saharon shelah- abstract elementary classes near ℵ1

8/3/2019 Saharon Shelah- Abstract Elementary Classes Near 1

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ABSTRACT ELEMENTARY

CLASSES NEAR 1SH88R

Saharon Shelah

The Hebrew University of JerusalemEinstein Institute of Mathematics

Edmond J. Safra Campus, Givat RamJerusalem 91904, Israel

University of WisconsinMadison, Wisconsin

Institute of Advanced StudiesJerusalem, Israel

Department of MathematicsHill Center-Busch Campus

Rutgers, The State University of New Jersey110 Frelinghuysen Road

Piscataway, NJ 08854-8019 USA

Abstract. We prove in ZFC, that no L1,[Q] have unique model of un-countable cardinality, this confirms the Baldwin conjecture. But we analyze this inmore general terms. We introduce and investigate a.e.c. and also versions of limitmodels, and prove some basic properties like representation by PC class, for anya.e.c. For PC0-representable a.e.c. we investigate the conclusion of having not too

many non-isomorphic models in 1 and 2, but have to assume 20 < 21 and even21 < 22 .

I would like to thank Alice Leonhardt for the beautiful typing.This research was partially supported by the United States Israel Binational Science Foundation(BSF) and the NSFFirst Typed - 04/May/18Latest version - 08/Apr/17

Typeset by AMS-TEX

1


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2 SAHARON SHELAH

0 Introduction

In [Sh 48], proving a conjecture of Baldwin, we show that (Q here stands for the

quantifier Qcar1 , there are uncountably many)

()1 no L1,(Q) has a unique uncountable model up to isomorphism

by showing that

()2 categoricity (of L1,(Q)) in 1 implies the existence of a model of of cardinality 2 (so has 2 non-isomorphism models).

Unfortunately, both ()1 and ()2 were not proved in ZFC because diamond on 1was assumed. In [Sh 87a] and [Sh 87b] this set theoretic assumption was weakened

to 20

< 21

; here we shall prove it in ZFC (see 3). However, for getting theconclusion from the weaker model theoretic assumption I(1, ) < 2

1 as there, westill need 20 < 21 .

The main result of [Sh 87a], [Sh 87b] was:

()3 if n > 0, 20 < 21 < . . . < 2n , L1,, 1 I(, ) < wd() for n, 1 (where wd() is usually 2 and always > 21 , see 0.5below) then has a model of cardinality n+1

()4 if 20 < 21 < . . . < 2n < 2n+1 < . . . and L1,, 1 I(, ) 0?

We think of this as a test problem and much prefer a model theoretic to a settheoretic solution. This is closely related to 0.1(4) above and to 3.11 (where we

assume categoricity in +, do not require 2 < 2+

but take = 0 or some similarcases) and 5.27(4) (and see 5.2 and 4.8 on the assumptions) (there we require

2 < 2+

, 1 I(+, K) < 2+

and = 0).Problem [Sh 576, 0.1] was stated a posteriori but is, I think, the real problem,

it says:

() Can we have some (not necessarily much) classification theory for reason-able non-first order classes K of models, with no uses of even traces ofcompactness and only mild set theoretic assumptions?

This is a revised version of [Sh 88] which continues [Sh 87a], [Sh 87b] but do notuse them. The paper [Sh 88] and the present chapter relies on [Sh 48] only when


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CLASSIFICATION OF NE CLASSES 5

deducing results on L1,(Q); it improves some of its early results and extendsthe context. The work on [Sh 88] was done in 1977, and a preprint was circulated.Before the paper had appeared, a user-friendly expository article of Makowsky

[\Mw85a ] represent, give background and explain the easy parts of the paper. In[Sh 88] the author have corrected and replaced some proofs and added mainly 6.See more in [Sh:F709].

We thank Rami Grossberg for lots of work in the early eighties on previousversions, i.e. [Sh 88], which improved this paper, and the writing up of an earlierversion of 6 and Assaf Hasson on helpful comments in 2002 and Alex Usvyatsovfor very careful reading, corrections and comments and Adi Jarden and Alon Sitonon help in the final stages.

On history and background on L1,,L, and the quantifier Q see [Ke71]. On(D, )-sequence-homogeneous (which 2.2 - 2.5 here generalized) see Keisler-Morley[KM67], this is defined in 2.3(5), and 2.5 is from there. Theorem 3.8 is similar to[Sh 87a, 2.7] and [Sh 87b, 6.3].

Remark. On non-splitting used here in 5.6 see [Sh 3], [Sh:c, Ch.I,Def.2.6,p.11] or[Sh 48].We finish 0 by some necessary quotation.

By [Ke70] and [Mo70],

0.2 Claim. 1) Assume that L1,(Q) has a modelM in which{tp(a, , M) :a M} is uncountable where L1,(Q) is countable, then has 2

1 pairwisenon-isomorphic models of cardinality 1, in fact we can find models M of ofcardinality 1 for < 21 such that {tp(a; , M) : a M} are pairwise distinctwhere tp(a ,A,M) = {(x, b) : (x, y) and M |= [a, b] and b >A}.2) If L1,(Q), L1,(Q) is countable and {tp(a, , M) : a

>M andM is a model of } is uncountable, then it has cardinality 20 .

Also note

0.3 Observation. Assume ( is a vocabulary and)

(a) K is a family of -models of cardinality

(b) >

(c) {(M, a) : M K and a M} has members up to isomorphism.


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6 SAHARON SHELAH

Then K has models up to isomorphisms (similarly for = ).

Proof. See [Sh:a, VIII,1.3] or just check by cardinal arithmetic. 0.3

Further

0.4 Claim. 1) Assume is regular uncountable, M0 is a model with countablevocabulary and T = ThL(M0), < a binary predicate from (T) and (P

M0 ,


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See more in [Sh 838, 0,9] and hopefully in [Sh:E45].

The following are used in 2.

0.6 Definition. 1) For a regular uncountable cardinal let I[] = {S : somepair (E, a) witnesses S I(), see below}.2) We say that (E, u) is a witness for S I[] if:

(a) E is a club of the regular cardinal

(b) u = u : < , a and a a = a

(c) for every E S, u is an unbounded subset of of order-type < (and is a limit ordinal).

By [Sh 420] and [Sh:E12]

0.7 Claim. Let be regular uncountable.1) If S I[] then we can find a witness (E, a) for S I[] such that:

(a) S E otp(a) = cf()

(b) if / S then otp(a) < cf() for some S E.

2) S I[] iff there is a pair (E, P) such that:

(a) E is a club of the regular uncountable

(b) P = P : < , where P {u : u } has cardinality <

(c) if < < and u P then u P

(d) if E S then some u P is an unbounded subset of (and is a limitordinal).


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1 Axioms and simple properties for classes of models

1.1 Context. 1) Here in 1-5, is a vocabulary, K will be a class of -models andK a two-place relation on the models in K. We do not always strictly distinguishbetween K and K = (K, K). We shall assume that K, K are fixed; and usuallywe assume that K is an a.e.c. (abstract elementary class) which means that thefollowing axioms hold.2) For a logic L let M L N mean M is an elementary submodel of N forthe language L(M) and M N, i.e., if (x) L(M) and a g

(x)M thenM |= [a] N |= [a]; similarly M L N for L a language, i.e. a set of formulasin some L(M). So M N in the usual sense means M L N as L is first orderlogic and M N means M is a submodel ofN.

1.2 Definition. 1) We say K is a a.e.c. with L.S. number (K) = LS(K) if:Ax 0: The holding of M K, N K M depend on N, M only up to isomorphism,i.e. [M K, M = N N K] and [ifN K M and f is an isomorphism from Monto the -model M, f N is an isomorphism from N onto N then N K M

].

Ax I: ifM K N then M N (i.e. M is a submodel of N).

Ax II: M0 K M1 K M2 implies M0 K M2 and M K M for M K.

Ax III: If is a regular cardinal, Mi(i < ) is a K-increasing (i.e. i < j < implies

Mi K Mj) and continuous (i.e. for < , M =

i


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1.3 Definition. The embedding f : N M is called a K-embedding if the rangeof f is the universe of a model N K M (so f : N N

is an isomorphism onto).

1.4 Definition. Let T1 be a theory in L(1), a set of types in L(1) for somelogic L, usually first order.1) EC(T1, ) = {M : M an 1-model of T1 which omits every p }.We implicitly use that 1 is reconstructible from T1, . A problem may arise onlyif some symbols from 1 are not mentioned in T1 and in , so we may writeEC(T1, , 1), but usually we ignore this point.2) For 1 we let PC(T1, , ) = PC(T1, ) = {M : M is a -reduct of someM1 EC(T1, )}.3) We say that K, a class of -models, is a PC or PC, class when for someT1, 1, 1 we have 1, T1 a first order theory in the vocabulary 1, 1 a set of

types in L(1), K = PC(T1, 1) and |T1| , |1| .4) We say Kis PC or PC, if for some (T1, 1, 1), (T2, 2, 2) as in part (3) we haveK = PC(T1, 1, ) and {(M, N) : M K N hence M, N K} = PC(T2, 2, )where = {P} 2, P a new one-place predicate, so || , || for = 1, 2.If = we may omit .5) In (4) we may say K is (, )-presentable and if = we may say K is-presentable.

1.5 Example: If T L(), a set of types in L(), then K := EC(T, ), K:=L

form an a.e.c. with LS-number |T| + || + 0, that is, satisfy the Axioms from1.2 (for LS(K) := || + 0).

1.6 Observation. Let I be a directed set (i.e. partially ordered by , such that anytwo elements have a common upper bound).

1) IfMt is defined for t I and t s I implies Mt K Ms then

sI

Ms K and

for every t I we have Mt K

sI

Ms.

2) If in addition t I implies Mt

K N then sI

Ms

K N.

Proof. By induction on |I| (simultaneously for (1) and (2)).

If I is finite, then I has a maximal element t(0), hence

tI

Mt = Mt(0), so there

is nothing to prove.


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So suppose |I| = and we have proved the assertion when |I| < . Let = cf()so is a regular cardinal; hence we can find I (for < ) such that |I| 0 that Fni (a) is F|u|i (a u), u = {i < n : ai / {aj :

j < i}.

