aa r X i v : . [ m a t h . L O ] J a n GOOD FRAMES WITH A WEAK STABILITY
ADI JARDEN AND SAHARON SHELAH
Abstract.
Let K be an abstract elementary class of models. Assumethat there are less than the maximal number of models in K λ + n (namelymodels in K of power λ + n ) for all n . We provide conditions on K λ ,that imply the existence of a model in K λ + n for all n . We do this byproviding sufficiently strong conditions on K λ , that they are inheritedby a properly chosen subclass of K λ + . Contents
1. Introduction 21.1. The required knowledge 21.2. The assumptions 31.3. Notations 32. Non-forking frames 102.1. non-forking with greater models 183. The decomposition and amalgamation method 223.1. The a.e.c. ( K ,bs , (cid:22) bs ) and amalgamations 223.2. Decomposition 263.3. A disjoint amalgamation 284. Uniqueness triples 295. Non-forking amalgamation 375.1. The axioms of non-forking amalgamation 375.2. The relation N F K λ + that is based on the relation N F ≺ + and saturated models 548. relative saturation 669. Non-smoothness implies non-structure 7110. a good λ + -frame 7411. Conclusions 8012. Comparison to [Sh 600] 81Acknowledgment 82References 82 This work is supported by Number 875 on Shelah’s list of publications. It is a part ofthe PHD thesis of the first author with the supervisor of Boaz Tsaban. Introduction
The book classification theory, [Sh:c], of elementary classes, i.e. classesof first order theories, presents properties of theories, which are so called“dividing lines” and investigates them. When such a property is satisfied,the theory is low, i.e. we can prove structure theorems, such as:(1) The fundamental theorem of finitely generated Abelian groups.(2) ArtinWedderburn Theorem on semi-simple rings.(3) If V is a vector space, then it has a basis B, and V is the direct sumof the subspaces span { b } where b ∈ B .(we do not assert that these results follow from the model theoretic analysis,but they merely illustrate the meaning of ‘structure’). But when such aproperty is not satisfied, we have non-structure , namely there is a witnessthat the theory is complicated, and there are no structure theorems. Thiswitness can be the existence of many models in the same power.There has been much work on classification of elementary classes, andsome work on other classes of models.The main topic in the new book, ([Sh:h]), is abstract elementary classes (In short a.e.c.). There are two additional books which deal with a.e.c.s([Ba:book] and [Gr:book]).From the viewpoint of the algebraist, model theory of first order theo-ries is somewhat close to universal algebra. But he prefers focusing on thestructures, rather than on sentences and formulas. Our context, abstractelementary classes, is closer to universal algebra, as our definitions do notmention sentences or formulas.As superstability is one of the better dividing lines for first order theories,it is natural to generalize this notion to a.e.c.s. A reasonable generalizationis that of the existence of a good λ -frame, (see Definition 2.1 on page 11),introduced in [Sh 600]. In [Sh 600] we assume existence of a good λ -frameand either get a non-structure property (in λ ++ , at least where 2 λ < λ + < λ ++ ) or derive a good λ + -frame from it. Our paper generalizes [Sh 600],weakening the assumption of a good λ -frame, or more specifically weakeningthe basic stability assumption.1.1. The required knowledge.
We assume basic knowledge in set the-ory (ordinals, cardinals, closed unbounded subsets and stationary subsets).In model theory, we just assume the reader is familiar with notions, everystudent in algebra knows (theory, model=structure, isomorphism and em-bedding). Especially we do not assume the reader is familiar with formulaand elementary substructure, as here we do not deal with those notions (ex-cept in one example). Of course, we do not assume the reader has read anypaper in abstract elementary classes, and if the reader prefers to translatea model as a group, he will not lose the main ideas. We sometimes refer toanother paper, for the following four tasks:
OOD FRAMES WITH A WEAK STABILITY 3 (1) To convince the reader that an assumption is reasonable, i.e. toprove that we can conclude something from its negation.(2) To give examples.(3) To compare it with [Sh 600].(4) To point out its continuations.The best way to read this paper is to read it until its end, before readingany reference.1.2.
The assumptions.
When we write a hypothesis, we assume it untilwe write another hypothesis, but usually we recall the hypothesis in thebeginning the following section. When we write ‘but we do not use localcharacter’, the reader may wonder why? The answer is that we want toapply theorems we prove here, in papers, in which local character is notassumed (for example [JrSh 940]). For the same reason, in Hypothesis 3.1we assume weak assumptions.1.3.
Notations.
We use the letters m, n, k, l for natural numbers or integernumbers, α, β, γ, i, j, ε, ζ for ordinal numbers, δ for a limit ordinal number, κ, λ, µ for cardinal numbers, p, q for types, P for a set of types, K for a classof models, k , s , U for specific uses. Definition 1.1 (Abstract Elementary Classes) . (1) Let K be a class of models for a fixed vocabulary and let (cid:22) = (cid:22) be a2-place relation on K . The pair ( K, (cid:22) ) is an a.e.c. if the followingaxioms are satisfied:(a) K, (cid:22) are closed under isomorphisms. In other words, if M ∈ K , M (cid:22) M and f : M → N is an isomorphism then N ∈ K and f [ M ] (cid:22) N .(b) (cid:22) is a partial order and it is included in the inclusion relation.(c) If h M α : α < δ i is a continuous (cid:22) -increasing sequence, then M (cid:22) [ { M α : α < δ } ∈ K. (d) Smoothness: If h M α : α < δ i is a continuous (cid:22) -increasing se-quence, and for every α < δ, M α (cid:22) N , then [ { M α : α < δ } (cid:22) N. (e) If M ⊆ M ⊆ M and M (cid:22) M ∧ M (cid:22) M , then M (cid:22) M .(f) There is a Lowenheim Skolem Tarski number, LST ( K, (cid:22) ), whichis the first cardinal λ , such that for every model N ∈ K and asubset A of it, there is a model M ∈ K such that A ⊆ M (cid:22) N and the cardinality of M is ≤ λ + | A | .(2) ( K, (cid:22) ) is an a.e.c. in λ if: The cardinality of every model in K is λ ,and it satisfies axioms a,b,d,e of a.e.c. (Definition 1.1.1), and axiom1.1.1.c for sequences h M α : α < δ i with δ < λ + . Remark 1.2.
ADI JARDEN AND SAHARON SHELAH (1) If K is a class of models for a fixed vocabulary, then ( K, ⊆ ) satisfiesaxioms b,d,e of a.e.c. (Definition 1.1.1).(2) Suppose ( K, (cid:22) ) is an a.e.c.. If ( K, ⊆ ) satisfies axiom 1.1.1.c, then( K, ⊆ ) is an a.e.c..(3) If ( K, (cid:22) ) is an a.e.c. and K ′ ⊆ K then ( K ′ , (cid:22) ↾ K ′ ) satisfies axiomsb,d,e of a.e.c. (Definition 1.1.1).We give some simple examples of a.e.c.s. One can see more examples in[Gr 21]. Example 1.3.
Let T be a first order theory. Denote K =: { M : M | = T } .Define M (cid:22) N if M is an elementary submodel of N . ( K, (cid:22) ) is an a.e.c.. Example 1.4.
Let T be a first order theory with Π axioms, namely axiomsof the form ∀ x ∃ yϕ ( x, y ) [For example ( ∀ x, y )( x + y = y + x ) is OK, as it isequivalent to the Π axiom ( ∀ x, y ) ∃ z ( x + y = y + x )]. Denote K =: { M : M | = T } . Then ( K, ⊆ ) is an a.e.c.. Example 1.5.
The class of locally-finite groups (the subgroup generated byevery finite subset of the group is finite) with the relation ⊆ is an a.e.c.. Example 1.6.
Let K be the class of groups. Let (cid:22) =: { ( M, N ) :
M, N aregroups, and M is a pure subgroup of N } ( M is a pure subgroup of N if andonly if N | = ( ∃ y ) ry = m implies M | = ( ∃ y ) ry = m for every integer r andevery m ∈ M ). ( K, (cid:22) ) is an a.e.c.. Example 1.7.
The class of models that are isomorphic to ( N , < ) with therelation ⊆ is not an a.e.c., as it does not satisfy axiom 1.1.1.c: S {{− n, − n +1 , − n + 2 .. , , ... } : 0 ≤ n } is isomorphic to ( Z , < ) although {− n, − n +1 , − n + 2 .. , , ... } is isomorphic to ( N , < ).But the class of models that are isomorphic to ( N , , < ) with the relation ⊆ is an a.e.c., (the relation ⊆ in this case is actually the equality, and thisa.e.c. has just one model). Example 1.8.
The class of
Banach spaces with the relation ⊆ is not ana.e.c., as it does not satisfy axiom 1.1.1.c. Example 1.9.
The class of sets (i.e. models without relations or functions)of cardinality less than κ , where ℵ ≤ κ and the relation is ⊆ , is not ana.e.c., as it does not satisfy axiom 1.1.1.c.The class of sets with the relation (cid:22) = { ( M, N ) : M ⊆ N and || N − M || >κ } where ℵ ≤ κ , is not an a.e.c., as it does not satisfy smoothness (axiom1.1.1.d). Definition 1.10.
We say M ≺ N when M (cid:22) N and M = N . Definition 1.11. K λ =: { M ∈ K : || M || = λ } , K <λ = { M ∈ K : || M || <λ } , etc.By the following proposition we can replace the increasing continuoussequence in axioms c,d in Definition 1.1 by a directed order. OOD FRAMES WITH A WEAK STABILITY 5
Proposition 1.12.
Let ( K, (cid:22) ) be an a.e.c., I be a directed order and sup-pose that for s, t ∈ I we have M s ∈ K and s ≤ I t ⇒ M s (cid:22) M t . Then:(1) M (cid:22) S { M s : s ∈ I } ∈ K. (2) If for every s ∈ I, M s (cid:22) N ∈ K , then S { M s : s ∈ I } (cid:22) N. Proof.
We prove the two items of the proposition simultaneously, by induc-tion on | I | . For finite I , there is nothing to prove, so assume I is infinite.There is an increasing continuous sequence of subsets of I , h I α : α < | I |i ,such that | I α | < | I | . Denote M I α := S { M s : s ∈ I α } and M I := S { M s : s ∈ I } . If α < β < | I | then by item (1) of the induction hypothesis, s ∈ I α ⇒ M s (cid:22) M I α . But as I α ⊆ I β , s ∈ I β , so M s (cid:22) M I β . So by item (2)of the induction hypothesis, M I α (cid:22) M I β . Hence the sequence h M I α : α < | I |i is increasing. But it is also continuous, as the sequence h I α : α < | I |i is con-tinuous. So by axiom c of Definition 1.1 M I α (cid:22) M I ∈ K . So as (cid:22) istransitive and M s (cid:22) M I α for s ∈ I α , we have M s (cid:22) M I ∈ K . Hence we haveproved item (1) of the proposition for the cardinality | I | . Now we prove item(2) of the proposition for | I | . If for every s ∈ I, M s (cid:22) N ∈ K , then by item(2) of the induction hypothesis, for α < | I | , we have M I α (cid:22) N ∈ K , hence wecan apply axiom (d) of Definition 1.1 for the increasing continuous sequence h M I α : α < | I |i , so S { M I α : α < | I |} (cid:22) N . But M I = S { M I α : α < | I |} . ⊣ Definition 1.13. ( K, (cid:22) ) up := ( K up , (cid:22) up ) where we define:(1) K up is the class of models with the vocabulary of K , such that thereare a directed order I , and a set of models { M s : s ∈ I } such that: M = S { M s : s ∈ I } and s ≤ I t ⇒ M s (cid:22) M t .(2) For M, N ∈ K up , M (cid:22) up N iff there are directed orders I, J andsets of models { M s : s ∈ I } , { N t : t ∈ J } respectively such that: M = S { M s : s ∈ I } , N = S { N t : t ∈ J } , I ⊆ J, s ≤ J t ⇒ N s (cid:22) N t , s ≤ I t ⇒ M s (cid:22) M t (cid:22) N t . Proposition 1.14.
If(1) ( K , (cid:22) ) , ( K , (cid:22) ) are a.e.c.s in λ .(2) K ⊆ K .(3) (cid:22) ↾ K is (cid:22) .Then K up ⊆ K up and ( (cid:22) ) up ↾ K up is ( (cid:22) ) up .Proof. Easy. ⊣ Fact 1.15 (Lemma 1.23 in [Sh 600]) . Let ( K, (cid:22) ) be an a.e.c. in λ . Then(1) ( K, (cid:22) ) up is an a.e.c..(2) ( K up ) λ = K .(3) (cid:22) up ↾ K is (cid:22) .(4) LST ( K, (cid:22) ) up = λ . Definition 1.16. (1) Let
M, N be models in K , f is an injection of M to N . We saythat f is a (cid:22) - embedding and write f : M → N , or shortly f is an ADI JARDEN AND SAHARON SHELAH embedding (if (cid:22) is clear from the context), when f is an injectionwith domain M and Im ( f ) (cid:22) N .(2) A function f : B → C is over A , if A ⊆ B T C and x ∈ A ⇒ f ( x ) = x . Definition 1.17. (1) K K, (cid:22) =: { ( M, N, a ) :
M, N ∈ K, M (cid:22)
N, a ∈ N } . When the class( K, (cid:22) ) is clear from the context we omit it writing K .(2) K λ := { ( M, N, a ) :
M, N ∈ K λ , M (cid:22) N, a ∈ N } . Definition 1.18. (1) E ∗ K, (cid:22) is the following relation on K K, (cid:22) : ( M , N , a ) E ∗ ( M , N , a )iff M = M and for some N ∈ K λ with N (cid:22) N there is anembedding f : N → N over M with f ( a ) = a .(2) E K, (cid:22) is the closure of E ∗ K, (cid:22) under transitivity, i.e. the closure to anequivalence relation.When ( K, (cid:22) ) is clear from the context we omit it writing E ∗ , E . Definition 1.19. (1) We say that ( K λ , (cid:22) ↾ K λ ) has amalgamation when: For every M , M , M in K λ , such that n < ⇒ M (cid:22) M n , there are f , f , M such that: f n : M n → M is an embedding over M , i.e. the diagram belowcommutes. In such a case we say that M is an amalgam of M , M over M . M f / / M M O O id / / M f O O (2) we say that K λ has joint embedding when: If M , M ∈ K λ , thenthere are f , f , M such that for n = 1 , f n : M n → M is anembedding and M ∈ K λ .(3) A model M in K λ is superlimit when:(a) If h M α : α ≤ δ i is an increasing continuous sequence of modelsin K λ , δ < λ + and α < δ ⇒ M α ∼ = M , then M δ ∼ = M .(b) M is (cid:22) -universal.(c) M is not (cid:22) -maximal.(4) M ∈ K is (cid:22) - maximal if there is no N ∈ K with M ≺ N . Proposition 1.20. (1) For every
M, N , N ∈ K λ , a ∈ N − M and b ∈ N − M , ( M, N , a ) E ∗ ( M, N , b ) iff there is an amalgamation N, f , f of N , N over M such that f ( a ) = f ( b ) .(2) E ∗ is a reflexive, symmetric relation. OOD FRAMES WITH A WEAK STABILITY 7 (3) If ( K λ , (cid:22) ↾ K λ ) has amalgamation, then E ∗ λ is an equivalence rela-tion.Proof. Easy. ⊣ Definition 1.21. (1) For every (
M, N, a ) ∈ K let tp K, (cid:22) ( a, M, N ), the type of a in N over M , be the equivalence class of ( M, N, a ) under E K, (cid:22) When the class( K, (cid:22) ) is clear from the context we omit it, writing tp ( a, M, N ) (Inother texts, it is called ‘ ga − tp ( a/M, N )’).(2) For every M ∈ K , S ( M ) := { tp ( a, M, N ) : ( M, N, a ) ∈ K } .(3) If p = tp ( a, M , N ) and M (cid:22) M , then we define p ↾ M = tp ( a, M , N ), Remark 1.22.
By the definitions of
E, E ∗ it is easy to check that p ↾ M does not depend on the representative of p . Proposition 1.23.
For every
M, N, N + ∈ K and a ∈ N − M with M S { a } ⊆ N (cid:22) N + , tp ( a, M, N ) = tp ( a, M, N + ) .Proof. Easy. ⊣ Definition 1.24.
Suppose M (cid:22) N .(1) For p ∈ S ( M ), we say that N realizes p if for some a ∈ N p = tp ( a, M, N ).(2) For P ⊆ S ( M ), we say that N realizes P if N realizes every type inP.(3) For p ∈ S ( M ) and a ∈ N − M , we say that a realizes p, when p = tp ( a, M, N ). Proposition 1.25.
Let
M, M ∈ K λ , M (cid:22) M . Suppose ( K λ , (cid:22) ↾ K λ ) hasamalgamation and LST ( K, (cid:22) ) ≤ λ . Let P be a set of types over M , | P | ≤ λ . Then there is a model N in K λ such that M (cid:22) N and N realizes P.Proof. Easy. ⊣ Definition 1.26.
Let
M, N ∈ K . M is said to be full over N when M satisfies S ( N ). M is said to be saturated in λ + over λ , when for everymodel N ∈ K λ , if N (cid:22) M then M is full over N . Remark 1.27.
This is the reasonable sense of saturated model we can usein our context, as we do not want to assume anything about K <λ , especiallynot stability and not amalgamation, (so a saturated model in λ + over λ maynot be full over a model N ∈ K <λ , N (cid:22) M ), see the following example from[BKS]. Example 1.28.
Let τ contain infinitely many unary predicates P n and onebinary predicate E . Define a first order theory T such that P n +1 ( x ) ⇒ P n ( x ), E is an equivalence relation with two classes, which are each rep-resented be exactly one point in P n − P n +1 for each n . Now let K be ADI JARDEN AND SAHARON SHELAH the class of models in T , that omit the type of two inequivalent pointsthat satisfy all the P n . Then a model M ∈ K is determined up to iso-morphism by µ ( M ) := |{ x ∈ M : ( ∀ n ) P n ( x ) }| . So K is categorical inevery uncountable powers, but has ℵ countable models (none of them isfinite). Now let (cid:22) be the relation of being submodel. Then ( K, (cid:22) ) isan a.e.c. with L.S.T. ( K, (cid:22) ) = ℵ . Let M , M , M ∈ K be such that µ ( M ) = 0 , µ ( M ) = µ ( M ) = 1 and M , M are not isomorphic over M .Then there is no amalgamation of M , M over M . Now if λ > ℵ thenevery model M ∈ K λ + is saturated (over λ ). But it is not saturated over ℵ , since it can not realize tp ( a , M , M ) , tp ( a , M , M ), (where a n is theunique element of M n − M of course). Definition 1.29.
Let M be a model in K λ + . M is said to be homogenous in λ + over λ if for every N , N ∈ K λ with N (cid:22) M ∧ N (cid:22) N , there is a (cid:22) -embedding f : N → M over N . Definition 1.30. A representation of a model M is an (cid:22) -increasing con-tinuous sequence h M α : α < || M ||i of models with union M , such that || M α || < || M || for each α and if || M || = λ + then || M α || = λ for each α .The following proposition is a version of Fodor’s lemma (there is no math-ematical reason to choose this version, but we think that it is comfortable). Proposition 1.31.
There are no h M α : α ∈ λ + i , h N α : α ∈ λ + i , h f α : α ∈ λ + i , S such that the following conditions are satisfied:(1) The sequences h M α : α ∈ λ + i , h N α : α ∈ λ + i are (cid:22) -increasingcontinuous sequences of models in K λ .(2) For every α < λ + f α : M α → N α is a (cid:22) -embedding.(3) h f α : α ∈ λ + i is an increasing continuous sequence.(4) S is a stationary subset of λ + .(5) For every α ∈ S , there is a ∈ M α +1 − M α M λ + − M α ) such that f α +1 ( a ) ∈ N α .Proof. Suppose there are such sequences. Denote M = S { f α [ M α ] : α ∈ λ + } .By clauses 4,5 || M || = K λ + . h f α [ M α ] : α ∈ λ + i , h N α T M : α ∈ λ + i arerepresentations of M . So they are equal on a club of λ + . Hence there is α ∈ S such that f α [ M α ] = N α T M . Hence f α [ M α ] ⊆ N α T f α +1 [ M α +1 ] ⊆ N α T M = f α [ M α ] and so this is an equivalences chain. Especially f α +1 [ M α +1 ] T N α = f α [ M α ], in contradiction to condition 5. ⊣ Proposition 1.32 (saturation = model homogeneity) . Let ( K, (cid:22) ) be ana.e.c. such that K λ has amalgamation, and LST ( K, (cid:22) ) ≤ λ . Let M be amodel in K λ + . Then M is saturated in λ + over λ iff M is a homogenousmodel in λ + over λ .Proof. One direction is trivial, so let us prove the other direction. Suppose M ∗ is saturated in λ + over λ , N , N ⊆ K λ , N (cid:22) N , N (cid:22) M ∗ and thereis no embedding of N to M ∗ over N . Construct by induction on α ∈ λ + atriple ( N ,α , N ,α , f α ) such that: OOD FRAMES WITH A WEAK STABILITY 9 (1) For n < h N n,α : α ∈ λ + i is a (cid:22) -increasing continuous sequence ofmodels in K λ .(2) N , = N , N , = N , f = id ↾ N .(3) For α ∈ λ + , N ,α (cid:22) M ∗ .(4) h f α : α ∈ λ + i is an increasing continuous sequence.(5) f α : N ,α → N ,α is an embedding.(6) For every α ∈ λ + there is a ∈ N ,α +1 − N ,α such that f α +1 ( a ) ∈ N ,α .Why can we carry out the construction?for α = 0 see 2. For α limit, take unions. Suppose we have chosen N ,α , N ,α , f α , how will we choose N ,α +1 , N ,α +1 , f α +1 ? f α [ N ,α ] = N ,α (otherwise f − α ↾ N is an embedding of N to M ∗ over N , in contra-diction to our assumption). Hence there is c ∈ N ,α − f α [ N ,α ]. As M ∗ is saturated in λ + over λ , there is a ∈ M ∗ such that tp ( a, N ,α , M ∗ ) = f − α ( tp ( c, f α [ N ,α ] , N ,α ). Now LST ( K, (cid:22) ) ≤ λ so there is N ,α +1 ∈ K λ ,such that N ,α S { a } ⊆ N ,α +1 (cid:22) M ∗ . So by axiom e of a.e.c. N ,α (cid:22) N ,α +1 . Hence f α ( tp ( a, N ,α , N ,α +1 )) = tp ( c, f α [ N ,α ] , N ,α ). By the def-inition of type and having amalgamation, for some model N ,α +1 and anembedding f ,α +1 the following hold: N ,α (cid:22) N ,α +1 , f ,α +1 ( a ) = c and f α ⊆ f α +1 : N ,α +1 → N ,α +1 . Hence we can carry out the construction.Now the conditions on the existence of the sequences h N ,α : α ∈ λ + i , h N ,α : α ∈ λ + i , h f α : α ∈ λ + i contradict Proposition 1.31 (requirement 5 in Propo-sition 1.31 is satisfied by requirement 6 in the construction here). ⊣ Now we prove the uniqueness of the saturated model, although we do notknow its existence.
