, 6,,, E 1M&,,l. As we can replace {a,: i < x+} by any subset of the same cardinality we can assume n, = n, fi = T . Let 6,,, = f,c,,,,(E,,.l) E,,, E M,,,,, hence Z,,, E Ma. As the number of complete types sequences from Ma realize over M1 is s 2IT1I, we can assume, if x 2 21T11, that tp(E,,,, lM1l) = g,. As the order p is atomically stable in x (see 2.8(6)) we can aasume the atomic type of @(j, I), . . . , p(j, n)) over w is constant, and p(j , 1) E w * b,,, = b,. So it suffices to show that under

CH. VII, 8 41 SEMI-DEFINABXUTY 43 1

a11 those conditions all the B, realize the same type over A. For this it suffices to show:

(*) If i(1) < * * * < i(m) < p,j(l) < * ' * < j ( m ) < p, 81 E 1M,(1)lY

Proof of (*). We prove it by induction on m. Form = 1, tp(si;, lM1l) = tp(7i1, lM1l) = tp(6,, lM1l) = tp (g , lM'1). Suppose we have proven for m, and we shall prove for m + 1. By the symmetry we can wume i(m + 1) 5 j ( m + 1). Now tp(7ik+l, 161'1) = tp(pm+l, 1M:l) is station-

U {IM,l: j < j ( m + 1))) are semi-definable over INI, hence the former is a subtype of the latter, and the latter does not split over INI, so by the induction hypothesis we get (*) and finish the proof.

(By checking that when h = A,+, A, 2 2ITil, we get the result easily.) Note that w.1.o.g. Mg $ N , hence llMll = p.

ary and tP(Za+lY p ' l u u {pq: j < i(m + 1))) tp(%+,, pfll u

EXERCISE 4.1 :Let Tbe the theory ofthe rational order, IMI = (0, l ) , a = 1, b = 2. Show there is no a', such that: tp (a ' ,Mub) extends tp(a,M) and is semi-definable over M and tp(b,M U a') is semi- definable over M.

EXERCISE 4.2: Show that Fg cannot (in general) satisfy more axioms than stated in Lemma 4.4. (Hint: Ax(II.3, 4): Choose A = B = 0, aEac1A - dclA.)

Ax(V): Let M = EM1(wly 0) where we get the theory T1 and @ as in Exercise 2.7. So M Ca < a,, implies: M Ca < a, for some i < 6, and a E acl{a,:j < 6) ({a,:j < w,} is the skeleton). Let B = acl{a,: i < w

or i > w + a}, A = acl{a,:i s w or i > w + w}, 7i = (a,,,), 6 =

(a*+7)- Ax(1V) : By the previous example, for A = B = C = acl{a,: i < w }

6 = (a,), z = (a,,,). Ax(X.l), (X.2): Choose A = dcl A = 0 (Ao = dcl Ao), g, = 5 = 5.

Ax(XI1): C1 = acl{a,: i < w } C, = acl{a,: w .2 < i < w.3) B = C, U

Cay A = acl(B u {ad), p = tp(a@+,, A) (in the notation above).]

432 CONSTRUCTION OR’ MODELS [a. VII, Q 5

VII.5. Hanf numbers of omitting types

K need not be infinite

DEFINITION 6.1: (1) We define p(A, K ) as the first cardinal p such that: if lLol 5 A, To a theory in Lo, r a set of (< No)-types in Lo, ’5 K ;

and for every x < p there is a model in EC( To, F) of cardinality 2 x, where EC( To, r) is the class of Lo-models of To omitting each p E r, then there are in EC( To, r) models of arbitrarily large cardinality.

(2) We define &A, K ) as the &at ordinal a such that: if lLol 5 A, < €Lo, To a theory in Lo, f a set of ( < N,)-types in Lo, If I < K and there is a model M in EC(To, r) such that (1611, <M) has order type 2 a, then there is M E EC(To, r ) which is not well-ordered by

Remark. The difFerence between the two definitions (“for every x < p” but not “for every S < S(h, K ) ” ) is inessential, sea 6.2.

LEMMA 6.1: (1) If in EC(To, r ) there i.3 a nzodel of cardinality A, I ToI 5 p s A, t h n in EC( To, r) thre is a model of cardinality p.

(2) If s A l , K s ~ 1 , thenp(h, 4 s Ah, 4, a d W, 4 5 W , ~ 1 ) .

(3 ) S(A, 1) 2 A+. (4) S(A, 1) = S(A, A). (6) &A, 0) = w.

Proof. (1) Because N < M , M e EC(To, r ) implies N E EC(To, r) using the downward Liiwenheim-Skolem theorem I, 1.4.

(2) Immediate. (3) For a < A+, let M , be (a, <, . . ., i,. . .),<,, T, = Th(M,),

p = {z # i: i < a}. Then every model in EC(Ta; {pa}) is well ordered and has order type a aa it is isomorphic to Ma. So S(A, 1) > a; and as this holds for every a < A+, 8(h, 1) 2 A+.

(4) By (2) S(A, 1) 5 S(A, A). To prove the converse, assume ]Lo! s A, To a theory in Lo, r a set of ( < #,)-types in Lo, I rl 5 A, M E EC( To, r ) has order type S(A, 1) and we shall prove that there is a non-well- ordered model in EC( To, r).

Let r = {pa: a < a, 5 A}, pa = {tpf(~,): i < i, 5 A}, pa an ma- type. Let $(a < ao, i < i,) be new individual constants, &, P, (a < ao)

CH. VII, 5 61 HANF NUMBER8 OF OrdITTINQ TYPES 433

new one place predicates, and for a < %, Fa a new ma-place function- symbol. Let L, = Lo U {Q} u {Pa, Fa: a < ao} u {q: i < i,, a < ao},

T I = To U {Q(cf), P,(cf): i < i,, a < ao} u {(VZ,)P,(F,(Z,)): a < ao}

u {(VZa)[F,(Z,) = cf + -,Q~Z,)]: i < i,, a < ao}

" {(W(Pa(4 -+ Q(4): a < a01 u {(wrp,(4 + 1p&41: a # B < ao},

p = {Q(z) A z # 4: i < i,, a < ao}.

Clearly every model in EC( To, r ) of cardinality 2 h can be expanded to a model of EC(T,,{p}), and the L,-reduct of every model of EC(T, ,b } ) belongs to EC(T, , r ) . So we can expand bl to W E EC(T,, { p } ) ; and then aa T,, p we in L,, lLll s A, and bl' haa order type S(h, l), there is N' = W, N'EEC(T, , { p } ) which is not well- ordered. The Lo-reduct of N', N, is the desired model.

( 5 ) If in (3) we take a < w, then every model in EC(T,, 0 ) is isomorphic to Ma, hence &(A, 0 ) > a; so S(h, 0 ) 2 w. By the compactness theorem if for every n < w To haa a model bl, of order tyoe 2 n, then To u {c ,+ , < c,: n < w} has a model, so To hae a non-well-ordered model.

LEMMA 6.2: 8wppse P, < E Lo(P--a one plccce predicate, < a two place predicate) lLol 5 A, To a theory in Lo, r a set of s K ( < N,)-typee. If for every a < 6 = S(h, K ) there is in EC(To, r) a model bl, such t?@ (P*u, 4%) laas order-type l a tkn in EC(To, r) there i s a model bl in which cM doecr not well-order PM.

Procf. For K = 0 this follows by the proof of 5.1(5), so let K > 0. Let P be the function such that F ( a ) = M a for a < &(A,K), w.1.o.g. the

are pairwise disjoint and F , be the function such that for each a < & ( A , K ) F,(a,z) is an isomorphism from (a, <) into (PMu, Let 93 be an elementary submodel of (R(E), E,

F, P,, 8, To, . . . , Q, . . . of cardinality 181, {i: i < S } = S E ] % I , and let # be a one-to-one function from 6 onto 193 I, and <* an order on 1931 such that # is an isomorphism from ( 8 , <) onto (1931, <*>. Let 93, = (99, <*, #) and T, = Th(99,). For Q = ~ ( f ) E Lo let Q, = 'pl(y, Z) be an L(S,)-formula saying that y is Ma = F(a) for some a < S and f is from lM,l and Q(Z) is satisfied in Ma. For p E I', p = p ( f ) let pl = {Q,: Q ~ p } , and r1 = { p l : p E r} u {{z E Lo A z # Q: Q E Lo}}. SO in EC(T,, rl) there is a model (99,) such that <* orders it in order-type

434 CONSTRUCTION OF MODELS [CH. VII, 8 6

6 = &A, K ) , lLll s A, lF1l s K (or if K < A, IF1( s A, so by 6.1(4) 8(A, IFll) = E ( A , K ) ) hence there is 9’ E EC(Tl, Fl) which is not well- ordered by <*. Hence ord(23*) = {a E IS*]: $* C “ a is an ordinal < 6”) is not well ordered by < (use a). Choose a E ord B* so that {b E ord 23*: b < a} is not well ordered. Then P(a) is, in fact, the required model.

THEOREM 6.3: Suppoee lLol s A, To a theory in Lo, r a get of I K

( < Ko)-typea in Lo and P a one-place predicate in Lo. If for every a < 6(A, K ) there irr H a € EC(To, r), IIM,II 2 aa(IPM“I) then for every p 2 ILol, there is H E E C ( T ~ , F), 11H11 = p , IP”I = ILol.

Remark. Notice that we can replace aa( lPM~l) by aa(lPMu1 + A) for K > 0, because ~ ~ ( 0 ) 2 A, SO for a 2 A + w they are equal.

Proof. We proceed as in the proof of 6.2. Let P be a function from 6 = 6(A, x), P(a) = Maandlet23, = (R(;), E, P, 8, To,. . . , tp,. . .)(PEL,,.

So again we can find 23 elementarily equivalent to 23, such that in 23 there are “ordinals” a,, < 6 , a,,+, < a,, and for each a < 8, P ( a ) is a model of To omitting each p E F. Now w.1.o.g. we can assume that To has Skolem functions and 93 k “a,,+, + n + 1 < a,,”. Let < be, in 23, a well ordering of P(ao). Now define by induction on n, elements X, of 23 such that 23 C “X, is an increasing sequence of elements from the universe ofF(a,), it has cardinality 3aJP(F(ao))l) , and is n-indiscernible over P(P(ao))” and 8 C “Xn+l is a subset of X,,”.

This is possible by the Erdos-Rado Theorem (2.6 of the Appendix) (more exactly-as 23 Hatisfies a first-order sentence saying it).

Now we define @ proper for (w, To) so that if p(zl, . . . , z,,) E @, then if 49 C “A:* , a, E X, A A::? a, < at+,” then 23 C “tp(al, . . . , a,,) is satisfied by P(ao)”. Clearly there is such @, and EM(& @) is the required model. (It does not realize types omitted by F(ao) because of Lemma 2.2; hence, it omits the set of types r.) THEOREM 6.4: p(A, K ) = ad(h,rc) for K > 0 ; and p(A, 0) = No, 6(h, 0 ) = 0.

Proof. For K = 0 this is by 6.1(6) and the compactness theorem, So let K > 0. By the last theorem it is easy to see that p(A, K ) 5 a6(A,rc). For the other direction assume that every model in EC(To, F) is well ordered by < ; To, Fare in Lo, lLol = A, IF( 5 K. Now let Q be a new one place predicate, for every # let $0 be # relativized to Q, TOO =

{p: + E ~ ~ 1 , = {p: 4 Ep), ro = {pa: p E r}.

CH. MI, 5 61 EANF NUMBERS OF OMITTIWO TYPES 435

Let E be a new two place relation, P a new one place relation, and P a new one place funotion symbol, and ct, i < KO new constant symbols. Let

(VZ, y ) [ - ~ p ( Z ) A - IP (Y) A ( v Z ) ( Z E 2 Z E y ) + X = 3 / 1 9

(WQ(m)), (VZY Y ) ( Z w + m ) < P(Y)))U Wet): i < So},

F, = YOU {{P(X) A 2 # C,: i < KO}}.

Clearly if iK E EC(Tl, F,) then its submodel with universe QM belongs to EC(To, r ) and if the order type of the latter is a, then lldlll 5 D ~ . Conversely, if some N E EC(To, F) has order type 2 a + 1, then some bl E EC(Tl, r,) has cardinality aa + I TII.

THEOREM 6.6: Let K > 0. (1) 8(hy K ) is a limit ordinal of cojdity > A.

( 3 ) If cf h > KO then 8(hy 1) > A+. (4) If of h = KO, p = ZX<’ 21 then S(h, 1) s p+.

(6 ) If h = No

(6) &(A, 2’) = (2’)+. (7) &(A, 1) < (29+ (and &(A, K ) < (2’)+ when 2“ < 2’)). (8) For a strong limit cardinal of cojinality > KO, 2’ < & ( A ) ; and

(2) h+ 8 ( h , K ) 5 (2’)’.

h i8 a &Ong limit cardinal O f W f i d i t y KO, t h 6(h, 1) = A + .

generally cf h > KO, (vp < h)(p”‘A < A) implies &(A, 1) > A”‘”.

Proof. (1) Easy, left as an exercise fo the reader. (2) We have already proved that 8 ( h , ~ ) 2 A+ (in 6.1(3)). For the

other inequality let lLol 5 A, To be in Lo, F a set of types in Lo, s K and suppose iK E EC(To, F) has order type 2 (2’)+ (ordered

by < ). We can assume Th(M) has Skolem functions, and let a, E IiKI, i < (2’)+ , be such that for i < j, dl t at < a,. Now we d e h e by induction on n < o, sets & E (2’)+, = (2”+, and for each i E&,

ordinals ai(n, 0) , . . . , a,(n, n - 1) for which i < al(n, n - 1) < - - - < al(n, 0 ) such that: if i E 8,, j eSm, n 5 m then tp((aa,,,,,,, . . . ,

For n = 0 let So = (2”)+, (we have no at(O, 1) to define). If we have defined for n; for each j < (2’)+ we choose an i, j < i €8, and let a,(n + 1 , l ) = at(n, 1) for 0 s 1 < n and a,(n + 1, n) = i. The number

aa,(n,n- I)), 0, = t~((aaj(m.o), * * 9 aa,(m,n- 1))s 0, Jf).

436 OON8TRU(rmON OB MODELS [a. VII, 5 6

ofpo=ibletypeep;+’ = tp(<au~n+l,o), * * * Y au,(”+l.n)), 0, M)is s 2lL0l s 2” c (2A) + w h e w the number of j ’ s is ( 2”) + . So there is a type pn +

such that Is,+, = {j c (2’)+: p;+’ = pn+l} has cardinality (2”)+. Now by the compactnw theorem and our construction there is a

model N , of Th(M), and in it elements any n c o, such that tp((ao, . . . , a”), 0, N , ) = p”+’. As Th(M) haa Skolem functions, we can wume that INl! is the closure of {an: n c u}. So for every 6 E lNll there are nmd7so tha tN1 C6 = ?(ao, ..., a ” ) s o i f i ~ & + , , 6’ = ?(aul(n,o)y...) (in M) then tp(& 0, N,) = tp(6’, 0, M) so N , omits every p E F 80

As for each i E&, q(n, 0) > q(n, 1) > - - - clearly N, C a,, > a,,,,, so N, is not well ordered. All this proves that 8(A, K ) s (2”)+.

(3) Let 49, = ( R ( i ) , E, c , A+ , A, i)‘< A ( c -the order between the ordinals s A + ) and let p = {z c A A z # i: i c A}, and F = {p} and T = Th(99,). Suppose 99 E EC(T, 0, and we shall prove that {z: 99 C c A + } is well ordered, thus by 6.2 proving that 8(A, 1) > A+. For

suppose 99 C z,, + , < x,, c A+. Clearly in 93 there is an element f such that 8 C“f is a one-to-one mapping from {z: 2 < z,,} onto {z: z < A}”.

Let 9 C “f(z,,) = a,,”. As of A > No, 49 omits p , and {a,,: n c W} is countable it is bounded by some a* c A. As b C “the set of x c A+ such that f(z) c a* has cardinality c A” there is g in 49 such that 53 C “g is an order-preserving function from {z c zo:f(s) c a*} into A” so 99 C “g(z,,+,) c g(z,,) c A” contradiction to the omitting of p by 9 (aa A is well-ordered).