1.9 Lemma. 1) K is (LS(K), 2LS(K))-presentable.2) There is a set of types inL(1) in fact complete quantifier free (where 1 isfrom Lemma 1.7) such thatK = PC(, ).3) For the from part (2), if M1 N1 EC(, ) and M, N are the -reducts ofM1, N1 respectively then M K N.4) For the from part (2), we have {(M, N) : M K N so N, M K} = {(M1 , N1 ) : M1 N1 are both from PC(, )}.

Proof. 1) By part (2) the first half of K is (LS(K), 2LS(K))-presentable holds. Thesecond part will be proved with part (4).2) Let n be the set of complete quantifier free n-types p(x0, . . . , xn1) in L(1)such that: if M1 is a 1-model, a realizes p in M1 and M is the -reduct of M1,

then Ma K and Mb K Ma for any subsequence b of any permutation of a; whereMc(c m|M1|) is the submodel of M whose universe is {Fmi (c) : i < LS(K)}.Clearly there are such submodels (when K = ).

Let be the set of p which, for some n, are complete quantifier free n-types (inL(1)) which do not belong to n. By 1.6(1) we have PC(, ) K and by 1.7K PC(, ).3) Similar to the proof of (2) using 1.6(2).


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4) The inclusion holds by part (3); so let us prove the other direction. GivenN K M we apply the proof of 1.7 to M, but demand further a

nN Ma N;simply add this demand to the choice of the Mas (hence of the F

ni s). We still

have a debt from part (1).We let n be the set of complete quantifier free n-types in

1 := 1 {P} (P a

new unary predicate), p(x0, . . . , xn1) such that:

() if M1 is an 1-model, a realizes p in M1, M the -reduct of M1, then

() Mb K Ma for any subsequence b of a where Mc (for c |M1|) is thesubmodel of M whose universe is {(Fmi )

M1(c) : i < LS(K)}, wherem = g(c) (and there are such models),

() b PM1 Mb PM1 for b a.

We leave the rest to the reader (alternatively, use PC1(T, ), T saying P is closed

under all the functions Fni ). 1.9

By the proof of 1.9(4).

1.10 Conclusion. The 1 and from 1.9 (so |1| LS(K)) satisfy: for any M Kand any 1-expansion M1 of M which is in EC1(, )

(a) N1 L M1 N1 M1 N1 K M

(b) N1 L N2 L M1 N1 N2 M1 N1 K N2

(c) if M K N then there is a 1-expansion N1 of N from EC1(, ) whichextends M1.

1.11 Conclusion If for every < (2LS(K))+,Khas a model of cardinality thenK has a model in every cardinality LS(K).

Proof. Use 1.9 and the classical upper bound on value of the Hanf number for: firstorder theory and omitting any set of types, for languages of cardinality LS(K) (see,e.g., [Sh:c, VII,5.3,5.5]). 1.11

1.12 Conclusion: Assume that K is an a.e.c., = |K| + LS(K) and for simplicityK or just K L, recalling L is the constructible universe of Gobel. If > and A (H(, ) and + 1 A and K A which means {(M, N) : M K N hasuniverse } A then:

(a) M K K M A K M


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(b) if M K N so both belongs to K and M, N A then M A K N A

(c) ifA B and [b


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2 Amalgamation properties and homogeneity

2.1 Context. K is an a.e.c.The main theorem 2.8, the existence and uniqueness of the model-homogeneous

models, is a generalization of Jonsson [Jo56], [Jo60] to the present context. The

result on the upper bound 220+||

for the number of D-sequence homogeneousuniversal-models of cardinality is of Keisler-Morley [KM67]. Earlier there wereserious good reasons to concentrate on sequence-homogeneous models, but here wedeal with the model-homogeneous case. From 2.13 to the end we consider what wecan say when we omit smoothness, i.e. AxIV of Definition 1.2.

2.2 Definition. 1) D(M) := {N/ =: N K M, N LS(K)}.2) D(K) := {N/ =: N K, N LS(K)}.

3) D(M) = {tpL(M )(a, , M) : a >M}.

2.3 Definition. Let > LS(K).1) A model M is -model-homogeneous when: if N0 K N1 K M, N1 < , f anK-embedding of N0 into M, then some K-embedding f

: N1 M extends f.1A) A model M is (D, )-model-homogeneous ifD = D(M) and M is a -modelhomogeneous.1B) Adding above means in K.2) M is -strongly model-homogeneous if: for every N K LS(K), every2 N KLS(K) is K-embeddable into M and for every N K


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5A) In (5) we omit D when D = {tpL(K)(a, , N) : a nN where n < and

M L N}.6) We omit the model/sequence, when which one is clear from the context,

i.e., if D is as in 2.2(3) = 2.3(5)(a), (D, )-homogeneous means (D, )-sequence-homogeneous: if D is as in Definition 2.2(1), (D, )-homogeneous means (D, )-model-homogeneous, if not obvious we mean the model version.7) M is -universal when every N K can be K-embedded into it. Similarly(< )-universal, ( )-universal.

2.4 Claim. Assume N is -model-homogeneous andD(M) D(N), (andLS(K) LS(K), then by 1.10 we can find M1 = M1 : < such that M1 =


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Proof of (ii). First, assume that = so we have proved (ii) for < andM1 = > M0, so LS(K) < = hence we can find M1 : < as in

the proof of (i) such that M01 = M0 and let () = M

1 . Now we define f

by induction on such that f is a K-embedding of M1 into N and f isincreasing continuous in and f0 = f. We can do this as in the proof of (i) by(ii)() for < .

Second, assume M1 < . Let g be a K-embedding of M1 into N, it existsby (i) which we have just proved. Let g be onto N

1 K N, and let g M0

be onto N0 K N1, and let f be onto N0 K N. So clearly h : N

0 N0 de-

fined by h(g(a)) = f(a) for a |M0|, is an isomorphism from N0 onto N0. SoN0, N

0, N

1 K N. As M1 < clearly N

1 < so (by the assumption N is

-model-homogeneous, see Definition 2.3(1)) we can extend h to an isomorphismh from N1 onto some N1 K N, so h

g : M1 N is as required. 2.4

2.5 Conclusion 1) If M, N are model-homogeneous, of the same cardinality (>LS(K)) and D(M) = D(N) then M, N are isomorphic. Moreover, if M0 K

M, M0 < M, then any K-embedding of M0 into N can be extended to anisomorphism from M onto N.

2) The number of model-homogeneous models from K of cardinality is 22LS(K)

;if in Definition 1.2, AxVI, in the definition of LS(K) we omit || LS(K), the

bound is 22LS(K)+|(K)|

.3) If M is -model-homogeneous and D(M) = D(K) then M is ( )-universal,i.e. every model N (in K) of cardinality , has a K-embedding into M. So if

D(M) = D(K

) then: M is -model universal homogeneous (see Definition 2.3(3))iff M is a -model-homogeneous iff M is (,D(K))-homogeneous.4) If M is -model-homogeneous then it is -universal for {N K : D(N) D(M)}.5) If M is (D, )-sequence-homogeneous, ( > LS(K)) then M is a -modelhomogeneous.6) For > LS(K), M is -model universal homogeneous iffM is -model-homogeneousand ( LS(K))-universal.

Proof. 1) Immediate by 2.4(1), using the standard hence and forth argument.2) The number of models (in K) of power LS(K) is, up to isomorphism, 2LS(K)

(recalling that we are assuming |(K)| LS(K)). Hence the number of possible

D(M) is 22LS(K)

. So by 2.5(1) we are done.3),4),5) Immediate. 2.5

2.6 Remark. The results parallel to 2.5(1)-(4) for -sequence homogeneous modelsand D(M) hold, too.


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2.7 Definition. 1) A model M has the (, )-amalgamation property (= am.p.,in K, of course) if: for every M1, M2 such that M1 = , M2 = , M K M1and M K M2, there is a model N and K-embeddings f1 : M1 N and f2 :

M2 N such that f1 |M| = f2 |M|. Now the meaning of e.g. the ( , < )-amalgamation property should be clear. Always , LS(K) (and, of course, ifwe use < , > LS(K)).1A) In part (1) we add the adjective disjoint when f1(M1) f2(M2) = M.Similarly in (2) below.2) Khas the (,,)-amalgamation property if every model M (in K) of cardinality has the (, )-amalgamation property. The (, )-amalgamation property for Kmeans just the (,,)-amalgamation property. The -amalgamation property forK is just the (,,)-amalgamation property.3) K has the (, )-JEP (joint embedding property) if for any M1 K, M2 Kof cardinality , respectively there is N K into which M1 and M2 are K-embeddable.4) The -JEP is the (, )-JEP.5) The amalgamation property means the (,,)-amalgamation property for every, ( LS(K)).6) The JEP means the (, )-JEP for every , LS(K).

Remark. Clearly in 2.7, parts (1), (2) first sentence, (3),(5), the roles of , aresymmetric.

2.8 Theorem. 1) If LS(K) < , =


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2.10 Remark. Also the corresponding converses hold.

2.11 Lemma. 1) If LS(K) and K has the -amalgamation property then

K has the (, +)-amalgamation property and even the (, +, +)-amalgamationproperty.2) If and K has the (, )-amalgamation property and the (, )-amalgamation property thenK has the (, )-amalgamation property. IfK has the(,,) and the (, )-amalgamation property, thenKhas the (,,)-amalgamationproperty.3) If i(i ) is increasing and continuous, LS(K) 0 and for every i < ,Khas the (i, + i, i+1)-amalgamation property then K has the (0, + 0, )-amalgamation property.4) If 1 and for every M, M = 1, there is N, M K N, N = , then

the (,,)-amalgamation property (forK

) implies the (, 1, )-amalgamationproperty (forK).5) Similarly with the disjoint amalgamation version.

Proof. Straightforward, e.g.3) So assume M0 K0 , M0 K M1 K+0 and M0 K M2 K and forvariety we prove for the disjoint amalgamation version (see part (5)). By e.g.1.10 we can find an K-increasing continuous sequence M2,i : i such thatM2,0 = M0, M2, = M2 and M2,i Ki for i .

Without loss of generality M1 M2 = M0. We now choose M1,i by induction on

i such that:

() (a) M1,j : j i is K-increasing continuous

(b) M1,i = M1 if i = 0

(c) M1,i K+i(d) M2,i K M1,i

(e) M2,i M1, = M1,i.