Theorem 1.33 (the uniqueness of the saturated model) . Suppose ( K λ , (cid:22) ↾ K λ ) has the amalgamation property and LST ( K, (cid:22) ) ≤ λ .(1) Let N ∈ K λ and for n = 1 , N (cid:22) M n and M n is saturated in λ + over λ . Then M , M are isomorphic over N .(2) If M , M are saturated in λ + over λ and ( K λ , (cid:22) ↾ K λ ) has the jointembedding property then M , M are isomorphic.Proof. (1) We use the hence and forth method. For n = 1 , h a n,α : α ∈ λ + i be an enumeration of M n without repetitions. We choose by inductionon α ∈ λ + a triple ( N ,α , N ,α , f α ) such that:(a) N n, = N, f = id .(b) N n,α (cid:22) M n .(c) The sequence h N n,α : α ∈ λ + i is an increasing continuous sequence ofmodels in K λ .(d) h f α : α ∈ λ + i is increasing and continuous.(e) f α : N ,α → N ,α is an embedding.(f) a n,α ∈ N n, α + n .Why can we carry out the construction?For α = 0 see a. Let α be a limit ordinal. For n = 1 , N n,α = S { N n,β : β < α } , f α = S { f β : β < α } . By axiom c of a.e.c. (i.e. the closure under increasing continuous sequences) for n = 1 , β < α ⇒ N n,β (cid:22) N n,α and by axiom 1.1.1.d (smoothness) N n,α (cid:22) M n . So there is no problem inthe limit case. Suppose we have defined N ,α , N ,α , f α . Suppose α = 2 β .As LST ( K, (cid:22) ) ≤ λ , there is a model N ,α +1 ∈ K λ such that N ,α S { a ,β } ⊆ N ,α +1 (cid:22) M . By the induction hypothesis (b) N ,α (cid:22) M . Now by axiom1.1.1.c (closure under increasing continuous sequences) N ,α (cid:22) N ,α +1 . Let f + α be an injection with domain N ,α +1 such that f α ⊆ f ,α +1 . Actuallyit is an isomorphism of its domain to its range. The relation (cid:22) is closedunder isomorphisms, so N ,α = f α [ N ,α ] (cid:22) f + α [ N ,α +1 ]. M is saturatedin λ + over λ and so by Lemma 1.32 it is model homogenous in λ + over λ . So there is an embedding g : f + α [ N ,α +1 ] → M over N ,α . Define f α +1 =: g ◦ f + α , N ,α +1 =: f α +1 [ N ,α +1 ]. f α ⊆ f α +1 and so (d) is satisfied.Requirement a is not relevant for the successor case. (b) is satisfied for n=1by the definition of N n,α +1 and for n=2 as g is (cid:22) − embedding. (c) is satisfiedfor n=1 by the construction and for n=2 as (cid:22) respects isomorphisms. (e)is satisfied by the definition of f α +1 . (f) is relevant only for n=1. Hence wecan carry out the construction in the α + 1 step for α even. The case α isan odd number is symmetric, so we have to change a, b . Hence we can carryout the construction.Now by (b),(f) S { N n,α : α ∈ λ + } = M n . Define f = S { f α : α ∈ λ + } . By(e) f : M → M is an isomorphism. By (a),(d) this isomorphism is over N .(2) For n = 1 , LST ( K, (cid:22) ) ≤ λ there is N n (cid:22) M n in K λ . K λ hasthe joint embedding property and so there is a model N and embeddings f n : N n → N . Let f + n an injection with domain M n such that f n ⊆ f + n .By Lemma 1.32 for n = 1 , g N : N → f + n [ M n ] over f n [ N n ]. Now f = g ◦ g − is an isomorphism and so there is an injection g + with domain f +2 [ M ] such that g ⊆ g + . By the definition of g , g [ N ] (cid:22) f +2 [ M ] and so as (cid:22) respects isomorphisms, g [ N ] = g [ g [ N ]] (cid:22) g + [ f [ M ]].By item a f +1 [ M ] , g + [ f +2 [ M ]] are isomorphic over g [ N ]. Hence M , M are isomorphic. ⊣ Non-forking frames
The plan.
Suppose we know something about K λ , especially that there isno (cid:22) -maximal model. Can we say something about K λ + n ? At least wewant to prove that K λ + n = ∅ . It is easy to prove that K λ + = ∅ [How? Wechoose M α by induction on α < λ + such that M α ≺ M α +1 and if α is limitwe define M α := S { M β : β < α } (by axiom 1.1.1.c M α ∈ K ). In the end M λ + ∈ K λ + ]. What about K λ +2 ? The main topic in this paper is semi-goodframes. If there is a semi-good λ -frame, then by Proposition 3.4.2 there isno (cid:22) -maximal model in K λ + . So K λ ++ = ∅ . Moreover, Theorem 11.1.1says that if s is a semi-good λ -frame with some additional assumptions and λ satisfies specific set-theoretic assumptions, then there is a good λ + -frame s + = ( K + , (cid:22) + , S bs, + , + S ), such that K + ⊆ K and the relation (cid:22) + ↾ K + is OOD FRAMES WITH A WEAK STABILITY 11 included in the relation (cid:22) ↾ K + . So K λ +3 = ∅ and so on. Thus we prove K λ + n = ∅ by induction on n < ω , (assuming reasonable assumptions).Definition 2.1 is an axiomatization of the non-forking relation in super-stable elementary class. If we subtract axiom 2.1.3.c., we get the basicproperties of the non-forking relation in ( K λ , (cid:22) ↾ K λ ) where ( K, (cid:22) ) is stablein λ .Sometimes we do not find a natural independence relation on all the types.So first we extend the notion of an a.e.c. in λ by adding a new function S bs which assigns a collection of basic types to each model in K λ , and then weadd an independence relation S on basic types.It is reasonable to assume categoricity in some cardinality λ for somereasons:(1) In Example 2.7 K is categorical in λ .(2) If K is not categorical in any cardinality, then we know { λ : K iscategorical in λ } , it is the empty set.(3) If there is a superlimit model in K λ , then we can reduce ( K λ , (cid:22) ↾ K λ )to the models which are isomorphic to it, and therefore obtain cat-egoricity in λ (see section 1 in [Sh 600]). However this case requiresstability.We do not assume amalgamation , but we assume amalgamation in ( K λ , (cid:22) ↾ K λ ). This is a reasonable assumption because it is proved in [Sh 88r] that ifan a.e.c. is categorical in λ and amalgamation fails in λ then under plausibleset theoretic assumptions there are 2 λ + models in K λ + . Definition 2.1. s = ( K, (cid:22) , S bs , S ) is a good λ -frame if:(1)(a) ( K, (cid:22) ) is an a.e.c..(b) LST ( K, (cid:22) ) ≤ λ .(c) ( K λ , (cid:22) ↾ K λ ) has joint embedding.(d) ( K λ , (cid:22) ↾ K λ ) has amalgamation.(e) There is no (cid:22) -maximal model in K λ .(2) S bs is a function with domain K λ , which satisfies the following axioms:(a) S bs ( M ) ⊆ S na ( M ) =: { tp ( a, M, N ) : M ≺ N ∈ K λ , a ∈ N − M } .(b) It respects isomorphisms.(c) Density of the basic types: If M, N ∈ K λ and M ≺ N , then there is a ∈ N − M such that tp ( a, M, N ) ∈ S bs ( M ).(d) Basic stability: For every M ∈ K λ , the cardinality of S bs ( M ) is ≤ λ .(3) the relation S satisfies the following axioms:(a) S is a subset of { ( M , M , a, M ) : M , M , M ∈ K, || M || = || M || = λ, M (cid:22) M (cid:22) M , a ∈ M − M and n < ⇒ tp ( a, M n , M ) ∈ S bs ( M n ) } . (b) Monotonicity: If M (cid:22) M ∗ (cid:22) M ∗ (cid:22) M (cid:22) M (cid:22) M ∗ , M ∗ S { a } ⊆ M ∗∗ (cid:22) M ∗ , then S ( M , M , a, M ) ⇒ S ( M ∗ , M ∗ , a, M ∗∗ ). See Remark2.2.(c) Local character: For every limit ordinal δ < λ + if h M α : α ≤ δ i is anincreasing continuous sequence of models in K λ , and tp ( a, M δ , M δ +1 ) ∈ S bs ( M δ ), then there is α < δ such that tp ( a, M δ , M δ +1 ) does not forkover M α .(d) Uniqueness of the non-forking extension: If p, q ∈ S bs ( N ) do not forkover M , and p ↾ M = q ↾ M , then p = q .(e) Symmetry: If M (cid:22) M (cid:22) M , a ∈ M , tp ( a , M , M ) ∈ S bs ( M ),and tp ( a , M , M ) does not fork over M , then there are M , M ∗ suchthat a ∈ M , M (cid:22) M (cid:22) M ∗ , M (cid:22) M ∗ , and tp ( a , M , M ∗ ) doesnot fork over M .(f) Existence of non-forking extension: If p ∈ S bs ( M ) and M ≺ N , thenthere is a type q ∈ S bs ( N ) such that q does not fork over M and q ↾ M = p .(g) Continuity: Let h M α : α ≤ δ i be an increasing continuous sequence. Let p ∈ S ( M δ ). If for every α ∈ δ, p ↾ M α does not fork over M , then p ∈ S bs ( M δ ) and does not fork over M . Remark 2.2. If S ( M , M , a, M ) and tp ( b, M , M ∗∗ ) = tp ( a, M , M ) thenby Definition 2.1.3.b (the monotonicity axiom) S ( M , M , a, M ). Thereforewe can say “ p does not fork over M ” instead of S ( M , M , a, M ).While in [Sh 600] we study good frames, so basic stability is assumed,here we assume basic weak stability so the following definition is central: Definition 2.3. s = (( K s , (cid:22) s , S bs, s , s S ) = ( K, (cid:22) , S bs , S ) is a semi -good λ -frame, if s satisfies the axioms of a good λ -frame except that instead ofassuming basic stability, we assume that s has basic weak stability, namelyfor every M ∈ K λ S bs ( M ) has cardinality at most λ + . s is said to be a semi-good frame if it is a semi-good λ -frame for some λ . Remark 2.4.
If for each M ∈ K λ S bs ( M ) = { tp ( a, M, N ) : M ≺ N, a ∈ N − M } then the continuity axiom is an easy consequence of the localcharacter.Can we define in our context independence, orthogonality and more thingslike in superstable theories? The answer is: See [Sh 705] (mainly sections5,6) and [JrSi 3].Now we give examples of good frames and examples of semi-good frames. Example 2.5.
Let T be a superstable first order theory and let λ be acardinal ≥ | T | + ℵ such that T is stable in λ . Let K T,λ be the class ofmodels of T of cardinality at least λ . Let (cid:22) denote the relation of being anelementary submodel. Let S bs ( M ) be the set of regular types over M . Let S be as usual. Then by Claim 3.1 on page 52 in [Sh 600] (or see [Sh 91])( K T,λ , (cid:22) , S bs , S ) is a good λ -frame. OOD FRAMES WITH A WEAK STABILITY 13
Example 2.6.
The same as Example 2.5 but the basic types are the non-algebraic types (see [HS 89]).
Example 2.7 (the main example) . An example of a semi-good λ -framewhich appears in section 3 of [Sh 600] and is based on [Sh 88r]: Let ( K, (cid:22) ) bean a.e.c. with a countable vocabulary, LST ( K, (cid:22) ) = ℵ , ( K, (cid:22) ) is P C ℵ (i.e. K is the class of reduced models to a smaller language, of some countableelementary class, which omit a countable set of types, and the relation (cid:22) isdefined similarly), it has an intermediate number of non-isomorphic modelsof cardinality ℵ and 2 ℵ < ℵ . Then we can derive a semi-good ℵ -framefrom it.How? For each M ∈ K ℵ define K M = { N ∈ K : N ≡ L ∞ ,ω M } , (cid:22) M = { ( N , N ) : N , N ∈ K M , N (cid:22) N , and N (cid:22) L ∞ ,ω N } . There is a model M ∈ K ℵ such that ( K M ) ℵ = ∅ . Fix such an M . For every N ∈ K M with || N || = ℵ define S bs ( N ) = { tp ( a, N, N ∗ ) : N ≺ M N ∗ ∈ K M , a ∈ N ∗ − N } . Define S := { ( M , M , a, M ) : M , M , M ∈ K M , || M || = || M || = ℵ , M (cid:22) M M ≺ M M and tp ( a, M , M ) is definable over somefinite subset A of M in the following sense: For every type q ∈ S bs ( M )if ‘ q ↾ A = tp ( a, A, M )’ [more precisely for some b, M ∗ , M , f , M (cid:22) M ∗ , tp ( b, M , M ∗ ) = q M (cid:22) M and f is an isomorphism of M ∗ to M over A with f ( b ) = a ] then q = tp ( a, M , M ). By the proof of Theorem 3.4 on page54 in [Sh 600] s := ( K M , (cid:22) M , S bs , S ) is a semi-good ℵ -frame [Why? Weassumed here assumptions ( α ) , ( β ) , ( γ ) of Theorem 3.4. So by Theorem 3.4.1for some M ∈ K ℵ we have ( δ − ) , ( ε ) too. So if ( δ ) (namely stability) holdsthen by Theorem 3.4.2 s is a good ℵ -frame. Here we have ( δ − ) (namelyweak stability) only. In the beginning of the proof of item 2 (on page 56) itis written ‘we assume ( δ − ) instead of ( δ )’. By the continuation of the proof,we see that s is a semi-good ℵ -frame] Definition 2.8.
Let p ∈ S ( M ) , p ∈ S ( M ). We say that p , p conjugate if for some a , M +0 , a , M +1 , f the following hold:(1) For n = 0 , tp ( a n , M n , M + n ) = p n .(2) f : M +0 → M +1 is an isomorphism.(3) f ↾ M : M → M is an isomorphism.(4) f ( a ) = a . Proposition 2.9.
Let p ∈ S ( M ) , p ∈ S ( M ) and assume that p , p conjugate.(1) If for n = 1 , , tp ( a n , M n , M + n ) = p n and there is an embedding f : M +0 → M +1 such that f ↾ M : M → M is an isomorphism and f ( a ) = a , then the types p , p conjugate.(2) Suppose that a , M +0 , a , M +1 , f are as in Definition 2.8, i.e. theywitness that p , p conjugate. If tp ( a ∗ , M , M ∗ ) = tp ( a , M , M +1 ) then for some M ∗∗ and f ∗ such that:(a) M ∗ (cid:22) M ∗∗ .(b) f ∗ : M +0 → M ∗∗ is an embedding. (c) a , M +0 , a ∗ , f ∗ [ M +0 ] , f ∗ witness that p , p conjugate.(3) Assume that ( p , p conjugate and) the types p , p conjugate. Then p , p conjugate.Proof. (1) Substitute f [ M +0 ] instead of M +1 in Definition 2.8.(2) Since tp ( a ∗ , M , M ∗ ) = tp ( a , M , M +1 ), there is an amalgamation( id M ∗ , g, M ∗∗ ) of M ∗ , M +1 over M such that g ( a ) = a ∗ . Now define f ∗ := g ◦ f . So f ∗ ( a ) = g ( f ( a )) = g ( a ) = a ∗ .(3) Suppose that a ∗ , M ∗ , a , M ∗ , g witness that p , p conjugate. Sincethe types p , p conjugate, there are witnesses for this. So by Propo-sition 2.9.2 for some a , M ∗∗ , f ∗ the following hold: M ∗∗ g + / / M +2 M +0 f ∗ = = zzzzzzzz id / / M ∗ id O O g / / M ∗ id O O M id O O f ∗ ↾ M / / M id O O g ↾ M / / M id O O (a) M ∗ (cid:22) M ∗∗ .(b) f ∗ : M +0 → M ∗∗ is an embedding.(c) a , M +0 , a ∗ , f ∗ [ M +0 ] , f ∗ witness that p , p conjugate.Since ( K λ , (cid:22) ↾ K λ ) has amalgamation, for some g + , M +2 the followinghold:(a) g + : M ∗∗ → M +2 is an embedding.(b) M ∗ (cid:22) M +2 .(c) g + ↾ M ∗ = g .Now define f := g + ◦ f ∗ . So a , M +0 , a , f [ M +0 ] , f witness that thetypes p , p conjugate. Why? We verify the conditions in Definition2.8:(1) tp ( a , M , M +0 ) = p because a , M +0 , a ∗ , f ∗ [ M +0 ] , f ∗ witness that p , p conjugate. tp ( a , M , M +2 ) = p because a ∗ , M ∗ , a , M ∗ , g witness that p , p conjugate.(2) f : M +0 → f ∗ [ M +0 ] is an isomorphism.(3) f ↾ M : M → M is or course an embedding, but why isit onto? Take z ∈ M . Since g ↾ M is an isomorphism (as a ∗ , M ∗ , a , M ∗ , g witness that p , p conjugate), there is y ∈ M such that g ( y ) = z . Since f ∗ ↾ M is an isomorphism (as a , M +0 , a ∗ , f ∗ [ M +0 ] , f ∗ witness that p , p conjugate), there is x ∈ M such that g ( x ) = z . Therefore g + ( f ∗ ( x )) = g ( f ∗ ( x )) = g ( y ) = z .(4) f ( a ) = g + ( f ∗ ( a )) = g + ( a ∗ ) = a . ⊣ OOD FRAMES WITH A WEAK STABILITY 15
Definition 2.10.
Let p = tp ( a, M, N ). Let f be a bijection with domain M . Define f ( p ) = tp ( f ( a ) , f [ M ] , f + [ N ]), where f + is an extension of f (andthe relations and functions on f + [ N ] are defined such that f + : N → f + [ N ]is an isomorphism). Proposition 2.11.
The definition of f ( p ) in Definition 2.10 does not de-pend on the representative ( M, N, a ) ∈ p .Proof. By Proposition 2.9.2. ⊣ Definition 2.12.
Let s be a semi-good λ -frame. We say that s has con-jugation when: K λ is categorical and if M , M ∈ K λ , M (cid:22) M and p ∈ S bs ( M ) is the non-forking extension of p ∈ S bs ( M ), then the types p , p conjugate. Proposition 2.13.
The semi-good frame in Example 2.7 has conjugation.Proof.
Assume that || M || = || M || = ℵ , M ≺ M M and p ∈ S bs ( M ) isdefinable over some finite subset A of M . We have to prove that the types p, p := p ↾ M conjugate. Since M ≺ L ∞ ,ω M , there is an isomorphism f : M → M over A . So f ( p ) does not fork over f [ M ]. But p does notfork over M too. p ↾ A = ( p ↾ M ) ↾ A = p ↾ A = f ( p ) ↾ A . Since f ( p ) , p are definable over A , f ( p ) = p . ⊣ Proposition 2.14 (versions of extension) . If for n < M n ∈ K λ , M (cid:22) M n , and tp ( a, M , M ) ∈ S bs ( M ) then:(1) There are M , f such that:(a) M (cid:22) M .(b) f : M → M is an embedding over M .(c) tp ( f ( a ) , M , M ) does not fork over M .(2) There are M , f such that:(a) M (cid:22) M .(b) f : M → M is an embedding over M .(c) tp ( a, f [ M ] , M ) does not fork over M .Proof. Easily by Definition 2.1.2. ⊣ Proposition 2.15 (The transitivity proposition) . Suppose s is a semi-good λ -frame. Then: If M (cid:22) M (cid:22) M , p ∈ S bs ( M ) does not fork over M and p ↾ M does not fork over M , then p does not fork over M .Proof. Suppose M ≺ M ≺ M , n < ⇒ M n ∈ K λ , p ∈ S bs ( M )does not fork over M and p ↾ M does not fork over M . For n < p n = p ↾ M n . By axiom f there is a type q ∈ S bs ( M ) such that q ↾ M = p and q does not fork over M . Define q = q ↾ M . By axiomb (monotonicity) q does not fork over M . So by axiom d (uniqueness) q = p . Using again axiom e, we get q = p , as they do not fork over M .By the definition of q it does not fork over M . ⊣ Proposition 2.16.
Suppose (1) s satisfies the axioms of a semi-good λ -frame.(2) n < ⇒ M (cid:22) M n .(3) For n = 1 , , a n ∈ M n − M and tp ( a n , M , M n ) ∈ S bs ( M ) .Then there is an amalgamation M , f , f of M , M over M such that for n = 1 , tp ( f n ( a n ) , f − n [ M − n ] , M ) does not fork over M .Proof. Suppose for n = 1 , M ≺ M n ∧ tp ( a n , M , M n ) ∈ S bs ( M ). ByProposition 2.14.1 there are N , f such that:(1) M (cid:22) N .(2) f : M → N is an embedding over M .(3) tp ( f ( a ) , M , N ) does not fork over M . M f / / N id / / N id / / f (cid:15) (cid:15) N N ∗ id / / id ; ; xxxxxxxxx f [ N ] id ; ; xxxxxxxx M id / / id O O id mmmmmmmmmmmmmmmm M id O O By axiom f (the symmetry axiom), there are a model N , N (cid:22) N ∈ K λ and a model N ∗ ∈ K λ such that: M S { f ( a ) } ⊆ N ∗ (cid:22) N and tp ( a , N ∗ , N ) does not fork over M .By Proposition 2.14.2 (substituting N ∗ , N , N , a which appear here insteadof M , M , M , a there) there are N , f such that:(1) N (cid:22) N .(2) f : N → N is an embedding over N ∗ .(3) tp ( a , f [ N ] , N ) does not fork over N ∗ .So by Proposition 2.15 (on page 15), tp ( a , f [ N ] , N ) does not fork over M . So as M (cid:22) f ◦ f [ M ] (cid:22) f [ N ] by axiom b (monotonicity) tp ( a , f ◦ f [ M ] , N ) does not fork over M . As f ↾ N ∗ = id N ∗ , f ( f ( a )) = f ( a ). ⊣ Theorem 2.17.