(4) Let M €EC(To, r ) have order type zp+, and F = {p}, p = {v‘(Z): i c i, s A}, Z(Z) = m. Let ?;(z0, . . ., $,,-,), a c A be a list of all terms from Lo where Z(f:) = m. Let for a E “1611, G(si) = min{i: M C 7cpi(i i)}. For A > No aa of A = No let A = z,, A,,, A,, c A,,+, c A, and let H(7i) = min{n: G(a) < A,,}.

The rest of the proof is just like that of (2), only p+ replaces (2”)+, and instead of tp((aul(,,,o), . . . , aul(n,n-l)), 0, M) we use when A = No

N, E EC(T,, r).

((1, FLY Q(~Lfr(aul(n.O)Y * - * Y ~u, (n ,I - l ) ) ) ) : 2 5 nY a 5 n}

and when A > KO,

OH. VII, 5 61 HANF NUMBEBB OB OMITTINO TYPES 437

At the end we choose Nl as a model of

Th(H) u {&+l < am: n < w}

u { l p p ( ~ ( U o l ..., an-’)) : p = G(q%*(t8,0p ... 1) for every i E&, n big enough}.

(6) It is immediate by parts (2) and (4). (0) By (2) it suffices to prove S(A, 2”) 2 (2”)+, or that for every

a < (2”)+, 6(A, 2”) 2 a. So let a < (2*)+ and choose for each /3 < a a subset SO that /3 # y * 86 # 8,. b t for every /3 < y < a

P6.7 = {+ c y)} u {P,(2)1f(fES0): i < A} u {P,(y)u(W: i < A} and for

To just “says” < is a linear order,

Of

every s c A p s = {P,(P)U(f€? i < A}. Let Lo = {Pf: i < A} u { < y =},

r = { & 7 : /3 < y < a} u bS: S c A, 8 # 86 for each /3 < a}

U {P # y A Pi(%) Pi($/): i < A}.

Clearly there is M, E EC( To, r ) of order type a (lHl = a, /3 E Pf 8 i E 86, < is the usual order) and every model in EC( To, r ) is a submodel of M , (up to isomorphiem) hence is well ordered.

(7) Immediate by cardinality considerations (there are essentially only 2* pairs (Tol {p}) and similarly for S ( ~ , K ) ) .

(8) Let K = cf A, and D be any H,-complete filter over K. For any function f from K to A (in fact, to the ordinals) we define its D-rank RD(f) : R D ( f ) 2 0 always, R D ( f ) 2 6 if for every a < 8, R D ( f ) 2 a,

and RD(f) 2 a + 1 if there is g : K 4 A. g/D < f/D (i.e., {a < K :

g (a) < f ( a ) } E D ) and R D ( g ) 2 a. NOW RD(f) = a if RD(f) 2 a but not R,(f) 2 a + 1; as D is 24,-complete, there is no descending sequence fn /D (n < w) SO RD(f) is always an ordinal. Let

a(A) = sup{RD(f):f: K + A}.

Let us define now a model H’: its universe is the disjoint union PM u QM u &f where ignoring trivialities:

PM = a(A), &M = {f: f a function. from K to A},

c M = the order on ordinals, F = a partial function such that F(f , i) = f(i) for i < K, f~ ah, i (i 5 A) = an individual constant, A (A c K ) = an individual constant,

&f = {A: A = K},

438 UONSTRUUTIOW OB MODEL8 [a. VII, $ 6

€M = {(i, A): i < K , A E K , i E A}, Qi = D, B = a partial function such that U ( f ) = BD(f) E P’.

Let T = Th(M) and p = {x < A A x # i: i < A} and let N be a model of T omittingp; so by renaming we can assume the interpretation of i (i 5 A) is i ifself and any member of QN is a function from K to A, and PV, i) = f(i), m d

QF = Qy, E N = { ( i , A ) : i c K,A E K , A E Q F , i , E A ) ,

so Qf = D. Now we prove PM is well ordered. If cn > c , + ~ (n < w) is a counterexample, then there is fo B ( f ) = co. (As (Vx E P) (3y cQ)(B(y) = 2) E T, as PM = a(A).) Similarly, by the rank’s definition, we can define fn E Q ~ , such that U(f, ,) = c,,, {i < K : f,,(i) > f,,+ l(i)} E Qf = D; and we get a contradiction. So in every model N E

EC(T, {p}), PN is well-ordered by <N. By 6.2 and T’s definition this implies a(A) < &(A).

Up to now we have not used any assumption on A, Kexcept K 5 A, cf K

> No, 2’5 < A; and if D is easily defined (e.g., the filter of closed unbounded subsets of K , or DEb the filter of co-bounded subsets of K ) the last restriction is not necessary.

Now we prove a(A) 2 Aofh, e.g., for D = DEb thus finishing. First note that there are Aof A functions f, : K --+ A such that (V i # j ) ( 3 a < K )

i < K ha be a one-to-one function from n,ca A, into A (exists as (Vp < A)(Vx < cf A)(px < A) ) . Let df:; i < AofA} be a list of all functions from K to A, and define fl(a) = ha((ft(y): y < a)). Thefl’s are as necessary. If a(A) < Ax, there are (29+ fl’s with the same D-rank, say cfl: i < (29+}. For any i < (29+, cf i = K, and a < K + , we define by induction on n < w, &(i, a) such that

(VB)(a f ib) ,

(iii) for any m < n, letting f = &(i, a), = &(i, a)

f d a ) > f , (a) +fc(a) ’ fM. By (iii) and (ii), fcncl,6(a)(n < w) is strictly decreasing hence for some n = n(i, a) &(i, a) is not defined. Let Sk = {&,(i, a): n < n(i, a)}, Si = Uac+ St, 80 lS1l 5 K, 9’ c i. As cf i = K + some h(i) < i bounds it, so by 1.3 of the Appendix, for some y {i < (29+ : h(i) = y } is stationary

OH. VII, f 61 HANF NUMBERS OF OMITTING TYPES 439

and also for some S : W = {i < (2")+ : h(i) = y , S6 = S} is stationary. We can find j < i, j, i E W such that for every 6 ES, t9 < K ,

fc(B) ' fdB) +fc(B) > f r ( B ) If for some a, f,(a) > h(a), j will satisfy the conditions on fna,a,(i, a), contradiction. So for every B f j ( j3) I f,(/3); but l{P: f j ( B ) = ff(/3)}I < K

hence {/3 c K : f j ( P ) c fi(/3)} E D( = D;Ib). But this implies f , /D < ff/D hence R,(f j ) < R,(f,), contradiction.

Remark. Barwise and Kunen [BK 711 have shown that it is consistent with ZFC that:

(1) A + < 2 A y 6(h, 1) < A + + , (2) A + < 2 A < 6(h, 1).

EXERCISE 5.1: Suppose lLol I h, To a theory in Lo, r a set of ( < No)- types in Lo, s K . Show that if for all a c 6(h, K ) there is a model Ma in EC(To, r) such that IP"aI 2 >a and llMall 2 > ( ~ P " U [ , a) then for all h 2 p 2 ITo[ there is a model M in EC(To, r) such that llMll = h, IP"I = p. If we omit " IPMuI 2 say' we can still get IP"1 I

PI. EXERCISE 5.2: Show that if in the definition of p(No, 1) we require To to be a complete theory and p ( F = {p}) to be a complete type the value of p( No, 1) is unaltered.

CHAPTER VIII

THE NUMBER OF NON-ISOMORPHIC MODELS IN PSEUDO-ELEMENTARY CLASSES

VIII.0. Introduction

The essence of this chapter is the construction of many non-isomorphic models in a pseudo-elementary class PC(T,, 2’) and we shall use methods developed in VII; the proofs have a combinatorial flavor.

At the beginning of Section 1 we make two important observations (Lemmes 1.3 and 1.4).

(1) If after adding K individual constants to T, there are p many ( > As) non-isomorphic models of cardinality A in the pseudo-elementary class, then the adding does not change this number.

(2) If we have h 2 lTll types in D(T) which are independent (i.e., for any subset of them, there is a model reelizing them but not the others) then I(h, TI, T) = 2” (aee Definition 1.1).

Now we can show that some properties of T imply the existence of large families of independent types, sometimes over a set A (1.6) and then by (1) and (2) we prove that I@, T,, T) is big (1.7). A more difficult theorem is 1.8 (the main crme is 2” = 2n0, ID(T)I = No).

THEOREM 0.1: If T ia countable, not Ko-8table, lTll = No, KO < A 5 2n0, then I(h, T, , T ) = 2”. If instea& of I TII = No we demand I TII < 2% we atill get a r d t when

lTll s h < P o : (1) If A satisjies the combinatorial condition (*) (“not AD(2No, A , A ,

No)”), then I (A , TI, T ) 2 2 H o (see 1.9). (2) If Martin Axiom (see [MS 701) holds, I(A, T, ,T) 2 2A (see

Exercise 2.6).

More interesting is the fact that we can generalize the theorem to higher cardinals, but the hypothesis is quite strong:

440

OH. m, 8 01 INTEODUOTION 44 1

THEOREM 0.2: 8ypptxe (i) T has A independent fornzulae, (ii) T E TI, A = IT11 = IT!, (iii) There is a A-Kurep tree with x branch (e.g., A strong limit, or

cf A = No, x = ANO),

Then for p 2 A, I (p , TI, T) 2 min{2xy 2”).

It would be nicer to get such a theorem aseuming only ID( T) I = x >

The main aim of Sections 2 and 3 is A =lT1l.

THEOREM 0.3: Any unsulperstable T , has 2A norc-isommy.~hk Cardinality A, for any A 2 IT1 + N,.

of

We prove it also for pseudo-elementary 01- in many

The easiest case is:

(e.g., countable T, T,) . The proof is by cams.

THEOREM 0.4: If T i s uwwperstabk, A is regzrlar and > lT1l, then in PC( T,, T) there is a family of 2” nzodele of cardinality A, no one ekmedarily embedduble in anothw.

In the proof note that if M = UleA Ml (1 = 0, l), Mt (i < A) in- increaaing and continuous, Aregular IliKll = A > IIMf 11 then{i: Mf = Mt} is a closed unbounded subset of A. Hence if P is a property of pairs of models then {i < A: (My M f ) satisfies P} is determined, mod D(A) by the isomorphism type of M ; and in fact we can make P to depend on more information. We urn the indiscernible tree of sequences we have constructed in VII, Section 3. Choose for each S < A , cfS = w , an inmawing sequence of ordinals of length w converging to it, 76, and for B G { S < A: of 8 = w} let I, = @’A u {q6: S €8). Now from the isomorphism type of EM(1,) we can reconstruct S/D(h) .

The other cams have each one its specific trick, and the proofs are somewhat complicated.

If p < A 5 pNo, 2” 2A, the method of Section 1, on independent types, applies. If A is singular, but no previous m e appliee we choose a proper p < A s 2’, and make a construction similar to the first one, but A-fold. If A is a singular strong limit cardinal, we use games. For an elementary clw, T stable, A = IT], use Fk-primclry models over indiscernible trees, and repeat the previous oaaw (all this in Section 2).

442 THE NUMBER OF NON-ISOMORPHIC MODEL8 [a. WII, 8 0

Then for A = I T , I, T unstable we note that skeletons of EhrenfeuchG Mostowski models rn N,-skeleton like (Definition 3.1 or below), and if in an insomorphism of one such model onto another, the intersection of one skeleton with the image of another is big, the orders of the skeletons are not contradictory (Definition 3.2). Proving the existenoe of large families of pairwise contradictory orders, we finish exoept when A satisfies a strong set theoretic condition, which implies it is < 2h or is high in the sequence of 8,’s (this is 3.2, case I). Note that M1 =

(A) (a,: 8 E I) is X,-skeleton like, i.e., for every 6 E lM1l there is a h i t e J E I, such that if 8 , t ~ I - J , (VUEJ) (8 < W E t < u) then 63,, gnat realizes the same type.

(B) Hence if I = I, + I,, then for every 6 E M,, for some t , E I, ,

The m e left is T unstable, 1 T,I = A, AD(2”, A, A, No); we can assume T is countable. For understanding the proof let us concentrate on showing I(A, T,, T) > 1 for Aregular > N,. Let N be a (A, N,)-eaturatd model (see Definition VII, 1.5), N = Ul<ol N,: and M = EM(I) , I = I , +12,11 z A , I, s o*, and let ( a 8 : s € I ) be the skeleton of M (order by < , of come) andf : M + N the isomorphism, and 6, = f(a,), so (6, : s €1) is an KO-skeleton like sequence in N. We want to show for some 6, C 6 < b, for 8 E I,, and c b, N,, for some 5 < o,,

sequence (Aa: a < A), increasing, continuous, 1A.I < A, U,<” A, = INC[. We can h d 8 < A, such that UaEIl 6, E A,. AS (6,: 8 E I) is H,-skeleton like all except < lA,l + + 8, members of {6,: 8 E J} realizes the same type over A,, let 6: be one of them and J, = (8 E J: tp(a,, A,) = tp&, A,)}. Now it suffices to prove some consistent type extending {Z < 6,: 8 E I,} U {6, < Z: 8 E J,} belong to St+,. But we can define such a type using N,, A,, and 6*:

E M l ( I ) eatisfies:

reah08 the mme type for all 8 E I Satisfeg tl < 8 < t,.

- J : UnErl b, G NC, UIEj 6, E Nc, J c I , , I JI = A. In Sc+l there is a

p = {Z < 6: 6 E A,, Nc C St < Q U (6 < 1: 6 E INC] , and for every 6’ E A,, Nc C 6: < 6’ implies Nc C 6 < 6’)

(every b i t e subset of p is realized by all a,: 8 E I2 except hitely many). In the actual proof we have to take care for getting 2A non-isomorphic models, and making them K-compact, for suitable K.

In Section 4 the reader can relax; here we mainly use previous results to prove theorems e.g., on categoricity, we prove:

CH. WII, 8 01 INTRODUCTION 443

Tireorem. If PC(T,, T ) is categorical in A > 1T11, then T is superstable, without the f.0.p. and stable in lT1l.

We also give a partial solution to the problem: doea the categoricity of PC(T,, T) in h imply its categoricity in p ? Show its equivalence to a two-cardinal theorem with omitting a type. We get similar results when we replace “categorical in A” by “each model in PC(T,, T ) of cardinality h is homogeneous ”.

Then we use the construction of VII, Section 4 to prove results on universality, i.e. :

THEOREM 0.5: If T ie unawperetable, A is regzllar >2IT1l t h in PC(T,, T ) there are arbitrarily lwqe ( < A)-univereal not h-universal nzode2e.

On the other hand we shall show that if of h = No the conclusion does

We also characterize the class of cardinalities in which a theory hw not hold.

saturated models, by using various previous results.

THEOREM 0.6: T hua a eaturated model of Cardinality A iff X = ID( T)I or T ie h-stahle.

PROBLEM 0.1: Though we get a complete reeult for elementary classes, this is not true for pseudo-elementary clmms, when we look for K-compact models K < K(T) , and when we look for 2* pairwise mon- elementary embeddable models; it will be interesting to complete them. PROBLEM 0.2: It will be desirable to simplify and unify the proofs. A possible way is in the proof of 2.1 to try to replace the filter of closed unbounded subsets of A, by a filter on S,(h) or on the family of increasing sequences of members of S,(h), of length S (see Kueker [Kk 771 and Shelah [Sh 7681).

PROBLEM 0.3: In many cams we have parallel proofs; in 2.7 an F&,-constructible set, replaces the Skolem Hull. Also clearly different proofa uae different sets of assumptions on the construction. Maybe an axiomatization is in order.

+

PROBLEM 0.4: For uncountable T,, we know little on I(h, T I , T ) when ,LA = JD(T)I > IT,I. It is natural to conjecture that for h > ITJ, l ( h , T,, T) 2 min{2A, W } , but it is reasonable to try also an independence proof.

444 “HE NUMBER OF NON-ISOMOBPHIU MODELS [OH. WII, 8 1

VIII.1. InaepenaenCe of tspes

DEFXNITION 1.1: (1) I ( A , T,, T ) is the number of non-isomorphic models in PC(T,, T ) of cardinality A.

(2) IE(A, T,, T ) is the maximal p such that there is a family of M f E PC(T,, T ) , i < p, llMtll = A, such that there is no elementary embedding of Mt into M, for i # j.