For i = 0 see clause (b), for i limit take union, for i = j + 1 apply the disjoint(j , + j , i)-amalgamation to M2,j , M1,j, M2,j+1. For i = we are done. 2.11

2.12 Conclusion. If LS(K) 1 < 2 and K has the -amalgamation propertywhenever 1 < 2 then K has the (,,)-amalgamation property whenever1 2, 2 and < 2.


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It may be interesting to note that even waiving AX IV we can say something.

2.13 Context: For the remainder of this section K is just a weak a.e.c., i.e., Ax IV

is not assumed.

2.14 Definition. Let M K have cardinality , a regular uncountable cardinal> LS(K). We say M is smooth if there is a sequence Mi : i < with Mi being

K-increasing continuous, Mi K M and Mi < for i < and M =

i LS(K)) is smooth and model-homogeneous and N Kis smooth, D(N) D(M) then N can be K-embedded into M.


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This can be proved in the context of universal classes (e.g. AxFr1 from [Sh 300b]).

2.19 Fact: 1) IfKi = (Ki,


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3 Limit models and other results

In this section we introduce various variants of limit models (the most importantare the superlimit ones). We prove that ifKhas a superlimit model M of cardinal-

ity for which the -amalgamation property fails and 2 < 2+

then I(, K) = 2

(see 3.8). We later prove that if L1,(Q) is categorical in 1 then it has modelin 2 see 3.18(2). This finally solves Baldwins problem (see 0). In fact we provean essentially more general result on a.e.c. and (see 3.11, 3.13).

The reader can read 3.3(1),(1A),(1B) ignore the other definitions, and continuewith 3.7(2),(5) and everything from 3.8 (interpreting all variants as superlimits).You may wonder can we prove the parallel to Baldwin conjecture in + if > 0;it is

ifKis -presentable a.e.c. with LS(K) = , categorical in + then K++ = .

This is false when cf() > 0.

3.1 Context. K is an a.e.c.

3.2 Example: Let be given and K= (K, K) be defined by

K = {(A,


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3.3 Definition. Let be a cardinal LS(K). For parts 3) - 7) but not 8), forsimplifying the presentation we assume the axiom of global choice (alternatively,we restrict ourselves to models with universe an ordinal < +).

1) M K is locally superlimit (for K) if:

(a) for every N K such that M K N there is M K isomorphic to M

such that N K M and N = M

(b) if < + is a limit ordinal and Mi : i < is K-increasing sequence and

Mi = M for i < then

i


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S1-medium limit, S1-limit

S1-weakly limit.

3.5 Lemma. 0) All the properties are preserved if S is replaced by a subset and ifK has the -JEP, the local and global version in Definition 3.3 are equivalent.1) If Si

+ for i < +, S = { < + : (i < ) Si} and Si i = for i < then: M is Si-strongly limit for each i < if and only if M is S-strongly limit.2) Suppose is regular and S { < + : cf() = } is a stationary set andM K then the following are equivalent:

(a) M is S-strongly limit

(b) M is (, {})-strongly limit

(c) M K is K-universal not


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Proof. 0) Trivial.1) Recall that in Definition 3.3(3), clause (b) we use F only on Mi+1; (see the proofof (2A) below, second part).

2) For (c) (a) note that the demands on the sequence are local, Mi+1 KF(Mi+1) K Mi+2, (whereas in part (4) they are global).2A) First assume that M is S-strongly limit and let F witness it. Suppose ,so we choose S with cf() = and let i : i < be increasing continuouswith limit , 0 = 0, i+1 a successor of a successor ordinal for each i < . We nowdefine F as follows: to define F(M) we define F, for by induction on . Let:

(a) if = 0 then F,0(M) = M

(b) if = + 1 then F,(M) = F(F,(M))

(c) if a limit ordinal then F,(M) = {F,(M) : < }.

Lastly, let F(M) be F,(M).Now suppose Ni : i is K-increasing continuous, Ni K and F(Ni+1) K

Ni+2 for i < and we should prove N = M. Now we can find Mj : j < + such

that it obeys F and Mi = Ni for i < ; so clearly we are done.Second, assume that for each , clause (c) of 3.5(2) holds and let F

exemplify this. Let : < () list so () < + and define F as follows. For

any M K choose M[] by induction on () as follows: M[0] = M, M[+1] =F(M[]) and for limit ordinal let M[] = {M[] : < }. Lastly, let F[M] =

M[()]. Now check.3) No new point.4) First note that (a) (b) should be clear. Second, we prove that (b) (a) solet F witness that clause (b) holds. Let E, u : < witness that S I[], i.e.

()1 (a) E a club of

(b) u and otp(u) for <

(c) if S E then = sup(u) and otp(u) =

(d) if \S E then otp(u) <

(e) if u then u = u .

We can add

()2 (f) if u then has the form 3+ 1.

Let : < list E in increasing order and without loss of generality 0 =0, 1+ is a limit ordinal (note that only the limit ordinals of S count).


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26 SAHARON SHELAH

To define F as required we shall deal with the requirement according to whether S is easy, i.e. / E so (, +1] for some <

+ so after we cantake care of it, or is hard, i.e. E so we use the u .

We choose e : S\E such that (, +1] S implies e = sup(e)and min(e) > , otp(e) = , e is closed and e = sup(e ) ( {3+ 2 : < }). If S E let e be the closure of u. Let , : < list ein increasing order.

We now define a function F so let Mj : j i+1 be given and let i < +1.We fix so (, +1) and now define F

(Mj : j i + 1) by induction on i [, +1) assuming that if j

+ 1 < i + 1 then F(Mj : j j + 1) K Mj+2

and further there is Nj+1 = Nj+1, : < +1 such that the following holds:

()3 Nj+1 is K -increasing continuous, Mj+1 K Nj+1,0 and Nj+1, K

Mj+2

()4 if (S\E)(+1\), j+1 = , (so necessarily j

+1 (, +1), j+

1 {3 + 2 : < }, is a successor ordinal) then let N,j = N,j, :

be the following sequence of length + 1, N,j, is N, , if

is a successor ordinal and is M, if is limit or zero, and we demand

F(N,j, : ) K Nj+1,+1

()5 if j + 1 u for some S E hence j + 1 {3 + 1 : < } and

= otp(uj+1) < and f is the one-to-one order preserving function from+ 1 onto c(uj+1 {j

+ 1}) and is a successor, then F(Mf() :

) K M+1.

This implicitly defines F. Now F is as required: Mi = M when i < , cf(i) = by ()4 when ()( < i < +1) and by ()5 when ()(i = ). 3.5

3.6 Lemma. Let T be a first order complete theory, K its class of models andK=L.1) If is regular, M a saturated model of T of cardinality , then M is (, {})-superlimit.2) If T is stable, and M a saturated model of T of cardinality then M is (, { :(T) and is regular})-superlimit (on (T)-see [Sh:c, III,3]). (Note thatby [Sh:c] if is singular and T has a saturated model of cardinality then T is

stable and cf() (T)).3) If T is stable, singular > (T), M a special model of T of cardinality , S { < + : cf() = cf()} is stationary and S I[] (see above 0.6, 0.7) then M is(, S)-medium limit.

Remark. See more in [Sh 868].


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Proof. 1) Because if Mi is a -saturated model of T for i < , cf() , then

i


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28 SAHARON SHELAH

I} K is superlimit (i.e., if Mi : i < is K-increasing, i < Mi K, a

limit ordinal < +, M = {Mi : i < } then

i


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there is a K-embedding f of M into N. So f is a function from + to +. Let

+

2, by the choice of F and of M : < + there is a closed unbounded

C + such that S C M = M, hence M fails the -amalgamation

property. Without loss of generality for C, M has universe . Now by0.5, if (f, C) :

+2 satisfies that for each +

2, f : + + and

C + is closed unbounded then for some =

+

2 and C S we have = , () = () and f = f .[Why? For every < +, 2 and f : + we define c(, f) 2 as follows:

it is 1 iff there is +

2 such that = & f = f & () = 0 and is

0 otherwise. So some +

2 is a weak diamond sequence for the colouring c andthe stationary set S. Now C, f are well defined and S

= { S : limit and() = c( , f )} is a stationary subset of+, so we can choose S C. If() = 0, then c( , f ) = 0 by the choice ofS but witness that c( , f )

is 1, standing for there. If () = 1 there is witnessing c( , f ) = 1, inparticular () = 0, so , , , are as required.]

Now as S C C it follows that M = M hence M fails the -amalgamation property. Also M has universe as C and M = M as = .

So f M = f = f = f M. So f M(+1), f M(+1)show that M(+1), M(+1), can be amalgamated over M contradicting clause(v)(b) of the construction, i.e. of. So there is no K-universal N K+ .

It takes some more effort to get 2+

pairwise non-isomorphic models (rather thanjust quite many).

Case A5: There is M K, M K M such that for every N satisfying M K

N K there are N1, N2 K such that N K N

1, N K N2 and N2, N1 cannot

be K-amalgamated over M (not just N). In this case we do not need M is

S-weakly limit.We redefine M, 2, <

+ such that:

2 (a) 2 M K M K:

(b) if = 0, M = M;

(c) if limit and 2 then M =


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30 SAHARON SHELAH

M, M cannot be amalgamated over M.

Obviously, the models M =

+

2 the function f can beextended to a K-embedding of M into M

(c) c(, ,f ) is zero iff it is defined but is = 1.

For each , as S is not small, by simple coding, for every < + there is h : S

{0, 1} such that:

() for every +

2, +

2 and f : + +, for a stationary set of S

c( , , f ) = h().

Now for every W + we define W +2 as follows:W() is h(), if W and S (note that there is at most one )W() is zero if there is no such .Now we can show (chasing the definitions) that

4 if W(1), W(2) +, W(1) W(2), then MW(1) cannot be K-embedded

into MW(2) .


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This clearly suffices.Why is 4 true? Suppose W(1) W(2), let W(1)\W(2) and toward contradic-tion let f be a K-embedding ofMW(1) into MW(2) , so E = { : MW(1), MW(2)

have universe and f is a K-embedding of MW(1) into MW(2)} is a clubof +. Hence by the choice of c and h there is E S such that

c(W(1) , W(2) , f ) = h() and Mw(1) is not an amalgamationbase.

Now the proof splits to two cases.