Suppose s satisfies conditions 1 and 2 of a semi-good λ -frame (so actually the relation S is not relevant).(1) Suppose:(a) h M α : α ≤ λ + i is an increasing continuous sequence of modelsin K λ .(b) There is a stationary set S ⊆ λ + such that for every α ∈ S andevery model N , with M α ≺ N there is a type p ∈ S ( M α ) whichis realized in M λ + and in N .Then M λ + is full over M and is saturated in λ + over λ .(2) Suppose:(a) h M α : α ≤ λ + i is an increasing continuous sequence of modelsin K λ . OOD FRAMES WITH A WEAK STABILITY 17 (b) For every α ∈ λ + and every p ∈ S bs ( M α ) there is β ∈ ( α, λ + ) such that p is realized in M β .Then M λ + is full over M and M λ + is saturated in λ + over λ .(3) There is a model in K λ + which is saturated in λ + over λ .(4) M ∈ K λ ⇒ | S ( M ) | ≤ λ + .Proof. Obviously 1 ⇒ ⇒
4. To show 2 ⇒
3, we construct a chainsatisfying the hypotheses of 2. Let cd be an injection from λ + × λ + onto λ + . Define by induction on α < λ + M α and h p α,β : β < λ + i such that:(1) h M α : α < λ + i is an increasing continuous sequence of models in K λ .(2) { p α,β : β < λ + } = S bs ( M α ).(3) M α +1 realizes p γ,β , where we denote: A α := { cd ( γ, β ) : γ ≤ α, p γ,β is not realized in M α } , ε α = Min( A α ) and ( γ, β ) = cd − ( ε α ).We argue that M λ + := S { M α : α < λ + } is saturated in λ + over λ . By 2 itis sufficient to prove that for every α ∈ λ + and every p ∈ S bs ( M α ) there is β ∈ ( α, λ + ) such that p is realized in M β . Towards a contradiction choose α ∗ so that p ∈ S bs ( M α ∗ ) is not realized in M λ + . There is β < λ + suchthat p = p α ∗ ,β . Denote ε := cd ( α ∗ , β ). For every α ≥ α ∗ ε ∈ A α , so A α is nonempty and ε α is defined. But ε α = ε , (as otherwise p is realized in M α +1 ), so ε α < ε . The function f : [ α ∗ , λ + ) → ε , f ( α ) = ε α is injectionwhich is impossible.It remains to prove item 1. Fix N , with M ≺ N . It is sufficient to provethat there is an embedding of N to M λ + over M . We choose ( α ε , N ε , f ε )by induction on ε < λ + such that: N id / / N ε id / / N ε +1 M f O O id / / M α ε f ε O O id / / M α ε +1 f ε +1 O O (1) h α ε : ε < λ + i is an increasing continuous sequence of ordinals in λ + .(2) The sequence h N ε : ε < λ + i is increasing and continuous.(3) h f ε : ε < λ + i is increasing continuous.(4) N := N , α := 0 and f = id M .(5) f ε : M α ε → N ε is an embedding.(6) For every α ∈ S there is a ∈ M α ε +1 − M α ε such that f ε +1 ( a ) ∈ N ε .By Proposition 1.31 we cannot carry out this construction. Where will weget stuck? For ε = 0 or limit we will not get stuck. Suppose we have defined( α ζ , N ζ , f ζ ) for ζ ≤ ε . If f ε [ M α ε ] = N ε then f − ε ↾ N is an embedding of N into M λ + over M , hence we are finished. So without loss of generality f ε [ M α ε ] = N ε . If α ε / ∈ S then we define α ε +1 := α ε + 1 and use theamalgamation in ( K λ , (cid:22) ↾ K λ ) to find N ε +1 , f ε +1 as needed.Suppose α ε ∈ S . By the theorem’s assumption, there is a type p ∈ S ( M α ε ) such that p is realized in M λ + and f ε ( p ) is realized in N ε . Define α ε +1 := Min { α ∈ λ + : p is realized in M α } . Take a ∈ M α ε +1 such that tp ( a, M α ε , M α ε +1 ) = p and take b ∈ N ε such that tp ( b, f ε ( M α ε ) , N ε ) = f ε ( p ).Then f ε ( tp ( a, M α ε , M λ + )) = tp ( b, M α ε , N ε ). By the definition of type (Def-inition 1.21.1 on page 7), there are N ε +1 , f ε +1 with N ε (cid:22) N ε +1 , f ε +1 is anembedding of M α ε +1 into N ε +1 , f ε ⊆ f ε +1 and f ε +1 ( a ) = b .Since the hypotheses of 5 applies to any cofinal segment of the sequence h M α : α < λ + i and any submodel of size λ lies in some M α we concludethat M λ + is saturated in λ + over λ . ⊣ non-forking with greater models. Now we extend our non-forkingnotion to include models of cardinality greater than λ . Definition 2.18. ≥ λ S is the class of quadruples ( M , a, M , M ) such that:(1) M (cid:22) M (cid:22) M .(2) λ ≤ || M || .(3) For some model N ∈ K λ with N (cid:22) M for each model N ∈ K λ , N (cid:22) N (cid:22) M ⇒ S ( N , a, N, M ). Remark 2.19. If S ( M , a, M , M ) then ≥ λ S ( M , a, M , M ). Definition 2.20.
Let M (cid:22) M and p ∈ S ( M ). We say that p does not forkover M , when for some triple ( M , M , a ) ∈ p we have ≤ λ S ( M , a, M , M ). Remark 2.21.
We can replace the ‘for some’ in Definition 2.20 by ‘for each’.
Definition 2.22.
Let M ∈ K >λ , p ∈ S ( M ). p is said to be basic whenthere is N ∈ K λ such that N (cid:22) M and p does not fork over N . Forevery M ∈ K >λ , S bs>λ ( M ) is the set of basic types over M . Sometimes wewrite S bs ≥ λ ( M ), meaning S bs ( M ) or S bs>λ ( M ) (the unique difference is thecardinality of M ).Now we present a weak version of local character, which is peripheral inthe continuation of this paper. Definition 2.23.
Let s be a semi good frame except local character. s is said to satisfy weak local character for ≺ ∗ -increasing sequences when: If α ∗ < λ + and h M α : α ≤ α ∗ + 1 i is an ≺ ∗ − increasing continuous sequenceof models, then for some element a ∈ M α ∗ +1 − M α ∗ and ordinal α < α ∗ , tp ( a, M α ∗ , M α ∗ +1 ) does not fork over M α . Definition 2.24.
Let s be a semi good λ -frame except local character. s issaid to satisfy weak local character for fast sequences when for some relation ≺ ∗ the following hold:(1) ≺ ∗ is a relation on K λ .(2) If M ≺ ∗ M then M (cid:22) M .(3) If M ≺ ∗ M (cid:22) M ∈ K λ then M ≺ ∗ M .(4) s satisfies weak local character for ≺ ∗ -increasing sequences. OOD FRAMES WITH A WEAK STABILITY 19 (5) If M ∈ K λ , M ≺ M ∈ K λ + , then there is a model M ∈ K λ suchthat: M ≺ ∗ M (cid:22) M . Remark 2.25. If s is a semi good λ -frame and ≺ ∗ is a relation on K λ such that M ≺ ∗ N ⇒ M (cid:22) N then s satisfies weak local character for ≺ ∗ -increasing sequences.The following theorem asserts that a non-forking relation in ( K λ , (cid:22) ↾ K λ )can be lifted to K ≥ λ with many properties preserved. Theorem 2.26.
Let s be a semi-good λ -frame, except local character.(1) Density: If s satisfies weak local character for fast sequences and M ≺ N, M ∈ K ≥ λ then there is a ∈ N − M such that tp ( a, M, N ) ∈ S bs ≥ λ ( M ) .(2) Monotonicity: Suppose M (cid:22) M (cid:22) M , n < ⇒ M n ∈ K ≥ λ , || M || > λ . If p ∈ S bs ≥ λ ( M ) does not fork over M , then(a) p does not fork over M .(b) p ↾ M does not fork over M .(3) Transitivity: Suppose M , M , M ∈ K ≥ λ and M (cid:22) M (cid:22) M . If p ∈ S bs ≥ λ ( M ) does not fork over M , and p ↾ M does not fork over M , then p does not fork over M .(4) About local character: Let δ be a limit ordinal. Suppose s satisfieslocal character or λ + ≤ cf ( δ ) . If h M α : α ≤ δ i is an increasingcontinuous sequence of models in K >λ , and p ∈ S bs>λ ( M δ ) then thereis α < δ such that p does not fork over M α .(5) Continuity: Suppose h M α : α ≤ δ + 1 i is an increasing continuoussequence of models in K ≥ λ . Let c ∈ M δ +1 − M δ . Denote p α = tp ( c, M α , M δ +1 ) . If for every α < δ, p α does not fork over M , then p δ does not fork over M .Proof. (1) Density: Suppose M ≺ N . Case 1: || M || = λ . Choose a ∈ N − M . LST ( K, (cid:22) ) ≤ λ and so thereis N ∗ ≺ N such that: || N ∗ || = λ and M S { a } ⊆ N ∗ . By axiom e of a.e.c M (cid:22) N ∗ But a ∈ N ∗ − M and so M ≺ N ∗ . By the existence axiom in s , thereis c ∈ N ∗ − M such that tp ( c, M, N ∗ ) is basic. So tp ( c, M, N ) ∈ S bs ( M ). Case 2: || M || > λ . We choose M n , N n by induction on n < ω such that: c ∈ N n id / / N n id / / N ω id / / NM nid O O id / / M n,c id / / M n +1 id O O id / / N ω id / / id O O M id O O (a) h N n : n ≤ ω i is an ≺ − increasing continuous sequence of models in K λ .(b) h M n : n ≤ ω i is an ≺ ∗ − increasing continuous sequence of models in K λ .(c) M n ≺ M (see the end of Definition 2.1).(d) N n ≺ N . (e) N * M .(f) For every c ∈ N n , M n,c ⊆ M n +1 where we choose M n,c ∈ K λ suchthat: If tp ( c, M n , N n ) ∈ S bs ( M n ) but does fork over M n then M n,c is awitness for this, namely M n ≺ M n,c ≺ M and tp ( c, M n,c , N ) forks over M n . Otherwise M n,c = M n .The construction is of course possible.Now we define M ω := S { M n : n < ω } and N ω := S { N n : n < ω } .By axiom 1.1.1.d (smoothness) M ω (cid:22) N ω . By the local character for ≺ ∗ -increasing sequences, for some element c ∈ N ω − M ω and there is n < ω suchthat tp ( c, M ω , N ω ) ∈ S bs ( M ω ) does not fork over M n . By the monotonicitywithout loss of generality c ∈ N n . We will prove that tp ( c, M, N ) does notfork over M ω . Take M ∗ with M ω ≺ M ∗ ≺ M . By way of contradictionsuppose tp ( c, M ∗ , N ) forks over M ω . By the monotonicity in s (axiom b), tp ( c, M ∗ , N ) forks over M n . So by the definition of M n,c , tp ( c, M n,c , N )forks over M n . Hence by axiom b (monotonicity) tp ( c, M ω , N ) forks over M n , a contradiction.(2) Monotonicity: We use the same witness.(3) Transitivity: N id / / N ∗∗ id / / M pN id / / N ∗ id / / id O O M id O O N id / / id O O id = = zzzzzzzz M id O O Suppose M ≺ M ≺ M , p ∈ S bs ( M ) does not fork over M and p ↾ M does not fork over M . We can find N ≺ M such that N witnesses that p ↾ M does not fork over M . We will prove that N witnesses that p does not fork over M . Let N ∈ K λ be such that N ≺ N ≺ M . Wehave to prove that p ↾ N does not fork over N . First we find a model N that witnesses that p does not fork over M . As LST ( K, (cid:22) ) ≤ λ thereis N ∗ ∈ K λ such that N S N ⊆ N ∗ (cid:22) M and there is N ∗∗ ∈ K λ suchthat N ∗ S N ⊆ N ∗∗ (cid:22) M . As N witnesses that p does not fork over M , p ↾ N ∗∗ does not fork over N . By the Definition 2.1.3.b (monotonicity), p ↾ N ∗∗ does not fork over N ∗ . N witnesses that p ↾ M does not forkover M , so p ↾ N ∗ does not fork over N . By the transitivity proposition(Proposition 2.15), p ↾ N ∗∗ does not fork over N . So by Definition 2.1.3.b(monotonicity), p ↾ N does not fork over N .(4) About local character: Let h M α : α < δ i be an increasing continuoussequence of models in K >λ . Let p ∈ S bs>λ ( M δ ) and N ∗ a witness for this,i.e. p does not fork over N ∗ ∈ K λ . Let h α ( ε ) : ε ≤ cf ( δ ) i be an increasingcontinuous sequence of ordinals with α ( cf ( δ )) = δ . OOD FRAMES WITH A WEAK STABILITY 21
Case a: λ + ≤ cf ( δ ). By cardinality considerations, there is ε < cf ( δ )such that: N ∗ ⊆ M α ( ε ) . By axiom 1.1.1.e N ∗ (cid:22) M α ( ε ) . As N ∗ witnessesthat the type p is basic, by Definition 2.18 N ∗ witnesses that p does not forkover M α ( ε ) . Case b: s satisfies local character and cf ( δ ) ≤ λ . Using LST ( K, (cid:22) ) ≤ λ and smoothness, we can choose N α ( ε ) by induction on ε ≤ cf ( δ ) such that: N ∗ id / / N δ id / / M δ pN α ( ε ) id / / id O O M α ( ε ) id O O (a) N α ( ε ) ∈ K λ .(b) h N α ( ε ) : ε ≤ cf ( δ ) i is an increasing continuous sequence.(c) M α ( ε ) T N ∗ ⊆ N α ( ε ) (cid:22) M α ( ε ) .By axiom 1.1.1.e N ∗ (cid:22) N δ (cid:22) M δ . Since p does not fork over N ∗ , bymonotonicity (Theorem 2.26.2) p does not fork over N δ . By local character,for some ε < cf ( δ ), p ↾ N δ does not fork over N α ( ε ) . By transitivity (Theo-rem 2.26.3), p does not fork over N α ( ε ) . By monotonicity (Theorem 2.26.2), p does not fork over M α ( ε ) . (5) Continuity: For every α ∈ δ denote p α := p ↾ M α . p does not forkover M . So for some N ∈ K λ , N (cid:22) M and p does not fork over N . Bymonotonicity (Theorem 2.26.2) and transitivity (Theorem 2.26.2) for every α < δ p α does not fork over N . We will prove that p does not fork over N .Take N ∗ ∈ K λ with N (cid:22) N ∗ (cid:22) M δ . We have to prove that p ↾ N ∗ does notfork over N . Let h α ( ε ) : ε ≤ cf ( δ ) i be an increasing continuous sequenceof ordinals with α ( cf ( δ )) = δ . Case a: λ + ≤ cf ( δ ). By cardinality considerations there is ε < cf ( δ )such that N ∗ ⊆ M α ( ε ) . But M α ( ε ) (cid:22) M δ and N ∗ (cid:22) M δ , so by axiom 1.1.1.e N ∗ (cid:22) M α ( ε ) . Since p α ( ε ) does not fork over N , by monotonicity (Theorem2.26.2) p ↾ N ∗ does not fork over N . Case b: cf ( δ ) ≤ λ + . We choose N α ( ε ) by induction on ε ∈ (0 , cf ( δ )] suchthat:(a) The sequence h N α ( ε ) : ε ≤ cf ( δ ) i is increasing continuous.(b) ε ≤ cf ( δ ) ⇒ N ∗ T M α ( ε ) ⊆ N α ( ε ) (cid:22) M α ( ε ) .(c) N α ( ε ) ∈ K λ .For every ε < cf( δ ), p α ( ε ) does not fork over N , so p ↾ N α ( ε ) does notfork over N . So by Definition 2.1.3.g (continuity) (in s ), p ↾ N δ does notfork over N . N ∗ ⊆ N δ , hence by axiom 1.1.1.e N ∗ (cid:22) N δ . Therefore byDefinition 2.1.3.b (monotonicity), p ↾ N ∗ does not fork over N . ⊣ The decomposition and amalgamation method
Discussion.
In section 2 we defined an extension of the non-forking notionto cardinals bigger than λ . But we did not prove all the good frame axioms.The purpose from here until the end of the paper is to construct a good λ + -frame, which is derived from the semi good λ -frame. In a sense, the mainproblem is that amalgamation in ( K λ , (cid:22) ↾ K λ ) does not imply amalgamationin ( K λ + , (cid:22) ↾ K λ + ). Suppose for n < M n ∈ K λ + , M (cid:22) M n and we want toamalgamate M , M over M . We take representation of M , M , M . Wewant to amalgamate M , M by amalgamating their representations. Forthis goal, we will find in section 5, a relation of “a canonical amalgamation”or “a non-forking amalgamation”. Sections 3,4 are preparations for section5. If the reader wants to know the plan of the other sections now, he maysee the discussion at the beginning of section 10. The decomposition and amalgamation method.
Suppose for n = 1 , M (cid:22) M n and we want to prove that there is an amalgamation of M , M over M which satisfies specific properties (for example disjointness or uniqueness,see below). Sometimes there is a property of triples, K , ∗ ⊆ K such that if( M , M , a ) ∈ K , ∗ and ( M , M , a ) (cid:22) ( M , M , a ) then the amalgamation M satisfies the required property. A classic example of this property in thecontext of fields is ‘ M is the algebraic closure of M S { a } . What can wedo, if there is no a ∈ M − M such that ( M , M , a ) ∈ K , ∗ ? Theorem 3.8says under some assumptions that we can decompose an extension of M over M by triples in K , ∗ . By Proposition 3.4.2 we may amalgamate M with the decomposition we have obtained. Applications of the decomposition and amalgamation method. (1) By Proposition 3.4(2) there is no (cid:22) -maximal model in K λ + .(2) By Proposition 3.13 the uniqueness triples are dense with respectto (cid:22) bs (see Definition 3.2.2). It enables to prove Theorem 3.14 (thedisjoint amalgamation existence), by the decomposition and disjointmethod.(3) By assumption 5.1 the uniqueness triples are dense with respectto (cid:22) bs . The density enables to prove Theorem 5.11 (the exitancetheorem for N F ).(4) Using again assumption 5.1, we prove Proposition 5.19 . But forthis, we have to prove Proposition 3.5, a generalization of 3.4, whichsays that we can amalgamate two sequences of models, not just amodel and a sequence.
Hypothesis . s is a semi good λ -frame, except basic weak stability andlocal character.3.1. The a.e.c. ( K ,bs , (cid:22) bs ) and amalgamations.Definition 3.2. OOD FRAMES WITH A WEAK STABILITY 23 (1) K ,bs =: { ( M, N, a ) :
M, N ∈ K λ , a ∈ N − M and tp ( a, M, N ) ∈ S bs ( M ) } .(2) (cid:22) bs is the relation on K ,bs defined by: ( M, N, a ) (cid:22) bs ( M ∗ , N ∗ , a ∗ )iff M (cid:22) M ∗ , N (cid:22) N ∗ , a ∗ = a and tp ( a, M ∗ , N ∗ ) does not fork over M .(3) The sequence h ( M α , N α , a ) : α < θ i is said to be (cid:22) bs -increasingcontinuous if α < θ ⇒ ( M α , N α , a ) (cid:22) bs ( M α +1 , N α +1 , a ) and thesequences h ( M α : α < θ i , h N α : α < θ i are continuous (and clearlyalso increasing). Proposition 3.3. ( K ,bs , (cid:22) bs ) is an a.e.c. in λ and it has no (cid:22) bs -maximalmodel.Proof. First we note that K ,bs is not the empty set [There is M ∈ K λ ,and as it has no (cid:22) -maximal model, there is N ∈ K λ with M ≺ N . Nowby Definition 2.1.3.f, there is a ∈ N − M such that tp ( M, N, a ) ∈ S bs ( M )].Why is axiom c of a.e.c. (defintion 1.1.1.c) satisfied? Suppose δ < λ + and h ( M α , N α , a ) : α < δ i is increasing and continuous. Denote M = S { M α : α < δ } , N = S { N α : α < δ } . By axiom c of a.e.c., M, N ∈ K λ , α < δ ⇒ M α (cid:22) M, N α (cid:22) N . By the definition of (cid:22) bs for every α < δ , tp ( a, M α , N α )does not fork over M . So by Definition 2.1.3.g (continuity), tp ( a, M, N ) isbasic and does not fork over M . By the smoothness, M (cid:22) N . By axiom cof a.e.c. M (cid:22) M and N (cid:22) N . So ( M , N , a ) (cid:22) bs ( M, N, a ) ∈ K ,bs . Whyis the smoothness satisfied? Suppose h ( M α , N α , a ) : α ≤ δ + 1 i is continuousand for α < β ≤ δ + 1, we have α = δ ⇒ ( M α , N α , a ) (cid:22) bs ( M β , N β , a ). So δ = α < β ≤ δ + 1 ⇒ M α (cid:22) M β . But by the continuity of the sequence h ( M α , N α , a ) : α ≤ δ + 1 i we have M δ = S { M α : α < δ } . So by the smooth-ness of ( K, (cid:22) ), M δ (cid:22) M δ +1 . In a similar way N δ (cid:22) N δ +1 . ( M , N , a ) (cid:22) bs ( M δ +1 , N δ +1 , a ), so by the definition, tp ( a, M δ +1 , N δ +1 ) does not fork over M . Therefore by Definition 2.1.3.b (monotonicity), tp ( a, M δ +1 , N δ +1 ) doesnot fork over M δ . Why does ( K ,bs , (cid:22) bs ) satisfy axiom 1.1.1.e? Suppose( M , N , a ) ⊆ ( M , N , a ) (cid:22) ( M , N , a ) , ( M , N , a ) (cid:22) bs ( M , N , a ). Bythe definition of (cid:22) bs we have M ⊆ M (cid:22) M and M (cid:22) M . Hence byaxiom 1.1.1.e we have M (cid:22) M . In a similar way N (cid:22) N . By the defini-tion of (cid:22) bs , tp ( a, M , N ) does not fork over M . By 2.1.3.b (monotonicity), tp ( a, M , N ) does not fork over M . So ( M , N , a ) (cid:22) bs ( M , N , a ). Whyis there no maximal element in ( K ,bs , (cid:22) bs )? Let ( M , N , a ) ∈ K ,bs . In K λ there is no (cid:22) -maximal element, and so there is M ≺ M ∗ ∈ K λ . By Propo-sition 2.16 there is N (cid:22) N ∈ K λ and there is an embedding f : M ∗ → N such that tp ( a, M , N ) does not fork over M where M := f [ M ∗ ]. Hence( M , N , a ) (cid:22) bs ( M , N , a ). ⊣ Proposition 3.4. N id / / N id / / N id / / N α id / / N α +1 id / / N θ M id / / id O O M id / / id O O M id / / id O O M α id / / id O O M α +1 id / / id O O M θid O O (1) Let h M α : α ≤ θ i be an increasing continuous sequence of models in K λ . Let N ∈ K λ with M ≺ N , and for α < θ , let a α ∈ M α +1 − M α , ( M α , M α +1 , a α ) ∈ K ,bs and b ∈ N − M , ( M , N, b ) ∈ K ,bs .Then there are f, h N α : α ≤ θ i such that:(a) f is an isomorphism of N to N over M .(b) h N α : α ≤ θ i is an increasing continuous sequence.(c) M α (cid:22) N α .(d) tp ( a α , N α , N α +1 ) does not fork over M α .(e) tp ( f ( b ) , M α , N α ) does not fork over M .(2) K λ + = ∅ , and it has no (cid:22) -maximal model.(3) There is a model in K of cardinality λ +2 .Proof. (1) First we explain the idea of the proof. If we ‘fix’ the models inthe sequence h M α : α ≤ θ i , then we will ‘change’ N θ times. So in limitsteps we will be in a problem. The solution is to fix N , and ‘change’ thesequence h M α : α ≤ θ i . At the end of the proof we ‘return the sequence toits place’. The proof itself:
We choose ( N ∗ α , f α ) by induction on α such that ( ∗ ) α holds where ( ∗ ) α is:(i) α ≤ θ ⇒ N ∗ α ∈ K λ .(ii) ( N ∗ , f ) = ( N, id M ).(iii) The sequence h N ∗ α : α ≤ θ i is increasing and continuous.(iv) For every α ≤ θ , the function f α is an embedding of M α to N ∗ α .(v) The sequence h f α : α ≤ θ i is increasing and continuous.(vi) For every α < θ tp ( f α ( a α ) , N ∗ α , N ∗ α +1 ) does not fork over f α [ M α ].(vii) For every α ≤ θ tp ( b, f α [ M α ] , N ∗ α ) does not fork over M .Now f θ : M θ → N ∗ θ is an embedding. Extend f − θ to a function withdomain N ∗ θ and define f := g ↾ N . Define N α := g [ N ∗ α ].(2) K λ + = ∅ , as we can choose an increasing continuous sequence ofmodels in K λ , h M α : α < λ + i , and so its union is a model in K λ + [As thereis no (cid:22) -maximal model in K λ and in limit step use axiom 1.1.1.c].Why is there no maximal model in K λ + ? Let M ∈ K λ + . Let h M α : α <λ + i be a representation of M . By the Definition 2.1.3.f (existence, on page11), for every α ∈ λ + there is an element a α ∈ M α +1 − M α (we do not use a α , but as we have written it in 1, for shortness, we have to write it here).As in K λ there is no maximal model, there is a model N such that M ≺ N ∈ K λ and without loss of generality N T M = M . By Definition 2.1.2.c(the density of basic types), there is b ∈ N − M such that tp ( b, M , N ) isbasic. Now by Proposition 3.4.1, there is an increasing continuous sequence OOD FRAMES WITH A WEAK STABILITY 25 h N α : α < λ + i and f such that f : N → N is an isomorphism over M andfor α ∈ λ + we have M α (cid:22) N α and tp ( f ( b ) , M α , N α ) does not fork over M .So by Definition 2.1, (on page 11), f ( b ) does not belong to M α for α ∈ λ + .So f ( b ) does not belong to M . But it belongs to N λ + , so M = N λ + , and forthis we defined b . But it is easy to see that M ⊆ N λ + and N λ + ∈ K λ + . Bythe smoothness (i.e. Definition 1.1.1.d on page 3) M (cid:22) N λ + . So M is not amaximal model.(3) We construct a strictly increasing continuous sequence of models in K λ + , h M α : α < λ +2 i . So its union is a model in K λ +2 . As by 2 there is nomaximal model in K λ + , there is no problem to choose this sequence. ⊣ Proposition 3.5 (a rectangle which amalgamate two sequences) . For x = a, b let h M x,α : α < θ x i be an increasing continuous sequence of models in K λ such that M a, = M b, and let h d x,α : α < θ x i be a sequence such that d x,α ∈ M x,α +1 − M x,α , and the type tp ( d x,α , M x,α , M x,α +1 ) is basic. Denote α ∗ = θ a , β ∗ = θ b . Then there are a “rectangle of models” { M α,β : α <α ∗ , β < β ∗ } and a sequence h f β : β < β ∗ i such that:(1) ( α < α ∗ ∧ β < β ∗ ) ⇒ M α,β ∈ K λ .(2) f β : M b,β → M ,β is an isomorphism.(3) M α, = M a,α .(4) f is the identity on M a, = M b, .(5) h f β : β < β ∗ i is increasing and continuous.(6) For every α, β which satisfies α + 1 < α ∗ and β < β ∗ , the type tp ( d a,α , M α,β , M α +1 ,β ) does not fork over M α, .(7) For every α, β which satisfies α < α ∗ and β + 1 < β ∗ , the type tp ( d b,β , M α,β , M α,β +1 ) does not fork over M ,β .(8) If S { Im ( f β ) : β < β ∗ } T S { M a,α : α < α ∗ } = S { M b,β : β <β ∗ } T S { M a,α : α < α ∗ } = M a, , then ( ∀ β ∈ β ∗ ) f β = id ↾ M b,β .(9) For all α (1) < α ∗ the sequence h M α (1) ,β : β < β ∗ i is increasing andcontinuous.(10) For all β (1) < β ∗ the sequence h M α,β (1) : α < α ∗ i is increasing andcontinuous. d a,α ∈ M α +1 , = M a,α +1 id / / M α +1 ,β id / / M α +1 ,β +1 M α, = M a,α id / / id O O M α,β id / / id O O M α,β +1 id O O M , = M a, = M b, id / / id O O M ,β = f β [ M b,β ] id / / id O O M ,β +1 = f β +1 [ M b,β +1 ] id O O Proof.