(3) If we omit T , this meam T , = T .

Remark. The “maximal” in (2) is inaccurate, as there may be none, and we may get, e.g., 8, < IE(A, T , , T ) < K, for any n.

Remember D( T ) = Urn Sm(0).

LEMMA 1.1: (1) 2” 2 I ( A , T,, T ) 2 IE(A, T,, T ) 2 1 for A 2 IT,!.

IE(A, T,, T ) s IE(A, T i , T’). (2) If T s T’ E Tl c T,, then I ( A , T,, T ) I I ( A , Ti , T’) and

Proof. Remember TI is consistent, and any model of T , is isomorphic to a model with universe {i: i < A}, and there are only

n 2(Am(”) (2”)“ = 2” RELl

such L,-models where m(R) is the number of places in R (we consider here m-place functions as (m + 1)-place relations). The rest is even more trivial.

Remark. We will be mainly interested in I and not in IE.

Proof. By VII, 2.9(1), for every p E D(T) there is a model M , E

PC(T1, 2’) of cardinality A which realizes p and is stable in lT1l. (Replace T, by T, u {v(E): v E p}, E a sequence of individual constants.) Now M, r Mq induces an equivalence relation over D(T) . Each equivalence clam has cardinality s I T,I as I{q E D(T): M q M,}l I I{q E D(T): q is realized in M,}1 s lTll (by the IT,I-stability of MP). Hence ID(T)I is at most ITl! plus the number of equivalence classes (by cardinal arithmetic). As ID(T)I > lTll, ID(T)I is equal to the number of equivalence classes. So there are 1D(T)1 non-isomorphic

OH. n I I , § 13 INDEPENDENCE OF TYPES 445

models M,, so I ( A , T,, T) 2 ID(T)I. Let g(p) = {q: q E D(T), M , has an elementary embedding into Mp}. So g(p) c D(T) and again Ig(p)l s ITJ; hence, when proving (2), by theorem 2.8 of the Appendix there is S c D(T), IS1 = JD(T)I, such that p # q E S implies p $ g ( q ) , hence IE(k T,, T) 2 1{2M,:pEs)l = ID(T)I.

PROBLEM 1.1: Does (2) hold when ID(T)I = ITl[ + ?

LEMMA 1.3: If A 2 1TlIy p = I ( A , T, u T(A) , T(A)) > AIAl then I(A, T,, T ) 2 p; where T ( A ) = T u { ~ ( a ) : si E A, Cp[si]) (the a E A serve also i n d i v i d d constants).

Proqf. Every model M, of T, of cardinality A may be expanded to a model of TI U T ( A ) in < hlAl forms, and we can get each model of T, u T(A) in this way, from just one M , (up to isomorphism) hence

p = I ( A , Ti U T(A) , T (A) ) 5 I ( & Ti, T) + But p > N A l , so our result follows.

Remark. Notice that we cannot use 1.2 for 1.3 a~ the estimation in 1.2 is too weak. Hence the following concept is interesting:

DEFINITION 1.2 : (1) If S is a family of types over A (in L), T s T,, then S or (8, A) is called (T,, T)-free (or free in (T,, T)) if for every S' c S there are models M , of arbitrarily large cardinality, A 5 M , (of come M, r L 4 6) such that p E S is realized in M , iff p E S' (hence in every A 2 lTll + IS'l there is such a model).

(2) S is free if it is (T,, T)-free for every T,. (3) The pair (T,, T) has (p, A)-freedom if there are A, IAI = A, 8,

IS1 = p satisfying (1). If this holds for any T, with the same A, S we omit T,.

LEMMA 1.4: (1) If (T,, T ) irae (p, x)-free&Om, p 2 A 2 lT1l, 2A > Ax, then I(A, T , , T ) = 2A.

(2) If (Tl, T) ' has (p, x ) - f r e e h , A 2 p, 2' > (p + p,p, then I ( A , T,, T) 2 2y.

( 3 ) If (S, 0 ) is (T,, T)-free, A 2 lTll, thenIE(A, T,, T) 2 min{21SI,2A}.

Proof. For ( l ) , we estimate I ( A , T, u T(A) , T (A) ) , from below, (where (S , ,A) is (T,, 2')-free, IS,l = p, IAI = x ) by looking at { ~ E S , : M realizes p}, and then use 1.3. Part (3) is immediate as by 1.6(2) of the

446 THE NUMBER OF NON-ISOMORPHIC MODELS [CH. VIII, 8 1

Appendix there are 2151 subsets of 8, no one is a subset of the other. We are left with (2); by assumption for every S E 8, there is a model M1(S) of T , of arbitrarily large cardinality, A E M1(S), each p ES is realized in M1(S) by as = a(p, S), and no p E So - 8 is realized in it. By VII, 6.3 proof there is a model Ma(&) of T , = T , u T ( A U {ap: p €8)) omitting each p E .So - IS, which is EM(A, 0) @ proper for (w, Ta). Let M3(S) be its L-reduct. By VII, 2.10 and 2.8(6) TP,(M3(S)) has cardinality 5 (p + I T,I)X which is < 2". As TP,(M3(IS)) depends on M3(S) only up to isomorphism and the number of possible 8's is 2', we finish.

THEOREM 1.6: (1) T has (A, p)-freedom if T is unstable and A < Ded p (i.e., there is an order of power A, with a dense subset of power p) .

(2) T has (A, p)-freedom if K < K ( T ) and there is a (A , +tree I , [In ASIC[ 5 p < [ I n AICl = A (e.g., if X , < K(T) , A 5 pwo).

( 3 ) T has (A , p)-freedorn if T has the independence property and p 5 A 5 211.

(4) (T, , T ) h (2b, X,)-freedom if lTll = No, T X,-umtable. It has (2%, 0)-freedcm when ID(T)I > 8,.

Proof. (1) Let J E I be orders, I JI = p , 111 = A, J dense in I. Let J c J, c I, and we can wume J, I are dense orders (by extending), and let I, be an order of type x. By VII, 2.4 there are (Z < 8) EL, and @ proper for (0, T, ) such that in M = E M ( J , + I , , @), M C 7is < at iff s < t . Clearly llMll 2 x which waa arbitrary. Let A = {a8: s E J } , S = {p8: s E I - J), where ps = {(Z < 7it)if(s<t): t E J } . In M ps is realized if s E J, (by 4); the converse is also true, for if si = 6(@) E My as s $ J, and J is dense, there are s(1) E J, 8 ( l ) - s(mod 8 ) for I = 1, 2, and s( 1) < s < s(2) hence (7i < a*,)) = (7i < si8(a)) so si does not realize p8. We only have to choose A independently of x, this can be done by VII, 2.6 or by:

CLAIM 1.6: If for every T I z T and x there are a pair (8, A ) , S E Um Sm(A), IS[ = A, IAl = p and for every s' E S a model My 11M11 2 x, M realizes p ES i f p €8' then T h (A , p)-freedonz.

Proof. We should prove that we can choose (8, A) independently of T, . Up to isomorphism (of Q ) there are s 2A such pairs, so if each such pair (8, A) fails for T,(S, A), s'(8, A ) and x(S, A) then (as T is complete) we can find T , z T , such that up to change of names of predicates and function symbols not in T, it extends each T,(S, A) (such T,, is consistent aa Q can be expanded to models of TI@, A) for any S, A by

CH. WII, 8 11 INDEPENDENCE OF TYPES 447

I , 1.12). For T, we get some suitable (8, A) for x = 2 h(S, A): possible (8, A)}; contradiction.

Continuation of the proof of 1.5. (2) Let I be the (A, K)-tree mentioned there, and x L A. T has a strongly uniform K-tree of the form ((q~~, ma): a successor, a < K ) and let @ be as in VII, 3.6( 1) and (2). Then let

A = {a,,: 7 E I n A<"},

8 = {p,,: 7) E I n A%}, p,, = {q~~(z , sinla): a < K successor}.

If s' c 8 let I(&') = x<' u (7: q E I n A", p,, €8') and M = EM(I(8') , @). Clearly lldlll L x, A G IMI, and if I),, E s', a,, E 1dl1 realizes p,,. If v E An ?(a,-) realizes p y , pv $ s' there is a < K such that q[Z] 1 a # v 1 a (or q[Z] has length <a). As M C pls(.i(E,), liyls) (/? = a + 1) for every i < x, letting p = (v 1 a)-(i) M C vs(?(iZ,,), 7ip) (by the indiscernibility of the indexed set {a,,: 7 E I(&")}). But ?(7i,) can satisfy such formulas only for finitely many i's, contradiction.

(3) Let x L p be arbitrary, T, z T, with Skolem functions, of course. As T has the independence property there are q(Z ; #) E L, and a,, such that for every w _c w {v(Z; ii,,)w(nsw): n < w } is consistent. So we can assume {a,, : n < w } is an indiscernible sequence. So as in VII, 2.4 there is @ proper for (w, T,) such that in M = EM(x; @) for w G x, {q(Z; a8)if(ssw): 8 E x } is consistent. Let A = {as: 8 E p} and {w(i): i < 2") be a family of subsets of p, such that for all distinct i,, i,, . . . , i,, w(i,) - U:=, w(il) is infinite (exists by 1.6(2) of the Appendix); and p, = {[v(Z; = +Z; ~ , a + l ) ] " a E * ' ~ : a < p} and 8 = {p,: i < 2u}. There are Skolem functions F,, = Fn(go, gl, . . . , #,,,-,) of (3Z) Ale,, [tp(Z; #,,)

+z; #m+1)1. SO Fn(aa(l), aa(1)+1, RzO), Ti,o)+ly . . .) realizes

{[d% aN1)) = l q ~ ( Z , ~ a ( l ) + d I : 2 < n)

(when, e.g., a(Z) # a(k) + 1 for 1 # k.) Let for i < 2", U G p, I Ul < KO 6,(U) = B',,(. . . ,a,, aaa+i , . . .)asw(t)n-u where n = Iw(i) n Vl . So for a E U; a E w, iff qJ[b,(U), a,,] = TqJ[b,(U), Let D be an ultrafilter over W = Sn0(p), such that for any finite w, s p, {w E W : wo s w} E D ; and let M* = Mw/D, making the natural identification of a and (. . .,a,. . . ) / D ; and for i < 2u let 6, = (. . ., 6,(U), . . .)&D. For 8' c S let M(S') be the Skolem closure of Idl] u {6,: p , E 8') in M*. Clearly A c M(s ' ) , and each p , E 8' is realized in M ( 8 ) . Suppose p , $ s', but 6 = .?(as, b,,,,, . . . , 6,,,,) realizes p,, I EX. By assumption on the w,'s, there is a $ UISn wj(l), 2a $ 8, 2a + 1 $8, a E wl.

-

-

448 TEE NUMBER OF NOW-IBOMOBPEIO MODELS [OH. n I , 8 1

We ca,n assume that for every U E W, 6[U] = T(~z,, 6,,,,[Uj,.. .) 80

if a E U E W 6[[u1=T(af) where i ~ { 2 p , 2/?+1: / ~ E w ~ ( ~ ) , ZGn} U 8,

hence 2a, 2a+ 1 $t; hence MCtp(.?(at), aZz,) = tp(?(af), aZt2.+J so {YE W: k&U], ZSu) = q(6[U] , Baa+,)} 2 {U: a E v) E D hence M* C q(b, aSU) E q(6, aSU+,), 80 6 does not redhe pfy contradiction.

(4) Suppose T, I> T hasSkolem functions, lTll = No. If ID(T)I '> No by 11, 3.16 there are formulas q, , (E) E L for r) E 2<" such that (3Z)q , (P) E

contradictory. We use theorem MI, 3.7, Exercise 3.1 on T,, q,, (and its notation), we let S = {p:: r) E 2"). Now (Is, 0) is (T,, T)-free, aa for any x and S' c Is let (Ml, . . . , b,,, . . . ),,<" satisfies the conclusion of MI, 3.7 where (in M,) {b,: i < w } is an indiscernible sequence over {an: r) E 2"}, and Ma an elementmy extension of M1 in which {bt: i < x } is an indiscernible sequence over {a": r ) E 2"). Then let M(Is') be the Skolem closure of {an: p: E S'} u {bt: i < x}. Clearly each p: E Is' is realized by a,, E M(S'), and by VII, 3.7(4) no p: $8' is realized in M(S'). If T is unstable in No we choose A, IS(A)I > IAI = No, and proceed as before with T(A).

T ; v Q 11 (W(V"(@ --+ VY(W E T and (Pv-<o)@), VV-<l>(Z) a m

Clearly 1.6 and 1.4 have many conclusions, e.g.

CONCLUSION 1.7: (1) If T b not eujperstuble, P o 2 p 2 h + IT,l, 2u > 2" t h n I (p , T,, T) = 2 Y .

(2) If T , z T , T , countable, T No-unstubk then h 2 P o implies I(h, T,, T) 2 2a'0y and 2" > 2'0 2 h implies I(h, T,, T ) = 2".

THEOREM 1.8: Suppoee T, 2 T are countable, T K,-unstable 2'0 2 h > KO, then I(h, T,, T) = 2".

Proof. We assume T, has Skolem functions. If 2" > 2'0 or ID( T)I > No the conclusion follows from 1.7(2), 1.6 and 1.4(3); if T is not superstable by the next sections (Theorem 2.1). So assume T superstable, No-UIIBt&ble, 2" = 2'0 and ID(T)I = No. The first two assumptions imply, essentially by 111, 6.1 that there am equivalence relations En@; jj) E L with finitely many equivalence classes (in a) En+, re- fining g,, and E,, r) E 2<" such that r) Q v, r) E 2,, implies En@,,, 4) but r ) # v E 2" implies 4 & ( Z V , 4). Let q,,(Z) = En@; 4) where 4 will be individual constmta of T,.

Using VII, 3.7 (to T,), and renaming, there is a model M of T, and a,,, r) E 2" satisfjdng (1)-(6) from that theorem, and by renaming we

CH. mII, 8 11 INDEPENDENCE OF TYPES 449

get h is the identity. We can dso wume (for (*2) by taking a subtree of 2<" and renaming)

(*) For every T&,. . . , Z,,,) E L, there is n, such that for every n 2 nf:

(1) There are in L, individual consbnta c? i < ni re- presenting the E,,-equivdence classes such that

~ I d V l Y - * * Y f), $1 E r,, (see VII, 3.7(5) for the F,,'s).

(2) If +j is a sequence of length m of distinct members of 2", and the same holds for 1, and there am k, n s k < w , and +jl, 1, such that Z(+jl) = Z(1,) = my +j[Q 4 +jl[Q E 2OP,

lEk(t(?ZGl)y T @ ~ J ) , then this holds for k = n, and any such +jl, 1,.

C[Q Q FJZ] E 2", and +j1[21] = Sl[Za] 0 +j[ZJ = F[Za] and

Remark. We can first take care of (2) then of (1). Note that M C E,,(sin, a,) (7, v E 2") 8 7 1 n = v 1 n.

Now for any 8 c 2" let M,(8) be the Skolem closure of {an: 7 €8) and N(8) its L-reduct. Suppose f : M ( 8 ) -+ M(8*) is an elementary embedding; 8, 8* c 2" uncountable. For any v E 8 there am T = T~~

n = n, and +j = +jv of length n such that M,(8*) b f ( a v ) = ?,(af). We can msume +j[Z] # +j[k] for 2 # k. As 8 is uncountable there is an uncountable 8, c 19 and t, no, k,,

+jo such that:

(**) For any v E 8,, T, = T , n, = no, and +jv[a ko = ijOIZ] 1 Eo and +jo[Zl] 1 ko # +j&] 1 ko for Zl # 2,. We can wume also that n, (from (*)) is 5 k, and that for 7 # v E 8,, h(7, v) > k,, where we let h(7, v ) = max{n:q r n = urn}, so q(n) # v(n) when n = h(7, v) .

Supposep # v E 8, h@, V ) = n, 80 B(8) C En@,, a,) A TE,,+~(~, , a,).

-En[f(q,),f(q,)l. Suppose for every Zh(qv[Z], qp[Z]) # n, so ~ , , [ l ] r n = qp[l] r n implies q,,[Z] r (n+ 1) = qp[Z] r ( n + 1).