Case 1: h() = 0.So W(1)() = 0 = W(2)() and by clause (b) of3 above, i.e., the definition of

c we have the objects W(1), W(2), f MW(1) = f MW(1)(+1) witness that

c(W(1) , W(2) , f ) = 1, contradiction.

Case 2: h() = 1.So W(1)() = 1, W(2)() = 0, c(W(1) , W(2) , f ) = 1. By the definition

of c, we can find such that (W(2) ) < 0 > +2 and a K-embedding g

of M(W(1)) into M .

For some (, +), f embeds MW(1)(+1) = M(W(1)) into MW(2)and g embeds M(W(1)) into M.

As W(2) < 0 > and W(2) < 0 > W(2) by clause (vii)above there are f1, g1 and N K such that

(a) MW(2) K N

(b) f1 is a K-embedding of MW(2) into N over MW(2)

(c) g1 is a K-embedding of M into N over MW(2).

So

(b) f1 f is a K-embedding of M(W(1)) into N

(c) g1 g is a K-embedding of M(W(1)) into N

(d) f1 f, g1 g extend f : MW(1) N (both).

So together we get a contradiction to assumption ()1(d). 3.8

3.10 Theorem. 1) Assume one of the following cases occurs:

(a)1 K is PC0 (hence LS(K) = 0) and 1 I(1,K) < 21

or

(a)2 Khas models of arbitrarily large cardinality, LS(K) = 0 and I(1,K) < 21 .


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32 SAHARON SHELAH

Then there is an a.e.c. K1 such that

(A) M K1 M K and M K1 N M K N and LS(K1) = LS(K)(= 0)

(B) if K has models of arbitrarily large cardinality then so does K1(C) K1 is PC0

(D) (K1)1 =

(E) all models of K1 are L,-equivalent and M K1 N M L, N &M K N and K1 is categorical in 0 and M (K1)0 K1 = {N K :N L,(K) M}

(F) if K is categorical in 1 then (K1) = K for every > 0; moreoverK1=K (K1)1 .

2) If in (1) we add LS(K

) names to formulas inL, (i.e. to a set of represen-tatios up to equivalence) then we can assume each member of K is 0-sequence-homogeneous. The vocabulary remains countable, in fact, for some countable firstorder theory T, the models of K are the atomic models of T (in the first ordersense) and K becomes (being a submodel).

Proof. Like [Sh 48, 2.3,2.5] (using 2.19 here for = 2). E.g. why, if K is cat-egorical in 1 then K1=K (K1)1? We have to prove that if M K N areuncountable then M L,(K) N. But there is M K0 such that K1 ={M K : M L, M} and (K1)1 = K1 = , so it suffices to proveM L1,(T) N, so assume this is a counterexample so for some (x, y) L1,()

and a g(y)M, b N we have N |= [b, a] but for no b M do we haveN |= [b, a] and without loss of generality the quantifier depth of (x, y), isminimal (for all such pairs (M, N)). Let = {(z) L1,(K) : hasquantifier depth } hence M K N

, M K>0 M N

. Alsowithout loss of generality M = N = 1. Now choose M K1 by induc-tion on < 2, which is K-increasing continuous (hence increases) and foreach there is an isomorphism f from N onto M+1 mapping M onto M, re-calling the categoricity. By Fodor lemma for some < we have f(a) = f(a),

so f1 (f(b)) contradict the choice of (x, y), b, a. 3.10

We arrive to the main theorem of this section.

3.11 Theorem. Suppose K and satisfy the following conditions:

(A) Khas a superlimit memberM of cardinality, LS(K), (ifK is categor-ical in , then by assumption (B) below there is such M; really invariantly


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+-strongly limit suffice if (d) of () of 3.12(2) below holds, see Definition3.3)

(B) K is categorical in +

(C) () K is PC0 , = 0 or() K= PC, = , cf() = 0 or

() = 1,K is PC0 or

() K is PC, (2)+; not useful for 3.11, still it too implies (), in3.12.

Then K has a model of cardinality ++.

3.12 Remark. 1) If = 0 we can wave hypothesis (A) by the previous theorem3.10.

2) Hypothesis (C) can be replaced by (giving a stronger theorem):

(),(a) K is PC and

(b) any L+, which has a model M of order-type +, |PM| = , has a

non-well-ordered model N of cardinality

(c) {M K : M = M} is PC (among models in K) and

(d) for some F witnessing M is invariantly -strongly limit, that is the class{(M, F(M)) : M K} is PC (if M

is superlimit this clause is notrequired as F = the identity on K is O.K.)

3) It is well known, see e.g. [Sh:c, VII,5] that hypothesis (C) implies (), frompart (2), see more [GrSh 259].

Proof. By 3.12(3) we can assume (), from 3.12(2).

Stage a: It suffices to find N0 K N1, N0 = +, N0 = N1.

Why? We define by induction on < ++ a model N K+ such that < implies N K N and N = N. Clearly N0, N1 are defined (without loss ofgenerality N1 = + as LS(K), also otherwise we already have the desired

conclusion), for limit < ++ the model


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34 SAHARON SHELAH

3.13 Theorem. Suppose the following clauses:

(A) K has an invariantly -strongly limit member M of cardinality , as ex-emplified by F : K K andK has the JEP (see Definition 3.3)

(B) I(+, K+) < 2+ or even just I(+, KF+) < 2

+ (or just IE(+, KF+) N the L-generic type of a in N is gtp(a; N, A; A; N) ={(x) L : N

K[a]}.

4) Let gtpL(a; N, A; A; N) where N K N K and L L(N, A; A) be {(x) :

L(N, A; A) and for some N

K


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44 SAHARON SHELAH

3) There are obvious implications, and forcing is preserved by isomorphism andreplacing N( K


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4.10 Claim. 1) If a N K0 and (x) L0,(

+0) (so a is a finite sequence)

then (N, N) 1K

[a] or (N, N) 1K

[a] (i.e. P is interpreted as N).

2) If (N, N) 1K

x p(x), where p(x) is a not necessarily complete n-type (n =

g(x)) in L where L L01,(+0) is countable, then for some complete n-type q inL extending p we have (N, N) 1

Kx q(x).

Proof. 1) Suppose not, for each S 1, we define by induction on , NS K0( (Nj2) and < } (see

Definition 4.3(4))


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(f) F(N2+1) K N2+2.

As 1 < 21 = 20 this is possible, i.e., in clause (e) we should find a type which is

not in a set of 1 || < 20

types, as the number of possibilities is 20

; let N ={Ni : i < 1} for < 20 , clearly N K1 . Now toward contradiction if M, M K0 , N K M and a materialize in (M, N) is 1

(g) if in clause (f) we get that there are 1 such types then I(1, K) 1

(h) let L1 := L0 L

11,

(+0) then the parallel clauses to (a)-(g) holds.

2) Clause (e) means that

(i) assume further that N0 K N K0 for = 1, 2 and a N and theL1


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Remark. We can prove clause (b) and the last sentence in clause (c) of 4.13 directlynot mentioning the L0-s.

Proof. Note that proving clause (e) we say repeat the proof of clause (a),(b),(c),(d)for L1,.

Clause (a): We choose L0 by induction on using 4.11. The second phrase isproved by induction on the depth of the formula using 4.10.

Clause (b): By iterating times, it suffices to prove this for each a N1, so againby iterating times it suffices to prove this for a fix a N1.

If the conclusion fails we can define by induction on n < for every n2, amodel M and (x) L01,(N) such that:

(i) M = N1

(ii) M K M K0 for = 0, 1

(iii) (M, N) 1K

(a)

(iv) (x) = (x).

Now for 2, let M =

n


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50 SAHARON SHELAH

1 if for = 1, 2, a >(N) materialize in (N, N) a complete L

0N

1 let P(N,M, a) = {gtpL0M; for (x) L01,

(N) of

quantifier depth < we have:(x) gtpL01,(N)

(a; N; M), iff for every q(x) P(N,M, a), (x) belong

to the type computed implicitly in , i.e. if q(x) = gtpL0


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Case A: 20 = 21 .We shall prove I(1, K) 20 , thus, (as 20 = 21) contradicting Hypothesis

4.8.

Let pi (for i < 2) be distinct complete L01,(

+0

)-types such that for each i, pi ismaterialized in some pair (M; N), so N K M K0 (they exist by the assumptionthat (f) fails). For each i < 2 we define Ni,, i, (for < 1) and ai, such that:

1 (i) Ni, K0 has universe (1 + ), N0,0 = N

(ii) Ni, : < 1 is K-increasing continuous

(iii) ai, Ni,+1, ai, materialize pi in (Ni,+1, Ni,)

(iv) for every < < 1 and a >(Ni,), the sequence a materialize

in (Ni,, Ni,) a complete L01,(+0)-type

(v) i, < 1 is strictly increasing continuous in

(vi) for < , Ni, is pseudo L0(Ni,)-generic, see 4.4(4) and take care

of Q, i.e., if < , p(y, x) a complete L0-type and

(Ni,, Ni,) 1K

(Qy) p(y, a)

then for some b Ni,+1\Ni, we have (Ni,+1, Ni,) 1K

p(b, a)

(vii) if < and a, b N1 materialize different L01,(Ni,)-types inNi,, then a, b realize different (L1,(

+0) L1i,+1)(N)-types inNi,

(viii) Ni = {Ni, : < 1}

(ix) if < for = 1, 2, < , n < and a1 n

(Ni,) then for somea2 n(Ni,) we have gtpL0 (a1; Ni,1 ; Ni,) =

gtpL0 (a2; Ni,2 ; Ni,)

(ix)+ moreover, if n < , 1 < 2 < , < , a n(Ni,) for = 1, 2and gtpL02

(a1; Ni,1; Ni,) = gtpL02(a2; Ni,2 ; Ni,) and

b1 Ni, then for some b2 Ni, we have gtpL01(a1b1; Ni,1 ; Ni,)

= gtpL01(ab2; Ni,2; Ni,).

This is possible by the earlier claims. By clause (e) of 4.13 clearly

2 the pair (Ni, N0) is L1


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52 SAHARON SHELAH

expanded to a -model, by predicates for K, K with QAi

1 = KH(2), QA

2 =

{(M, N) : M K N both in H(2)}, cAi

0 , . . . , cAi

4 being {Ni, : < 1}, i, : < 1, {ai, : < 1}, Ni and {i} respectively.