We define by induction on β < β ∗ f β , { M α,β : α < α ∗ } such that theconditions 1-6 and 8,9 are satisfied. For β = 0 see 3,4. For β a limitordinal, we define f β = S { f γ : γ < β } , M α,β = S { M α,γ : γ < β } . Why does 6 satisfy, i.e. why for every α , does tp ( d a,α , M α,β , M α +1 ,β ) notfork over M α, ? By the induction hypothesis 6 is satisfied for every γ <β , i.e. tp ( d a,α , M α,γ , M α +1 ,γ ) = tp ( d a,α , M α,γ , M α +1 ,γ ) does not fork over M ,γ . By Definition 2.1.3.b (monotonicity) and Definition 2.1.3.g (conti-nuity) tp ( d a,α , M α,β , M α +1 ,β ) does not fork over M α, . So condition 6 issatisfied. For β = γ + 1 use Proposition 3.4.1. So we can carry out theinduction. Now without loss of generality condition 7 is satisfied too. ⊣ Decomposition.
When we speak about tp ( a, M, N ) the order of N is peripheral. Now we consider classes K , ∗ of triples ( M, N, a ) where theorder of N is very important. For example N is the algebraic closure of M S { a } , where ( K, (cid:22) ) is the class of fields with the partial order of beingsub-field. Definition 3.6.
Let K , ∗ ⊆ K ,bs be closed under isomorphisms.(1) K , ∗ is dense with respect to (cid:22) bs if for every ( M, N, a ) ∈ K ,bs thereis ( M ∗ , N ∗ , a ∗ ) ∈ K , ∗ such that ( M, N, a ) (cid:22) bs ( M ∗ , N ∗ , a ∗ ).(2) K , ∗ has existence if for every ( M, N, a ) ∈ K ,bs there are N ∗ , a ∗ such that tp ( a ∗ , M, N ∗ ) = tp ( a, M, N ) and ( M, N ∗ , a ∗ ) ∈ K , ∗ . Inother words if p ∈ S bs ( M ) then p T K , ∗ = ∅ . Definition 3.7.
Let K , ∗ ⊆ K ,bs be closed under isomorphisms. We saythat M ∗ is decomposable by K , ∗ over M , if there is a sequence h d ε , N ε : ε <α ∗ i ⌢ h N α ∗ i such that:(1) ε < α ∗ ⇒ N ε ∈ K λ .(2) h N ε : ε (cid:22) α ∗ i is increasing and continuous.(3) N = M .(4) N ,α ∗ = M ∗ .(5) ( N ε , N ε +1 , d ε ) ∈ K , ∗ .In such a case we say that the sequence h d ε , N ε : ε < α ∗ i ⌢ h N α ∗ i is adecomposition of M ∗ over M by K , ∗ . The main case is K , ∗ = K ,uq (which we have not defined yet), and in such a case we may omit it. Theorem 3.8 (the extensions decomposition theorem) . Let K , ∗ ⊆ K ,bs be closed under isomorphisms.(1) Suppose s has conjugation. If K , ∗ is dense with respect to (cid:22) bs thenit has existence.(2) Suppose K , ∗ has existence. If N ∈ K λ and p = tp ( a, M, N ) ∈ S bs ( M ) then there are N ∗ , N + such that ( M, N ∗ , a ) ∈ K , ∗ T p, N (cid:22) N + , N ∗ (cid:22) N + .(3) Suppose K , ∗ has existence and M ≺ N . Then there is M ∗ (cid:23) N such that M ∗ is decomposable over M by K , ∗ . Moreover, letting a ∈ N − M , tp ( a, M, N ) is basic, we can choose d = a , where d isthe element which appears in Definition 3.7.Proof. (1) Suppose p = tp ( M, N, a ) ∈ S bs ( M ). As K , ∗ is dense withrespect to (cid:22) bs , there are M ∗ , N ∗ , b with ( M, N, a ) (cid:22) bs ( M ∗ , N ∗ , b ). As s OOD FRAMES WITH A WEAK STABILITY 27 has conjugation, p ∗ =: tp ( M ∗ , N ∗ , b ) conjugate to p . K , ∗ is closed underisomorphisms and so p T K , ∗ = ∅ .(2) K , ∗ has existence and so there are b, N ∗ such that: tp ( b, M, N ∗ ) = p, ( M, N ∗ , b ) ∈ K , ∗ . By the definition of a type (i.e. the definition ofequivalence between triples in a type), there are a model N + , N (cid:22) N + and an embedding f : N ∗ → N + over M such that f ( b ) = a . Denote N ∗∗ = f [ N ∗ ]. Now as K , ∗ respects isomorphisms, ( M, N ∗∗ , a ) ∈ K , ∗ . M (cid:22) N ∗∗ (cid:22) N + .(3) Assume toward a contradiction that M ≺ N and there is no M ∗ asrequired. We try to construct M α , a α , N α by induction on α ∈ λ + such that(see the diagram below):(a) M = M, N = N .(b) ( M α , M α +1 , d α ) ∈ K , ∗ .(c) M α (cid:22) N α .(d) For every α ∈ λ + , d α ∈ M α +1 T N α − M α .(e) The sequence h M α : α < λ + i is increasing and continuous.(f) The sequence h N α : α < λ + i is increasing and continuous. N id / / N id / / N α M id / / id O O M id / / id O O M α id / / id O O M α +1 ∋ a α We cannot succeed because otherwise substituting the sequences h M α : α ∈ λ + i , h N α : α ∈ λ + i , h id M α : α ∈ λ + i in Proposition 1.31 we get acontradiction. So where will we get stuck? For α = 0 there is no problem.For α limit take unions. 3 is satisfied by (smoothness) (Definition 1.1.1.d).What will we do for α + 1, (assuming we have defined ( M α , N α , d α )? If N α = M α then N α is decomposable over M by K , ∗ and the proof has reachedto its end. Otherwise by the existence of the basic types (2.1), there is d α ∈ N α − M α such that ( M α , N α , d α ) ∈ K ,bs (and for the “more over” take d = a if α = 0). By assumption K , ∗ has existence, so there are d ∗ α , M ∗ α +1 such that: ( M α , M ∗ α +1 , d ∗ α ) ∈ K , ∗ , tp ( d ∗ α , M α , M ∗ α +1 ) = tp ( d α , M α , N α ). Bythe definition of a type, there are N α +1 , N α (cid:22) N α +1 and an embedding f : M ∗ α +1 → N α +1 over M α such that f ( d ∗ α ) = d α . Denote M α +1 = Im ( f ).We have N α (cid:22) N α +1 , M α +1 (cid:22) N α +1 and ( M α , M α +1 , d α ) ∈ K , ∗ . So 2,3,4are guaranteed. ⊣ Proposition 3.9 (existence of decomposition over two models) . If n < ⇒ M n (cid:22) N then there is M ∗ such that: N (cid:22) M ∗ and M ∗ is decomposable over M and over M .Proof. Choose an increasing continuous sequence h M n : 2 (cid:22) n ≤ ω i suchthat:(1) N (cid:22) M .(2) For every n ∈ ω , M n +2 is decomposable over M n . The construction is possible by Theorem 3.8. Now by the following propo-sition M ω is decomposable over M and M . ⊣ Proposition 3.10 (the decomposable extensions transitivity) . Let h M ε : ε ≤ α ∗ i be an increasing continuous sequence of models, such that for every ε < α ∗ , M ε +1 is decomposable over M ε . Then M α ∗ is decomposable over M .Proof. Easy. ⊣ A disjoint amalgamation.
The following goal is to prove the exis-tence of a disjoint amalgamation. For this we are going to prove the densityof the reduced triples.
Definition 3.11.
The amalgamation f , f , M of M , M over M is saidto be disjoint when f [ M ] T f [ M ] = M . Definition 3.12.
The triple (
M, N, a ) ∈ K ,bsλ is reduced if ( M, N, a ) (cid:22) bs ( M ∗ , N ∗ , a ) ⇒ M ∗ T N = M . We define K ,r := { ( M, N, a ) ∈ K ,bs :( M, N, a ) is reduced } . Proposition 3.13.
The reduced triples are dense with respect to (cid:22) bs : Forevery ( M, N, a ) ∈ K ,bsλ there is a reduced triple ( M ∗ , N ∗ , a ) which is (cid:22) bs -bigger than it.Proof. Suppose towards contradiction that over (
M, N, a ) there is no re-duced triple. We will construct models M α , N α by induction on α < λ + such that:(i) ( M , N , a ) = ( M, N, a ).(ii) For every α ∈ λ + , ( M α , N α , a ) (cid:22) bs ( M α +1 , N α +1 , a ).(iii) For every α ∈ λ + , M α +1 T N α = M α .(iv) The sequence h ( M α , N α , a ) : α < λ + i is continuous, (see Definition 3.2on page 22).Why can we carry out the construction? For α = 0 see clause (i) ofthe construction. For limit α see clause (iv). Suppose we have defined h M β , N β , a ) : β ≤ α i . By Proposition 3.3 ( K ,bs , (cid:22) bs ) is closed under in-creasing union. So by clauses (i),(ii),(iv) ( M, N, a ) (cid:22) bs ( M α , N α , a ). So bythe assumption ( M α , N α , a ) is not a reduced triple, i.e. there are M α +1 , N α +1 which satisfies clauses (ii),(iii). Hence we can carry out this construction.Now we have:(1) The sequences h M α : α < λ + i , h N α : α < λ + i are increasing (byclause (ii) and the definition of (cid:22) bs ).(2) These sequences are continuous (by clause (iv)).(3) For α ∈ λ + , M α ⊆ N α (by the definition of K ,bs ).(4) For every α ∈ λ + , M α +1 T N α = M α (by clause (iii)).We got a contradiction to Proposition 1.31. ⊣ OOD FRAMES WITH A WEAK STABILITY 29
Theorem 3.14 (The disjoint amalgamation existence theorem) . Assumethat:(1) s has conjugation.(2) M , M , M ∈ K λ , M (cid:22) M and M (cid:22) M .Then there are M , f such that f : M → M is an embedding over M , M (cid:22) M , and f [ M ] T M = M . Moreover if a ∈ M − M and tp ( a, M , M ) ∈ S bs ( M ) then we can add that tp ( a, f [ M ] , M ) does not fork over M .Proof. If M = M then the theorem is trivial. Otherwise by the density ofbasic types (see Definition 2.1, page 11) there is an element a ∈ M − M suchthat tp ( a, M , M ) ∈ S bs ( M ). So it is sufficient to prove the “moreover”.By Proposition 3.13 the reduced triples are dense with respect to (cid:22) bs . So byTheorem 3.8 (the extensions decomposition theorem), as s has conjugation,there is a model M ∗ such that M (cid:22) M ∗ and M ∗ is decomposable over M by reduced triples, i.e. there is an increasing continuous sequence h N ,α : α ≤ δ i of models in K λ such that: N , = M , M ,δ = M ∗ and thereis a sequence h d α : α < δ i such that ( N ,α , N ,α +1 , d α ) is a reduced tripleand d = a . By Proposition 3.4.1 there is an isomorphism f of M over M and there is an increasing continuous sequence h N ,α : α ≤ δ i suchthat: N ,α (cid:22) N ,α , f [ M ] = N , and tp ( d α , N ,α , N ,α +1 ) does not forkover N ,α . So for α < δ, ( N ,α , N ,α +1 , d α ) (cid:22) bs ( N ,α , N ,α +1 , d α ). Butthe triple ( N ,α , N ,α +1 , d α ) is reduced, so N ,α T N ,α +1 = N ,α . Hence N , T N ,δ = N , [Why? let x ∈ N , T N ,δ . Let α be the first ordinalsuch that x ∈ N ,α . α cannot be a limit ordinal as the sequence is continuous.If α = β + 1 then x ∈ N ,α T N ,β = N ,β , in contradiction to the definitionof α as the first such an ordinal. So we must have α = 0, i.e. x ∈ N , ].Hence M T f [ M ] = N , = N . Denote M = N ,δ . ⊣ Uniqueness triples
Hypothesis . s is a semi-good λ -frame. Definition 4.2.
Suppose(1) M , M , M ∈ K λ , M (cid:22) M ∧ M (cid:22) M .(2) For x = a, b , ( f x , f x , M x ) is an amalgamation of M , M over M .( f a , f a , M a ) , ( f b , f b , M b ) are said to be equivalent over M if there are f a , f b , M such that f b ◦ f b = f a ◦ f a and f b ◦ f b = f a ◦ f a , namely the following diagram commutes: M b f b / / M M f b > > |||||||| f a / / M a f a O O M id O O id / / M f a = = zzzzzzzz f b O O We denote the relation ‘to be equivalent over M ’ between amalgamationsover M , by E M . Proposition 4.3.
The relation E M is an equivalence relation.Proof. Assume ( f a , f a , M a ) E M ( f b , f b , M b ) and ( f b , f b , M b ) E M ( f c , f c , M c ).We have to prove that ( f a , f a , M a ) E M ( f c , f c , M c ). Take witnesses g , g ,M a,b for ( f a , f a , M a ) E M ( f b , f b , M b ), and witnesses g , g , M b,c for ( f b , f b ,M b ) E M ( f c , f c , M c ). Take an amalgamation ( h , h , M ) of M a,b and M b,c over M b . Now we will prove that h ◦ g , h ◦ g , M witness that that( f a , f a , M a ) E M ( f c , f c , M c ), i.e. to prove that the following diagram com-mutes: M c h ◦ g / / M M f c = = {{{{{{{{ f a / / M a h ◦ g O O M id M O O id M / / M f a = = zzzzzzzz f c O O i.e. to prove that the following diagram commutes: M c g / / M b,c h / / M M b g O O g / / M a,b h O O M f c G G (cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14) f b mmmmmmmmmmmmmmmmm f a / / M a g O O M id O O id / / M f c O O f b E E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f a llllllllllllllllll ( h ◦ g ) ◦ f c = h ◦ g ◦ f b = h ◦ g ◦ f b = ( h ◦ g ) ◦ f a and similarly( h ◦ g ) ◦ f c = ( h ◦ g ) ◦ f a . ⊣ OOD FRAMES WITH A WEAK STABILITY 31
Definition 4.4. K ,uq = K ,uq s is the class of triples ( M , N , a ) ∈ K ,bs such that if M (cid:22) N ∈ K λ then up to equivalence over M there is a uniqueamalgamation ( f , f , N ) of M , N over M such that tp ( f ( a ) , f [ N ] , N )does not fork over M . Equivalently, if for n = 1 , M, N, a ) (cid:22) bs ( M ∗ n , N ∗ n , a )and f : M ∗ → M ∗ is an isomorphism over M , then for some f , f , N ∗ thefollowing hold: f n : N ∗ n → N ∗ is an embedding over N , and f ↾ M ∗ = f ↾ M ∗ ◦ f .A uniqueness triple is a triple in K ,uq . Proposition 4.5. (1) If p , p are conjugate types and in p there is a uniqueness triple,then also in p there is such a triple.(2) If s has conjugation, then every uniqueness triple is reduced.Proof. (1) Suppose p = tp ( a, M, N ) , ( M, N, a ) ∈ K ,uq . Let f be an isomor-phism with domain M , such that f ( p ) = p . K, (cid:22) are closed underisomorphisms, so it is easy to prove that ( f [ M ] , f + [ N ] , f + ( a )) ∈ K ,uq , where f ⊆ f + , dom ( f + ) = N . But ( f [ M ] , f + [ N ] , f + ( a )) ∈ p .(2) Suppose ( M , N , a ) ∈ K ,uq and ( M , N , a ) (cid:22) bs ( M , N , a ). ByTheorem 3.14 (the existence of a disjoint amalgamation), there are f, N such that f : M → N is an embedding over M , N (cid:22) N , f [ M ] T N = M and tp ( a, f [ M ] , N ) does not fork over M .By Definition 4.4, there are f , f , N ∗ such that: f n : N n → N ∗ andembedding over N and f ↾ M = f ◦ f . Let x ∈ M − M . Then x / ∈ N [Why? otherwise f ( x ) ∈ f [ M ] − M , so f ( x ) / ∈ N , so f ( x ) = f ( f ( x )) / ∈ N and hence x / ∈ N ]. ⊣ Definition 4.6.
Let s be a semi good λ -frame.(1) s is weakly successful in the sense of density , if K ,uq is dense withrespect to (cid:22) bs .(2) s is weakly successful if K ,uq has existence. Proposition 4.7. (1) If s is weakly successful in the sense of density and it has conjugationthen it is weakly successful.(2) Let s be weakly successful. If p = tp ( a, M, N ) ∈ S bs ( M ) , then thereis a model N ∗ such that ( M, N ∗ , a ) ∈ K ,uq T p .Proof. (1) Substitute K , ∗ := K ,uq in Theorem 3.8.1 (page 26).(2) By Theorem 3.8.2. ⊣ Now the reader can believe that the assumption that s is weakly successfulis reasonable and jump to section 5 or to read the rest of this section (whichis based on [Sh 838]). Hypothesis . s is (a semi-good λ -frame and) not weakly successful in thesense of density. Discussion toward defining nice construction frame:
Every model M ∈ K µ + can be represented as S { M β : β < µ + } where each M β is in K µ (and thesequence is increasing and continuous). Now we can represent each M β as S { M α,β : α < µ } where each M α,β is in K <µ . So we can approximate amodel M in K µ + by a “rectangle” { M α,β : α < µ, β < µ + } of models in K <µ , where h M α,β : α < β i is an increasing continuous sequence of modelsin K <µ , h S { M α,β : α < µ } : β < µ + i is an increasing continuous sequenceof models in K µ and S { M α,β : α < µ, β < µ + } = M .Now we want to violate this rectangle. For n = 1 , F R n such that ( ∀ α, β )[( M α,β , M α +1 ,β , I α,β ) ∈ F R ∧ ( M α,β , M α,β +1 , J α,β ) ∈ F R , where I α,β and J α,β are witnesses for the extensions, namely I α,β ⊂ M α +1 ,β − M α,β and J α,β ⊂ M α,β +1 − M α,β . So essentially, F R n is a relationon extensions.We have to violate also the pairs of such pairs, i.e. (( M α,β , M α +1 ,β ) , ( M α,β +1 ,M α +1 ,β +1 )). In other words, we have to define 2-dimensional relations ≤ , ≤ on F R , F R respectively. Definition 4.9. U = ( µ, k u , F R , F R , ≤ , ≤ ) is a nice construction frameif: (1) ℵ < µ is a regular cardinal.(2) k U = ( K U , (cid:22) U ) is an a.e.c. in < µ . The vocabulary of K U willdenoted τ U .(3) For n = 1 , F R n is a class of triples ( M, N, J ) such that:(a)
M, N ∈ K U , M (cid:22) U N, J ⊆ N − M .(b) For every M ∈ K U there are N, J such that: J = ∅ and( M, N, J ) ∈ F R n .(c) If M (cid:22) U N, then ( M, N, ∅ ) ∈ F R n .(4) “( F R n , ≤ n ) satisfies some axioms of a.e.c. and disjointness”:(a) ≤ n is an order relation of F R n .(b) The relations F R n , ≤ n are closed under isomorphisms.(c) If ( M , , M , , J ) ≤ n ( M , , M , , J ) then ( n ≤ n < ∧ m ≤ m < ⇒ M n ,m (cid:22) U M n ,m and M , T M , = M , .(d) Axiom c of a.e.c.: For every δ < µ and an ≤ n -increasing con-tinuous sequence h ( M α , N α , J α ) : α < δ i we have( M , N , J ) ≤ n ( S { M α : α < δ } , S { N α : α < δ } , S { J α : α <δ } ).(5) U has disjoint amalgamation (at first glance one can think that thedisjointness is in the assumption, but it is in the conclusion, see 4c):If ( M , M , J ) ∈ F R , ( M , M , J ) ∈ F R and M T M = M then there are M , J ∗ , J ∗ such that for n = 1 , M n (cid:22) U M and( M , M n , J n ) ≤ n ( M − n , M , J ∗ n ).A way to force an amalgamation to be disjoint, is to replace the equal-ity relation by an equivalence one. This is the role of E in the followingdefinition. Definition 4.10.
Let U be a nice construction frame. Let ( K, (cid:22) ) be ana.e.c. with a vocabulary τ , such that τ ⊆ τ U and there is a 2-place predicate E ∈ τ U − τ (in the main case τ U = τ S { E } ), such that for M ∈ K U wehave:(1) E M is an equivalence relation.(2) If R is a predicate in τ U different from = and xE M y then R M ( x , ...,x i − , x, x i +1 , ...x n ) iff R M ( x , ..., x i − , y, x i +1 , ...x n ).Similarly for function symbols.We write ( K, (cid:22) ) = ( U/E ) τ when:( K, (cid:22) ) is an a.e.c. and K <µ = { N : ( ∃ M ∈ K U )( N = M/E ) } , where M/E is defined by the following way: Its world is the set of equivalence classesof E M , its vocabulary is τ and it interprets the predicates and functionsymbols by representatives of the equivalence classes.Now we are going to define approximations of cardinality µ , by the ap-proximations of cardinality < µ . Definition 4.11. (1) K qt = K qt,U := { ( ¯ M , ¯ J , f ) : ¯ M = h M α : α < µ i , ¯ J = h J α : α <µ i , f ∈ µ µ, α < µ ⇒ ( M α , M α +1 , J α ) ∈ F R } ( f plays a role in therelation ≤ qt ).(2) ≤ qt is a relation on K qt . ( M , J , f ) ≤ ( M , J , f ) iff there is a club E of µ such that for every δ ∈ E and α ≤ f ( δ ) we have:(a) f ( δ ) ≤ f ( δ ).(b) M ,δ +1 ≤ M ,δ +1 .(c) ( M ,δ + α , M ,δ + α +1 , J ,δ + α ) ≤ ( M ,δ + α , M ,δ + α +1 , J ,δ + α ).(d) M ,δ + α T S { M ,ε : ε < µ } = M ,δ + α . Definition 4.12.