If for every Z h(+j,[Z], ij,[Z]) < n then by ( ~ 2 ) 74,+lLf(a,),f(~,)] implies

Now define a sequence +j: if +j,[Q I n # +j,[Z] I n t h +j[Q = +j,[Q, and otherwise +j[Q = +jv[Q. Let 6 = .?(a,); notice that by the suppoai- tion above q[Z] (n + 1) = +j,[l] 1 (n + l), hence by (*1) M(Pi*) C E,,+,(6,f(aD)), hence M(8*) C lE,,+1(6,f(Bv)). Now 6 = ~( iZ6) , f ( iZ , ) = ?(a;") and for any Z+j[Z] = +j$] or h(+j[Z],+j,[Q) < n (by +j's definition)

450 THB: NUMBER OF WON-ISOMORJ?EIC MODELS [CH. WI, f 1

and ko < 12, +$41 t ko # +a1 t k, S V [ 4 1 t ko # .)?v[JaI t ko for 4 + la (by (**)). SO by (*2), 4 , , + , ( b , f ( G i Y ) ) implies TEn(6, f (a;)) . But kE,,[6, f (a,,)] A EnV(a,),f(7iv)] contradiction.

So we proved that for any v # p ~8,, for some 1 h(fjv[Z], + j , , [ Z ] ) = h(v, P I .

Choose infinite subsets w,, i < 2N0, of o, such that for i # j Iw, n wjl c KO, and let 8, c {r] E 2@: $13 = 0 when I $ w,}, be a set of cardinality A; and M, = M(8,) . Clearly IIM,)I = A. Suppose i # j and there is an elementary embedding f : M, --+ M,. By what we have proved till now there is an uncountable 8; E 8‘ such that for r] # v €8; there are r]’ # v’ E 8 j such that h(r], v ) = h(r]’, v’). NOW w = {h(r], v) : r ] # v E S ~ } is infinite (as 8; is) and w is a subset of w, (by the definition of a) and is a subset of {h(r], v ) : r ] # v E 8j} (by the above) which is a subset of wj (by the definition of wj) . Hence Iw, n wjI 2 IwI = KO, contradiction. so

I(A, T,, T) 2 IE(A, TI, T) = 2n~ .

But we have restricted ourselves to the w e 2N0 = 2” in the beginning, so we finish.

QUE8TION 1.2: Is thereaparallelfor 1.8 when 2No 2 A > ITl! > No? (Add i f n e c e s q ID(T)I > lT1l).

THEOREM 1.9:8uppoae

(*I 2wo > A > No, and there is no family of 2No aubaeta of A, euch of power A, the intersection of any two of which iajnite. (This is just rtot AD(2N0, A, A, KO) , )

If T , 2 T, T superatable but not totally transcendental A 2 lTll, them I(A, T,, T) L 2%

Remarks: (1) Baumgartner [Ba 761 proved the consistency of “ A = K, satisfies (*)” and of “ A = K, does not satisfy (*)” with ZFC.

(2) The msumption on superstability is needed only for proving the existence of the En.

Proof. Let p 5 2no be regular. As in 1.8 we can find epivalence relations E,,(%,g) (n < o) and a,, r] ~2~ such that M , kE,,(ii,,,av) iff 7 n = v n; in some model M, of T,. Assume for simplicity, l(ii,,) = 1 so a,, = a,,.

Let, for B c 2”, M ( 8 ) be the Skolem closure of {a,,: r] ~ r 9 ) . As in 1.8 choose w, c w for i c P o , w, infinite but Iw, n w,I < KO for i # j , and

CH. VIII, 8 13 INDEPENDENCE OF TYPES 45 1

let Sf E {q E 2O: $11 = 0 for I # w,}, of caxdinality A, Mt = M(Sf). So IIMiII = A, Y: a model of T,, and let M, be the L-reduct of M i .

Suppose I(A, T, , T ) < 2No; as p s 2 h is regular, for some i W =

{ j: M j M,} has cardinality 2 p, and let fj : M, j. bl, be the isomorphism (we use only its being an elementary embedding). Let, for j E W, A, = {f,(a,): q EL$}. Clearly IA,I = A, A, G lMfl where llMfll = A; the number of the A,'s is p, and for a # p A, n A8 is h i t e (otherwise this contradicts Iw, n w81 < So). If 2No is regular choose p = 2No and we get a contradiction. Otherwise combine our conclusion on all p and by 3.1 of the Appendix we get a contradiction to (*).

THEOREM 1.10: 8uppoee T s T,, ITl[ = A, and there is a family {tp,(Z): i < A} of A independent fomnulae of L(T) (i.e., for every $nite w E A and h : w j. (0, l}, T I- (33) AfEW tpf(3)h(f)). Suppose J is a subtree of 2sA (closed under initial eqm.enta) such that, letting J , = J n 2", for EL < A, IJ,I c A, and x = IJAI (for x > A , the interesting m e , this is a Kurep tree). T h

(1) For eweryp 2 A,IE(p, T,, T ) 2 min{2",2X}.Moreower i f (3u)(u < A < 21"1 A 1011<"(O) = loll), there are, for each p = p<"(O) 2 A, min{2",2X} L(T)-models, the reducts of ~(0)-compact models of T,, of cardinality p, m one elementarily embeddable in another.

(2) There are M, C T, , g,, E M l ( q E JA), zf E M , (i < A ) such that (A) For each formula P)(z~, ..., Z~~~),~,...,X~(~))EL(T~) there is

a, < A such that if v,, . . . , vn(,), ql, . . . , qn(,) E JA, p 2 a, and vl 1 fl =

71 rpbutforl # k, vl rp # vk lg, then

M1 C ~ @ v , , - * * 9 #vn(l), 21, * - * 9 zn(aJ tp&ql, * * - 9 #vncl), 21, - * - 9 zn(,)l. ( U ) {zl : i < p} is an indiscernible sequence ower {Yq : 7 E JA}. ( C ) For any v E J,, a < A, there ie tpv(Z) E {tpf(3): i < A} such that

( D ) For every term T(Z, ..., z,,, xl, ..., xm) there is a 01, < A, such that if rll, . . . , qn+, E JA, p 2 % and q, rp ( I = 1, n + 1) arepairwiee distinct, then ~(g,,, . . . , &, zl, . . . , z,) doea not realize I ) , , ~ + ~ , where

v Q 7 E JA implies Mi C V~&,J"(~).

p,, = {cpf(P)t: Ml != Vf@,It, i < A, t E (0, 1)).

( E ) For each K < A such that 2" = A choose a good ultraJlter D, ower K , and assume (Vu < A ) ( ~ U ~ < ~ ( O ) < A ) (and ~ ( 0 ) is regular). We can assume that i fS E JA, IS1 < K ( O ) , M; is the Skolem Hull of {q , , :v~S} U {q:i < K ( O ) } , V E J n - S then pv is not realized in MICs(O), where M i ( K < ~ ( 0 ) ) is increasing and continuous, M", = M,, M"! is isomorphic to (M,)"/D, ower M,.

452 THE WUWBER OF NON-ISOMORPHI0 MODELS [OH. an, 5 1

Remark. When does such a tree exist? If A is strong limit (i.e., a*) J = 2sA wi l l serve, 80 x = 2A. Clearly J c 2s* is not really ncessary, it suffices J c asB, cfg = cfh, y < g+ lJYl < A, x = IJ,I. So if of A = No, J = n,, A,,, where A,, < A = z,,<a A,, is sufficient, 80

x = A h > A.

Instead assume J is a A-Kurepa tree (i.e., a < A => I J,I < A), i t suffices to assume : there are Ji c J,(a < A), I u, < JiI < A for ,8 < A, and for every distinct ql, ..., qn E J,, {a < A: ql la, .. ., vn ra are in c} is unbounded (for part (E) we should replace n < o by K < ~ ( 0 ) ) . For x = 2A this follows from 0, (A regular), and if A = ACA it holds with x = A+.

Proof. ( 1 ) Follows by taking Skolem hulls of {tj,:qcq U { z i : i < p} for J c JA, using (2) (b) (the K(0)-compact case by (2) (E)), using 1.4.

( 2 ) Case I: There is K, K < A < 2".

Let 8 = {w(i): i < A} be an independent family of subsets of p (see 1.5 of the Appendix). For j < K let a, realize {q+(~)if~Ew(i)):i < A}, let I = {a, :j < K}, and let M , be a model of T, , a5 E (2M,(, llM,II = p and {z, :i < p} be an indiscernible sequence over U,,,it, (inM,). We define by induction on a < A, for each r] E J n 2" a filter D, over K, and a set a,, E 8 independent mod D,,, (see Definition VI, 3.1) such that 18 - 8,J < K + 11(7)1; and v 4 7 implies D, E D,, S, G S,. For Y E JA let D, be a completion of A D,, I (I to an ultrafilter. Let M , be a (x + A) + - saturated model of T,, No 4 M I , and for each r) E JAY a,, will realize Av(Di, B,,) where B,, = U {&: v < r ) (in lexicographic order) v E JA} U U5<r a, u {zt: i < p}, D,, = {{a,:j~ W): W E D,,}. It should be clear that the only nontrivial point is (and parts (C) (D) (E) are left to the reader) :

CLAIM 1 . 1 1 : Let A, K, S, I, M,, be as above, and let D,, . . . , D, be $filters over K, S, independent modD,, S, c S, and FEW,^, cpcL(T1). Then we can find fltm8 D:, and families 8: m h that

(A) D, c 0:; (B) Si C S,, IS,-Sil < K ; and for 1 = l ,n ,S t is independent

(C) There is tE{O, l } , such that { ( i ( l ) ,..., i ( n ) ) : i ( l ) , ..., i (n) < K,

mod 0: ;

MI t v[Gf,,), . . . , at,,,); ZJt} E D: x 0: x * * x 0:.

CH. VIII, Q 13 INDEPENDENCE OF TYPES 453

Prmf. We prove by induction on n.

induction on a d K , Dt,a, St,a for 1 d I < n such that For n = 1 this follows by VI, 3.3. For m = n + 1, we define by

(1) Dl E DlSB E Dl,, for B < a. (2) (3) Sl,, is independent Dl,,. (4) There is t(a) E (0, 1) such that

E Sl,p E S,, lSl-Sl,al < K for B < a.

{<i(i), ..., i (n)) :i(i), ..., < K , &?I kq[a,(,,, ..., a,(,,, 8a,f?~lf'u')

E Dl, , x * - * x D,,,.

We can do it by the induction hypothesis on n. Let D! = Dl,B (1 I I 5 n), and let Dk,Sk be such that D, C Dk,Sk E S,, (S,-Skl d K , Sk is independent modD;, and for some te{O, i}, {a < K : t(a) = t}ED;. Clearly this proves the olaim, henoe Claim I.

Case 11: A > No, not I; so h is strong limit. Here in the induction step- a, we have a set r, (a < A) of formulas of L, in the variables gfl ( r ) E J,) zf(i < w ) and set U, c A, s 1.1 + No, a model M! of T,, and sequenoes 6; ( r ) E J,, V s A - U,) and cf E My (i < A) such that:

(i) for every p)(gfln,, . . ., g,,,, z,, . . ., 2,) E Fa, and V(l), . . ., V(n) = A - U,, Mf k p)[6ri1), . . . , 6;>), o,, . . . , c,]; and M! C p)f(6fl)u(fev) for i E A - u,, v s A - u,,

(ii) {cf: i < w } is indiscernible over {E:: r ) E J,, V E A - U}. The connection between r, and r, (a < 8) is that p)(g,,,,, . . . , &, zl,. . . , z,) E r, the ql distinct, q1 Q v1 EJ, implies p)(gVl,. . . , &,, z,, . . . , z,) E r,. Again the point is the assertion parallel to claim. More exactly we want to add p)(Zfll, . , , , Zflnn, zl, . . . , z,) or its negation, extend U, by sx = 1.1 + No indices, and preserve (i) and (ii). Let N be the model (E1((2"+), E) expanded by individual constant for M? (assuming w.l.0.g. 11JfyII I 2')s ((7, $,): 7 E J,}, {(v, J', 6;): - u,}, {zf: i < A}, r. Our desire can be expressed by an (L(N)),+,,+-sentenoe, hence if it fails, it fails in some N , < N , llNlll = 2,, llNl n All = 2 5 and b s lNll A lbl s x =. b E N , . But then we can apply the claim, and find the truth value of p), and the subset of A - U, of cardinality 5 x we want, contrctdiction.

E Jh, v =

Case 111: A = No. Thie follows by 1.6.

THEOREM 1.11: Swppwe (*) from 1.9, A 2 IT,l, T stable not q e r - stable. Then I (& TI, T) 2 2Ko.

454 THE NUMBER OF NON-ISOMORPHIC MODELS [OH. WII, 6 1

Remurk. We a n omit "T stable" by 3.4 and replace 2% by 2" by 1.7(1), and replace "not superstable" by "not totally transcendental" by 1.9.

Proof. Let M = EM (AscD, @) where @ is as in 1.6(2) and w.1.o.g. we can aasume that if A = u {a,: q E A<@}, A, = UnccD a,,,, dim(p,,, A,, M') is A, where M' = EH(A<OD u {q}, @), p,, = stp(a,,,A,) for 7 E A@, and I),, is stationary, and let I,, c itl be an indiscernible set based over I),,. We can assume also that L(2') = UntcD A , ( A , finite and incredg) and

Iw, n w,l c No, and let Si be a subset of { ~ , ~ ~ 2 ~ : n # w , + r ] [ n ] = 0) of cardinality A, and let Mi = EM(A<(O u S,, 0). Suppose A < p 5; 2*0, p regular and

tp, ( ~ , , , E M ( A < ~ , = tp,,(av,v,qA<y a)) iffq rn = v rn. dhoose w, E o (i < 2N0) i # j

u = {j < 2%: N, 2 M,(,)} has cardinality 2 p, and let f, : 111, + M,(,, be the isomorphism. Let M,,,, = U,,ABa, B" increasing, 11BU11 < A . For each j~ V there is a ( j ) < A such thatf, maps A, ( q ~ 2 < " ) intoB"(-", provided that cfA > M,, hence for some /3, U' = {j E U: a(j) = /3)} has cardinality ~ p . For each j e U' and r ] ES, choose q ~ f , ( I , , ) such that tp(8, lBll) does not fork overf,(A,,). So for somej(1) # j(2) E U'

{6(1): 9 E B,,,)} n {5{(%): 9 E 8j,2)}

is infinite. But clearly for ~ E U ' , tp,,(?$B!) = tp(&Bl)iffq rn = v n, hence we get a contradiotion as in 1.8 and 1.9. So aasume cf A = X,, A = Znem A,,, A,, < A. For each n, j there is a(j, n) < A such that

l{q €8,: Ifj(In) n dla('sn)I 2 Xo}l 2 A,,.

We oan also find P(n) such that Un = { j E U: a(j, n) = /3(n)} has cardinality zp. So as before we get AD&, A, A,,, No), and aa thie holds for each n, p it is easy to prove AD(Po, A, A, X,), i.e., not (*) of 1.9.

QUE8TION 1.3: Can we in 1.4 deduce something about I E ?

EXERCISE 1.4: Prove in 1.9 and 1.11 that if 2No is a singular oardinal then IB(A, TI, T) 2 2% and if 2b is regular, x < 2w0, and in (*) there is no such family of cardinality x then IE(A, T,, T) 2 2Wo.

QUESTION 1.5; Can we prove in 1.8 that IE(A, TI, T) = 2A? (The remaining case is: T superstable, KO-unstable, 2N1 > 2'0, and ID(T)I = KO.)

QUESTION 1.6: Try to improve 1.7(2) to results on IE(p, TI, T).

EXERCISE 1.7: Suppose T has (A, p)-freedom by one of the cases 1.5 A 2 IT,\, A = 2 p = A". Then IE(A, T I , T ) = 2A.

Remark. Instead 1.5 it may suffice to assume that in Definition 1 . 1 , there is a model M , 2 A, and a P € M realizes p, {b , :i < w ) is indiscernible over A U {a, : p E S ) , and for each p, p is not realized in the Skolem Hull of A U {ag : q # p) U {b, : i < w).

EXERCISE 1.8: Rewrite the proof of 1.8 where you prove only that for any v + p E S1 for some lh(v, p) 5 h(lj,[l]), .ii,[l]) I; h(v, p) + 100, assume (*1) holds for even n's, (*2) for odd n. Is it simpler ?