Let Ai be a countable elementary submodel of Ai so |Ai| 1 is an ordinal(i) < 1. It is also clear that c

Ai3 is Ni,(i) as c

Ai

3 = Ni. As Ai is definedfor i < 2, for some unbounded S 2 and < 1, for every i S, (i) = and for i, j S, some sequence from Nj materializes pi in the pair (Nj , Nj,(j))

iff i = j. For i S let Di = {p : p is a complete L1(i)

-type materialized in

(Ni,(i), Ni,0)}. Because of the i,s choice and 2 the pair (Ni,, N0) is (Di, 0)-

homogeneous and Di is a countable set of complete L1 -types. Note that by the

choice of S, i = j( S) Di = Dj .Let = {D : D a countable set of complete L1 -types, such that for some model

A

=AD of

iS Th

L, (

Ai), with {a :

AD |= a countable ordinal} = (and the

usual order) we have D = {{(x) : (x) L1 and AD |= (N; N0) 1K

[a]} :

a N where N = cAD3 }}.So Di for i < 2, hence is uncountable.By standard descriptive set theory (is an analytic set hence) has cardinality

continuum. So let D() be distinct for < 20 . For each , let A0D() be as inthe definition of . We define by induction on < 1,A

D() such that

() AD() is countable

() < AD() L, A

D()

() for limit we have AD() =


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Let M, be the d,-th member of the 1-sequence of models in A

D()for >

(remember cAi

0 = Ni, : < 1). Let M =

when < . Now if

a, b >(NAD()

d2) and [ < gtpL0 (a; N

AD()

d1; N

AD()

d2) = gtpL0 (b; N

AD()

d1; N

AD()

d2)]

then also AD() satisfies this but A

D() thinks that the countable ordinals are

well ordered hence for some d,AD() |= d is a countable ordinal > for each

< and we have AD() |= gtpL0d(a; Nd1 ; Nd2) = gtpL0d(a; Nd1 ; Nd2). Hence if

AD() |= d < d then for every a NAD()

d2 for some b NAD()

d2 we have

AD() |= gtpL0d(aa; Nd1 ; Nd2) = gtpL0d(bb; Nd1 ; Nd2)

hence gtpL0 (aa; NAD(); N

AD()

d2) = gtp(bb; N

AD()

d1; N

AD()

d2).

Also we can replace L0 by L1 . By clause (x) of1 the set {gtpL0(a; N

AD()

d1; N

AD()

d2) :

a >(NAD()

d2)} is Di.

So (NAD()

d2, N

AD()

d2) is (Di, 0)-homogenous.

So from the isomorphism type of M we can compute D(). So = M M. As M K1 we finish.

Case B: 20 < 21 .By 3.8, K has the 0-amalgamation property. So clearly if N K M K0 , a

M, then a materializes in (M, N) a complete L01,(+0)-type. We would now like

to use descriptive set theory.We represent a complete L01,(

+0)-type materialized in some (N, M) by a real,by representing the isomorphism type of some (N,M, a), N K M K0 , a M.The set of representatives is analytic recalling K is PC0 , and the equivalence

relation is 11. [As (N1, M1, a1), (N2, M2, a2) represents the same type if and onlyif for some (N, M), N K M K0 , there are K-embeddings f1 : M1 M, f2 :M2 M such that f1(N1) = f2(N2) = N and f1(a) = f2(a).]

By Burgess [Bg] (or see [Sh 202]) as there are > 1 equivalence classes, there isa perfect set of representation, pairwise representing different types.

From this we easily get that without loss of generality that their restriction tosome L0 are distinct, contradicting part (a).


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54 SAHARON SHELAH

Clause (g): Easy by the proof of clause (f), Case A above but much simpler as in4.12.

Clause (h): As in the proof of clause (e).

2) Should by clear by now. 4.13

4.15 Remark. 1) Note that in the proof of Clause (f) of 4.13, in Case (A) we getmany types too but it was not clear whether we can make the N to be genericenough, to get the contradiction we got in Case (B) but this is not crucial here.2) We may like to replace L01, by L

11,

in 4.10, 4.11 and 4.13 (except that,

for our benefit, in 4.13(e), we may retain the definition of L1(N)). We lose theability to build L-generic models in K1 (as the number of (even unary) relationson N K0 is 2

0 , which may be > 1). However, we can say a materializes inN K0 the type p = p(x) which is a complete type in L

11,

(Nn, Nn1, . . . , N 0);where N

0K . . . K N

nK N, N

countable).

[Why? Let some N1, a1 be as above, a1 materialize p in (N1, Nn, . . . , N 0) then thisholds for (N, a) iff for some N, f we have N K N

K1 and f is an isomorphismfrom N1 onto N mapping a1 to a and N to N for n. If there is no such pair(N1, a1) this is trivial.]We can get something on formulas.

This suffices for 4.10.

4.16 Concluding remarks for 4. 0) We can get more information on the case1 I(1, K) < 21 (and the case 1 I(1, KF1) < 2

1 , etc.).

1) As in 3.8, there is no difficulty in getting the results of this section for the classof models of L1,(Q) because using (K, K) from the proof of 3.18(2) in allconstructions we get many non-isomorphic models for appropriate F, as in 4.9(2).2) For generic enough N K1 with K-representation N : < 1, we havedetermined the Ns (by having that without loss of generality K is categorical in0). In this section we have shown that for some club E of 1, for all < from E the isomorphism type of (N, N) essentially

6 is unique. We can continuethe analysis, e.g., deal with sequences N0 K N1 K . . . K Nk K0 such thatN+1 is pseudo L

0(N, N1, . . . , N 0)-generic. We can prove by induction on k that

for any countable L L01,(+k) for some , any strong L-generic N K1 is

L-determined. That is, for any N

: < 1

, N

K N countable K-increasingcontinuous with union N, for some club E for all 0 < . . . < k from N theisomorphic type ofNk , Nk, . . . , N 0 is the same; i.e., determining for L,(aa).3) We can do the same for stronger logics, let us elaborate.

Let us define a logic L. It has as variable

6why only essentially? as the number of relevant complete types can be 1; we can get rid ofthis by shrinking K


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variables for elements x1, x2...andvariables for filters Y1,Y2...

The atomic formulas are:

(i) the usual ones

(ii) x Dom(Y).

The logical operations are:

(a) conjunction, negation

(b) (x) existential quantification where x is individual variable

(c) the quantifier aa acting on variables Y so we can form (aaY)

(d) the quantification (x Dom(Y))

(e) the quantification (fx Dom(Y)).

It should be clear what are the free variables of a formula . The variable Yvary on pairs (a countable set, a filter on the set). Now in x[,Y], (x Dom(Y)), (fx Dom(Y)), x is bounded but not Yand in aaY,Yis bounded.

The satisfaction relation is defined as usual plus

() M |= (x Dom(Y)(x,Y, a) if and only if for some b from the domainofY, M |= [b,Y, a]

() M |= fx Dom(Y)(x,Ya) if and only if{x Dom(Y) :|= (x,Y, a)}

Y() M |= (aaY, a)(Y) if and only if there is a function F from >([M]


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56 SAHARON SHELAH

5 There is a superlimit model in 1

Here we make

5.1 Hypothesis. Like 4.8, but also 20 < 21 .(Note that we can assume that K0 is the class of atomic models of a first order

complete countable theory).This section is the deepest (of this paper = chapter). The main difficulties are

proving the facts which are obvious in the context of [Sh 48]. So while it was easy toshow that every p D(N) is definable over a finite set (D(N) is defined below),it was not clear to me how to prove that if you extend the type p to q D(M)where N K M K0 , by the same definition, then q |= p (remember p, q are typesmaterialized not realized, and at this point in the paper we still do not have the

tools to replace the models by uncountable generic enough models). So we ratherhave to show that failure is a non-structure property, i.e., implies existence of manymodels.

Also symmetry of stable amalgamation becomes much more complicated. Weprove existence of stable amalgamation by four stages (5.26,5.27(3),5.30,5.32). Thesymmetry is proved as a consequence of uniqueness of one sided amalgamation, (soit cannot be used in its proof). Originally the intention was the culmination ofthe section to be the existence of a superlimit models in 1 (5.39). This seems anatural stopping point as it seems reasonable to expect that the next step shouldbe phrasing the induction on n, i.e., dealing with n and P(n )-diagrams ofmodels of power as in [Sh 87a], [Sh 87b]; (so this is done in Chapter III). But

less is needed in Chapter II.

5.2 Definition. We define functions D, D with domain K0 .1) For N K0 let D(N) = {p : p is a complete L

01,

(N)-type over N such that forsome a M K0 , N K M and a materializes p in (M, N)}, (i.e. the membersofp have the form (x, a), (x finite and fixed for p) a a finite sequence from N and L01,(N)).2) For N K0 let D

(N) = {p : p a complete L01,(N; N)-type such that forsome a M K0 , N K M and a materializes p in (M, N; N)}.3) For p(x, y) D(N) let p(x, y) x D(N) be defined naturally; i.e. if forsome M, N K M K0 and a b

g(xy)M materializing p(x, y) such thatg(x) = g(a), the sequence a materializes p(x, y) x D(N). Similarly forpermuting the variables.

5.3 Explanation: 0) Recall that any formula in L01,(N) has finitely many freevariables.


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1) So for every finite b N and (x, y) L01,(N), if p D(N), then (x, b) por (x, b) p.2) But a formula from p D(N) may have all c N as parameters whereas a

formula from p D(N) can mention only finitely many members of N.

5.4 Lemma. 1) K has the 0-amalgamation property.2) If N K N K0 , Ai N for i n then for every sentence L1,(N, An, . . . , A1; A0) we have

N1K

or N1K

.

3) IfN K M K0, then everya M materializes in(M, N; N) one and only onetype fromD(N) and also materializes in(M, N) one and only one type fromD(N).

Also for every N K M K0 and q D(N) for some M, M K M K0 andsome b M materializes q in (M; N).4) For every N K0 and countable L L

01,

(N; N) the number of complete

L(N; N)-types p such that N 1K

(x) p is countable; note that pedanticallyL L1,(

+ {c : c N}) and we restrict ourselves to models M such thatPM = |N|, cM = c.5) For N K0 there are countable L

0 L

01,

(N; N) for < 1 increasingcontinuous in , closed under finitary operations (and subformulas) such that:

() for each complete L0-type p we have

[N1K

x p p L0+1].