We say that almost every ( ¯
M , ¯ J , f ) ∈ K qt satisfies theproperty pr when: There is a function g : K qt → K qt such that if h ¯ M α :¯ J α , f α i is an ≤ qt -increasing continuous (in the sense which is defined in[Sh 838] and not here) and sup { α ∈ δ : g (( ¯ M α , ¯ J α , f α )) = ( ¯ M α +1 , ¯ J α +1 , f α +1 ) } = δ ), then ( ¯ M δ , ¯ J δ , f δ ) ∈ pr . Definition 4.13. (1) Let U be a nice construction frame. We say that U satisfies the weakcoding property for ( K, (cid:22) ) if almost every ( ¯ M , ¯ J , f ) ∈ K qt satisfiesthe weak coding property.(2) We say that ( ¯ M , ¯ J , f ) ∈ K qt satisfies the weak coding property when: There are α < µ and N , J such that ( M α , N , J ) ∈ F R , N T M = M α where M := S { M α : α < µ } , and thereis a club E of µ such that for every α ∈ E and every N , J , whichsatisfy ( M α , N , J ) ≤ ( M α , N , J ) ∧ N T M = M α , there is α ∈ ( α , µ ) and for n = 1 , N ,n , J ,n such that:(a) ( M α , N , J ) ≤ ( M α , N ,n , J ,n ).(b) N , , N , are incomparable amalgamations of M α , N over M α ,i.e. there are no N, f , f such that f n is an embedding of N ,n into N over N S M α . N , N id / / N id / / id : : vvvvvvvvv N , M α (0) id / / id O O M α (1) id / / id O O M α (2) id / / id O O id ; ; xxxxxxxx M Definition 4.14. µ unif ( µ + , µ ) := M in {| P | : P is a family of subsets of µ + (2 µ ) with union µ + (2 µ ) and for each A ∈ P there is a function c withdomain S { α (2 µ ) : α < µ + } such that for each f ∈ A , the set { δ ∈ µ + : c ( f ↾ δ ) = f ( δ ) } is not stationary } . Proposition 4.15. µ unif ≤ µ + .Proof. Easy. ⊣ Remark 4.16. µ unif ( µ + , µ ) is “almost 2 µ + ”: If i ω ≤ µ , then µ unif ( µ + , µ )= 2 µ + , and in any case it is not clear if µ unif ( µ + , µ ) < µ + is consistent.There are propositions which say that it is “a big cardinal”.The following theorem is written in [Sh 838], and we bring it without aproof. Theorem 4.17.
Let U be a nice construction frame which satisfies the weakcoding property for ( K, (cid:22) ) . Suppose the following set theoretical assump-tions:(1) θ = 2 <µ < µ .(2) µ < µ + .(3) The ideal W dmId ( µ ) is not saturated in µ + .Then µ unif ( µ + , µ ) ≤ I ( µ + , K ) , where I ( µ + , K ) is the number of non-isomorphic models in K µ + . Now we are going to study a specific nice construction frame. From now( K, (cid:22) ) will denote the a.e.c. of s . Definition 4.18.
Define U = ( µ, ( K u , (cid:22) u ) , F R , F R , ≤ , ≤ ):(1) µ = λ + . OOD FRAMES WITH A WEAK STABILITY 35 (2) The vocabulary of K U is τ U := τ S { E } where E is a new predicate.(3) K U := { M : || M || = λ, M/E ∈ K λ } . ( M/E is well defined only if E M is a congruence relation on | M | , see Definition 4.10. So if not,then M does not belong to K U ).(4) (cid:22) U := { ( M, N ) :
M/E (cid:22)
N/E ∧ M ⊆ N } .(5) F R n := { ( M, N, J ) :
M, N ∈ K U , J = ∅ ⇒ ( ∃ a )[ J = { a } ∧ ( M/E, N/E, a/E ) ∈ K ,bs ] } .(6) For n = 1 , ≤ n is defined by the relation (cid:22) bs in thesame way we defined F R n . Proposition 4.19.
Almost every ( ¯
M , ¯ J , f ) ∈ K qt,U satisfies: S { M α /E : α < λ + } is a saturated model in λ + over λ .Proof. See [Sh 838]. ⊣ Theorem 4.20. If ¯ M = h M α : α < λ + i , ¯ a = h a α : α < λ + i , ( ¯ M , ¯ a, f ) ∈ K qt and S { M α /E : α < λ + } is saturated in λ + over λ , then ( ¯ M , ¯ a, f ) satisfies the weak coding property.Proof. For distinguishing between models in K λ to models in K U , we addto the names of models in K λ subscript e , unless they are written in theform M/E . For example: M e , M ,e . Similarly for isomorphisms. Lemma 4.21. (1) Let N ∈ K U , N ,e ∈ K λ be such that N /E (cid:22) N ,e . Then there is N ∈ K U such that:(a) N /E = N ,e .(b) N (cid:22) U N .(c) N is embedded in every model which satisfies 1,2.In this case we call N the canonical completion of N , N ,e . Thereis exactly one such a model up to isomorphism. Clearly every [ x ] ∈ N − N is a singleton.(2) Suppose:(a) N (cid:22) U N , N (cid:22) U N .(b) g e : N /E → N /E is an embedding over N /E .(c) N is the canonical completion of N /E, N .Then there is an embedding g : N → N over N such that ( ∀ x ∈ N )( g ( x ) ∈ [ g e ( x/E )]) .(3) Suppose for n < , N n ∈ K U , N /E (cid:22) N n /E (cid:22) N ,e ∈ K λ and N T N = N . Then there is N ∈ K U such that N /E = N ,e andfor n = 1 , N n (cid:22) N .Proof. (1) Trivial.(2) Use the axiom of choice [For x ∈ N − N g ( x ) choose an arbitraryelement in g e ([ x ])].(3) Trivial. ⊣ Now we prove that ( ¯
M , ¯ a, f ) satisfies the weak coding property, by thefollowing steps: Step a:
Denote α (0) = 0. M /E ∈ K λ . So by the categoricity in K λ and non-weak successfulness, there are N ,e ∈ K λ and a ∈ N ,e such that( M /E, N ,e , a ) ∈ K ,bs and every triple which is (cid:22) bs -bigger from it isnot a uniqueness triple. Without lose of generality N ,e T M/E = M /E .Let N ∈ K U be the model with world N ,e , E N is the equality, and N /E = N ,e . λ + is of course a club of λ + . Let α (1) ∈ ( α (0) , µ ), andlet N ∈ K U such that N T M = M α (1) , ( M , N , a ) ≤ n ( M α (1) , N , a ). Wehave to find α (2). Step b: ( M α (1) /E, N /E, a ) is not a uniqueness triple. So for n < M ,n,e , N ∗ ,n,e ∈ K λ and an isomorphism g e : M , ,e → M , ,e over M α (1) /E such that ( M α (1) /E, N /E, a ) (cid:22) bs ( M ,n,e , N ∗ ,n,e , a ) and thereare no g ,e , g ,e , N ,e such that g n,e : N ∗ ,n,e → N ,e ∈ K λ an embedding over N /E and g ,e ◦ g e = g ,e . We choose new elements for N ∗ ,n,e − ( M α (1) /E ),i.e. without loss of generality M/E T N ∗ ,n,e = M α (1) /E . By item 1 in thelemma for n < M ,n which is canonical over M α (1) , M ,n,e .By item 3 of the lemma for n < N ∗ ,n ∈ K U such that M ,n (cid:22) U N ∗ ,n , N (cid:22) N ∗ ,n and N ∗ ,n /E = N ∗ ,n,e . N id / / N id / / N ∗ ,n,e M id / / id O O M α (1) id / / id O O M ,n,eid O O Step c:
M/E is saturated in λ + over λ , so by Lemma 1.32 (the saturation= model homogeneity lemma), there is an embedding f ,e : M , ,e → M/E over M α (1) /E . So by item b of the lemma over, there is an embedding f : M , → M over M α (1) . Define f = f ◦ g − e . Now for n < f n : M ,n → M is an embedding. N ,e M , ,eg ,e ; ; g e / / M , ,eg ,e c c GGGGGGGG
Step d:
For n < h n be a function with domain N ∗ ,n that extends f n by the identity. So h n ↾ N is the identity. N id / / h n [ N ∗ ,n ] M α (1) id / / id O O f n [ M ,n ] id / / id O O M OOD FRAMES WITH A WEAK STABILITY 37
Step e:
Define α (2) := Min { α ∈ λ + : f [ M , ] (cid:22) M α (2) } . Step f:
For n < N ,n ∈ K U such that ( f n [ M ,n ] ,h n [ N ∗ ,n ] , a ) (cid:22) ( M α (2) , N ,n , a ). N id / / h n [ N ∗ ,n ] id / / N ,n M α (1) id / / id O O f [ M , ] id / / id O O M α (2) id O O By the transitivity of the relation ≤ , we have ( M α (1) , N , a ) ≤ ( M α (2) ,N ,n , a ). Step g: N , , N , witness that α (2) is as required [Toward contradictionassume that there are N ,e ∈ K λ and embeddings g ,e , g ,e such that g n : N ,n /E → N is an embedding over M α (2) /E S N /E Define an isomor-phism g ∗ n,e : N ∗ ,n,e → N ,e by g ∗ n,e ( x ) := g n,e ([ h n ( x )]). This is an embed-ding over N /E and it includes f n,e . This contradict the way we chose M ,n,e , N ∗ ,n,e in step b]. Hence the triple ( ¯ M , ¯ a, f ) satisfies the weak codingproperty. ⊣ Corollary 4.22. U satisfies the weak coding property.Proof. By 4.19,4.20. ⊣ Corollary 4.23.
Let s be a semi-good λ -frame which is not weakly successfulin the sense of density. Then I ( λ +2 , K ) ≥ µ unif ( λ +2 , λ + ) .Proof. By 4.17,4.22. ⊣ Non-forking amalgamation
Hypothesis . s is a weakly successful semi-good λ -frame with conjugation,but we do not use local character in this section.5.1. The axioms of non-forking amalgamation.
Introduction:
We want to find a relation of a canonical amalgamation (seethe discussion in the beginning of section 3). In Definition 5.2 we define theproperties this relation has to satisfy.
Definition 5.2.
Let
N F ⊆ ( K λ ) be a relation. We say N NF when thefollowing axioms are satisfied:(a) If N F ( M , M , M , M ) then n ∈ { , } ⇒ M (cid:22) M n (cid:22) M and M ∩ M = M .(b) Monotonicity: If N F ( M , M , M , M ) and N = M , n < ⇒ N n (cid:22) M n ∧ N (cid:22) N n (cid:22) N , ( ∃ N ∗ )[ M (cid:22) N ∗ ∧ N (cid:22) N ∗ ] then N F ( N , N , N , N ).(c) Existence: For every N , N , N ∈ K λ if l ∈ { , } ⇒ N (cid:22) N l and N T N = N then there is N such that N F ( N , N , N , N ) . (d) Weak uniqueness: Suppose for x = a, b N F ( N , N , N , N x ). Then thereis a joint embedding of N a , N b over N S N .(e) Symmetry: N F ( N , N , N , N ) ⇔ N F ( N , N , N , N ).(f) Long transitivity: For x = a, b let h M x,i : i ≤ α ∗ i an increasing continu-ous sequence of models in K λ . Suppose i < α ∗ ⇒ N F ( M a,i , M a,i +1 , M b,i ,M b,i +1 ). Then N F ( M a, , M a,α ∗ , M b, , M b,α ∗ )We give another version of weak uniqueness: Proposition 5.3.
Suppose(1) N NF .(2) N F ( M , M , M , M ) and N F ( M , M ∗ , M ∗ , M ∗ ) .(3) For n = 1 , there is an isomorphism f n : M n → M ∗ n over M .Then there are M, f such that:(1) For n < f ↾ M n = f n .(2) M ∗ (cid:22) M .(3) f [ M ] (cid:22) M .Proof. M T M = M , so there is a function g with domain M such that f S f ⊆ g . So g [ M ] = M ∗ and g [ M ] = M ∗ . Hence N F ( M , M ∗ , M ∗ ,g [ M ]) and N F ( M , M ∗ , M ∗ , M ∗ ). Therefore we can use the weak unique-ness in Definition 5.2. ⊣ Roughly speaking the following proposition says that finding a relation
N F that satisfies clauses a,c,d of Definition 5.2 is equivalent to assigning toeach triple ( M , M , M ) ∈ D := { ( M , M , M ) : M , M , M ∈ K λ , M (cid:22) M , M (cid:22) M } a disjoint amalgamation (see Definition 3.11) ( f , f , M ) of M , M over M up to E M (see Definition 4.2. Proposition 5.4.
Let
N F be a relation that satisfies clauses a,c,d of Defi-nition 5.2. Then:(1) There is a function G with domain D := { ( M , M , M ) : M , M , M ∈ K λ , M (cid:22) M , M (cid:22) M } which assign to each triple ( M , M , M ) an amalgamation ( f , f , M ) of M , M over M , such that N F ( M , f [ M ] ,f [ M ] , M ) (in this item we do not use clause d).(2) If G , G are two functions as in item 1 (with respect to N F ), thenfor every ( M , M , M ) ∈ D , G (( M , M , M )) E M G (( M , M , M )) .(3) If G is a function with domain D := { ( M , M , M ) : M , M , M ∈ K λ , M (cid:22) M , M (cid:22) M } which assign to each triple ( M , M , M ) a disjoint amalgamation, then the relation R := { ( M , M , M , M ) : M T M = M , G (( M , M , M )) E M ( id M , id M , M ) } satisfies clausesa,c,d of Definition 5.2.Proof. We leave to the reader. ⊣ Definition 5.5.
Suppose N NF . N F is said to respect the frame s when: if N F ( M , M , M , M ) and tp ( a, M , M ) ∈ S bs ( M ) then tp ( a, M , M ) doesnot fork over M . OOD FRAMES WITH A WEAK STABILITY 39
The relation
N F . First we define a relation
N F ∗ and then we definea relation N F as its monotonicity closure. Theorem 5.27 asserts that therelation
N F is the unique relation which satisfies N NF and respects theframe s . Definition 5.6.
Define a relation
N F ∗ = N F ∗ λ on ( K λ ) by: N F ∗ ( N ,N , N , N ) if there is α ∗ < λ + and for l=1,2 there are an increasing contin-uous sequence h N l,i : i ≤ α ∗ i and a sequence h d i : i < α ∗ i such that: N = N , id / / N ,i id / / N ,i +1 id / / N ,α ∗ = N N = N , id / / id O O N ,i id / / id O O N ,i +1 id O O id / / N ,α ∗ = N id O O (a) n < ⇒ N (cid:22) N n (cid:22) N .(b) N , = N , N ,α ∗ = N , N , = N , N ,α ∗ = N .(c) i ≤ α ∗ ⇒ N ,i (cid:22) N ,i .(d) d i ∈ N ,i +1 − N ,i .(e) ( N ,i , N ,i +1 , d i ) ∈ K ,uq .(f) tp ( d i , N ,i , N ,i +1 ) does not fork over N ,i .In this case, h N ,i , d i : i < α ∗ i ⌢ h N ,α ∗ i is said to be the first witness for N F ∗ ( N , N , N , N ), d i is said to be the i -th element in the first wit-ness for N F ∗ and h N ,i : i ≤ α ∗ i is said to be the second witness for N F ∗ ( N , N , N , N ). Definition 5.7.
N F = N F λ is the class of quadruples ( M , M , M , M )such that M (cid:22) M (cid:22) M , M (cid:22) M (cid:22) M and there are models N , N , N ,N such that: N = M , l < ⇒ M l (cid:22) N l and N F ∗ ( N , N , N , N ). Proposition 5.8.
The relations
N F ∗ , N F are closed under isomorphisms.Proof. Trivial. ⊣ Proposition 5.9.
Suppose for x = a, b ( f x, , f x, , M x, ) is an amalgamationof M , M over M . If ( f a, , f a, , M a, ) E M ( f b, , f b, , M b, ) , then N F ( M , f a, [ M ] , f a, [ M ] , M a, ) ⇔ N F ( M , f b, [ M ] , f b, [ M ] , M b, ) Proof.
Easy. ⊣ Proposition 5.10. (1) Every triple in K ,uq is reduced.(2) If N F ∗ ( N , N , N , N ) then N T N = N .(3) If N F ( N , N , N , N ) then N T N = N .Proof. (1) Suppose ( N , N , d ) (cid:22) bs ( N , N , d ) , ( N , N , d ) ∈ K ,uq . By Hy-pothesis 5.1 and Proposition 3.14 (page 29) there is a disjoint amalgamationof N , N over N , such that the type of d does not fork over N , so by thedefinition of uniqueness triple (definition 4.4), N is a disjoint amalgamationof N , N over N . (2) Let x ∈ N T N . We will prove that x ∈ N . Let h N ,α , d α : α <α ∗ i ⌢ h N ,α ∗ i , h N ,α : α ≤ α ∗ i be witnesses for N F ∗ ( N , N , N , N ). Let α be the first ordinal such that x ∈ N ,α . α is not a limit ordinal, be-cause a first witness for N F ∗ is especially a continuous sequence. we provethat α is not a successor ordinal, so we conclude that α = 0. Suppose α = β + 1. By Definition 5.6.e ( N ,β , N ,β +1 , d β ) ∈ K ,uq . By Definition5.6.f tp ( d β , N ,β , N ,β +1 ) does not fork over N ,β . So by Proposition 5.10.1 N ,β +1 T N ,β = N ,β . But x ∈ N ,β +1 T N ⊆ N ,β +1 T N ,β , so x ∈ N ,β incontradiction to the assumption that α is the minimal ordinal with x ∈ N ,α .(3) By 2. ⊣ Theorem 5.11 (the existence theorem for
N F ) . Suppose that for n = 1 , N (cid:22) N n and N T N = N .(a) For some model N ∈ K λ N F ( N , N , N , N ) .(b) Moreover, if N is decomposable over N by K ,uq then for some N ∈ K λ N F ∗ ( N , N , N , N ) .(c) Moreover, letting a ∈ N − N we can choose a as the first element inthe first witness for N F ∗ .Proof. (a) By Theorem 3.8.3 (the extensions decomposition theorem, page 26),(and assumption 5.1), there is a model N ∗ with N (cid:22) N ∗ which is de-composable over N , i.e. there is a sequence h N ,α , d α : α < α ∗ i ⌢ h N ,α ∗ i ,such that: N = N , , ( N ,α , N ,α +1 , d α ) ∈ K ,uq , N (cid:22) N ,α ∗ = N ∗ .Therefore we can use item b.(b) Let h N ,α , d α : α < α ∗ i ⌢ h N ,α ∗ i be an increasing continuous sequencewith N , = N and N ,α ∗ = N . By Proposition 3.4.1 there is a se-quence h N ,α : α ≤ α ∗ i which is a corresponding second witness for N F ∗ ( N , N ,α ∗ , N , N ,α ∗ ).(c) By the ‘more over’ in Theorem 3.8.3 (the decomposing extensions theo-rem, page 26). ⊣ The following theorem is a private case of Theorem 5.25, i.e. the longtransitivity theorem.
Proposition 5.12.
For x = a, b let h M x,α : α ≤ α ∗ i be an increasing con-tinuous sequence of models. Suppose α < α ∗ ⇒ N F ∗ ( M a,α , M a,α +1 , M b,α ,M b,α +1 ) . Then N F ∗ ( M a, , M a,α ∗ , M b, , M b,α ∗ ) .Proof. Concatenate all the sequences together. ⊣ Proposition 5.13 (the monotonicity theorem) . (1) If N F ∗ ( N , N , N , N ) and N (cid:22) M (cid:22) N , then N F ∗ ( N , N , M ,N ) .(2) If N F ( M , M , M , M ) then we can find N , N such that N F ∗ ( M ,N , M , N ) and M (cid:22) N (cid:22) N ∧ M (cid:22) N . OOD FRAMES WITH A WEAK STABILITY 41 (3)
N F ∗ ( M , M , M , M ) ∧ M (cid:22) M ∗ ⇒ N F ( M , M , M , M ∗ ) .(4) The relation N F satisfies monotonicity (in the sense of Definition5.2.b).Proof. (1) Let h N ,α , d α : α < α ∗ i , h N ,α : α < α ∗ i be witnesses for N F ∗ ( N , N ,N , N ). Then h N ,α : α < α ∗ i , h M i ⌢ h N ,α : 0 < α < α ∗ i are witnessesfor N F ∗ ( N , N , N , N ) (notice that by Definition 2.1.3.b (monotonicity) tp ( d , M , N , ) does not fork over N ).(2) By the definition of N F (Definition 5.7) and item 1.(3) a ∈ M ∗ f / / M ∗∗ M id / / id O O M id / / M ∗ id O O M id / / id O O M id O O Take p ∈ S bs ( M ), and take M ∗ , a such that ( M , M ∗ , a ) ∈ p T K ,uq . ByDefinition 2.1.1.3.f (on page 11) there is an amalgamation ( f, id M ∗ , M ∗∗ )of M ∗ , M ∗ over M such that tp ( a, f [ M ∗ ] , M ∗∗ ) does not fork over M . So N F ∗ ( M , f [ M ∗ ] , M ∗ , M ∗∗ ). Hence by item 1, N F ∗ ( M , f [ M ∗ ] , M , M ∗∗ ).Now by Proposition 5.12 N F ∗ ( M , M ∗ , M , M ∗∗ ). So the definition of N F (Definition 5.7),
N F ( M , M , M , M ∗ ).(4) Suppose M ∗ = M , < n < ⇒ M ∗ (cid:22) M ∗ n (cid:22) M ∗ , M ∗ n (cid:22) M n , M ∗ (cid:22) M ∗∗ , M (cid:22) M ∗∗ , N F ( M , M , M , M ). M ∗∗ f / / M ∗∗∗ N id / / N id O O M id O O id / / M id O O id sssssssssss M ∗ id / / id O O M ∗ id G G (cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15) M id / / id O O M ∗ id / / id O O M = N id O O By item 2, for some N , N , N F ∗ ( M , N , M , N ), M (cid:22) N (cid:22) N and M (cid:22) N . Take an amalgamation ( f, id N , M ∗∗∗ ) of M ∗∗ and N over M (so over M ∗ S M ∗ ). By item 3 N F ( M , N , M , M ∗∗∗ ). So by the definition of N F (Definition 5.7),
N F ( M , M ∗ , M ∗ , f [ M ∗ ]). But the relation N F isclosed under isomorphisms, so
N F ( M , M ∗ , M ∗ , M ∗ ). ⊣ Weak Uniqueness.
We want to show that
N F satisfies weak unique-ness and long transitivity. Proposition 5.17 is a key point. To emphasize theexact hypotheses involved in the proof, we extract from the axioms N R , asmaller set N − R . Definition 5.14.