EXERCI8E 1.9: Show that in Definition 1 .l, B and A can be chosen independently of the cardinality.

Our main theorem here is

THEOREM 2.1: I(A, TI, T) = 2"owidd that t h following condition holds:

(*I T is urcswperstable, A 2 ITII + K1 and at leaat one of the following me8 occurs: (i) A > I TI/; (ii) Po = A; (iii) A =

x,,, A,, A, < n, A> = A,.

Proof. By the subsequent theorems. If A is regular by 2.2 for p = A. Also if there is p < A, 2" = 2% 2.2 implies our conclusion. If there is p < A 5 pNo, 2" < 2Qhe result follows from 1.7(1). If none of the previous cases occur end there is p < A, 2Y 2 A, 2.3 gives our result. If p < A =- 2' < A, but A is singular, there is alwaya X, of A 5 x < A, x = KO or x a strong limit cardinal of cofinality KO. Noticing that Aof" Theorem 2.6(1) gives our result. We covered all possibilities, thus prove the theorem.

When (*) (from 2.1) holds there is a model No of cardinality ITII, proper for (us", TI), such that the skeleton of EM1(ws", No) is a uniform KO-tree of the form <(cpn, 1): n < w), vn E L (by VII, 3.6(2) and 3.5(2)); let cp,, = cpn(x; gn) = vn(J; 8). In Case (ii) we can aaaume

456 TEE NUMBER OF NON-ISOMORPHIC MODELS [a. VID, 8 2

N o is K,-compact; (by I, 1.7, and as we can replace N o by any elementarily equivalent model of the same cardinality). In Case (iii), by VII, 1.11 we can assume there are P, E INol, (n < w ) P, E P,+,, lNol = Un<o, P, such that N , = (No, Po, P,, . . .) satisfies: if a countable type over N, is finitely satisfiable in P,, then it is realized in P, (i.e., by a finite sequence from P,). In Cases (i) and (ii) let P, = INol, N , = (No, P o , . . .). In all cases let L, be a countable sublanguage of L(N,), such that P, E L,, and the formulas Re,cp,r EL, when 0 E L(w*”), tp E {tp,: n < o}, T E LE. We use those conventions in this section, except in 2.7.

THEOREM 2.2: Suppocre T is not auperstuble. (1) If A 2 p > lT1l, p regular then IE(A, T,, T ) 2 2’. (2) If (*) (ii) or (iii) ho& No < p 5 A, pregular, t 7 m I(A, T, , T ) 2 2”.

Proof. For every ordinal 8 < p cf 8 = No choose a (strictly) increasing sequence of ordinals 7s = $8) whose limit is 8. For every w E p let I ,

cardinal addition, so in (ii) and (iii) the third psrt disappears). Let ilfk = EM1(Iw, No) and M, be the L-reduct of Mt, (we use the notation of 1.1). By 1.3 of the Appendix there are pairwise disjoint stationary subsets of p, u, E (6 < p: cf 8 = KO} (i < p), and by 1.5 of the Appendix there are B(a) E p, (a < 2’) such that #(a) G 8(p) * a = p. Let ~ ( a ) = u,. Our family of models is {MWca,: a < 2’). To prove that it exemplifies our conclusion, it s d c e s to prove: supposing w , u c { & < p : c f 8 = No},w-uu-warestationary,andf:ilf,+ilfu is an isomorphism for (2), and for (1) an elementary embedding, we get a contradiction (“ u - w stationary ” is not needed for (1) and can be weakened to “A-w stationary” for (2)).

Clearly for every q E I , there are T,, E LE, F(q) = F, E p”” n I , and

be the KO-tl’Wp<” U {qd: 8 E W } u {(a): p 4- IT11 5 a < A} (p -I- lT1l-

I ( q ) = I, E {(a): p + IT11 s a < A}, En E No such that

Now we can define by induction on i c p, A, c INol, a ( i ) < p and X, _C { (p): p + ITl] 5 p < A} such that:

(A) X,, A, are increasing (by G ) and continuous sequence in i.

(C) a( i ) is a (strictly) increasing and continuous sequence, a(i) 2 i. (D) I f y E l , n uB<I/3Go, thenc9[Z]EXi,c9EA,, and F9[Z]~Up<a(o /3SW.

(B) 1-q < p, 141 < p, and if IT11 < tc, A, = INol.

CH. MII, 8 21 UNSUPERSTABLE THEORIES 457

(E) If r ) ~ a ( i + l)<* and f l IT11 =- % = %-<Y>*

(/3) i+ N $n<y> modX1 (in I,, so, in fact, in the ordered set

( y ) fi,,n^ N +<y> mod a(i)*".

It is easy to define a ( i ) inductively; for i + 1 notice that the demand is only that in the Skolem closure ofA,+l U { q , : v d w n ul<o(t+l)jqw} U X,+, there will be some < y elements, and as y > KO is regular, clearly such a(i+ 1) exists; for E) notice that I, is X-atomically stable for any 2.

Now clearly S = {i < p: j + + IT11 5 a < w. (6) +,,n^ = + V < Y )

Prooffor (1). (p > lT1l). Choose 6 E (w - u) n S. As 6 4 u, r), 4 I,, hence if u1 = ij,,,[Z] E p", then (as it is increasing)

there is at < 6 such that vt[n] < al e v1[n] < 6 e- n < nt where nl s w.

If v1 = ind[l] ~ p < * there is a, < 6 such that q[n] < a, ov l [n] < 6. Let a* = g(6) be < 6, 2 at and such that x1 = min{z E x d : 2 anna[l]} E X,. or x1 = a0 (exists as Xd = u{ < ,, X1). By the dehition of S there me 8', n; a* < a( i ) < 6, r ) d [ r n ] < a( i ) for rn < n but ?la[?&] > a(i + 1). Let p = v8 n, /3 = v8[n], then by (E) there are, for m < .w, distinct y(m), u* < a( i ) 5 y(m) < a(i + 1) < fl , such that ZDn<y(m)> - gDn mod Xi,

= ZDn<B) and Spn<v(m)> N ijD-<,,> mod a(i)$* and +p<ycm)) =

T ~ - < ~ > ; hence SDn<y(mn)) - 3D-<B) mod S,,, (by the definition of a*). Hence, by the indiscernibility of the skeleton M,~~,+l[f(~v~),,),f(~~-~ycm,>)l for m < w, hence Mw C g~,,+~[ii,,,; iiDA<y(m)J (as f is an elementary embedding); contradicting the definition of a uniform KO-tree of the form ((v,, 1):n < w). So we proved (1) as no sequence in the model realizes Q ) ~ + ~ ( Z ; ~ ~ - < ~ ) ) for infinitely many y's (see VII, 3.6(4)).

Proof of (2). As p 5 I T,I clearly Xi = 0. We define g as in the proof of (1). As (w-u) n SisastationarysubsetofyandforeachSinitg(8) < 6, by 1.3 of the Appendix for some a* < y , wo = { 6 : 6 ~ ( w - u ) n S, g(6) = q} is stationary. As the number of r ) E (a*)'* n I,, is <p; and if we partition a stationary set to < p parts, at leaat one is stationary (see 1.2 of the Appendix), there is a stationary w1 E w,, such that for all 8~ w1 : = T is constant (as Tv(8) E LE, LE is countable) ; ( F7(8)[Z]) r n is constant or always 4 (a*)s*; the similarity type of $,,(a) is constant;

458 T E E NUMBER OB NCYN-ISOMORPHIC MODEL8 [OH. VIII, 8 2

and there is a constant n(6) = n(*) such that ZMa, E Pno,(N,). Clearly we can find a closed unbounded 8, G 8 such that

(1) if a 7 E a<” and there is 6 E w1 such that 7 Q 7 6 then there is such 6 < a.

(2) if a E S1, 7 E a<@ and (6 E w1 : 7 Q 76) is s t a t ~ a n q then there are v E a<”, with arbitrarily large (<a) last element such that (6 < p: 7 - y Q vd} is stationary (the existence of such v’s follows from 1.3 of the Appendix).

(3) If SES, n wl,r]Ea<W,r]Q rs, then SEW,:^] Q q6} is stationary. So for each limit point S* of S,, cf S* = No, we can easily find

a strictly increasing 7 of length w with limit 6*, and S(n) < a*, 6(n) E

wl, and S(n) < P(n) < S(n + 1 ) where P(n) €8, ; such that 7 r n =

r]b(n) r n (define S(n), /3(n) simultaneously by induction on n w , using the properties ofS,). Choose also S > S*, SE wl. Then we can check that

are similar, by a*’s definition). Now choose E E P,(*)(N,) which satisfies every formula of L, all but finitely many Zv(d(n)) satisfy (a exists by N1’s definition). So clearly ~(7i~(,,,,,,,, E) realizes {rp,(E, f (8v,m)): m < w}. As in (2) f is an isomorphism, in M , {vm(E; Bqrm): m < w } is realized. Hence S*EW. As S* was arbitrary S, G w. So we finish the proof of (2) too.

?(&(,d)), Zv(d(n))) satisfies f (%$tI)): < (as Gvlmhfiv(d), FnrmhGMd(n))

THEOREM 213: Suppose (*) from 2.1, and that p 5 h is regular; h > IT11 * p > lTll; 2 Y 2 A, p > KO; and x 0 ) ( ~ [ n ] < p)} u (7: 7 E A”; n > O-r[n] < v[n+i] < p ; 7’s limit is 6 and S ~ t @ [ r ] [ O l ] } .

Let Mi = EM1(&,, No) and MG its L-reduct. Choose for each 6 < p, cf 6 = Ho a (striotly) increasing sequence of ordinals of length w, 78, whose limit is 6, and 7d[o] = 0. For a sequence 7 define q* such that 7 = (7[0])-7*. Let u,, i < p, be pairwise disjoint stationary subsets of (6 < p: cf 6 = No}, and {#(a): a < 2”) be a family of subsets of p, no one of which includes an intersection of finitely many others. (See 1.3 and 1.5 of the Appendix for existence). Let {ui: i c 2A} be a family of subsets of A, no one a subset of the other. Let a, be a sequence of length h whose range is u,: a < A, a E u,}. Our family is {ME,,,: i < 2”).

So it suffices to prove that iff is an isomorphism from ME onto Mg then for every i < h there are n < w ; j,, . . . , j, < A and closed un-

CH. n I I , 8 21 UNSUPERSTABLE THEORIES 459

bounded8 E p such that @[i] 2 @[jl] n . - . n @[j,,] n 8. For simplicity let i = 0, p > IT,\ and let f(a7) = T7(ti&(7),~7) for YE&,, (so i$ = F(7) E I,, Z,, E No) . Now define by induction on i < p, ordinals a( i ) and sets Xi such that:

(A) Xi is an increasing and continuous (by G ) sequence of subsets of h of cardinality <p.

(B) a ( i ) is a (strictly) increasing and continuous sequence of ordinals <P.

(C) If 7[0] = 0, 7 ~ 1 ~ n U,,,pSu then (F7[Z])[O]~X,, and (F#])*E B“” (this is possible as x < p * X’O < p) . We know (see 1.2 and

3 of the Appendix) that if w c p is stationary then if g is a function from w to a set of cardinality < p then for some t , {OIE w:g(a) = t} is stationary ; and if g is a function from w, g(a) EX, ;X, is an increasing continuous sequence of sets of cardinality < p then for some t, {OIE w:g(a) = t} is stationary. By successive use of this we can find a stationary w1 E 301 and a( *) < p such that :

(a) For d 6 E w1 T,,(d) == ?, Z,,(d) = c. (p) For all SE wl, v7(&) has the same similarity type. (7) For all ~ E W , for each Z,Bj = (FvO[Z])[O] is constant and EX=(*)

or always it # Xd, but min {fi E xd: f l > fij} is fixed (maybe as GO). Let

(8) Let 4 = (fiMd)[i])*. Then for some n,, vf I n , is constant and E a(*)s”, and vi[n,], if defined, is 2 8.

Now, &B in the proof of 2.2(2) we can find a closed unbounded 8 E p, such that for every 8 E 8 n Qj,] n. . n @[jn] for some 7 E pa, 7[0] = 0, 7 strictly increasing with limit S, {vn(~,f(lZvr,J) : n < w } is realized in Mu, hence 8 EW[O], what we want to prove (if p s lTll the changes are like the proof of 2.2(2) and we can have llN,II = JTJ’o).

pi: 0 n Xam be (jl, - * ’ , jn}.

DEFINITION 2.1 : For models M, N cardinal x and ordinal a we define a game GE;(M, N) [GI;(M, N)] between the players I and I1 as follows: in the @h move player I chooses i, < x and af E M i < i,, and then player I1 chooses bf E N, for i < i,. The play ends after a moves, and then player I1 wins if tp*({bf:i < ifl,/3 < a},f3,N) = tp,({uf:i < i,,

[TPP({bf: i < i,, f i < a}, 0, N ) = TPSX({af: i < i,, /3 < a}, 0, M ) ]

and player I wins otherwise. A strategy (of a player) is a sequence of functionsf, (fi < a) which “tells” him what to do (f, for the fih move) depending only on the previous choices in the play. A winning strategy

p < 4, B,M),

460 "HE NUMBER OF NON-ISOMOBPHIO MODELS [OH. WII, 8 2

is a strategy such that in any play in which the player "behaves" according to it, he wins. A player wins in the game if he has a winning strategy.

Remark. Sometimes we denote that player I chooses sequences Zf instead of elements.

LEMMA 2.4: (1) If there is an elementary embedding f of M into N ; then player I1 wins in QE,O(M, N ) .

(2) If M , N are isomorphic, pluyer I1 wins in QI;(M, N ) . ( 3 ) In the gamea from DeJinitim 2.1, at most one player wins. (4) If player I1 wins in GX$(M,N) he wins in GXI;(M, N) where

( 5 ) If X E {By I } , player I1 wins in QX;(N,, M I + , ) 1 = 1, 2 then /3 2 a, X E { I , E} .

player I1 wins in QX:(M,, Ma).

Proof. (1) In the pth move player I1 chooses be = f (af) . Also the other proofs are immediate.

To clarify the proof of 2.6, we prove first:

LEMMA 2.5: Suppose A = Ano 2 I Tl l , Mi are models of T of cardinality 5 A, for i < A. 8uppo8e T i s unswperetable. Then there is N E PC( T,, T) such thut player I Win8 in C#I&(N, Mi) for m h i c A.

Proof. Let g be a one-to-one function from {(i, a): i < A, si E lMil} onto A. Let g(g l (a ) , g,(a)) = a for a < A, and

I = A'" U {r] E A": gl(q[n]) = i for every n, and

g0(7[n1) E 1 ~ ~ 1 and {vn(s; ~ ~ ( q r n i ) ) : < w>

is (well defined and) omitted by M,}

and let N = E M ( I , N o ) (No-proper for (us", T,) and of cardinality lTll). Let us show how player I wins in GI$(N, Mi): for p = 0 player I chooses the sequence a,, r ] = qo = ( ); for p = n + 1, if in the mth move player I1 chooses 6m for 11c 5 n, player I now chooses a,,, r ] = v,+, = ( g ( i , 6O), . . . , g(i, 6")). If in such play player I1 wins, necessarily N realizesp, = {v,(z; a,,,): n < w } iff Mi realizes {v,,(s; gn): n < w}, (letting Y E A", r ] , 4 Y); then N realizes pv iff Y E I iff Mi omits {v,(z;gz(v[n])) : n < o} = {q,(z; 6"): n < o}, contradiction.

CH. VIII, Q 21 UNSUPERSTABLE THEORIES 46 1

THEOREM 2.6: S p e A 2 p = fl = 21, (VK < A ) ( K X s A) and A 2 lTll. Then:

(1) I ( A y Ti, T ) 2 A". (2) There are models Mh E PC(T1, T ) fOr h : p --+ A M h that if fop 8Orne

i hl(i) < h,(i) then in aI:(Mhly Mh1) player I wim.