Hence for everyL01,(N; N) formula (x) for some n(x)


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58 SAHARON SHELAH

9) If f is an isomorphism from N1 K0 onto N2 K0 then f induces a one toone function from D(N1) onto D(N2) and from D

(N1) onto D(N2).

Proof. 1) By 3.8.2) By 1).3) By 2) and 1).4) Like the proof of 4.11 (just easier).5) Like the proof of 4.13(a).6) Like the proof of 4.13(f) (recalling 0.3).7) Clear as in p D(N) we allow more formulas than for q D(N).8),9) Easy, too. 5.4

We shall use from now on a variant of gtp (in Definition 4.3(4) we define gtpL(a; N,

A; A; N).5.5 Definition. 1) If N0 K N1 K0 , a N1, gtp(a, N0, N1) is the p D(N0)such that (N1, N0)

1K

p[a]. So a materializes (but does not necessarily re-alize) gtp(a, N0, N1). We may omit N1 when clear from context. We definegtp(a, N0, N1) D(N0) similarly.2) We say p = gtp(b, N0, N1) is definable over a N0 if gtp(b, N0, N1) = p :={(x, a) p : (x, y) L01,(N0) and a

g(y)(N0) >(N0)} is definable over a(see Definition 5.7 below, note that p p is a one-to-one mapping from D(N0)onto D(N0) by 5.9(1) below). So stationarization is defined for p D(N0), too,after we know 5.9(1).

5.6 Claim. 1) Eachp D(N) does not(L01,(+0),L1,())-split (see Definition

5.7 below; also see more below) over some finite subset C of N, hence p is definableover it.Moreover, letting c list C there is a functiongp satisfyinggp((x, y)) is p,(y, z) L1,() such that for each (x, y) L

01,

(N) and a N we have [(x, a) p N |= p,(a, c)], (in particular, Q is not necessary).2) Every automorphism ofN maps D(N) onto itself and eachp D(N) has at most0 possible images; we may also call them conjugates. So if g is an isomorphismfrom N0 K0 onto N1 K0 then g(D(N0)) = D(N1).

3) If N0 K N1 K N2 K0 and a N1 then gtp(a, N0, N1) = gtp(a, N0, N2).

Before we prove 5.6:

5.7 Definition. Assume

(a) N is a model


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(b) 1 is a set of formulas (possibly in a vocabulary N) closed under negation

(c) 2 is a set of formulas in the vocabulary = N

(d) p is a (1, n)-type over N (i.e., each member has the form (x, a), a from

N, (x, y) from 1, x = x : < n; no more is required (we may allowother formulas but they are irrelevant)

(e) A N.

0) We say p is a complete 1-type over B when:

(i) B N

(ii) (x, b) p b A (x, y) 1

(iii) if (x, y) 1 and b g(y)A then (x, b) p or (x, b) p.

The default value here for 1 is L1,(K).1) We say that p does (1, 2)-split over A when there are (x, y) 1 andb, c g(y)N such that

() (x, b), (x, c) p

() b, c realize the same 2-type over A.

2) We say that p is (1, 2)-definable over A when: for every formula (x, y) 1there is a formula (y, z) 2 and c g

(z)A such that

(x, b) p N |= [b, c]

(x, b) p N |= [b, c]

(in the case p is complete over B, b B we get iff).3) Above we may write 2 instead of (1, 2) when this holds for every 1 (equiv-alently 1 is {(x, y) : (x, a) p}).

5.8 Observation. Assume

(a), (b), (c), (d), (e) as in 5.7 and in addition(d)+ p is a complete (1, n)-type over N, i.e., if

(x, y) 1, d g(y)N, x = x : < n

then (x, d) p or (x, d) d.

Then the following conditions are equivalent:

() p does not (1, 2)-splits over A


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60 SAHARON SHELAH

() there is a sequence of g(x,y) : (x, y) 1 of functions such that:

(i) g(x,y) is a function with domain including {tp2(b ,A,N):

b g(y)N}

(ii) the values ofg(x,y) are truth values

(iii) if (x, y) 1, b g(y)N and q = tp2(b ,A,N) then:

(x, b) p g(x,y)(q) = true, and

(x, b) p g(x,y)(q) = false.

Proof of 5.8. Reflect on the definitions.

Proof of 5.6. 1) Clearly the second sentence follows from the first, so we shallprove the first. Assume this fails. Let (M, a) be such that N K M K0the sequence a M materializes p and clearly for every b M, (M, N) q[b]for some q(x) D(N) and let b : < list N. We choose by induction onn, C0 , C

1 , f, a

0, a

1 :

n2 such that

(a) C is a finite subset of N for < 2, n2

(b) f is an automorphism of N mapping C0 onto C

1

(c) {bg()} C0 C

1 C

0 C

1 for = 0, 1

(d) a

0

, a

1

N realize in N the same L1,()-type over C0

C

1

{b

g()} in(M, N) but aa0, aa1 do not materialize the same L

01,

(+0) in (M, N)(this exemplifies splitting), so (x, y) belongs to the first, (x, y) be-longs to the second (where g(x) = g(a), g(y) = g(a

0))

(e) f(a0) = a

1, f(a

1) = a

1

(f) f C0 f for = 0, 1

(g) a0a1 C

0 C

1.

For n = 0 let C0 , C1 = , f = idN. Recall that K0 is categorical in 0 and

N is countable, hence if n < ,b

,b

n

N realize the same L1,()-type over afinite subset B of N, then some automorphism of N over B maps b to b by atheorem of Scott (see [Ke71]). If (C0 , C

1 , f) are defined and satisfies clauses (a),

(b) we recall that by our assumption toward contradiction as C0 C1 {b

g()}

is a finite subset of N, there are a0, a1

>N as required in clause (d) again.

So clearly there are automorphisms f, f extending f C0 such that

f(a0) = a

1, f(a

1) = a

1 as required in clause (e), (f).


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Lastly, choose C0 = C0 C

1 f

1(C

0) {b

g(), f

1(b

g()),

a0a1, f

1(a

0a

1)} and C

1 = f(C

0).

Having carried the induction, for every 2 clearly f = {fn C0 : n < }

is an automorphism of N.[Why? As fn C

0n : n < is an increasing sequence of functions by

clauses (b) + (c) + (f), the union f is a partial function; as in addition eachf is an automorphism of N by clause (b), also f is a partial automorphism ofN. Recalling b : < n list N, clearly f have domain N by clause (c) andas fn(C

0n) = C

1n the union f has range N by clause (c).] Hence for some

M K0 there is an isomorphism f+ from M onto M extending f. Now for some

p D(N), f(a) materialize p in (M, N). Choose a countable L L01,(+)

which include {(x, y) : >2}. Easily if

2 for = 0, 1 then(x, a1) p0, (x, a

1) p1. So =

2 p L = p L by clauses (d) +

(e), contradiction to 5.4(4) as we can use 0 formulas to distinguish.2) Follows.3) Trivial. 5.6

5.9 Claim. 1) Suppose N0 K N1 K0 and N1 forces that a, b (in N1) real-ize the same L01,(N0)-type over N0, then N1 forces that they realize the sameL01,(N0; N0)-type; (the inverse is trivial).1A) Suppose N0 K N K0 and a

>(N) for = 1, 2 and gtp(a1, N0, N1) =gtp(a2, N0, N1) then we can find (N

+1 , N

+2 , f) such that N1 K N

+1 K0 , N2 K

N+2 K0 and f is an isomorphism from N

+1 onto N

+2 over N0 mappnig a1 to a2.

2) IfN0 K N1 K N2 K0 and a, b N2 (rememberN2 determines the completeL01,(N1)-generic types of a, b) then from the L

01,

(N1)-generic type of a over N1we can compute the L01,(N0)-generic type of a over N0 (hence if the L

01,

(N1)-

generic types of a, b over N1 are equal, then so are the L01,(N0)-generic types ofa, b over N0).3) For every Na K0 there is a one-to-one function f from D(N) onto D

(N)such that: ifN K M K0 anda

>M thenf(gtp(a ,N,M)) = gtpL1,(N;N)(a; N; N; M).

Remark. 1) So there is no essential difference between D(N) and D(N).2) Recall that in a formula ofL01,(N0; N0) all c N0 may appear as individualconstants.

Proof. 1) We shall prove there are N2 such that N1 K N2 K0 and an automor-phism of N2 over N0 taking a to b; this clearly suffices; and we prove the existence


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62 SAHARON SHELAH

of such N2, of course, by hence and forth arguments. We shall use 5.4(2) freely. Soby renaming and symmetry, it suffices to prove that

() if m < , N 0 K

N0 and a,b

m

(N1) materialize the same L0

,(N0)-type over N0 then for every c N1, there are N2 and d N2 such thata < c >, b < d > materialize the same L01,(N0)-type over N0.

However, by the previous claim 5.4 for some a >(N0) the L01,(N0)-typeover N0 that a < c > materialize in (N1, N0) does not L01,(

+0)-split over a.

Now a, b materialize in (N1, N0) the same L01,(N0)-type over N0 hence aa, ab

materialize in (N1, N0) the same L01,(N0)-type. Hence there is N2, N1 K N2 K0 and an automorphism f of N2 mapping N0 onto N1 and mapping a

a toab (but possibly f N0 = idN0), this holds by the last sentence in 4.13(c). Letd = f(c), hence if a < c >, b < d > materialize the same L01,(N0)-type in(N2, N0) then they materialize the same L01,(N0)-type over N0 in (N2, N0).1A) Similarly to part (1).2) Clearly it suffices to prove the hence part. By the assumption and proof of5.9(1) there are N3 satisfying N2 K N3 K0 and f an automorphism ofN3 overN1 taking a to b. Now the conclusion follows.3) Should be clear. 5.9

5.10 Definition. 1) We say that D is a K-diagram function when

(a) D is a function with domain K0 (later we shall lift it to K)

(b) D(N) D(N) and has at least one non-algebraic member for N K0(c) if N1, N2 K0 and f is an isomorphism from N1 onto N2 then f maps

D(N1) onto D(N2), this applies in particular to an automorphism ofN K0 .