Let R ⊆ ( K λ ) be a relation. We say N − R when:(1) If R ( M , M , M , M ) then n ∈ { , } ⇒ M (cid:22) M n (cid:22) M .(2) Weak Uniqueness: Suppose for x = a, b ( f x , f x , N x ) is an amalga-mation of N and N over N and R ( N , f x [ N ] , f x [ N ] , N x ). Then( f a , f a , N a ) E N ( f b , f b , N b ).(3) If R ( M , M , M , M ) and f : M → M is an embedding, thenthere is an amalgamation ( g, id M , M ) of M , M over M such that R ( f [ M ] , g [ M ] , M , M ). M id / / M g / / M M id O O id / / M id O O f / / M id O O Definition 5.15.
N F uq := { ( M , M , M , M ):there is a ∈ M − M suchthat ( M , M , a ) ∈ K ,uq and tp ( a, M , M ) does not fork over M } . Proposition 5.16. (1) N − NF uq .(2) For every relation R , N R ⇒ N − R .Proof. (1) By the definition of K ,uq (Definition 4.4), Definition 2.1.3.f andDefinition 2.1.1.d (to get M ).(2) By axioms d,f in Definition 5.2 and by Proposition 2.15. ⊣ Proposition 5.17 (the transitivity of the weak uniqueness) . Suppose(1) N − R .(2) α ∗ ≤ λ + .(3) For every α < α ∗ N ,α , N a ,α , N b ,α ∈ K λ .(4) h N ,α : α ≤ α ∗ i , h N a ,α : α ≤ α ∗ i , h N b ,α : α ≤ α ∗ i are increasingcontinuous sequences.(5) N a , = N b , .(6) For every α ≤ α ∗ f aα : N ,α → N a ,α and f bα : N ,α → N b ,α .(7) ( α < α ∗ ∧ x ∈ { a, b } ) ⇒ R ( f xα [ N ,α ] , f xα +1 [ N ,α +1 ] , N x ,α , N x ,α +1 ) .Then ( f aα ∗ , id N a , , N a ,α ∗ ) E N , ( f aα ∗ , id N a , , N b ,α ∗ ) . OOD FRAMES WITH A WEAK STABILITY 43
Proof.
We choose N ,α , g a,α , g b,α by induction on α ≤ α ∗ , such that for x = a, b and α ≤ α ∗ the following hold:(i) g x,α : N x ,α → N ,α is an embedding such that g a,α ◦ f aα = g b,α ◦ f b,α .(ii) N , = N x , , g x, = identity .(iii) h N ,α : α ≤ α ∗ i is an increasing continuous sequence.(iv) h g x,α : α ≤ α ∗ i is an increasing continuous sequence.If we can construct this, then the following diagram commutes: N a ,α ∗ g a,α ∗ / / N ,α ∗ N ,α f aα ∗ < < yyyyyyyyy f bα ∗ / / N b ,α ∗ g b,α ∗ ; ; xxxxxxxxx N , id O O id / / N , id O O id ; ; [By clause (i) g a,α ∗ ◦ f aα ∗ = g b,α ∗ ◦ f bα ∗ and by clauses ( ii ) , ( iv ) g x,α ∗ ⊇ g x, = id N , ].Therefore ( g a,α ∗ , g b,α ∗ , N ,α ∗ ) witnesses that ( f aα ∗ , id N , , N a ,α ∗ ) E N , ( f bα ∗ ,id N , , N b ,α ∗ ).Why can we construct this? For α = 0 only clause (ii) is relevant. For α limit ordinal, take unions, and by the smoothness, g x,α is (cid:22) -embedding.What will we do for α +1? By clause 7 for x = a, b R ( f xα [ N ,α ] , f xα +1 [ N ,α +1 ] ,N x ,α , N x ,α +1 ). By clause (i) g x,α [ N x ,α ] (cid:22) N ,α and by clause 1 N − R . So byDefinition 5.14.3 we can find g x , N x such that the following hold:(1) g x : N x ,α +1 → N x is an embedding.(2) g x,α ⊂ g x .(3) R ( g x ◦ f aα [ N ,α ] , g x ◦ f aα +1 [ N ,α +1 ] , N ,α , N x ). N a h a / / N ,α +1 N a ,α +1 g a jjjjjjjjjjjjjjjjjjjj N bh b O O N ,α +1 f bα +1 / / f aα +1 : : uuuuuuuuu N b ,α +1 g b kkkkkkkkkkkkkkkkkkk N a ,α g a,α / / id O O N ,α id D D (cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10) id O O N ,α f bα / / f aα : : uuuuuuuuu id O O N b ,αid O O g b,α ; ; Hence by Definition 5.14.2 ( g a ↾ f aα +1 [ N ,α +1 ] , id N ,α , N a ) E f aα [ N ,α ] ( g b ↾ f bα +1 [ N ,α +1 ] , id N ,α , N b ). So there is a joint embedding ( h a , h b , N ,α +1 ) of N a , N b such that for x = a, b id N ,α ⊆ h x and h a ◦ g α ◦ f aα +1 = h b ◦ g b ◦ f bα +1 .Now we define g x,α +1 := h x ◦ g x . ⊣ The following proposition asserts that we have weak uniqueness over firstwitness for
N F ∗ . Proposition 5.18.
If for x = a, b N F ∗ ( N , N , N , N x ) and they have thesame first witness, then there is a joint embedding of N a , N b over N S N .Proof. By Proposition 5.16.1, N − NF uq . Hence it follows by Proposition 5.17. ⊣ The following proposition is similar to weak uniqueness for
N F ∗ , butnotice to the order of N , N in the two quadruples. Proposition 5.19 (the opposite uniqueness proposition) . Suppose
N F ∗ ( N ,N , N , N a ) and N F ∗ ( N , N , N , N b ) . Then there is a joint embedding of N a and N b over N S N .Proof. Let h N aα , d aα : α < α ∗ i ⌢ h N aα ∗ i be a first witness for N F ∗ ( N , N , N ,N a ) and let h N bβ , d bβ : β < β ∗ i ⌢ h N bβ ∗ i be a first witness for N F ∗ ( N , N , N ,N b ). By Proposition 3.5 (page 25), there is a rectangle { M α,β : α ≤ α ∗ , β ≤ β ∗ } such that:(1) M α, = N aα .(2) M ,β = N bβ .(3) tp ( d aα , M α,β , M α +1 ,β ) does not fork over M α, .(4) tp ( d bβ , M α,β , M α,β +1 ) does not fork over M ,β . OOD FRAMES WITH A WEAK STABILITY 45 N a f a / / N a, ∗ g a / / N ∗ N = N ,α ∗ id / / id hhhhhhhhhhhhhhhhhhhhhhhhhh id - - [[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[ M α ∗ ,β ∗ id / / id O O N b, ∗ g b O O d aα ∈ N aα +1 id / / id O O M α +1 ,β id / / M α +1 ,β +1 id pppppppppppp N b f b O O N aα = M α, id / / id O O M α,β id / / id O O M α,β +1 id O O N a = M , id / / id O O M ,β id / / id O O M ,β +1 id O O N = M , id / / id O O N bβ = M ,β id / / id O O N bβ +1 id / / id O O N = M ,β ∗ id F F (cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14) id O O id U U +++++++++++++++++++++++++++++++++++++++++++ By clauses 1,3 h d aα , N aα : α < α a i is a first witness for N F ∗ ( N , N , N ,M α ∗ ,β ∗ ). But by definition this is also a first witness for N F ∗ ( N , N ,N , N a ). So by Proposition 5.18, there is a joint embedding ( id M α ∗ ,β ∗ , f a , N a, ∗ )of M α ∗ ,β ∗ , N a over N S N . Similarly by clauses 2,4 there is a joint em-bedding ( id M α ∗ ,β ∗ , f b , N b, ∗ ) of M α ∗ ,β ∗ , N b over N S N . Since ( K λ , (cid:22) ↾ K λ )has amalgamation, there is an amalgamation ( g a , g b , N ) of N a, ∗ , N b, ∗ over M α ∗ ,β ∗ . N is an amalgam of N a , N b over N S N . ⊣ Theorem 5.20 (the weak uniqueness theorem) . Suppose for x = a, b N F ( M , M , M , M x ) . Then there is a joint embedding of M a , M b over M S M .Proof. First note that since M T M = M , the conclusion of the theoremis equivalent to ( id M , id M , M a ) E M ( id M , id M , M b ). Case a:
N F ∗ ( M , M , M , M x ) and M is decomposable over M . In thiscase, by Theorem 5.11.b (the existence theorem for N F ) there is M c suchthat N F ∗ ( M , M , M , M c ). By Proposition 5.19 for x = a, b id M , id M , M x ) E M ( id M , id M , M c ). But the relation E M is an equivalence relation, so it istransitive. The general case:
Since
N F ( M , M , M , M a , ) by Proposition 5.13.5there are N a , N a, − such that N F ∗ ( M , N a , M , N a, − ) and M (cid:22) N a (cid:22) N a, − ∧ M a (cid:22) N a, − . Similarly there are N b , N b, − such that N F ∗ ( M , N b , M ,N b ) and M (cid:22) N b (cid:22) N b, − ∧ M b (cid:22) N b, − . By Theorem 3.8 (the extensionsdecomposition theorem) there is a model M +2 (cid:23) M which is decomposableover M . Without loss of generality for x = a, b M +2 T N x, − = M . So by Theorem 3.8.3 (the extensions decomposition theorem) there is N x (cid:23) N x, − such that N F ∗ ( M , N x , M +2 , N x ). N a, + N g a ; ; xxxxxxxx g b / / N b, + N a f a = = {{{{{{{{ id / / N ag a O O N b f b O O id / / N bg b O O M a id E E (cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11) M id O O id F F (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) id < < zzzzzzzz id / / M bid O O M id O O id / / M +2 id O O id H H (cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17) By Proposition 3.9 there is an amalgamation ( f a , f b , N ) of N a , N b over M such that N is decomposable over N a and over N b . Hence for x = a, b there is an amalgamation ( g x , g x , N x, + ) of N , N x over N x such that N F ∗ ( g x ◦ f x [ N x ] , g x [ N ] , g x [ N x ] , N x, + ). So for x = a, b by Proposition5.13.8 (a private case of transitivity), since N F ∗ ( M , N x , M +2 , N x ) and N F ∗ ( N x , N x , N , N x, + ) it follows that N F ∗ ( M , N , M +2 , N x, + ). So by casea ( g a , g a ↾ M +2 , N a, + ) E M ( g b , g b ↾ M +2 , N b, + . Therefore ( g a ↾ M , g a ↾ M , N a, + ) E M ( g b ↾ M , g b ↾ M , N b, + ). ⊣ Proposition 5.21. N − NF .Proof. We have to check clauses 1,2,3 of Definition 5.14 (on page 42).1. Trivial.2. By Theorem 5.20.3. Suppose
N F ( M , M , M , M ) and f : M → M is an embedding. Wehave to find a model M and an embedding g : M → M over M such that N F ( f [ M ] , g [ M ] , M , M ). By Theorem 5.13.2 we can find N , N such that N F ∗ ( M , N , M , N ) and M (cid:22) N (cid:22) N ∧ M (cid:22) N . By Theorem 5.11.b(the existence theorem for N F , on page 40) we can find a model M with M (cid:22) M and an embedding h : N → M such that N F ∗ ( M , M , N , M ).Hence N F ( M , M , M , M ). Now we define g := h ↾ M . ⊣ OOD FRAMES WITH A WEAK STABILITY 47
Theorem 5.22 (the symmetry theorem) . N F ( N , N , N , N ) ⇔ N F ( N ,N , N , N ) .Proof. By the monotonicity of NF, i.e. Propositon 5.13.3, It is sufficient toprove
N F ∗ ( N , N , N , N ) ⇒ N F ( N , N , N , N ). Suppose N F ∗ ( N , N ,N , N ). By Theorem 3.8 (the extensions decomposition theorem) there is N +2 (cid:23) N which is decomposable over N . By Theorem 5.11.b there isan amalgamation ( id N , f, N +3 ) of N , N +2 over N such that N F ∗ ( N , N ,f [ N +2 ] , N +3 ). So N T f [ N +2 ] = N . Hence by Theorem 5.11.b, there isa model N ∗ such that N F ∗ ( N , f [ N +2 ] , N , N ∗ ). By Proposition 5.19 (theopposite uniqueness proposition) there is a joint embedding id N +3 , g, N ∗∗ of N +3 and N ∗ over N S f [ N +2 ]. Since N F ∗ is closed under isomorphisms, N F ∗ ( N , f [ N +2 ] , N , g [ N ∗ ]). Now we have to use the monotonicity of N F twice. Since N (cid:22) N (cid:22) f [ N ∗ ] it follows that N F ∗ ( N , N , N , g [ N ∗ ]). Since N (cid:22) N ∗ (cid:22) N ∗∗ (cid:23) g [ N ∗ ], it follows that N F ( N , N , N , N ). ⊣ Theorem 5.23.
N F respects s (see Definition 5.5)Proof. Suppose
N F ( M , M , M , M ) , tp ( a, M , M ) ∈ S bs ( M ). We haveto prove that tp ( a, M , M ) does not fork over M . Without loss of gener-ality N F ∗ ( M , M , M , M ) [Why? see the Definition 2.1.3.b (monotonic-ity)]. By the definition of N F ∗ , M is decomposable over M . By of NF,(Theorem 5.11.c (the existence theorem for N F ), there is M ∗ such that N F ∗ ( M , M , M , M ∗ ) and the first element in the first witness is a. M ∗ a ∈ M id / / id ; ; vvvvvvvvv M M id / / id O O M id O O id = = zzzzzzzz By the definition of a first witness, tp ( a, M , M ∗ ) does not fork over M .By the weak uniqueness theorem (Theorem 5.20) there are f, M ∗∗ suchthat M (cid:22) M ∗∗ , and f : M ∗ → M ∗∗ is an embedding over M S M .So tp ( a, M , M ) = tp ( a, M , f [ M ∗ ]) = tp ( a, M , M ∗ ) does not fork over M . ⊣ Long transitivity.Proposition 5.24.
Let h M ε : ε ≤ α ∗ i be a ≺ -increasing continuous sequenceof models in K λ .(a) There is an ≺ -increasing continuous sequence of models in K l ambda h N ε : ε ≤ α ∗ i such that: N = M , M ε (cid:22) N ε , N F ( M ε , M ε +1 , N ε , N ε +1 ) and N ε +1 is decomposable over N ε and over M ε +1 . (b) Suppose M ∗ ∈ K λ , M ∗ ≻ M and M ∗ T M α ∗ = M . Then there is an ≺ -increasing continuous sequence of models in K λ h N ε : ε ≤ α ∗ i such that: M ∗ (cid:22) N , M ε (cid:22) N ε , N F ( M ε , M ε +1 , N ε , N ε +1 ) , N is decomposableover M and N ε +1 is decomposable over N ε and over M ε +1 .Proof. (a) We choose a pair ( N ε , f ε ) by induction on ε ≤ α such that:(1) h N ε : ε ≤ α i is an increasing continuous sequence of models in K λ .(2) f ε : M ε → N ε is an embedding.(3) f = id M .(4) The sequence h f ε : ε ≤ α i is increasing and continuous.(5) For ε < α ∗ , N F ( f ε [ M ε ] , N ε , f ε +1 [ M ε +1 ] , N ε +1 ).(6) For ε < α ∗ , N ε +1 is decomposable over N ε and over f ε +1 [ M ε +1 ].Why can we carry out this construction? For ε = 0 or limit there is no prob-lem. Suppose we chose ( N ε , f ε ), how will we choose ( N ε +1 , f ε +1 )? By Theo-rem 5.11.a we can find N − ε +1 and f ε +1 such that N F ( f ε [ M ε ] , N ε , f ε +1 [ M ε +1 ] ,N − ε +1 ). Now by Proposition 3.9 we can find N ε +1 such that N − ε +1 (cid:22) N ε +1 and N ε +1 is decomposable over N ε and over f ε +1 [ M ε +1 ]. Therefore we cancarry out this construction.Now, as in the proof of Proposition 3.4, without loss of generality f ε = id M ε for every ε ≤ α ∗ (because we can extend f − α ∗ to a bijection g of N α ∗ and take the sequence h g [ N ε ] : ε ≤ α ∗ i ).(b) It demands a tiny change in the proof: In the construction M ∗ (cid:22) N and it is decomposable over M . ⊣ Theorem 5.25 (the long transitivity theorem) . For x = a, b let h M x,ε : ε ≤ α ∗ i be an ≺ -increasing continuous sequence of models in K λ . Suppose ε <α ∗ ⇒ N F ( M a,ε , M a,ε +1 , M b,ε , M b,ε +1 ) . Then N F ( M a, , M a,α ∗ , M b, , M b,α ∗ ) . Similarly to the proof of Proposition 2.15 (the transitivity proposition),we use the existence and weak uniqueness theorems to prove the long tran-sitivity. But here the proof is more complicated, and it is divided to fourcases, each one is based on its previous and generalizes it.
Proof. Case a: ε < α ∗ ⇒ N F ∗ ( M a,ε , M a,ε +1 , M b,ε , M b,ε +1 ). Concatenate allthe sequences together.In the other cases we are going to use the following claim: Claim 5.26.
It is enough to find ( N b,ε , f ε ) for ε ≤ α ∗ such that:(1) M b, (cid:22) N b, .(2) h N b,ε : ε ≤ α ∗ i is an increasing continuous sequence of models in K λ .(3) f ε is an embedding of M a,ε to N b,ε .(4) f = id M a, .(5) h f ε : ε ≤ α ∗ i is an increasing continuous sequence.(6) For ε < α ∗ , N F ( f ε [ M a,ε ] , f ε +1 [ M a,ε +1 ] , N b,ε , N b,ε +1 ) .(7) N F ( M a, , f α ∗ [ M a,α ∗ ] , N b, , N b,α ∗ ) . OOD FRAMES WITH A WEAK STABILITY 49
Proof.
Suppose we found ( N b,ε , f ε ) for ε ≤ α ∗ such that clauses 1-7 are satis-fied. By Proposition 5.21, N − NF . Therefore by Proposition 5.17 (the transi-tivity of the uniqueness) ( id M a,α ∗ , id M b, , M b,α ∗ ) E M a, ( f aα ∗ , id M b, , N b,α ∗ ) [Sub-stitute h M a,ε : ε ≤ α ∗ i , h M b,ε : ε ≤ α ∗ i , h N b,ε : ε ≤ α ∗ i , h id M a,ε : ε ≤ α ∗ i , h f ε : ε ≤ α ∗ i in place of h N ,α : α ≤ α ∗ i , h N a ,α : α ≤ α ∗ i , h N b ,α : α ≤ α ∗ i , h f aα : α ≤ α ∗ i , h f bα : α ≤ α ∗ i ] . By clause 7 N F ( M a, , M a,α ∗ , N b, , N b,α ∗ ).So by Proposition 5.9 N F ( M a, , M a,α ∗ , M b, , M b,α ∗ ). ⊣ Case b:
For every ε , M a,ε +1 is decomposable over M a,ε . In this case wechoose ( N b,ε , f ε ) such that clauses 1-6 of Claim 5.26 are satisfied: For ε = 0we define N b, := M b, . In successor step we use Theorem 5.11.a. For ε limitwe define N b,ε := S { N b,ζ : ζ < ε } , f ε := S { f ζ : ζ < ε } . Now clause 7 issatisfied by case a of the proof. Case c: α ∗ ≤ ω . In this case we apply Claim 5.26 with f ε = id M a,ε . N b, id / / N b, id / / N b, id / / N b,ε id / / N b,ε +1 id / / N b,α ∗ M b, id O O M a, id / / id O O N a, id / / id O O N a, id / / id O O N a,ε id / / id O O N a,ε +1 id / / id O O N a,α ∗ id O O M a, id / / id O O M a, id / / id O O M a, id / / id O O M a,ε id / / id O O M a,ε +1 id / / id O O M a,α ∗ id O O By Proposition 5.24.a, there is an increasing continuous sequence h N a,ε : ε ≤ α ∗ i such that: N a, = M a, , M a,ε (cid:22) N a,ε , N a,ε +1 is decomposable over N a,ε and over M a,ε +1 and ε < α ∗ ⇒ N F ( M a,ε , M a,ε +1 , N a,ε , N a,ε +1 ). Since α ∗ ≤ ω , by Proposition 5.24.b, there is an increasing continuous sequence h N b,ε : ε ≤ α ∗ i such that N b, ≻ M b, , for ε ≤ α ∗ , N b,ε is decomposable over N a,ε and N F ∗ ( N a,ε , N a,ε +1 , N b,ε , N b,ε +1 ).Now it is enough to prove that h ( N b,ε , id M a,ε ) : ε ≤ α ∗ i satisfies clauses1-7 of Claim 5.26. Clauses 1-5 are satisfied trivially. We check clauses 6,7.6. First assume ε >
0. As
N F ( M a,ε , M a,ε +1 , N a,ε , N a,ε +1 ), N F ( N a,ε ,N a,ε +1 , N b,ε , N b,ε +1 ), N a,ε is decomposable over M a,ε and N b,ε is decom-posable over N a,ε , by case b (for α ∗ = 2), N F ( M a,ε , M a,ε +1 , N b,ε , N b,ε +1 ).Second assume ε = 0. As N F ( N a, , N a, , N b, , N b, ), N a, = M a, and M a, (cid:22) N a, , by the monotonicity of N F , N F ( M a, , M a, , N b, , N b, )7. By case b, we have N F ( N a, , N a,α ∗ , N b, , N b,α ∗ ). By the smoothness M a,α ∗ (cid:22) N a,α ∗ . So by the monotonicity of N F , N F ( M a, , M a,α ∗ , N b, , N b,α ∗ ). The general case:
By the proof of case c. We have only one problem:For ε limit it is not clear why does N F ( M a,ε , M a,ε +1 , N b,ε , N b,ε +1 ), where weknow N F ( M a,ε , M a,ε +1 , N a,ε , N a,ε +1 ) ∧ N F ( N a,ε , N a,ε +1 , N b,ε , N b,ε +1 ). Herewe cannot use case b, because we do not know if N b,ε is decomposable over N a,ε and N a,ε is decomposable over M a,ε . But we can use case c with α ∗ = 2. ⊣ Theorem 5.27.
N F = N F λ is the unique relation which satisfies N NF and respects s .Proof. N F satisfies N NF : Clause a is clear. Clause b (the monotonicity)by Theorem 5.13.4. Clause c (the existence) by Theorem 5.11.a. Claused (weak uniqueness) by Theorem 5.20. Clause e (symmetry) by Theorem5.22. Clause f (long transitivity) by Theorem 5.25. By Theorem 5.23 N F respects s .Suppose the relation R satisfies N R and respects s . First we prove N F ( M , M , M , M ) ⇒ R ( M , M , M , M ). case a: Take a ∈ M − M with ( M , M , a ) ∈ K ,uq . As N F respects s , tp ( a, M , M ) does not fork over M . So as R respects s , by the definitionof unique triples (see Definition 4.4 on page 31), R ( M , M , M , M ). case b: N F ∗ ( M , M , M , M ). As R satisfies long transitivity, and bycase a, R ( M , M , M , M ). The general case:
Since R satisfies monotonicity, by case b, R ( M , M , M ,M ). So we have proved that the relation N F is included in the relation R . conversely : Suppose R ( M , M , M , M ). We have to prove that N F ( M , M , M , M ).As N R , R satisfies disjointness. So M T M = M . By N NF , for somemodel M N F ( M , M , M , M ). But by the first direction of the proof N F ( M , M , M , M ) ⇒ R ( M , M , M , M ), so R ( M , M , M , M ). As N R , R satisfies weak uniqueness, R ( M , M , M , M ) and R ( M , M , M , M ),it follows that ( id M , id M , M ) E M ( id M , id M , M ). Therefore by Proposi-tion 5.9 N F ( M , M , M , M ) implies N F ( M , M , M , M ), so N F ( M , M , M , M )as required. ⊣ A relation on K λ + that is based on the relation N F
Remember that we want to derive from s a good λ + -frame. So first wehave to define an a.e.c. in λ + with amalgamation. Definition 6.4 presentsthe relation on models of this a.e.c. in λ + . Hypothesis . s is a weakly successful semi-good λ -frame with conjugation. Definition 6.2.