Proof. Clearly (2) implies (1). By the assumptions on A we can assume No,Nl are as mentioned in 2.l(ii) or (iii) except that maybe lliV0II > IT,I (but llNoII s A). We can assume x = zncU xn, xto = xn or x = KO and let xn = n. We define by induction on a < A pairwise disjoint well- ordered sets JL (i < p), lJ6.1 = IaIx + p s A, and for a > 0 the universe of JL is the set of quadruples 8 = (a, 5, u, y ) where 5 < x and u is a subset of (J;)"" of cardinality < x and y < a, and let gl(s) = a, g, (s) = 5, g3(8) = u, g4(s) = y. For a = 0, we let J i = ((0, g, i, - 1) : 5 < x} and, to simplify notation, iet distinct Ji's have distinct empty subsets. The well ordering is arbitrary. For ~ E ( J ; ) ~ " let His(q) be the smallest set S such that ~ E S , and V E S , n < Z ( v ) , p ~ g ~ ( v [ n ] ) = s - p ~ S . Let His*(q) = {~[n]: Y E S , n < w } . Let DP(q) be the order type of T(q) = {p:His*(q) n J i # (4 for some i or gJs) = /3 for sEHis*(q)}. Clearly His*(q), His(q) have cardinality < x hence DP(q) < x+.

WeshalldefinesetsSI c xmfori < p,y < X+.Nowlet,forh:p+A,

and g d r ) ) E sr where y = DP(r))}, where we define ga(7) by ga(q)[Q = J h = &<pJk({)YandIh = uf<# ( Jk({) )c"~{r) : for~om~~ < /-br)E(Jk(t))mY

g,(r)[l]); and let Mh = E B ( I h y No) . SUppOSe hl ( i ) < h2(i) = a0 and We describe the winning Stl'abgY Of player 1 in aIj;)(Mhl, Mhl).

In the nth move player I chooses {a,,: r) E u,,} which is defined by induction as follows: let player I1 choose in the rnth move {&,: V,I E am}; and let 6,, = .fn(&m, E,,), F,, E Ih,y En E No, and let

v,,, = (J;,ct,)sa n {;,,[I]: 1 < I(;,,), E u,,,}.

Let uo = { ( (Uo , t, 0, hdi))): 5 < XO}, and

un+1= (7 -(<a09 CS,vn, h(i))) CS < Xn+1, VEUn}.

Clearly the strategy is well defined and the nth move of the player I depends only on what player I1 haa done in the previous moves. Let the history R of this play be (J {His(Sn[Q): r) E Uncm u,, I < I(; , ,)}; (Clearly lEIl 5 x) R* = U {His*(;,,[l]): r) E Uncm u,,, I < I(;,,)). So it suffices to prove we can define the Sr's so that those strategies me winning strategies for player I.

Suppose we have another such play, (where player I plays by the

strategy we mention) with hi, . . . ,Ek, . . . , H ' , instead of h,, . . . , En, . . . ,H. We call the two plays isomorphic if there are functions f : H + H' fY(y < p) with domain {a: JL n H , # 0 or a = g&) for 8 E Ha}, f*: H*+H*,gn:un+u6 such that:

(1) f, fa, g,, are one-to-one and onto H', Ha, u;, resp., (2) If 8 E Js, f(8) E J ~ , ( I ) and f, is inCreeg. (3) If s = (B, 5, u, a> E JJ then fb) = (fy(B), S,f*(u),fy(a)> where

(4) (f(s))[nl = f*(r)bI)Y f @ o ) = 4Y i = i'. ( 5 ) s0(((a0, 6,6, h,( i )>>) = ((a;, 5, B , h ; m > , and

(6) If r) E u,, f*(%$I) = q l where p = g"(r)).

CM,,), E;(o)- * - * h-'

f* (4 = {f* (0: t E 4.

gn+l(rlh<(a0, 5, V,,, hdi)))) = gn(Vr<<4Y 5 , 4 , Y(i')>>.

(7) If q(Z) EU,(~) Z s n then; denoting p(Z) = g,,,(ll(r)(Z)); EM0,^. - - (8) f* r J; : J; -+ JJYu) is order preserving ; moreover if sl, s2 E J; n

cp(,,) realize the same type in the L2-reduct of N,. n-

Domf,, then

&{&Y I{t EJ?: 81 S t < 82}1} = &{&I, [{t EJSI0:f*(81) 4 t < f*(82)}l}.

Clearly, by the dewtion of the Ih's, the winner in two isamorphic games is the same (as the La-reduct of iVl is 8,-homogeneous). It is also olear that the number of isomorphiam types of such playa is s 2x = p. Let {at: j < p} be a set of representatives. We define by induction on j, a set I', of non-oontradictory "requirements " E 8r, $ 8r, euoh that II', - I'ol s lj I + x and I',+, c'ensure"' the victory of player I in the play @.

is eventuaJly zero}. If I', is defined, let us use for at the notation in the definition of the strategy of player I. Chooae r ) E n,,<, x,, suoh that r ) does not "appeaz" in I', (there is one by cardinality wnsiderahions). Let r)o be defined by yo 1 n E

Ua, and ( . . . , g2(r)O[m]), . . . ) = r), and let p = {V,,(Z; 7in,r,,): n < o}, and

Choose, if possible, 8 t ' s so that they saw the requirement of I', and SO that q is well defined and *, in a6,, say by ~(6, 5); note 7 0 1 71 E Ihl- Then define, when y = DP(q0) I',+ 1 = I', U {q $ &} U @ E 8t: for some m < Z(3), Z(C[m]) = o, p = ga(F[m]), f l = DP(G[m]) and P[m] E Uaeh (J:)<,}. There is no contradiction tw we can aaume, in the third term of the union above, that 6 = i =s fl < y beculuse if Z(f[m]) is w, g,(fi[m]) E 8f then ( ~ [ m ] ) t Z E Uk<, uk for every Z < w [otherwise we wn make P[m] eventually zero but stil l T ( ~ , E ) will

Let To be (7 E&: r ) E xm,

q { ~ n ( ~ s &o,n): n < w}, and Y = DP(to), i = B ~ ( ~ [ o I ) *

OH. WII, 8 21 UNSUPERSTABLE THEORIES 463

r e h e q] hence by the strategy definitions, Hie*(r)) z Hb+(i@]), a. = max T(q) 4 Hk*(c[m]), so /3 < y.

So F,+, ensurea q is realiz;ed in H,,,, but p is not realized in M,,, hence player I wins. But if there are no such #$'a, let F,,, = F, u (7 E For limit 6, F, = Uje6 F,; so Fu shows us how to define the 8's properly:

s; = {?j:{T/€AS;}Eq}.

Pro$ The proof is similar to that of 2.2.

By 111, Exercise 4.13 we can find a,, E

(1) ForqEpm,vEpn, (la > 0) C 9 7 , , ~ , , ; t Z v 3 i E v 4 r ) .

(2) For q E pn, the Q,, + ,(Z; (3) For r) cpSm, tp(tZ,,, A,') does not fork over A: where A: =

U {G,,,,,: ta < Z(r))}, Al, = u (8": v # q but not q Q v}.

Let, for S < y, cfS = w , 7, be an increasing sequence of length w of ordinals with limit 6, and for w E {S < p : cfS = KO} let M , be a F&-constructible model over A, = Ck of cardinality A, where CL={cT7: ~ ] E U ( " or 7 =yd , ~ E W , &<a}. Let the construction- sequence of M , be {ar:i < i(0) < A+}. (See table in IV, 2 and IV, 3.1). Let A: = (a?: i < a}, and let E A, U A: be a finite set over which tp(a,",A, U A;) does not fork. So if (Va < a)@," c Cb, U A",, 6 < p, cf 6 = o, 6 4 w, r), € p S w , q, t n, E S S W but n, = o or r), 1 (n,+l)$Scu for Z < m ; then C = U { a v : v Q ~ z ~ n z , Z < m , or v = r ] , n, = w } is finite, and tp(Eq,Cb,) does not fork over C, where r] = (q0 , ..., T ~ - . ~ > , hence, by 111, 4.13, and as B," c Cb, U A;, also tp(cTq,Cb, U Ab,) does not fork over C. Now by IV, 3.2 if ~ E A , U Ag there is finite B c A, U Ab, such that tp(b,A, U Ab,) does not fork over B, and there is ~ E , U ( O such that B n A, c aq, %nd there is finite C as above; so tp,(6 U B, Cb, U A;) , hence tp(b, C: U A:) does not fork over C U (B n (Cb, U A;)) . So for every ~ E A , U AL tp(b,Cb, U Ab,) is F&-isolated. Ae in 2.2 it s&ces to prove that i f f : &Iw -+ Mu is an elementary

embedding, then for some closed unbounded8 s p, w n 8 E u n 8. By renaming we can aasume f maps A, u A5 onto A, u At (see IV, 3.3), hence 8, = (6: 6 < p, f maps Cd, u A: onto 0: u A:}, 8 = (6 ~ 8 ~ :

for r] ~ y ~ " and pn E L such that:

i < p are pairwiae contradiofory.

464 THE NUMBER OF NON-ISOMORPHIC MODELS [CH. nII , 8 3

6 a limit point of S,} are closed and unbounded. If 6 E w n S but 6 $ u, then 6 = .f(iZnc6J will give us a contradiction, as tp(6, C$ u Ad,) splits strongly over any CE u A4, for any a c 6 (because {ii,,-(t>: 6 I i c p} is indiscernible over Ck u A&, also {f(iZ,,-): 6 I i c p} is indiscernible over Ci u At for 6 E 8).

EXERCISE 2.1: Suppose A = lTll = (VK < ~ , ) ( K ~ O < K,) A 2 = = 2%; /3 < x+ or 2”3l I 2%; T not superstable, then I(A, T,, T ) 2 2’.

PROBLEM 2.2: Uniformize the proofs (maybe use the filters from Kueker [Kk 771 and Shelah [S 761 Section 3).

CONJECTURE 2.3: IE(A, T , , T ) = 2A for T unsuperstable, A > IT, I. EXERCISE 2.4: Show that for A regular > IT], T not superstable the partial order (P(A), G ) can be embedded into {M: M C T, llMll = A} quasi-ordered by elementary embeddability.

EXERCISE 2.5: For each cardinal A = P o , show there is no Boolean algebra M, of cardinality A, such that any other Boolean algebra M of cardinality I A has an embedding in Mo, preserving countable inter- sections. (Hint: See 2.5, Th. 12, Grossberg and Shelah [GSh 831.)

EXERCISE 2.6: Generalize VII, 3.7 and 1.7 to the cme IT,] c 2b, assuming MA (Martin Axiom).

VIII.3. Saturated models and the case A = ITl[

DEFINITION 3.1: The indexed-set (4: 8 E I) (I an index-model) is called K-skeleton-like in (a model) M when: (1) It is indiscernible. (2) For any E E M there is J c III, I JI c K such that if By 8 E I ,

B N t mod J , then for any p E L(M) M C p[@, E] E p[?if, El.

DEFINITION 3.2 : The orders I , , I , are called K-contradictory if there is no model M y nz c w , an antisymmetric formula in L(M) [ E < j’j](Z(E) = 1( j j ) = m), an order J with lower cohality 2 K and sequences iZ8 E &I for 8 E I , u I , u J (we wume for notational simplicity I , , I,, J are disjoint) such that:

OH. =, 8 31 SATITBATED MODELS 465

(1) (a*: 8 E I1 + J ) , (g8 : 8 E I , + J ) are K-skeleton like in M . (2) For 1 = 1, 2, s, t E I , + J , M t= [a, < iLt]u(*rt).

LEMMA 3.1: (1) 8wppose a,, M are d e b of T,, K is regular 1M1 =

M,. If the type each a, realizes over lMol U {a,: j < i} ie F~-isolaterl, or even F~-isolated, then (a8: s E I ) is ~-8keketOn like in M .

(2) The skeleton of EM1(I , @), or of EM1(I , N ) are ~-8kehdt-m like, for every K 2 8,.

(3) If (a8: s E I ) is the skeleton of M = EM1(I , N ) , M , t k subntodel of MAID (D ultraJilter over A) whose universe is {h/D: for 8ome J E I , IJ( < K , {h( i ) : i < A} E dcl{gS:sEJ)} (clearly it exists), then ( g S : s € I ) is K-skeleton like in M , and M , < MA/D(dcl in L,).

l d l o l U {a,: i < a}, 4 E lM,l for 8 E I and (a8: 8 E I ) i8 ~-8kehton like in

Proof (1) Prove for CE POI U {a, : i < p} by induction on 8. (2) and (3) immediate.

THEOREM 3.2: If A 2 lTll + K + , A<n = A, T umtable, K regular then there are M , E PC(T,, T ) for i < 2A such that M, h cardinality A and

(1) For i # j, M,, M , are not i8omorphic. (2) M , ie K-wm‘paCt, and ~-hmmgeneoue and if ID(T)I 5 A, M , ie

(3) M , is the L-reduct of M t , M i ia ~-c~ml ) (cc t , ~ - h m m g e n ~ u ~ ; and i f K-8dUrded.

ID(T1)I I A, &O K - 8 d W ’ d d .

DEFINITION 3.3: (*)! means that there is a family of 2” subsets of A, each of mdinality A, the intersection of any two is < K.

Remark. This is just not AD(2”, A, A, K ) (see Definition V I I , 1.11). A related property appears in 1.9. On independence results concerning it see Baumgwtner [Ba 761.

Proof. Let 0. be proper for (w, T,), such that the antisymmetric formula, [Z < 83 E L orders the skeleton of EM(w, @). We assume the assumptions from VII, Section 2.

Case I : not (*)!. By 3.3 of the Appendix there axe 2A pairwise K-contradictory orders I,, a < 2“; each of cardinality A. Let, for a < 2A, i 5 A If, be an order isomorphic to the converse of I,, and sf, E I,. Let, for a < 2”, J , = z,sAIf,. Let N: = EM1(J,, @), SO llNiII = A (T, has

Skolem functions). Clearly (as: 8 E J,) is the skeleton of N;. Let N i < M i , M i satisfy (2) and (3) from the theorem, but such that (as: 8 E J,) is still K-skeleton like (this can be done by 3.1 and VII, 4.4), and h = IIMi(I (by cardinality considerations this is possible). Suppose the number of non-isomorphic M,'s is < 2A and p 5 2A is regular, where Ma is the L-reduct of M i . Then for some M S = {a: Ma z M) has cardinality 2 p and let fa : Ma + M be such an isomorphism. Let & = f,(a8) for s E J,, a E S. So (g : s E J, ) is K-skeleton like in M. Let W, = {@: 8 = sf , i < A}. Suppose a # p; a , p E S , [ W a n W,l 2 K; then, as h is well ordered, there are for < K i (6) < A, j ( f ) < h such that 8&) = 6tCe) where a(& = @), t ( f ) = g e ) , and i ( f ) , j ( f ) are strictly increasing.

h t &I) = SUp{i(f): f < K} 5 h, 8(2) = SUp{j(&: 5' < K } I h. NOW clearly

are r-skeleton like in M. Looking at Definition 3.2 this clearly proves I,, I, are not K-contradictory ; contradiction.

So a # /3~8* IW,n W,I < K ; also IW,l = A, and all those W, me subsets of ]MIrn for suitable m, which has cardinality A. If 2A is regular, choose p = 2", so we have proved (*)f, hence finished, otherwise use 3.1 of the Appendix to get the same contraxiiction.

Owe 11: h = p+, hand p me regular. Let I, be the set {(i,j, y) : i < A,

j, > j2, or i , = i,,j, = jz, y1 < y2. We identify (i,j, 0) with ( i , j ) . For every I c I, let M,(I) = EM1(I, @), M,(I) = Mo(I)A/D where D is a regular, good ultrafilter over A (exists by VI, 3.1). Let d,((i,j, y ) ) = {i,j,y},andforJ c I , ,d , (J) = (JseJd,(s).ForeveryI G I 0 , I t / D ~ M 1 ( I ) let dl(h) = {s E I: for some i < A, h(i) = T(@, 8 minimal, and 8 = t[Z] for some 11, (note w.1.o.g. Cis uniquely chosen). Now for I G I,, let M2(I) be the submodel ofM,(I) whose universe is {h /D : Jd,(h)J < y}. By 3.1(3) M,(I) < Ml(I) m d (a,: 8 € 1 ) is p-skeleton-like in Ma(I) . It is also Clem that M,(I) is K-saturated because if J c I, IJI < p then M 2 ( J ) = M , ( J ) < M 2 ( I ) is even y-saturated (by VI, 2.3 and VI, 2.11). In the same way we can prove that for such J, if p is an m-type over M2(I ) finitely satisfiable in M 2 ( J ) , Ipl d h then p is realized

j < A, y < p} ordered by: (il, jl, yl) < ( i s , j,, 72) iff il < ia OT il = i a ,

in N,(J). Let 93 = (R(k), IS), and define 93, for a 5 p such that:

OH. VIII, 8 31 SATURATED MODELS 467

(1) p a l l = A, %<9. (2) If a E 193,l and a has cardinality 5 A (in R(k)) , then b E a; e-

b E I%l* (3) @ n : B (4) L, I,, D, Mo(Io) , J H M l ( J ) , (a8: 8 E I,) all belong to 9,; and

I E 9 1 .