1A) Such D is called weakly good when:

(d) () D(N) is closed under subtypes, that is: if p(x) D(N), x = x : < m, is a function from {0, . . . , m1} into {0, . . . , n1} then some (nec-essarily unique) q(x0, . . . , xn1) D(N) is equal to {(x0, . . . , xn1) :

(x(0), . . . , x(m1)) p(x)}() if N K M K0 , a1, b1

>N, a2 g(a1)M and (M, a1) = (M, a2)

and gtpL1,(+)(a2; N; M) D(N) then for some M+, b2 we have M K

M+ K0 , b2 g(b1)(M+), (M+, a1b1) = (M

+, a2b2) and gtpL1,(+)(a2b2; N; M+)

() ifN K M K0 , a >M and b >N and gtpL1,(+)(a; N; M)

D(N) then gtpL1,(+)(ab; N; M) D(N).


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64 SAHARON SHELAH

(f)+ if : is increasing continuous sequence of countable ordinals, > 0and N : is K-increasing continuous, N K0 , for every a N+1,gtp(a, N, N+1) D(N) and for every < for some [, ), N+1 is

(D(N), 0)

-homogeneous then N is (D(N0), 0)

-homogeneous(g) N1 is (D(N0), 0)

-homogeneous if and only ifN1 is (D(N0), 0)

-homogeneouswhere N0 K N1 K0

(h) D is a very good countable K-diagram function.

2) If D is very good then clauses (d),(e),(f),(f)+ hold for it (and also (g), definingD as f(D), f from 5.16(3).

5.13 Remark. 1) We can add

(h) ifK,


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5.14 Definition. Assume N0 K N1 K0 and D is a K-diagram.1) We say that (N1, N0) or just N1 is (D(N0), 0)-homogeneous over N0 (but wemay omit the over N0) if:

(a) every a N1 materializes in (N1, N0) over N0 some p D(N0) and everyq D(N0) is materialized in (N0, N1) by some b N1

(b) if a, b N1, a, b materialize in (N1, N0) the same type over N0 and c N1then for some d N1 sequence a < c >, b < d > materialize in (N1, N0)the same type from D(N0).

2) Similarly for (D(N0), 0)-homogeneity, pedantically we have to say (N1, N0; N0)

is (D(N), 0)-homogeneous, but normally say N1 is.

5.15 Remark. 1) Now this is meaningful only for N K M K0 , but later itbecomes meaningful for any N K M K.2) Uniqueness for such countable models hold in this context too.

Now by 5.9.

5.16 Conclusion. If (N1, N0) is (D(N0), 0)-homogeneous then N1, i.e. (N1, N0, c)cN0

is (D(N0), 0)-homogeneous.

Proof. This is easy by 5.9(1) and clause (g) of 5.12. 5.16

5.17 Lemma. There is N K1 such that N =


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66 SAHARON SHELAH

Uniqueness: For = 0, 1 let N, D ( < 1) be as in 5.12, 5.17 and we should prove


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and let M0 K0 be (D(M+0 ), 0)

-homogeneous so M+0 K M0 , exists by

5.12(1)(e), hence M0 K0 is (D(M0), 0)-homogeneous by 5.12(1)(f) because

0. Now N is (D(N), 0)-homogeneous by 5.12(1), so as 1 is follows

that N is (D(M1), 0)

-homogeneous.By 5.12(1)(d),(e) we can extend f to an isomorphic g from M0 onto N , so

g M+0 is a K-embedding of M+0 into N.

We can deduce N is a model-homogeneous directly; let M0, M1 K N be

countable, and f is isomorphic from M0 onto M1. Let < 1 be such thatM0, M1 K N , Let be such that {gtp(a, M, N) : a

>(N)} D(M)for = 0, 1 and let = max{, 0, 1} + 1. As above N is (D(M), 0)

-homogeneous, and now we choose an automorphism f of N increasing with and extended f for [, 1) by induction. Now {f : (, 1)} is anautomorphism of N extending f. 5.18

5.19 Definition. 1) IfN0 K N1 K0 and p D(N) for = 1, 2 and they aredefinable in the same way (see Definition 5.7 (and 5.6), so in particular both donot split over the same finite subset of N0), then we call p1 the stationarization ofp0 over N1.2) For p D(N) for = 0, 1 let p1 |= p0 mean that N0 K N1 and ifN1 K N2 K0 and a N2 materializes p1 then it materializes p0.

5.20 Remark. It is easy to justify the uniqueness implied by the stationarization.

Observe5.21 Claim. If p = gtp(a, N, N2) for = 0, 1 and N0 K N1 K N2 K0 thenp1 |= p0.

Proof. Easy. 5.21

5.22 Claim. 1) Suppose N0 K N1 K N2 K0 , ai Ni, (for i = 0, 1, 2),a0 a1 a2, i.e. the ranges increase, gtp(a1, N0, N1) is definable over a0 andgtp(a2, N1, N2) is definable over a1. Then gtp(a2, N0, N2) is definable over a0.Moreover, the definition depends only on the definitions mentioned previously.2) If N0 K N1 K N2 and p D(N) for = 0, 1, 2 and p+1 is the station-arization of p over N+1 for = 0, 1, then p2 is the stationarization of p0 overN2.

Proof. 1) So we have to prove that gtp(a2, N0, N2) does not split over a0. Letn < and b, c nN0 realize the same type in N0 over a0 (in the logic L1,(K),


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68 SAHARON SHELAH

or even first order logic when every N K0 is atomic). Now also ba1, ca1materialize the same L1,(N0)-type in N1 hence they realize the same L1,(K)-type (recall 5.4(8)). Hence b, c realize the same L1,(K)-type in N1 over a1 in N1.

But gtp(a2, N0, N2) does not split over a1, so by the previous sentence we get thatba2, ca2 materializes the same L1,(N0)-type in N2.2) Easy. The moreover is proved similarly. 5.22

5.23 Lemma. Suppose N0 K N1 K0 , p D(N) and p1 is a stationarizationof p0 over N1, then p1 |= p0, i.e., every sequence materializing p1 materializes p0in any N2 such that N1 K N2.

Remark. 1) In [Sh 48], [Sh 87a], [Sh 87b] and [Sh:c] the parallel proof of the claims

were totally trivial, but here we need to invoke I(1, K) < 21

.2) A particular case can be proved in the context of 4.

Proof. So suppose N0, N1, p0, p1 contradict the claim and let a N0 be such that

p0 is definable over a so p1, too. By 5.12(e)+(f) there are < 1 and N2 K0

satisfying N1 K N2 such that N2 is (D(N), 0)

-homogeneous for = 0, 1. Wecan find p2 D(N2) which is the stationarization of p0, p1. It is enough to provethat p2 |= p1.[Why? First, note that there is an automorphism f of N2 which maps N1 onto N0and f(a) = a hence f(p2) = p2, f(p1) = p0 hence p2 |= p0. Now assume that

N1 K N+1 K0 and a1 N+1 materializes p1 clearly we can find N+2 , a2 suchthat N2 K N

+2 K0 and a2 N

+2 which materializes p2, as we are assuming

p2 |= p1 it also materializes p1 hence there are N3, f such that N+1 K N3 K0

and f is a K-embedding of N+2 into N3 over N1 mapping a2 to a1. But p2 |= p0

(see above) hence f(a2) = a1 materializes p0 and p1, too.]So without loss of generality for some

N1 is (D(N0), 0)

-homogeneous over N0.

For N K0 , N0 K N, let pN be the stationarization of p over N; so

1 if N0 K N K0 then pN is definable over a.

Without loss of generality the universes of N0, N1 are , 2 respectively.Now we choose by induction on a model N K0 ( < 1), |N| = (1 +

), [ < N K N]; N0, N1 are the ones mentioned in the claim anda N+1 materializes the stationarization p D(N) of p0 over N andfor < , N is (D

(N), 0)-homogeneous (see 5.12(f),(f)

+). Recalling that K is


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categorical in 0 (and uniqueness over N0 of (D(N0), 0)-homogeneous models)we have > (N, N) = (N1, N0) so, recalling clearly a does not materi-alize pN (in N+1). Let N = {N : < 1}. Let B be (H(2), ) expanded

by N, K H(2), K H(2) and anything else which is necessary. Let B

bea countable elementary submodel ofB to which N : < 1, N belong and let() = B 1. For any stationary co-stationary S 1, let BS be a model whichis

() BS an elementary extension ofB

() BS is an end extension ofB for 1, that is, ifBS |= s < t are countable

ordinals and t B then s B

() among the BS-countable ordinals not in B there is no first one

() the set of countable ordinals ofBS is IS , IS =

and (, 1)does a sequence from NS materialize both p = pNS and its stationarization

pNS over NS in N

S (again remember N

S = N

BS

s()because S)

and similarly

(c) for a closed unbounded set of > , NS is (D(N

S ), 0)

-homogeneous.


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70 SAHARON SHELAH

We shall prove that every < 1,

if / S then cannot satisfy the statement (c) above.

This is sufficient because if S(1), S(2) 1 are stationary co-stationary, f is anisomorphism from NS(1) onto NS(2) mapping a to itself, then for a closed un-

bounded set E 1, for each < 1 the function f maps NS(1) onto N

S(2) , hence

the property above is preserved, hence S(1) E = S(2) E. But there is a sequenceSi : i < 21 of subsets of 1 such that for i = j the set Si\Sj is stationary. So by

0.3 we have I(1, K) = 21 , contradiction.

So suppose 1\S, p = pNS and clause (c) above hold; but obviously (c)

(a), recalling p0 D(N0) hence pNS D(NS ) so let a N

S materialize p in NS

and we shall get a contradiction.There are elements 0 = t(0) < t(1) < . . . < t(k) of IS and a0 N0 =

NBSt(0)

, a+1 NBSt()+1 such that a ak, a a0, a a+1 and gtp(a+1, NBSt() , N

BS

t(+1))

is definable over a and if t( + 1) is a successor (in IS) then it is the successor oft() and if limit in IS then a = a+1.[Why do they exist? Because of the sentence saying that for every a we can findsuch k, t()( k), a( k) as above is satisfied by B and involve parameterswhich belong to B hence to BS, etc., so BS inherits it (and finiteness is absolute

from BS)]. It follows that gtp(a, NBS

t(), NBSt(k)

) is definable over a for each < k.