Define a 4-place relation d N F on K by d N F ( N , N , M , M ) iff the following hold:(1) n < ⇒ N n ∈ K λ , M n ∈ K λ + .(2) There is a pair of increasing continuous sequences h N ,α : α <λ + i , h N ,α : α < λ + i such that for every α, N F ( N ,α , N ,α , N ,α +1 ,N ,α +1 ) and for n < N ,n = N n , M n = S { N n,α : α < λ + } . Theorem 6.3 (the d N F -properties) . (a) Disjointness: If d N F ( N , N , M , M ) then N T M = N . OOD FRAMES WITH A WEAK STABILITY 51 (b) Monotonicity: Suppose d N F ( N , N , M , M ) , N (cid:22) N ∗ (cid:22) N , N ∗ S M ⊆ M ∗ (cid:22) M and M ∗ ∈ K λ + . Then d N F ( N , N ∗ , M , M ∗ ) .(c) Existence: Suppose n < ⇒ N n ∈ K λ , M ∈ K λ + , N (cid:22) N , N (cid:22) M , N T M = N . Then there is a model M such that d N F ( N , N ,M , M ) .(d) Weak Uniqueness: If n < ⇒ d N F ( N , N , M , M ,n ) , then there are M, f , f such that f n is an embedding of M ,n into M over N S M .(e) Respecting the frame: Suppose d N F ( N , N , M , M ) , tp ( a, N , M ) ∈ S bs ( N ) . Then tp ( a, N , M ) does not fork over N .Proof. (a) Disjointness: Let h N ,ε : ε < λ + i , h N ,ε : ε < λ + i be witnesses for d N F ( N , N , M , M ). Especially ε < λ + ⇒ N F ( N ,ε , N ,ε , N ,ε +1 , N ,ε +1 ).So by Theorem 5.10.3 ε < λ + ⇒ N ,ε T N ,ε +1 = N ,ε . So by the proof ofProposition 5.10.2 N T M = N . Let x ∈ N T M . So there is ε < λ + such that x ∈ N ,ε . Denote ε := Min { ε < λ + : x ∈ N ,ε } . ε cannot be alimit ordinal as the sequence h N ,ε : ε < λ + i is continuous. If ε = ζ + 1 then x ∈ N ,ζ +1 T N ⊆ N ,ζ +1 T N ,ζ = N ,ζ , in contradiction to the minimalityof ε . So ε must be equal to 0. Hence x ∈ N , = N .(b) Monotonicity: Let h N ,ε : ε < λ + i , h N ,ε : ε < λ + i be witnessesfor d N F ( N , N , M , M ). Let E be a club of λ + such that 0 / ∈ E and ε ∈ E ⇒ N ,ε T M ∗ (cid:22) N ,ε [Why do we have such a club? Let E be aclub of λ + such that 0 / ∈ E and ε ∈ E ⇒ N ,ε T M ∗ (cid:22) M ∗ . By the as-sumption M ∗ (cid:22) M . So ε ∈ E ⇒ N ,ε T M ∗ (cid:22) M . Now as N ,ε (cid:22) M ,by axiom 1.1.1.e ε ∈ E ⇒ N ,ε T M ∗ (cid:22) N ,ε ]. We will prove that thesequences h N i ⌢ h N ,ε : ε ∈ E i , h N ∗ i ⌢ h N ,ε T M ∗ : ε ∈ E i witness that d N F ( N , N ∗ , M , M ∗ ). First, they are increasing [Why ε < ζ ∧ { ε, ζ } ⊆ E ⇒ N ,ε T M ∗ (cid:22) N ,ζ T M ∗ ? By the properties of E, N ,ε T M ∗ (cid:22) N ,ε .But N ε (cid:22) N ζ . So N ,ε T M ∗ (cid:22) N ,ζ . In the other side again by theproperties of E, N ,ε T M ∗ ⊆ N ,ζ T M ∗ (cid:22) N ,ζ . So by axiom 1.1.1.e N ,ε T M ∗ (cid:22) N ,ζ T M ∗ ]. Second, we will prove that if ε < ζ, { ε, ζ } ⊆ E then N F ( N ,ε , N ,ε T M ∗ , N ,ζ , N ,ζ T M ∗ ). Fix such ε, ζ . By Theorem5.25, (the long transitivity theorem), N F ( N ,ε , N ,ε , N ,ζ , N ,ζ ). By theproperties of E and axiom 1.1.1.e, N ,ε (cid:22) N ,ε T M ∗ (cid:22) N ,ε , N ,ζ S ( N ,ε T M ∗ ) ⊆ N ,ζ T M ∗ (cid:22) N ,ζ . Now by Theorem 5.13.5 (the monotonicity ofNF), we have N F ( N ,ε , N ,ε T M ∗ , N ,ζ , N ,ζ T M ∗ ).(c) Existence: By Proposition 5.24.b.(d) Weak Uniqueness: Since N NF holds, it follows by Proposition 5.16.2and Proposition 5.17. But we give another proof using section 7: By Propo-sition 7.12.f, there is a model M +1 ,n such that M ,n ≺ + M +1 ,n . By Theo-rem 7.13.c, there is an isomorphism f : M +1 , → M +1 , over M S N . So M +1 , , id M , , f ↾ M , is a witness as required.(e) Let h N ,ε : ε < λ + i , h N ,ε : ε < λ + i a witness for d N F ( N , N , M , M ).There is ε such that a ∈ N ,ε . By Definition 6.2 (the definition of d N F ), we have N F ( N , N , N ,ε , N ,ε ). So the proposition is satisfied by Theorem5.23 (the relation N F respects the frame). ⊣ Definition 6.4. M (cid:22) NF M when: there are N , N such that d N F ( N , N ,M , M ). Proposition 6.5. ( K λ + , (cid:22) NF ) satisfies the following properties:(a) Suppose M (cid:22) M , n < ⇒ M n ∈ K λ + . For n < let h N n,ε : ε < λ + i be a representation of M n . Then M (cid:22) NF M iff there is a club E ⊆ λ + such that ( ε < ζ ∧ { ε, ζ } ⊆ E ) ⇒ N F ( N ,ε , N ,ζ , N ,ε , N ,ζ ) .(b) (cid:22) NF is a partial order.(c) If M (cid:22) M (cid:22) M and M (cid:22) NF M then M (cid:22) NF M .(d) It satisfies axiom c of a.e.c. in λ + , i.e.: If δ ∈ λ +2 is a limit ordinal and h M α : α < δ i is a (cid:22) NF -increasing continuous sequence, then M (cid:22) NF S { M α : α < δ } and obviously it is ∈ K λ + .(e) It has no (cid:22) NF -maximal model.(f ) If it satisfies smoothness (Definition 1.1.1.d), then it is an a.e.c. in λ + ,(see Definition 1.1, page 3).(g) LST for (cid:22) NF : If M (cid:22) NF M , n < ⇒ ( A n ⊆ M n ∧ | A n | ≤ λ ) ,then there are models N , N ∈ K λ such that: d N F ( N , N , M , M ) and n < ⇒ A n ⊆ N n .Proof. (a) One direction:
Let E be such a club. So h N ,ε : ε ∈ E i , h N ,ε : ε ∈ E i witness that M (cid:22) NF M . conversely: Let h M ,α : α < λ + i , h M ,α : α < λ + i be witnesses for M (cid:22) NF M . Let E be a club such that ( n < ∧ ε ∈ E ) ⇒ M n,α = N n,α .Suppose ε < ζ ∧ { ε, ζ } ⊆ E . We will prove N F ( N ,ε , N ,ε , N ,ζ , N ,ζ ), i.e. N F ( M ,ε , M ,ε , M ,ζ , M ,ζ ). The sequences h M ,α : ε ≤ α ≤ ζ i , h M ,α : ε ≤ α ≤ ζ i are increasing and continuous. So by Theorem 5.25 (the longtransitivity theorem) N F ( M ,ε , M ,ε , M ,ζ , M ,ζ ).(b) The reflexivity is obvious. The antisymmetry is satisfied by the anti-symmetry of the inclusion relation. The transitivity is satisfied by item a,Theorem 5.25 and the evidence that the intersection of two clubs is a club.(c) For n < h M n,α : α < λ + i be a representation of M n suchthat α < λ + ⇒ N F ( M ,α , M ,α +1 , M ,α , M ,α +1 ). Let E be a club of λ + such that α ∈ E ⇒ M ,α (cid:22) M ,α (cid:22) M ,α . By the monotonicity of N Fα ∈ E ⇒ N F ( M ,α , M ,α +1 , M ,α , M ,α +1 ). The representations h M ,α : α ∈ E i , h M ,α : α ∈ E i witness that M (cid:22) NF M .(d) Without loss of generality cf ( δ ) = δ , so δ ≤ λ + . Denote M δ := S { M α : α < δ } . For α < δ let h M α,ε : ε < λ + i be a representation of M n . By item a for every α there is a club E α, ⊆ λ + such that ( ε <ζ ∧ { ε, ζ } ⊆ E α, ) ⇒ N F ( M α,ε , M α,ζ , M α +1 ,ε , M α +1 ,ζ ). Let α be a limitordinal. S { M α,ε : ε < λ + } = M α = S { M β : β < α } = S { S { M β,ε : ε < λ + } : β < α } = S { S { M β,ε : β < α } : ε < λ + } . Every edge of thisequivalences’s sequence is a limit of an ⊆ -increasing continuous sequence ofsubsets of cardinality less than λ , and it is equal to M α [Why is the sequence OOD FRAMES WITH A WEAK STABILITY 53 in the right edge, h S { M β,ε : β < α } : ε < λ + i continuous? Let ε < λ + be alimit ordinal. Suppose x ∈ S { M β,ε : β < α } . Then there are ζ, β such that x ∈ M β,ζ . So x ∈ S { M β,ζ : β < α } ]. So there is a club E α, ⊆ λ + such that ε ∈ E α, ⇒ M α,ε = S { M β,ε : β < α } . For α limit define E α := E α, T E α, ,and for α not limit define E α := E α, . Case a: δ < λ + . Define E := T { E α : α < δ } . If ε ∈ E then for α < δ , ε ∈ E , so N F ( M α,ε , M α,Min ( E − ( ε +1)) , M α +1 ,ε , M α +1 , Min( E − ( ε +1)) ).So be Theorem 5.25 (the long transitivity theorem), ε ∈ E ⇒ N F ( M ,ε ,M , Min( E − ( ε +1)) , M δ,ε , M δ, Min( E − ( ε +1)) ). Hence M (cid:22) NF M . Case b: δ = λ + . Let E := { ε ∈ E : ε is a limit ordinal, α < ε ⇒ ε ∈ E α } .Denote N ε := S { M α,ε : α < ε } . M id / / M α id / / M ε id / / M ζ id / / M λ + M ,ζ id / / id O O M α,ζ id / / id O O M ε,ζ id / / id O O N ζid O O M ,ε id / / id O O M α,ε id / / id O O N εid O O M ,α id / / id O O N αid O O M , id O O Claim 6.6.
For every ε ∈ E the sequence h M α,ε : α < ε i ⌢ h N ε i is increasingand continuous (especially N ε ∈ K ),Proof. If ε ∈ E is limit, then α < ε ⇒ ε ∈ E α, , so the sequence h M α,ε : α <ε i is continuous. So it is sufficient to prove that α < ε ⇒ M α,ε (cid:22) M α,ε +1 .Suppose α < ε . ε ∈ E , so ε ∈ E α, . Hence N F ( M α,ε , M α +1 ,ε , M α, Min( E − ( ε +1)) ,M α +1 , Min( E − ( ε +1)) ), and especially M α,ε (cid:22) M α +1 ,ε . ⊣ Claim 6.7.
The sequence h N ε : ε ∈ E i is (cid:22) -increasing.Proof. Suppose ε < ζ, { ε, ζ } ⊆ E . By (*), the sequences h M α,ε : α <ε i ⌢ h N ε i , h M α,ζ : α ≤ ε i are increasing and continuous. For every α ∈ ε thesequence h M α,β : β < λ + i is a representation of M α , and especially it is (cid:22) -increasing. So ( ∀ α ∈ ε ) M α,ε (cid:22) M α,ζ . Hence by the smoothness N ε (cid:22) M ε,ζ .But by (*), M ε,ζ (cid:22) N ζ , so N ε (cid:22) N ζ .] ⊣ Claim 6.8.
The sequence h N ε : ε ∈ E i is continuousProof. Suppose ε = sup ( E T ε ). Let x ∈ N ε . By the definition of N ε thereis α < ε such that x ∈ M α,ε . ε is limit and the sequence h M α,β : β ≤ ε i is continuous. So there is β < ε such that x ∈ M α,β . ε = sup ( E T ε ), so thereis ζ ∈ ( β, ε ) T E . x ∈ M α,ζ but by (*), M α,ζ ⊆ N ζ , so x ∈ N ζ ]. ⊣ Claim 6.9. S { N ε : ε ∈ E } = M δ Proof.
Clearly S { N ε : ε ∈ E } ⊆ M δ . The other inclusion: Let x ∈ M δ .Then there is α < δ such that x ∈ M α . So ( ∃ α, β ) x ∈ M α,β . So as sup ( E ) = δ , There is ζ ∈ ( β, δ ) T E . So x ∈ M α,ζ which by (*) is ⊆ N ζ . So x ∈ N ζ ]. ⊣ Claim 6.10. If ε < ζ, { ε, ζ } ⊆ E then N F ( M ,ε , N ε , M ,ζ , N ζ ) Proof.
By the definition of E, ( ∀ α ∈ ε ) { ε, ζ } ⊆ E α . So ( ∀ α ∈ ε ) N F ( M α,ε ,M α +1 ,ε , M α,ζ , M α +1 ,ζ ). By (*), the sequences h M α,ε : α < ε i ⌢ h N ε i , h M α,ζ : α ≤ ε i are increasing and continuous. So by Theorem 5.25 (the long transi-tivity theorem), N F ( M ,ε , N ε , M ,ζ , M ε,ζ ). But by Claim 6.6 M ε,ζ ≺ N ζ , so N F ( M ,ε , N ε , M ,ζ , N ζ )]. ⊣ By Claims 6.7,6.8,6.9, the sequence h N ε : ε < δ i is a representation of M δ .The sequence h M ,ε : ε < λ + i is a representation of M . Hence, by Claim6.10 and item a, they witness that M (cid:22) NF M δ .(e) By Proposition 6.3.c. Derived also by the existence proposition of the ≺ + -extension, (Proposition 7.12.f), which we will prove later.(f) We have actually proved it, (for example: axiom 1.1.1.e by item c hereand axiom 1.1.1.c. By item d here).(g) Let h N ,ε : ε < λ + i , h N ,ε : ε < λ + i be witnesses for M (cid:22) NF M .By cardinality considerations there is ε ∈ λ + such that for n < A n ⊆ N n,ε . But for every ε < λ + , d N F ( N ,ε , N ,ε , M , M ) . ⊣ ≺ + and saturated models Hypothesis . s is a weakly successful semi-good λ -frame with conjugation. Definition 7.2. K sat is the class of saturated models in λ + over λ .Now we study the class ( K sat , (cid:22) NF ↾ K sat ). Note that in the followingtheorem there is no any set-theoretic hypothesis beyond ZF C . Theorem 7.3.
If ( s is a weakly successful semi-good λ -frame with conju-gation and) ( K sat , (cid:22) NF ↾ K sat ) does not satisfy smoothness (see Definition1.1.1.d), then there are λ +2 pairwise non-isomorphic models in K λ +2 .How can we prove this theorem? First we find a relation ≺ + on K λ + suchthat:(*) For every model M in K λ + there is a model M such that M ≺ + M .(**) If for n = 1 , M ≺ + M n then M , M are isomorphic over M .(***) If h M i : i ≤ α ∗ i is an increasing continuous sequence, and i < α ∗ ⇒ M i ≺ + M i +1 then M ≺ + M α ∗ . OOD FRAMES WITH A WEAK STABILITY 55
In section 7 we study the properties of ≺ + . Sections 8,9 are preparationsfor the proof of Theorem 7.3. A key theorem is Theorem 9.7: Supposethat there is an increasing continuous sequence h M ∗ α : α ≤ λ + 1 i of modelsin K sat such that: α < β < λ + ⇒ M ∗ α ≺ + M ∗ β ∧ M ∗ α (cid:22) NF M λ + +1 and M ∗ λ + (cid:14) NF M ∗ λ + +1 . Then for every S ∈ S λ +2 λ + := { S : S is a stationary subsetof λ +2 and ( ∀ α ∈ S ) cf ( α ) = λ + } , there is a model M S in K λ +2 such that S ( M S ) = S/D λ +2 . So there are 2 λ +2 pairwise non-isomorphic models in K λ +2 .Note that while ≺ + is a priori defined on K λ + , Proposition 7.6 shows thatany ≺ + extension is saturated in λ + over λ , so in K sat . Definition 7.4. ≺ + is a 2-place relation on K λ + . For M , M ∈ K λ + , we say M ≺ + M iff: there are increasing continuous sequences h N ,α : α < λ + i , h N ,α : α < λ + i , h N ⊕ ,α : α < λ + i , and there is a club E of λ + such that:(a) For n = 1 , M n = S { N n,α : α < λ + } .(b) α ∈ E ⇒ N ,α (cid:22) N ,α (cid:22) N ⊕ ,α .(c) If α < β and they are in E , then N F ( N ,α , N ⊕ ,α , N ,β , N ,β ).(d) For every α ∈ E , and every p ∈ S bs ( N ,α ), there is an end-segment S of λ + such that for every β ∈ S T E the model N ⊕ ,β realizes the non-forkingextension of p to N ,β .In such a case h N ,α : α < λ + i , h N ,α : α < λ + i , h N ⊕ ,α : α < λ + i , E are saidto be witnesses for M ≺ + M . M id / / M N , id / / id O O N , id / / N ⊕ , id > > }}}}}}}}}}}}}}}}} N , id / / id O O N , id / / N ⊕ , id O O N , id / / id O O N , id / / N ⊕ , id O O N , id / / id O O N , id / / N ⊕ , id O O By the following proposition if M ≺ + M then we can find witnesses forit, with E = λ + . Proposition 7.5.
If(1) h N ,α : α < λ + i , h N ,α : α < λ + i , h N ⊕ ,α : α < λ + i , E are witnessesfor M ≺ + M .(2) For α ∈ E M ,otp ( α T E ) = N ,α , M ,otp ( α T E ) = N ,α , M ⊕ ,otp ( α T E ) = N ⊕ ,α .then h M ,β : β < λ + i , h M ,β : β < λ + i , h M ⊕ ,β : β < λ + i , λ + are witnessesfor M ≺ + M .Proof. Easy, so we prove Definition 7.4.c only. Suppose γ < γ . Wehave to prove that N F ( M ,γ , M ⊕ ,γ , M ,γ , M ,γ ). There is a unique or-dinal α ∈ E with otp ( α T E ) = γ . So M ,γ = N ,α ∧ M ⊕ ,γ = N ⊕ ,α .Similarly there is a unique β ∈ E such that M ,γ = N ,β ∧ M ,γ = N ,β . Now by clause b in the assumption N F ( N ,α , N ⊕ ,α , N ,β , N ,β ), namely N F ( M ,γ , M ⊕ ,γ , N ,γ , N ,γ ). ⊣ Proposition 7.6. If h N ,α : α < λ + i , h N ,α : α < λ + i , h N ⊕ ,α : α < λ + i , E are witnesses for M ≺ + M and E − is a club of λ + with E − ⊆ E then h N ,α : α < λ + i , h N ,α : α < λ + i , h N ⊕ ,α : α < λ + i , E − are witnesses for M ≺ + M .Proof. Trivial. ⊣ Proposition 7.7.
Suppose:(a) For n = 1 , N F ( M , , M , , M n, , M n, ) .(b) M , (cid:22) N , M , (cid:22) N .(c) N T M , = M , .Then for some model N with N F ( M , , M , , N , N ) we can assign toeach n ∈ { , } an embedding f n : M n, → N over M , S M n, such that N F ( M n, , f n [ M n, ] , N , N ) . N id / / N M , id ; ; xxxxxxxx id / / M , f ; ; xxxxxxxx M , id O O id / / M , f O O M , id O O id < < xxxxxxxx id / / M , id O O id < < xxxxxxxx Proof.
For each n ∈ { , } by Theorem 5.11 (the existence theorem for N F ),we can find an amalgamation ( id N , g n , N n, ) of N , M n, over M n, with OOD FRAMES WITH A WEAK STABILITY 57
N F ( M n, , N , g n [ M n, ] , N n, ). But N F ( M , , M n, , M , , M n, ). So by The-orem 5.25 (the long transitivity theorem) N F ( M , , N , g n [ M , ] , N n, ). Byassumption c N T M , = M , . So by Theorem 5.20 (the weak uniquenesstheorem) we can find h , h , N such that the following hold:(1) h n : N n, → N is an embedding.(2) h n ↾ N = id N .(3) h ◦ g ↾ M , = h ◦ g ↾ M , = id M , .Now we define for n = 1 , f n := h n ◦ g n . Why is f n over M , S M n, ?By clause 3 x ∈ M , ⇒ f n ( x ) = x . Let x ∈ M n, . Then g n ( x ) = x . Byassumption b M n, ⊆ N , so x ∈ N . So by clause 2 h n ( x ) = x . Hence f n ( x ) = h n ( g n ( x )) = h n ( x ) = x . Claim 7.8.
N F ( M n, , f n [ M n, ] , N , N ) .Proof. N F ( M n, , N , g n [ M n, ] , N n, ). So by clauses 1,2 N F ( M n, , N , f n [ M n, ] , h n [ N n, ]). But h n [ N n, ] (cid:22) N , so N F ( M n, , N , f n [ M n, ] , N ). ⊣ Claim 7.9.
N F ( M , , M , , N , N ) .Proof. Since
N F ( M , , M , , N , N ), by Theorem 5.25 (the long transitivitytheorem) it is enough to prove that N F ( M , , M , , M , , f [ M , ]). But f n is over M , S M , . Hence it follows by assumption a. ⊣⊣ Proposition 7.10. (a) If M ≺ + M then M ≺ NF M .(b) If M ≺ + M then M ∈ K sat .(c) If M (cid:22) NF M ≺ + M then M ≺ + M .(d) If M ≺ + M ≺ + M then M ≺ + M .Proof. (a) If h N ,α : α < λ + i , h N ,α : α < λ + i , h N ,α : α < λ + i , E witnessthat M ≺ + M then h N ,α : α ∈ E i , h N ,α : α ∈ E i witness that d N F ( N , , N , , M , M ). So M (cid:22) NF M .(b) By Theorem 2.17.2 (page 16).(c) Easy.(d) By items a,c. ⊣ Definition 7.11.