(6 ) In fact, 93, depends on I so when confusion may arise we write 8, = S,(I), but sSo(I) = SO(I0).

Let for a 5 p, M i ( I ) be Ma(I) as interpreted in 3,. Clearly Ni(1) < M a V ) and JfiV) = U;<a J f h ( 0

Let N;(I) be the L-reduct of Mg(I). Notice that when cf 6 2 K, Ng(I) satisfies conditions (2) and (3) from the theorem.

For w E A let I (w) = {(i, j , y ) E I,: i $20 * j < p} . Suppose wl, w2 c (6: 6 < A, of 6 = p}, w = w1 - w, is stationary and there is an isomorphism f from N t onto N{ where Nf = N;(I(w,)) (for I = 1,2) . We shall get a contradiction, and this is sufficient t o prove the theorem in this caae (as in the proof of 2.2).

Let 68 = f(a8). We can find a stationary w' E w, and 5 < p such that for a E w', 6(,,,) E% and U, = {i < A : t?(a,t) E@} is a stationary subset of A. Let, for a E Ml(I0) , d2(a) be a set J c I of minimal cardinality such that d,(lr) c J where a = h/D (we can aasume the function

I%+il, I%l = U i < d 19d.

d , belong to So) . Also, for E Ml(IO), da(a) = Ul da(a[Q). SO for each E dla(Io), Id&)] < p. Let d&) = (a < A: for some 8, (a, 8) E &(a)

or (8, a) E da(a)}. SO clearly, for a E Ma(Io), d,(@ is a subset of A of cardinality <p. Hence by a double use of 1.8 of the Appendix, there are w", U&, f o and V,, V such that:

(1) V c A, I VI < p; w' E w', U: c U,; and w', U; are stationary, for aEw" and V, c A , IV,l < p.

(2) Let? @ E w", a < @. (i) d3(b<a,O>) n da(6<~,o>) E V , V a n Vn E V , and V E VaY Van

(ii) if i E dg(6<,,,>), i 4 V then i 2 a, (iii) if i ~'d3(6<, ,0>) , then i < 8. (3) L e t a E w " , i , j E U ~ , i < j.

p = V ,

(i) &3(6<rr.i>) n d3@<,,,>) = Va, (5) if f E d3(5<a,t>), f 4 V,, then t 2 i, (iii) if f E d3(6<,,;>), then i, f < j. (4) If i E U;, then i > sup V , and i > a + a, and a > p. ( 5 ) There are sets B , E ~ $ + ~ (a < A ) , B,GN2, IB,l < A ; P E W '

8 < a =- 6<,,,, E B,, and B, = Used B, (this ody because 9, ~93,+1).

468 THE: NUMBER OF WON-ISOMORPHIC MODEL8 [OH. WII, 5 3

(6) a E w", c E B, =- d3(c) c a.

Cliooee 6 E w" which is an accumulation point of w", (so bigger than lo, and than p). So 6 E wl, 6 # wa, cf(6) = p and (6, i) E I(w1) o i < A

Define J = J, u ((6, lo + n): 0 < n < w} u {(a, 6, lo + n): n < w,

a E V , a E wa}. Clearly I JI < p, J E I (W2) .

(7) l o > SUP(V n p), l o < p.

but (6, i ) E I (Wa) o i < p. Let J o = {(i,j, y): (i,j, y ) E 1 0 ; i,j, 7 E V } .

Let (for a < 8 )

A = {a: U € N 5 , , d 2 ( U ) E { (a , i , y ) :a = s*i <&, y < &}},

A,= {a E A : d&) c a v ( A - a)}, p = {'p($; E): E E A , 'p E L, and for all sufficiently large a < 6,

'% ' d b < K , O > ; '11, PO'p I A K '

As f is an isomorphism, (b8: 8 E I (w, ) ) is p-skeleton-like in ivg, so p is a complete type over A. We now show that p is realized in N2(I(w2)), andeveninM,(J).As IJI < p, IpI s h,its~cestoprovepiehitely satisfiable in H,(J); so as p = u a < d pa, pa inoreasing it suffices to prove any finite q s pa is realized in it. By the choice of 6, p there is B E W " , a < f l < 6, B > p such that 6<8,,) realizes q and let it be (ho/D, . . . , hn.,-JD) where dl(hl) s cEa(6<<B,o)) for Z < n. NOW define hb,. . ., hkql such that: If h(i) = T(@) (minimal 8) then x'(i) = .t(iZj) and t satisfies the

following (here x( i ) = (hl( i ) : 1 < n)): (a) t - 8,

( y ) if 8[m] E J,, then t[m] = arm], (6) if B[m] = <i, j, y>, i > 8, then i[m] E ((6, to + k): 0 < k < a}, (8) if 9[m] = (i, j, y>, i < B (so i E V), 9[m] # J, (so /3 s j < 6) then

t[m] ~ { ( i , 6, lo + k): k < w}.

Clearly this can be done, and 3 - i mod {(i,j, y): i = 6 j < to, and i < 6 * i s a} hence if 6 = (hb/D, . . . , hk-JD) then 6 O realizes q, and by the above 6 c M 2 ( J ) .

SO by previous remarks there is 6 ~ M , ( l ( w ~ ) ) realizing p. As (6#: 8 E I(wl)) is p-skeleton-like in N;, l B d l < h, there is &1) < h such that all 6<d,') ([(I) 5 i < h) redhe the same tSpe (in N$) over &. h t

(B) Z E J ,

q = {E < z: E E B d , N& k E < 6<d,E(l)))

{z < E: A , tp(E, Bd, N!l) = t P ( ~ < ~ , ~ ( l ) > , B d , %)}*

CH. VIII, 31 SATURATED MODELS 469

It is easy to check that q E p (by (6 ) ) so 6 realizes q, and q E 8, + , . So, as 99 C "in M2(I(wa)) there is an element realizing p"

also CZJ,+, satisfies this, so some 6' E Nb+1 realizes q, so (&: 8 E I(w,)) is not p-skeleton-like (as U; is an unbounded subset of A, a d i E U; =- 6<d,i> E A). Contradiction. Cme 111: A = A, is a limit cardinal, or A = A,+ , A, singular; and for some p < A,, 2' = 2A. We can choose regular p < A, p > K, p+ # of A,, 2'= 2A. The proof is similar to the previous case. W.1.o.g. h =

Define I,, M,(I) , M , ( I ) as in Case 11. Let, for J E 10 p r l ( J ) = {a: for some 8, (a, 8) E J } and M,(1) will be the submodel of M l ( l ) with universe {h/D: pr,[d,(h)] has cardinality e p , d,(h) has cardinality sp}. 23,, M;(I) , are defined aa in case 11. For any function g : p+ + A, such that each g ( i ) is a regular cardinal > p+ , < Al, let

lEM(P, @)I.

I(g) = {(i,j) E l o :j < g(i), i < p'}.

Now we suppose g,, 9, are such functions, and for every x .c A, {i < p+: cf i = p, gl(i) > g , ( i ) > x} is stationary, and f is an isomorphism from iVt onto N t , and get a contradiction, and this is sufficient [where iVF = N;(I(gI))].

Easily we can find 5 < p, B E B,, p+ < IBI < A, and S = {a < p+: 6<,,,> E B, cf a = p}

is stationary, and let i3 = pa: a < p+} where pa is increasing (so 8, 2 a). Let w' = (6 < p+ : cf 6 = p, gl(S) > ga(6) > IBI} (so w' is stationary) and let w", V be such that (exists by 1.8 of the Appendix):

(1) w" is stationary, w" E w'.

(3) If a E w", i ~ p r ~ [ d ~ ( 6 , ~ , , , ~ ) ] , i 4 V , then i 2 a. (4) If a 6 w", i E V , then i < a. (5) Let B, = B n {c: c EN;, p l [ d 2 ( c ) ] E a} so B, E 23,+,, B, is in-

( 6 ) If 6 E w", a < 6, then 6<8a,o> E B,. Choose 6 E w", 6 an accumulation point of w" (so cf(6) = p, gl(S) >

ga(6) > IBI). Let J, = {(i,j): (i,j) €IO, i E V; and for some a E w", a < 6, (i,j) ~d,(6<~,,,>)}. Now there are 6, c gJ6) and U E gl(S),

Let J = Jo u ((6, (, + n): n < w}. Clearly IJI I p, Iprl(J)I H,). Let A = {a: u EN$, d,(u) n ((6, i): So 5 i < ga(6)} = 0}.

470 THE NUMBER OF NON-ISOMORPEIU YODEL8 [OH. m, 8 3

A,, p , pa are defined as in Case 11, and so is the existence of 6 M2(I(g2)) which realizes p (only part (e) falls). The resi is like Caee II- we prove q is realized in N$ and so get the contradiction.

So it suffices to prove that Cases 1-111, exhaust all possibilities. Suppose not (*)R, and it suffices to prove that for some regular p,

p+ I A, 2y = 2A. By 3.2(1) of the Appendix AK = 2A, so if A I 2K, 2'5 = 2A so p = K will suffice. If A > 2=, by 3.2(2) of the Appendix for some X , K + < x+ < h , ~ " 2 h so x" = 2A) and 2X = 2A. So we finish.

COROLLARY 3.3: If A 2 I T,I + N,, T uwtable, then I(A, T , , T) = 2A.

Proof. Take K = No in 3.1.

COROLLARY 3.4: If A 2 lTll + N,, T not Bulperatuble, then I(A, T ) = 2 A .

Proof. By 3.3, 2.1, 2.7 and 1.7(1).

THEOREM 3.6: 8 m e A 2 p 2 lTll + K + , Aca = A, K < K(T), K , p areregular;andx < p=+x<IC < p.8upposealsotlurtp > lTll&x ITJ, then no M, is elementarily embeddable in M, for i # j . (3) A¶, ia K - c o T ~ ~ & , ~-liomogeneous, and, i f lD(T)l < p, ~-edureted.

8imilarly for w. Proqf. For simplidty we aasume p > IT,!. Using VII, 3.6(3) and 3.6 we can find @ proper for (us", T,) such that the skeleton of EH(osa , 0) has the form ((cp,, 1): a < K a successor). For 8 < p, cf 8 = K choose qs e p K increasing with limit 8, and I, = psa U {qd: S E w}, and let Nt, = EM1(I,, a), Hr a IFb,-primq model over NL with constructing sequence {ur: i < A}, eatiafying (3). (For proving (2) when A s 2ITd we ehouldaeaumethatiffhetypeofaroverBp 3: { a v : r ) ~ I , , , } u { u ~ ; j < i } is rAv(DfO, dd Bfo) ,

Let Hw be the L - d u c t of i& f : MW + H,, an isomorphism (for 2) elementary embedding). For (2) let p = A and the proof is aa in 2.2. For (1) there is V c A, I Vl = p such that f maps {itn: r) E I,] u {ap: i E p) onto {a,,: q E I,,} u {ur: i E V).

Let V = {i,: a < p}, Then there is a closed unbounded 8 E p such

< K, then I{p: B f o = B}I < A.)

OH. Vm, 8 41 CATEQOBICITY, SATURATION AND HOMOQENEITY 471

that for a E S f maps {a,,: r) E I,, r) E UBeK 8'") u {a?: i = i, E V , 8 < a} onto {a,,: 7 E I,, r ) E Ubcr p'"} U {a;: i = i, E V , 8 < a}. The rest is like 2.7.

THEOREM 3.6: 8-e A 2 p + IT1 + K + ; A<' = A, K < ~ ( 2 ' ) ; K , p are regdur, x < p =- x c u < p. Then T has 2' n o n - k m p h i c models of cardinality A, which are ~-lumtogeneowr, K - C O ~ Z ) ( C C ~ , and i f ID(T)I s A also K-saturated.

Proof. Similar to 3.6 and 2.7.

CONJECTURE 3.1: (1) If A = A"K 2 lTll + K + , K < K(T) , K is regular, then there are 2" non-isomorphic models M E PC(T,, T ) of cardinality h satisfying (2) and (3) from Theorem 3.2.

(2) Moreover, if h > I T,I, no one of them is elementarily embedded in any other.

EXERCISE 3.2: Prove conjecture 3.1(1) for A strong limit cardinal > D ~ . (Hint: look at 2.6.)

EXERCISE 3.3: Prove conjecture 3.1(1) when there is p, pcK = p < A, 2' = 2".

EXERCISE 3.4: Prove conjecture 3.1(1) when K < K ~ ~ ( T ) and A ie regular, or p < A s p", 2fi < !P.

PROBLEM 3.6: Axiomatize the proofs in Sections 2 and 3 (to include a180 1.7(1)).

Remark. Clearly in 2.6, 6 and 1.4, 7 we have less than in, e.g., 2.2. Of course, it will be d d a b l e to use fewer axioms in each proof.

VIII.4. Categoricity, saturation and homogeneity up to a cardinslity

THEOREM 4.1: Suppose PC(T,, T ) i s categorical in A. (1) If A 2 lTll + K,, then T k8tabkandif A > 2No lrae nd the f.c.p. (2) If A > lTll then T k eulperstubZe without the f.c.p., and dabk in

every p 2 lT1l.

Proof. (1) T is stable by 3.3; hence hae not the f.c.p. by VII, 3.4. (2) T is superstable by 2.1. By VII, 2.7 there is M E PC(T,, T) of

cardinality A, stable in lT1l. If T is not stable in ITII, there is M E

PC(T,, T ) unstable in ITl!, 11M11 = A, contradicting the categoricity. So T should be stable in I T , I, hence by 111,O. 1 stable in every p 2 I T, I . By VII, 3.4, T is without the f.c.p.

THEOREM 4.2: (1) If T is not mperstable, then for every A > lTll there is a nvn-I T,I +-model--- model in PC(T,, T ) of c a r d ~ d d y A ( 8 0 i t k n o t I T , I + - l u r m o g e n e o u e , ~ - ~ ~ ~ ~ ) . SM below.

(2) If T i8nOteulpW8tabk, IT11 < p 5 A,p*@rY (VX < p)(xWO 5 A), then there are rnodels Mt of T , of cardinality A, for i < 2y

the L-reduct of Mi,

cardinality < p doea not t-kpnd on i.

thd (i) for i # j, the L-redwt of 211: cannot be elernentady ernbedded into

(ii) the set of ieommphhni typerr of elementary submodels of Mi of

Remark. M is A-model-homogeneous if M, < M y lldl, 11 < A, f : dl, Ma, Ml < N , < My llNIII < A, then for some f ' extending f, and N,, M2 < Na 4 N , f ' : N , Na.

Proof. (1) We use the terminology of 2.2. Dehe a(i) for i < I!l',l+ as follows: a(0) = 0, a(i + 1) = a(i) + i , a(&) = Uleaa(i). We SSY u = /I + 9 i f u [ i ] = /I + q [ i ] for any i < l(u) = l(q). Let

I , = A<" u {a(in-l)+ ... +a(i,)+7ja: s < a(i,) < ... < a(in-l) < ITII+},

I , = IT,J<" U(9a: 8 < p l l + } Y Ml = EM1(PY No).

As a(i) is an inrreaaing and continuous function (6 < I T,J + : there is 9 E Io, r ) E a', sup,, ~ [ n ] = 6) is not stationary (each a( i ) does not belong to it). As in 2.2 this implies that M1 1 L cannot be elementarily embedded

into Mo L. Let, for 7 < lT1l+, I : = I 1 n y s m , 131: = EM1(I i ,No) . Clearly every elementary submodel of M1 of cardinality 5 I T,I is an elementary submodel of some Mi, but Mi a n be elementarily em-

those two facts imply No

YEP for some p < a(i)}. The rest is like 2.2 and 4.2(1).

bedded into Mo, by the embedding induced by a,, H t5crtr+l)+n. Easily, L is not I T1 I +-model homogeneous.