Clearly t(0) = 0 I but t(k) / I (otherwise t(k)+1 I hence a NBS

t(k)+1 K

NS , impossible as p is a non-algebraic type over NBS ). Hence for some we have

t() I, t( +1) / I. By the construction t( + 1) is limit (in IS

) hence a+1 = a.As / S we can choose t() IS\IS , t() < t( + 1). As we are assuming (towardcontradiction) that , p satisfy clause (c), for some S, s() is well defined ands() > t(k) (on the definition of s() for S see clause () above) and NS is

(D(NS ), 0)

-homogeneous. Now NBSs() = N

S , N

BS

t(+1) are isomorphic over Nt()

(being both (D(NBS

t()), 0)-homogeneous by the choice ofBS, see above).

So as NS K NBS

t(+1) K NBS

s() = NS and, as said above, N

S is (D

(N

S ), 0)

-

homogeneous also NBSt(+1) is (D

(N

S ), 0)

-homogeneous, too, hence (NBSt(+1), N

S , a

) =

(N1, N0, a).

As, by above, clearly NS

, NBS

t() are (D

(NBS

t()+1)), 0)

-homogeneous there isan isomorphism f0 from N

S onto N

BS

t() over NBS

t()+1. As NBS

t(+1) is (D(N

BS

t()), 0)-

homogeneous and (D(NS ), 0)

-homogeneous by the previous paragraph (where

we use ) we can extend f0 to an automorphism f1 of NBS

t(+1). Let S E

satisfy s() t(k)+1. As gtp(ak, NBS

t(+1), NS ) is definable over a = a+1 and a =

f0(a) = f1(a) (as a NBS

t()+1) and NS+1 is (D

(N

BS

t(+1)), 0)-homogeneous, we


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can extend f1 to an automorphism f2 of NS satisfying f2(ak) = ak.

Notice that by the choice of a : k and t() : k it follows that for anym < k, gtp(ak, Nt(m), Nt(k)+1) does not split over am hence is definable over it by

5.22, and recall that we know that a = a+1.So there is in NS a sequence materializing both gtp(a, NS , N

S ) = pNS and its

stationarization over NSt(+1): just a( ak) (so use f2).

This contradicts the assumption as (N1, N0, a) = (NBSt(+1), N

S , a

). 5.23

The following claim 5.24(5)-(9) and Definition 5.25 are closely related.

5.24 Claim. 1) If a N0 K N1 K N2 K0 , b N2, p1 = gtp(b, N1, N2) isdefinable over a N0, then p0 = gtp(b, N0, N2) is definable in the same way overa, hence gtp(b, N1, N2) is its stationarization.

2) For a fixed countable M K0 to have a common stationarization in D(N

) for some N satisfying M K N or N K M is an equivalence relation over {p:

for some N K M, p D(N)} (and we can choose the common stationarizationin D(M) as a representative). So if N0 K N1 K N2 K0 , p D(N) for = 0, 1, 2 and p1, p2 are stationarizations of p0 then p2 |= p1.3) If N K0 ( + 1) is K-increasing and continuous and a N+1 then for some n < , for every k we have: n < k implies gtp(a, N, N+1) isthe stationarization of gtp(a, Nk, N+1).4) If N K M K, N K0 , a M then for all M

K0 , satisfying a M, N K M

K M, gtp(a ,N,M) is the same, we call it gtp(a ,N,M) (the new

point is that M is not necessarily countable. This is compatible with Definition5.25(c) being a special case).5) Suppose N0 K N1 (in K), a N1, then there is a countable M K N0, suchthat for every countable M satisfying M K M

K N0 we have gtp(a, M, N1) is

the stationarization of gtp(a ,M,N1). Moreover there is a finite A N0 such thatany countable M K N0 which includes A is O.K. So gtp(a, N0, N1) from 5.25(c)is well defined and D(N0) and is definable over some finite A N0.6) The parallel of Part (3) holds for N K, too, and any limit ordinal instead of. That is if N : + 1 is K-increasing continuous and a N+1, then forsome < and countable M K N we have: M K M

K M gtp(a, M, M)

is the stationarization of gtp(a ,M,M); similarly for every p D(N).

7) IfN0 K N1 K N2 K N3 K N4 and a N4 and gtp(a, N3, N4) is the station-arization ofgtp(a, N0, N4) thengtp(a, N2, N4) is the stationarization ofgtp(a, N1, N3).Also if b satisfies Rang(b) Rang(a) and gtp(a, N2, N4) is the stationarizationof gtp(a, N1, N4) then this holds also for b. We can replace gtp(a, N3, N4) byp D(N4).8) If N0 K N1 K N2 K0 and p D(N) for = 0, 1, 2 and p+1 is thestationarization of p for = 0, 1 then p2 is the stationarization of p0.


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9) If M : + 1 is K-increasing continuous, a limit ordinal and a >(M+1) then

(a) for some < we have gtp(a, M, M+1) is the stationarization ofgtp(a, M, M+1)whenever [, )

(b) if gtp(a, M, M+1) is the stationarization of gtp(a, M0, M+1) for every < then this holds for = , too.

10) IfM : is K-increasing continuous, a limit ordinal and p D(M),then for some < there is p D(M) such that p is the stationarization ofp.11) Those definitions in 5.25 are compatible with the ones for countable models.12) gtp(a ,N,M) (where a M, N K M are both is K) is the stationarizationover N of gtp(a, N, M) for every large enough countable N K N, see 5.24(5).

Proof. 1) As we can replace N2 by any N2 satisfying N2 K N

2 K0 ,. without

loss of generality for some , N2 is (D(N0), 0)

-homogeneous and (D(N1), 0)-

homogeneous. Let p2 D(N2) be the stationarization of p1 over N2.So by 5.23 we get p2 |= p1. On the other hand, clearly there is an isomorphism

f0 from N0 onto N1 such that f0(a) = a; and by the assumption above on N2, f0can be extended to an automorphism f1 of N2.

Note that f1 maps p0 = gtp(b, N0, N2) to p0 := gtp(f1(b), f1(N0), N2) and maps

p2 to itself as f0(a) = a.

Now p1 |= p0 (by the choices of p1, p0) and p2 |= p1 by 5.9(1), together p2 |=p0. As f1(p2) = p2, f1(p0) = p

0 it follows that p2 |= p

0. As also p2 |= p1 and

p0, p1 D(N1) it follows that p0 = p1 hence p1, p

0 have the same definition over

a, but now also p0 D(N0), p0 D(N1) have the same definition over a (usingf1), together also p1, p0 have the same definition over a, which means that p1 is thestationarization ofp0 over N1 and we are done.2) Trivial.3) By part (1).4) Easy.5) By (3) and (4).

6)-12) Easy by now. 5.24

5.25 Definition. By 5.24(5) the type gtp(a ,M,N) can be reasonably defined whenM K N, a >N and we can define D(N) and D(N), gtp(a ,N,M) and sta-tionarization for not necessarily countable N and N K M K. Everything stillholds, except that maybe some ps are not materialized in any K-extension of N.

More formally


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(a) ifN K M and N K0 and p D(N) then the stationarization of p overM is {q : N1 K0 satisfies N1 K N1 K M and q is the stationarizationof p D(N1)}

(b) ifM K then D(M) = {q: for some countable N K M and p D(N) thetype q is the stationarization of p over M}, similarly for D, a K-diagram

(c) ifN K M and a >M then gtp(a ,N,M) is defined as {gtp(a, N, M) :

N0 K N K M

K0 , M K M, N

K N} for every countable largeenough N0 K N; it is well defined and belongs to D(N) by 5.24(5) and wesay a materializes gtp(a ,N,M) in M

(d) if N K, N K M and p D(N) is definable over the countable N0 K Nequivalently is the stationarization of some p D(N0), then the station-arization of p over M is the stationarization of p over M, see clause (a),equivalently {p

M0: N

0K M

0K M, M

0is countable} where p

M0is the

stationarization of p D(N0) over M0; it belongs to D(N0)

(e) if p(x, y) D(M) then p(x, y) x D(M) is naturally defined; 5.2(3)similarly for permuting the variables

(f) for N K M we say that M is (D(N), 0)-homogeneous when for every

p(x, y) D(N) and a g(x)(M) materializing p(x, y) x in M there isb g(y)M such that ab materializes p(x, y) in M.

Remark. Claim 5.26 below strengthens 3.8, it is a step toward non-forking amalga-mation.

5.26 Claim. Suppose N0 K N1 K0 , N0 K N2 K0 , a N1. Then wecan find M, N0 K M K0 and K-embeddings f of N into M over N0 (for = 1, 2) such thatgtp(f1(a), f2(N2), M) is a stationarization ofp0 = gtp(a, N0, N1)(so f1(a) / f2(N2)).

Proof. Let p2 D(N2) be the stationarization of p0. Clearly we can find an < 1 (in fact, a closed unbounded set of s) and N1, N2 from K0 which are(D(N0), 0)

-homogeneous and N K N (for = 1, 2) and some b N

2 materi-

alizing p2. But by 5.23, b materializes p0 hence there is an isomorphism f from N1

onto N2 over N0 satisfying f(a) = b, recalling 5.9(1A). Now let M = N2, f1 = f

N1, f2 = id. 5.26


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74 SAHARON SHELAH

5.27 Claim. 1) For any N0 K N1 K1 so N0 K1 , there is N2 such thatN1 K N2 K1 and N2 is (D(N0), 0)

-homogeneous.2) Also 5.26 holds for N2 K1 (but still N0, N1 K0).

3) IfN0 K N1 K0 andN0 K N2 K1 then we can findM K1 and K-embeddings f1, f2 ofN1, N2 into M overN0 respectively such thatgtp(f1(c), f2(N2), M)is a stationarization ofgtp(c, N0, N1) for everyc N1, hence f1(N1)f2(N2) = N0.4) K2 = .

Remark. 1) Note that 5.27(3) is another step toward stable amalgamation.2) Note that 5.27(3) strengthen 5.27(2) hence 5.26.

Proof. 1) As we can iterate K-increasing N1 in K1 , it is enough to prove that:

if p(x, y) D(N0) and a N1 materializes p(x, y) x in (N1, N0) then for someN2 K1 , N1 K N2 and for some b N2 the sequence ab materializes p(x, y)in (N2, N0). Let M0 K N0 be countable and q

saharon shelah- abstract elementary classes near ℵ1

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