(2) For w c p let I, = A'" U {vd: 8 E w} u (I. + a( i ) + 7: i < p,

CH. n I I , $41 CATEGORICITY, SATURATION AND HOMOGENEITY 473

THEOREM 4.3: The following assertions on the cardinals, p, A, x, where A > x, p > x, are equivalent:

( 1 ) If ILI I x, L(T) E L, p a 1-type in L, P a one place predicate in L, and T haa a nzodel M omitting p , 11M(1 = A > I P(M)I, then T hae a

(2) If T c TI, (TJ < x, PC(T,, T ) is categorical in p, then PC(T,, T ) is categorical in A:-

(3) If T c T,, ITl! 5 x, and every model in PC(T,, T ) of cardinality p is h g e n e o u s , then every model in PC(T,, T ) of cardinality A is hornogeneowr.

(4) If T E T,, ITl] I; x, p a 1-type, IpI s x and every model of T , of cardinality p omitting p has homogeneous L-reduct then every model of T , omitting p of cardinality p haa hmnogeneozls L-reduct.

mo&el N omittingp, llNll = p > IP(N)I.

CONCLUSION 4.4: If A = ad, ( V C ~ < S)(a + S(x ) s S), A, p > x then (1)-(4) from 4.2 h7rolds.

Proof of 4.4: By V I I , 6.3, ( 1 ) holds in this w e .

Proof of 4.3. (1 ) =+ (4) Suppose M1 is a model of T,, omitting p, of cardinality A, with a non-homogeneous L-reduct M. So there are K < A, a, E M (i K) b, E N (i < K) such that

tp*((b,: i < K), M) = tp*((a,: i < K), M)

but for no b E M

tp*((b,: i < K ) ^ ( b ) , Y) = tp*((ai: i I; K), M).

Let M* = ( M , P, P, c) where P = {a,: i < K} (one place relation), P(a,) = b, for i < K and P(a) = c for a # P, c = a,. So by ( 1 ) there is a model N* elementarily equivalent to M*, ((N*I( = p > (P(N*)( and omitting the types p and

Y = { (vY~,.. - - Y J [ A P(Y:) -+

i

(because omitting p and q is equivalent to omitting r = {~(z) v #(z): ~ ( z ) €21, #(x) E q}). Clearly (by q) N is not homogeneous. So we prove (4).

(4) * (3) Immediate.

474 THlD NUMB= OF AON-ISOMORPHI0 MODlDLS [UE. mI, 5 4

(1) 3 (2) Suppose PC(T,, T) is categorical in p. By 4.1 T is stable in every X 2 IT1[, hence in p , hence by 111, 3.12 T haa a saturated model in p, so by I, 1.13 PC(T,, T) has a saturated model in p, hence every model in PC(T,, T) of cardinality p is saturated. Suppose M1 is a model of T, of cardinality A and its L-reduct H is not saturated. So there me A E lMl, andp €&'(A, M); p is omitted by H, IAI c A. For every 'p(z; 5) E L let R, = {a: 'p(z; 7i) ~ p } , and P = A; let M* = (M, P, . . . , R,, . . .),EL. By (1) there is N* elementarily equivalent to M*; IN*] = p > IP(N*)I and it omits

Q = W5W,(B) -+ 9J@, 0)l: 'p E L}.

Clearly N* L E PC( T,, T) and it is not saturated as it omits p' =

{'p(z, a): N* t R,[7i], 'p EL, E IN*I} (p' is consistent as N*, H* a m elementarily equivalent). Contradiction. So every ME PC(T,, T) of power A is saturated. Hence (by I, 1.11) PC(T,, T) is categorical in A.

not (1) 3 not (2), not (3). Suppose L, T, p, P form a counterexample to (1) and M is a model of T omittingp, 11M)1 = A > IP(M)I, and let I, = {vi(z): i -c x}. Let P be a one-to-one function from 1M1 x onto M - P ( M ) , with converses P,, Pa [i.e., for a, b E IMI, c E lM1 - P ( M ) , P,(P(a, 6 ) ) = a, P,(P(a, 6 ) ) = b, P(Pl(c), Pp(c)) = c]. Let Qi =

{a E

Th(M*), L* = {Q: i < x}, T* = Tf n L*. Clearly eat& Q, hee cardinality A or is empty, and 13.21 - Ui<,Qi = P(M). Be IP(Mf)l c A = IMf I, the L-duct of Mf is not saturated nor homogeneous. (If P ( M ) = {hi: i < K), let at = a, = b,, and then we m o t find 8 euitable b,.)

Suppose now Nf is a model of Tf of cardinality p. Being a model of Tf, each Q,(Nf) is empty or has cmdimlity IINf 11 (use P, Pl, 3'2). By aesumption Nf realizes p or IP(Nf)l = A. In eaoh 0&88 A = Nf - Ui<,Qi(Nf) has oardinality A (in the first ( ~ ~ 8 8 , if a realizee p , as {P(a, c) : c E Nf} E A; and the second ( ~ ~ 8 8 as P(Nf) E A). As Tf has elimination of quanti6ers, it is easy to check that the L-reduct of Nf is eaturated, hence homogeneous. So T?, T* provides the countmexample for (2) and (3).

: a 4 P ( M ) and M I= 'p,[P,(a)] for j c i, but M I= 7v,[Pl(a)]}. Let Mf = (M, J', FI, FS,. . .,Qi,. . . ) {ex and Lf = L(M*), Tf

Be we prove (1) * (4) * (3) * 1, (1) (2) * (1) we hi&.

THEOREM 4.6: T?MM i8 Q 4 1 Ml of Tl of CMClinaljty p which is ( < A)-univemd, but its L-reduct i8 not A-univereat, if:

( 1 ) T unsuperstable, 2IT1l < A, 2<A < p ; and A is regular or An, = A ; or

Ca. WII, 8 41 OATEOORICJITY, SATUEATION AND HOMOUENEITY 475

(2) T = T , is stable but not superstable, IT1 Q A , 2cA Q p ; and A is regular or hNo = A.

Proof. Our model M , will be the model constructed in VII, 4.6. Clearly it ie ( < A)-universal. We should show only that its L-redud,

h f o is not A-universal. (1) If A is regular, the proof is similes to that of 2.2. If P o = A,

o h m @, cpn aa in the proof of 2.2, and let No = EM(Asm, @), so N is a model of T of oesdinalify A, henoe it s d m s to show No is not elementarily embedded into M0. Suppose f is suoh an embedding, and 8, = f ( i i , ) for q E As@. W e mn 6nd finite w, c p suoh that v Q r )

w, c w,, We now define by induotion q,, = q(n) E An, suoh that r),, Q qn+l and

- b, = q $ . o , - * * Y Z,,k(,)), 4 . 1 E J f & , I ) , w, = {i(r), 0: 1 < k(7)).

(7 E An+': 7, = TMn+l), k(q) = k(~n+d;

01, * - * > ( ih+l , 0)s * * * > mod WMn);

i&, ZM,,+l),l realize the same type over M1, moreover if

a =I u {z,,~,,: i c ny j 5 k(7 1 i), i(7 1 i, j) = ~ [ n l } ,

then 7i"Z,,1, anZM,,+l),l rea.lizee the same type over lM1l}

haa oardidity 2 No. (Of o o m , 2, # 1, * i(q,Z1) # i(q, la)).

Let 7 E A@, q,, 4 q for every n. Then {v,,(Z, 8,,,,): n < w} is realized by b,, but oannot be realized in Mo by the dehition of q, oontradiction. (2) A similar proof.

LEMMA 4.6: (1) There are complete countable t h r i e a T I z T , T un- etable, 8uch that: if M E PC(Tl, T ) i8 ( < A)-univered, A etrong limit, cf A = KO then hf ie A-univered.

( 2 ) There are complete countable theories T , 1 T , T etable but not ezclper8tableY 8uch that: i f bl E PC(Tl, T ) iS ( c A)-universa2, A etrong Jim$, cfA = KO then M is h-universal.

Proof. (1) Let T be the theory of the rational order, and T 1 eays that any two intervals are ieomorphio. We leave the proof to the reader.

(2) Define the relation E,, over w@: 71,,v o 9 r n = v 1 n. Let M = (wcu, 1 0 , 4,. . ., 4,. . .). Let P,,(v, p) = ( ~ [ o ] , . . ., v[n - I), p[n], p[n + 11,. . .), and MI = (M, Po, P,, . . .). Let T = Th(M), T , = Th(M,). We leave the proof to the reader.

476 THE NUMBER OF NON-ISOMORPHIC MODELS [CH. WII, $ 4

THEOREM 4.7: The folknning cunditiom on A, T are equivalent: (1) A = A < A + ID(T)I or T is stable in A. (2) T laas a saturated nzodel of cardinality A. (3) If T, z T, lTll 5 A t h n there is a saturated bl E PC(T,, T) of

cardinality A.

Proof. (1) * (2) If A = A < A + ID(T)I, the proof of I , 1.7 shows that (2) holds. If T is stable in A 111, 3.12 shows that (2) holds.

(3) + (2) This is immediate. (2) + (3) This is I , 1.13 (in its proof we use (2) * (1) only, so no

vicious circle arises). (2) (1) Let M be the saturated model of T of cardinality A. So

clearly A = llblll 2 ID(T)I, and if IAI < A, I,S(A)I 5 A (as A2 is universal w.1.o.g. A E My now bl is saturated). Now we prove by cams.

Case I : T unstable, A < 2<”. By 11, 2.2(6) there are Q, and 7 E A>2 such that for each r ) E A2 {~ (z , 7i,r,),[al: a < A} is consistent. Now we define 6, E bl by induction on 1(7), such that if q0 4 7, 4 - - - 4 7, then G,(0)hGMl)h- - - a,(,,), and 6,(0)hb,(l~n- . --6,(,,) realize the same type. (This is possible as the only requirement on 6, is that it realizes a certain type over U {6,: v Q r)}. The type is consistent by the induction hypothesis, and is realized aa M is A-saturated). So for every 7 E ”’2, some c, E M realizes &, = {I&, ii,ra),caJ: a < l(7)). If l(7) = l(v) then c, # c, [for if a = min{a: y[a] # v[a ] } , p = 7 a = Y a, then ~(c , , G,,) = ~ c p ( c V , ~ J ] . So for every a < A, h = llMll I ( c , : 7 ~ 2 ” } = 127, hence A 2 2<”, contradiction.

- n-

Case 11: T unstable A 2 2<A, A regular. It is etley to check A = A<”, so we finish.

Case 111: T unstable, A 2 2<A, A singular. It is easy to check that p < A - 2” < h (otherwise 2~+cPA > A , ,u + cf h < A ) . So it suffices to prove (as a saturated model is universal by I, 1.9(3)).

LEMMA 4.8: 8uwo8e A 2 I T,I, A 8trong limit cardinal and cf A c K ( T ) . Then thre i8 H E P C ( T , , T), lldlll = A, mch that no elementary ez- temion N Of bl O f Cardinality is (Of h)+-&UTded.

Proof of Lmmu 4.8. Let K = of A. We can define suitable M and ii, E M for q E x > A y such that for every 7 E “A, q, = {Q@, Gv)u(n9v): v E x>A} is

CH. VIII, 8 41 CATEGORICITY, SATURATION AND HOMOGENEITY 477

consistent (by 111, 7.6, 7 and the definition of the independence “property”). Suppose M < N, llNll = A, so let IN1 = U f e e A f , !A,! < A, hence I.S(Af)I s 2141 c A. Now defme by induction r),, u, for a s K such that a c and tP(av(a+1), A,) = tp(liv(a+l), A,), hence no element of A, realizes

a < K } is cqvcrc1, hence consistent, but is not realized in N &B IN1 = (JaerC A,. So N is not K+-saturated.

=- r)“ Q 78, l(v,) = a, Z(r),) = a, r), Q

F J h %)b+l)) A T ( Z , %+l)). Clearly Q = M . 9 av(a+l))9 lQ)(Z,

Case IV: T stable, A<K(T) = A. So A = + ID(T)I. If T is stable in A we finish, otherwiae, by 111,5.16 it follows that A c 2n0, and for some countable A, IS(A)I 2 2n0, contradiction.

Case V: T stable, A<c(T) > A. It suffices to prove the following lemma.

LEMMA4.9:SuppoaeA 1 lTll,Ax > AandK < K ( T ) < o o o r T h m t h e atrict order property or K < KrOdt(T). Then there is M E PC(T,, T ) , 11M11 = A auch that ru) elementury extension N of M of cardinality A ia K + -Saturated.

Remrk. In fact if K < K ( T ) < 00 or T has the strict order property, then K c mdt(T).

Proof of Lemma 4.9. We can aasume K = minb: A# > A}, hence A = A<’. So there is H E PC(T1, T ) , tp, E L, and a,, E bl, for 7 E ‘“A such that:

(1) {tpa(it; av)il(v*V): a < K , r ) E “A, a successor} ie consistent, for every v E “A.

(2) If r ) E “A, i c j < A {~.I,+,(?z; i~ , - (~>): 5 E (4, j}} is inconeistent. (This is by 111, 7.7, 111, 7.6(1) and (5) . )

Suppose M i N , I(N(1 = A, and for ~ E ‘ A let p , = { F J ~ ( z , @ ~ ~ ~ ) : a < Z(r)), a a successor}.

If N is K + -saturated, each p n is realized in N; but the p i s me pairwise contradictory. So IlNll = Arc contradiction. So we finish the proof of Theorem 4.6.

THEOREM 4.10: .Suppo8e A 2 lTll + K,, and every model M E

PC(Tl, T ) of wrdiml i ty A ia K,-homogeneoua. Then T is swperatable.

Proof. Our assumption implies that every M E PC(T,, T ) of cardinality

478 THE NUMBER OF NON-ISOMORPHIC MODELS [OH. VIII, 8 4

2 A is K,-homogeneous [Because if (a,: i II w), (bi: i < w) is a counterexample, i.e., tp,((a,: i < 0)) = tp,((b,: i < w>) but for no b, E M tp*(<a,: i I; w)) = tp,((b,: i I; w)) and M is the L-reduct of M,, M , a model of T,, then by any elementary submodel M i of M , of cardinality A such that a,, b, E IMiI, we arrive at a contradiction]. But there is a strong limit cardinal p > A of cofinality No; so by 4.8 T is stable, and by 4.9 superstable (remember I, 1.9(4)).

CONJECTURE 4.1: If PC(T,, T) is categorical in A, A 2 lTll + N,, then T is superstable (but see [Sh 801).

PROBLEM 4.2: Can we improve 4.2(2) (i.e., weaken the conditions on p) ?

PROBLEM 4.3: Close the gap between 4.5 and 4.6. That is characterize the cardinals A, p , x such that: if I T I ] 5 x, T 5 T,, T unstable [stable but not superstable] [has the independence property], then there is a model M , of T , of cardinality p which is (<A)-universal but its L( T)-redwt ie not A-univereal.

PROBLEM 4.4: Characterize the T, A, K such that:

(*I For every model M of T of cardinality A there is N, M < N , IlNll = A, N K-saturated (note 4.7,4.8, I, 1.7 and Exercise 4.5) (but see [Sh 80al).

EXERCISE 4.5:Suppose pn = p, p I; A s 2’, T -- Tina (see 11, 4.8). Then T , h satisfies (*) from Problem 4.4. [Hint: See the solution in [Sh 811.1

PROBLEM 4.6: (1) Characterize the cardinals p, A, K , x such that for every unstable T, T c T,, ITl] II x, there is a model M E PC(T,, T), lldlll = p, M is (< A)-universaI but not A-universal, M is K-saturated but not K + -saturated.

(2) Replace “unstable” by K c K(T) < 00, or by “has the independence property ”.

EXERCIBE 4.7: Solve those cases of 4.6 which can be solved by inessential changes in the proofs here. E.g., assume G.C.H., p, A, K are successor cardinals.

CONJECTURE 4.9: Remove the exception in 3.4.

CONJECTURE 4.10: In 3.5 waive the condition (Vx < A)(x‘“ < A) .

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