The ≺ + -game is a game between two players. It lasts λ + moves. In any move the players choose models in K λ with the followingrules:The 0 move: Player 1 chooses models N , , N , ∈ K λ with N , (cid:22) N , and player 2 does not do anything.The α move where α is limit: Player 1 must choose N ,α := S { N ,β : β <α } and Player 2 must choose N ,α := S { N ,β : β < α } .The α + 1 move: Player 1 chooses a model N ,α +1 such that the followinghold: (1) N ,α (cid:22) N ,α +1 .(2) N ,α +1 T N ,α = N ,α .After player one chooses N ,α +1 , player 2 has to choose N ,α +1 such thatthe following hold:(1) N ,α (cid:22) N ,α +1 .(2) N F ( N ,α , N ,α , N ,α +1 , N ,α +1 ).In the end of the game, player 2 wins the game if S { N ,α : α < λ + } ≺ + S { N ,α : α < λ + } .A strategy for player 2 is a function F that assigns a model N ,α +1 toeach triple ( α, h N ,β : β ≤ α + 1 i , h N ,β : β ≤ α i ) that satisfies the followingconditions:(1) α < λ + .(2) h N ,β : β ≤ α + 1 i , h N ,β : β ≤ α i are increasing continuous se-quences of models in K λ .(3) N F ( N ,α , N ,α , N ,α +1 , N ,α +1 ) for β < α .(4) N ,α +1 T N ,α = N ,α .A winning strategy for player 2 is a strategy for player 2, such that if player2 acts by it, then he wins the game, no matter what does player 1 do. Proposition 7.12. (a) For every M ∈ K λ + there is M with M ≺ + M .(b) If M ∈ K λ + , n < ⇒ N n ∈ K λ , N ≺ M , N ≺ N , N T M = N ,then there is M such that M ≺ + M and d N F ( N , N , M , M ) .(c) Player 2 has a winning strategy in the ≺ + -game.Proof. (a) By c.(b) By c.(c) We describe a strategy: For α = 0 player 2 has nothing to do, but he takesa paper and writes for himself: I define N temp , := N , . For α limit player 2chooses N ,α := S { N ,β : β < α } and writes for himself N temp ,α := N ,α . Inthe α + 1 move, he writes for himself 3 things:(i) A model N temp ,α +1 with N F ( N ,α , N ,α , N ,α +1 , N temp ,α +1 ). By Theorem5.11.a (on page 40) it is possible.(ii) A sequence of types h p α,β : β < λ + i that includes S bs ( N temp ,α ).Now player 2 chooses a model N ,α +1 such that the following hold:(1) N temp ,α +1 (cid:22) N ,α +1 .(2) For each type in p γ,β with γ < α, β < α , N ,α +1 realizes the non-forking extension of p γ,β over N temp ,α +1 .By Proposition 1.25 (page 7) it is possible.Why shall player 2 win the game? Substitute the sequences h N ,α : α <λ + i , h N temp ,α : α < λ + i , h N ,α : α < λ + i which appear here instead of thesequences h N ,α : α < λ + i , h N ,α : α < λ + i , h N ⊕ ,α : α < λ + i in Definition7.4, and substitute E = λ + . ⊣ OOD FRAMES WITH A WEAK STABILITY 59
Roughly the following theorem says that:(a) The ≺ + -extension is unique.(b) Locality: Every type over a model in K λ + is determined by its restric-tions to submodels in K λ .(c) A preparation for symmetry. Theorem 7.13.
Suppose for n = 1 , M ≺ + M n then:(a) M , M are isomorphic over M .(b) For every a ∈ M , a ∈ M if for each N ∈ K λ with N (cid:22) M tp ( a , N, M ) = tp ( a , N, M ) then there is an isomorphism f : M → M over M with f ( a ) = a .(c) Let N ∗ ∈ K λ , N (cid:22) N ∗ . If for n = 1 , d N F ( N , N ∗ , M , M n ) , thenthere is an isomorphism f : M → M over M S N ∗ .The plan of the proof: We prove the three items at once. The proof issimilar to that of the uniqueness of the saturated model in λ + over λ .Suppose h N ,ε : ε < λ + i , h N ,ε : ε < λ + i , h N ⊕ ,ε : ε < λ + i , λ + wit-ness that M ≺ + M . So h N ,ε : ε < λ + i is a representation of M and h N , , N ⊕ , , N , , N ⊕ , , ...N ,ω , N ⊕ ,ω ... i is a representation of M . Suppose inaddition that h N ,ε : ε < λ + i , h N ,ε : ε < λ + i , h N ⊕ ,ε : ε < λ + i , λ + wit-ness that M ≺ + M . We amalgamate M , M over M in λ + steps. Ineach step we amalgamate the corresponding models in the representationsof M , M over the corresponding model in the representation of M . Nowif ( f , f , M ) is an amalgamation of M , M over M and f , f are onto M , then f − ◦ f is an isomorphism of M into M over M as required. Inodd steps we choose the amalgamations such that in the end f , f will beonto M , see requirement 8 below. In even steps we choose amalgamationswith N F , see requirement 4 below.
Proof.
Roughly, the following claim says that one representation of M canserve as a part of the witness to M ≺ + M and M ≺ + M together. Claim 7.14.
There are sequences h N ,ε : ε < λ + i , h N ,ε : ε < λ + i , h N ⊕ ,ε : ε < λ + i , h N ,ε : ε < λ + i , h N ⊕ ,ε : ε < λ + i such that for n = 1 , h N ,ε : ε <λ + i , h N n,ε : ε < λ + i , E = λ + , h N ⊕ n,ε : ε < λ + i witnesses that M ≺ + M n (so S { N ,ε : ε < λ + } = M and for n = 1 , S { N n,ε : ε < λ + } = S { N ⊕ n,ε : ε < λ + } = M n ).Proof. For n = 1 , h N temp ,n,ε : ε < λ + i , h N tempn,ε : ε <λ + i , h N ⊕ ,tempn,ε : ε < λ + i , E n for M ≺ + M n . Take a club E of λ + suchthat E ⊆ E T E and ε ∈ E ⇒ N temp , ,ε = N temp , ,ε . By Proposition 7.6 for n = 1 , h N temp ,n,ε : ε < λ + i , h N tempn,ε : ε < λ + i , h N ⊕ ,tempn,ε : ε < λ + i , E are witnesses for M ≺ + M n . Define N ,otp ( ε T E ) := N temp , ,ε ) . For n = 1 , ε ∈ E , define N n,otp ( ε T E ) := N tempn,ε . By Proposition 7.5 for n = 1 , h N ,ε : ε < λ + i , h N n,ε : ε < λ + i , E = λ + , h N ⊕ n,ε : ε < λ + i witness that M ≺ + M n . ⊣ For item b, we require in addition that a n ∈ N n, and tp ( a , N , , N , ) = tp ( a , N , , N , ). For item c, we require in addition that N F ( N , N ∗ , N , ,N n, ).Define by induction on ε ≤ λ + a triple ( N ε , f ,ε , f ,ε ) such that:(1) h N ε : ε ≤ λ + i is an increasing continuous sequence of models in K λ and for every ε < λ + N ε T M = N ε +1 T M = N ,ε .(2) For item c we add: f n, ↾ N ∗ is the identity.(3) For item b we add: f , ( a ) = f , ( a ).(4) ε < λ + ⇒ N F ( N ,ε , N ε +1 , N ,ε +1 , N ε +2 ).(5) For n = 1 , h f n,ε : ε ≤ λ + i is increasing and continu-ous.(6) For ε < λ + , f n, ε is an embedding of N n,ε to N ε and f n, ε +1 is anembedding of N ⊕ n,ε to N ε +1 .(7) f n, ε ↾ N ,ε = f n, ε +1 ↾ N ,ε and it is the identity on N ,ε .(8) For every ε < λ + if for some n ∈ { , } ( ∗ ) n,ε holds then for some m ∈ { , } ( ∗∗ ) m,ε holds, where:( ∗ ) n,ε There is p ∈ S bs ( N n,ε ) such that p is realized in N ⊕ n,ε and f n, ε ( p ) is realized in N ε .( ∗∗ ) m,ε , f m, ε +1 [ N ⊕ m,ε ] T N ε = f m, ε [ N m,ε ].Note that requirement 4 is essentially a property of N ε +2 and ( ∗∗ ) m,ε isessentially a property of f m, ε +1 . OOD FRAMES WITH A WEAK STABILITY 61 N ⊕ ,ε +1 f , ε +3 / / N ε +3 N ,ε +1 id : : uuuuuuuuu f , ε +2 / / N ε +2 id O O N ,ε +1 id B B (cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4) id / / N ,ε +1 id / / f , ε +2 nnnnnnnnnnnnnnnnnnnnnnnnnn N ⊕ ,ε +1 f , ε +3 G G (cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15) N ⊕ ,εid O O f , ε +1 / / N ε +1 id O O N ,εid O O id : : uuuuuuuuu f , ε / / N εid O O N ,εid O O id B B (cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4) id / / N ,εid O O id / / f , ε nnnnnnnnnnnnnnnnnnnnnnnnnnnn N ⊕ ,εid O O f , ε +1 G G (cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15) Why can we carry out the construction?
For ε = 0 let ( f , , f , , N ) bean amalgamation of N , , N , over N , , such that N T M = N , (i.e. wechoose new elements for N − N , ). In the proof of item b, by the definitionof the equality between types without loss of generality f , ( a ) = f , ( a ), so3 is satisfied. In the proof of item c, by Theorem 5.20 (the weak uniquenesstheorem of NF), there is a joint embedding f , , f , , N of N , , N , over N , S N ∗ . So 2 is satisfied.For limit ε define N ε = S { N ζ : ζ < ε } , f n,ε = S { f n,ζ : ζ < ε } . 5 issatisfied. 1 is satisfied by axiom 1.1.1.c. 6 is satisfied by the continuity ofthe sequence h N n,ε : ε < λ + i , and by the smoothness (Definition 1.1.1.d).Clearly 7 is satisfied. 4,8 are not relevant in the limit case. The successor case:
How can we construct N ε +1 , f , ε +1 , f , ε +1 and N ε +2 , f , ε +2 , f , ε +2 , assuming we have constructed N ε , f , ε , f , ε ? The construction of N ε +1 , f , ε +1 , f , ε +1 : Without loss of generality forsome n ∈ ,
2, we have ( ∗ ) n,ε [Otherwise requirement 8 is not relevant andwe can use the existence of an amalgamation in ( K λ , (cid:22) )]. Fix n ∗ with( ∗ ) n ∗ ,ε . We are going to find N ε +1 , f n ∗ , ε +1 , f − n ∗ , ε +1 with ( ∗∗ ) n ∗ ,ε , namely f n ∗ , ε +1 [ N ⊕ n ∗ ,ε ] T N ε = f n ∗ , ε [ N n ∗ ,ε ]. Let p be a witness for ( ∗ ) n ∗ ,ε , so forsome a, b tp ( a, N n ∗ ,ε , N ⊕ n ∗ ,ε ) = p, tp ( b, f n ∗ , ε [ N n ∗ ,ε ] , N ε ) = f n ∗ , ε ( p ). So tp ( f n ∗ , ε ( a ) , f n ∗ , ε [ N n ∗ ,ε ] , f n ∗ , ε [ N ⊕ n ∗ ,ε ]) = tp ( b, f n ∗ , ε [ N n ∗ ,ε ] , N ε ). Hence bythe definition of equality of types, for some N temp ε +1 , f tempn ∗ , ε +1 the followinghold:(1) N ε (cid:22) N temp ε +1 .(2) f tempn ∗ , ε +1 : N ⊕ n ∗ ,ε → N temp ε +1 is an embedding.(3) f n ∗ , ε ⊆ f tempn ∗ , ε +1 (4) f tempn ∗ , ε +1 ( a ) = b . N ⊕ − n ∗ ,ε f − n ∗ , ε +1 / / N ε +1 a ∈ N ⊕ n ∗ ,εf n ∗ , ε +1 ssssssssss f tempn ∗ , ε +1 / / N temp ε +1 id O O N − n ∗ ,ε id rrrrrrrrrr f − n ∗ , ε / / id O O N ε ∋ b id O O N ,ε id : : vvvvvvvvv id / / N n ∗ ,ε f n ∗ , ε rrrrrrrrrr id O O Claim 7.15. f tempn ∗ , ε +1 [ N ⊕ n ∗ ,ε ] T N ε = f tempn ∗ , ε [ N n ∗ ,ε ] . OOD FRAMES WITH A WEAK STABILITY 63
Proof. b ∈ N ε . p is a basic type so a non-algebraic one. So a ∈ N ⊕ n ∗ ,ε − N n ∗ ,ε .Hence b = f tempn ∗ , ε +1 ( a ) ∈ f tempn ∗ , ε +1 [ N ⊕ n ∗ ,ε ] − f tempn ∗ , ε +1 [ N n ∗ ,ε ]. Therefore b ∈ f tempn ∗ , ε +1 [ N ⊕ n ∗ ,ε ] T N ε − f tempn ∗ , ε [ N n ∗ ,ε ]. ⊣ As ( K λ , (cid:22) ) satisfies amalgamation, there are N ε +1 , f − n ∗ , ε +1 such that N temp ε +1 (cid:22) N ε +1 and f − n ∗ , ε +1 : N ⊕ − n ∗ ,ε → N ε +1 is an embedding thatincludes f − n ∗ , ε . Now we define f n ∗ , ε +1 : N ⊕ n ∗ ,ε → N ε +1 by f n ∗ , ε +1 ( x ) = f tempn ∗ , ε +1 ( x ). By Claim 7.15 ( ∗∗ ) n ∗ holds, so requirement 8 is satisfied. As for m = 1 , f m, ε +1 includes f m, ε , requirement 7 is satisfied.Without loss of generality requirement 1 is satisfied. Requirement 4 is notrelevant in this case. Requirements 5,6 are satisfied. The construction of N ε +2 , f n, ε +2 : By Proposition 7.7, there are N ε +2 ,f , ε +2 , f , ε +2 such that: N F ( f n, ε +1 [ N ⊕ n,ε ] , f n, ε +2 [ N n,ε +1 ] , N ε +1 , N ε +2 ),and the reduction of f n, ε +1 to N ,ε is the identity [Let f + n, ε +1 be an in-jection of N n,ε +1 , f n, ε +1 ⊆ f + n, ε +1 , and the reduction of f + n, ε +1 to N ,ε +1 is the identity. Substitute the models N ,ε , N ,ε +1 , f n, ε +1 [ N ⊕ n,ε ] , N ε +1 , f +2 ε +1 [ N n,ε +1 ] , N ε +2 which appear here, instead of the models M , , M , , M n, , N ,M n, , N which appear in Proposition 7.7 respectively. Assumption a ofProposition 7.7 (i.e. N F ( N ,ε , N ,ε +1 , f n, ε +1 [ N ⊕ n,ε ] , f + n, ε +1 [ N n,ε +1 ])), is sat-isfied by Definition 7.4.a (remember that f + n, ε +1 is an isomorphism over N ,ε +1 and N F respects isomorphisms). Assumption b of Proposition 7.7is satisfied by requirement 6 of the induction hypothesis. Assumption c ofProposition 7.7 is satisfied by requirement 4 of the induction hypothesis].Hence we can carry out the construction.
Why is it sufficient?
By clause 7 for n = 1 , f n,λ + : M n → N λ + is anembedding over M . Claim 7.16. f ,λ + [ M ] = f ,λ + [ M ] = N λ + .Proof. Toward a contradiction suppose there is n ∈ { , } such that f n,λ + [ M n ] = N λ + . By Density (Theorem 2.26.1), there is an element b such that tp ( b, f n,λ + [ M n ] , N λ + ) is basic. h f n, ε [ N n,ε ] : ε < λ + i is a representation of f n,λ + [ M n ], so by Definition 2.18 there is ε < λ + such that for every ζ ∈ ( ε, λ + ) the type q ζ := tp ( b, f n, ζ [ N n,ζ ] , N λ + ) does not fork over f n, ε [ N n,ε ].We choose this ε such that b ∈ N ε , (remember: b ∈ N λ + = S { N ε : ε <λ + } ). So q ζ is basic. Define p ζ := f − n, ζ ( q ζ ). So p ε ∈ S bs ( N n,ε ). For every ζ ∈ ( ε, λ + ) , q ζ is the non-forking extension of q ε , so p ζ is the non-forking ex-tension of p ε . Hence by Definition 7.4, there is an end segment S ∗ ⊆ λ + suchthat for ζ ∈ S ∗ , p ζ is realized in N ⊕ ζ . But q ζ = tp ( b, f n, ζ [ N n,ζ ] , N ζ ). So forevery ζ ∈ S ∗ we have ( ∗ ) n,ζ ( p ζ is a witness for this). So by clause 8 thereare m ∈ { , } and a stationary set S ∗∗ ⊆ S ∗ such that for every ζ ∈ S ∗∗ we have ( ∗∗ ) m,ζ , (there are no two non-stationary subsets which their unionis an end segment of λ + ). The sequences h N ζ : ζ ∈ S ∗∗ i , h N m,ζ : ζ ∈ S ∗∗ i , h f m, ζ : ζ ∈ S ∗∗ i are increasing and continuous. But by ( ∗∗ ) m,ζ , we have f m, ζ +1 [ N ⊕ m,ζ +1 ] T N ζ = f m, ζ [ N m,ζ ], in contradiction to Proposition1.31. ⊣ By Claim 7.16 f − ,λ + ◦ f ,λ + is an embedding of M onto M over M . Inthe proof of item b we have to note that f − ,λ + ◦ f ,λ + ( a ) = f − , ◦ f , ( a ) = a (by clause 3). In the proof of item c we have to note that f − ,λ + ◦ f ,λ + ↾ N ∗ = f − , ◦ f , ↾ N ∗ and by clause 3 it is the identity. ⊣ OOD FRAMES WITH A WEAK STABILITY 65
Corollary 7.17. (a) ( K λ + , (cid:22) NF ↾ K λ + ) has amalgamation. So ( K sat , (cid:22) NF ↾ K sat ) has amal-gamation.(b) Locality: Let M , M , M be models in K λ + , such that M (cid:22) M , M (cid:22) M . Suppose there is N ∈ K λ such that: N ≺ M and for every N ∈ K λ , [ N (cid:22) N (cid:22) M ] ⇒ tp ( a , N, M ) = tp ( a , N, M ) . Then tp ( a , M , M ) = tp ( a , M , M ) . [The version we actually use: Supposethere is N ∈ K λ such that tp ( a n , M , M ) does not fork over N and tp ( a , N , M ) = tp ( a , N , M ) . Then tp ( a , M , M ) = tp ( a , M , M ) ].Proof. (a) Suppose for n = 1 , M ≺ NF M n . By Proposition 7.12.a, there is M + n such that M n ≺ + M + n . By Proposition 7.12.a M ≺ + M + n . Soby Theorem 7.13.c (the uniqueness of the ≺ + -extension), there is anisomorphism f : M +1 → M +2 over M . Hence ( f ↾ M , id M , M +2 ) is anamalgamation of M , M over M . Now Proposition 7.10.a.(b) Locality: By Proposition 7.12.a there is M + n such that M n ≺ + M + n . ByTheorem 7.13.b there is an isomorphism f : M +1 → M +2 over M , suchthat f ( a ) = a . So ( f ↾ M , id M , M +2 ) witnesses that tp ( a , M , M ) = tp ( a , M , M ). ⊣ Theorem 7.18. (a) ( K sat , (cid:22) NF ↾ K sat ) satisfies axiom c in λ + (1.1.2.c).(b) If ( K sat , (cid:22) NF ↾ K sat ) satisfies smoothness, then it is an a.e.c. in λ + .(c) ( K sat , (cid:22) NF ↾ K sat ) has the amalgamation property.Proof. (a) Let j < λ +2 and h M i : i < j i be an (cid:22) NF -increasing continuousof models in K sat . Let M j be the union of this sequence. We prove that M j ∈ K sat by induction on j . Let N be a model in K λ such that N ≺ M j . Case a: λ < cf ( j ). In this case for some i < j N ≺ M i . Since M i is fullover N , of course M j is. Therefore M j ∈ K sat . Case b: cf ( j ) ≤ λ . Without loss of generality cf ( j ) = j . So | j | = j = cf ( j ) ≤ λ . Let h N i,α : α ∈ λ + i be a representation of M i . Forevery i < j let E i be a club of λ + such that for α ∈ E i , N F ( N i,α , N i +1 ,α ,N i,α +1 , N i +1 ,α +1 ) and if i is a limit ordinal, then N i,α = S { N ε,α : ε < i } .So E := T { E i : i < j } is a club set of λ + (because | j | ≤ λ ). Define N j,α := S { N i,α : i < j } . h N j,α : α ≤ λ + i is a representation of M j . Take α ∗ ∈ E such that N ⊆ N j,α ∗ . By axiom 1.1.1.e N (cid:22) N j,α ∗ , so it is sufficientto prove that M j is saturated over N j,α ∗ . Let q ∈ S bs ( N j,α ∗ ). We will provethat q is realized in M j . By the definition of E the sequence h N i,α ∗ : i < j i is increasing and continuous, so by Definition 2.1.3.c (the local character)there is an ordinal i < j such that q does not fork over N i,α ∗ . M i is saturatedin λ + over λ , so there is a ∈ M i such that tp ( a, N i,α ∗ , M i ) = q ↾ N i,α ∗ . ByDefinition 6.2 we have d N F ( N i,α ∗ , N j,α ∗ , M i , M j ), so by Theorem 6.3.e ( d N F respects s ) tp ( a, N j,α ∗ , M j ) does not fork over N i,α ∗ . Hence by Definition2.1.3.d (the uniqueness of the non-forking extension) tp ( a, N j,α ∗ , M j ) = q . (b) The first part of Axiom c of a.e.c. in λ + is item a here. Axioms b,eand the second part of axiom c follows by Proposition 6.5.f.(c) By Corollary 7.17.a. ⊣ relative saturation Discussion:
This section is, like its previous, a preparation for the proof ofTheorem 7.3. We study the relation (cid:22) ⊗ , a kind of relative saturation. Thisrelation is similar to ‘closure of (cid:22) NF under smoothness’ (see Proposition8.3.b). Theorem 9.13 says that non-equality between the relations (cid:22) NF , (cid:22) ⊗ is equivalent to non-smoothness and also to a strengthened version of non-smoothness. Hypothesis . s is a weakly successful semi-good λ -frame with conjugation. Definition 8.2. (cid:22) ⊗ := { ( M , M ) : M , M ∈ K sat , M ≺ M and for every N , N ∈ K λ , if M (cid:23) N (cid:22) N (cid:22) M and p ∈ S bs ( N ) does not fork over N , then for some element d ∈ M tp ( d, N , M ) = p } . Proposition 8.3. (a) If M ∈ K sat and M (cid:22) NF M then M (cid:22) ⊗ M .(b) If h M ε : ε ≤ δ i is an (cid:22) NF -increasing continuous sequence of models in K sat and for every ε ∈ δ, M ε (cid:22) NF M δ +1 , then M δ (cid:22) ⊗ M δ +1 .Proof. (a) Suppose M (cid:22) NF M and M ∈ K sat . Let N , N be models K λ with M (cid:23) N (cid:22) N (cid:22) M and let p be a type S bs ( N ) that does notfork over N . We have to find an element d ∈ M with tp ( d, N , M ) = p .By Proposition 6.5.g (LST for (cid:22) NF ) for some N +0 , N +1 ∈ K λ N (cid:22) N +0 , N (cid:22) N +1 and d N F ( N +0 , N +1 , M , M ). By axiom 1.1.1.e N (cid:22) N +0 and N (cid:22) N +1 . Let q be the non-forking extension of p to N +1 . Since M ∈ K sat for some d ∈ M tp ( d, N +0 , M ) = q ↾ N +0 . By Proposition 2.15 q does notfork over N , so by Definition 2.1.3.b (monotonicity) q does not fork over N +0 . By Theorem 6.3 d N F respects s , so tp ( d, N +1 , M ) does not fork over N +1 . So by Definition 2.1.3.b (uniqueness) tp ( d, N +1 , M ) = q . Therefore tp ( d, N , M ) = p .(b) Suppose N , N ∈ K λ , M δ (cid:23) N (cid:22) N (cid:22) M δ +1 and p ∈ S bs ( N ) doesnot fork over N . We have to find an element d ∈ M δ that realizes p . Forevery α ≤ δ + 1 there is a representation h N α,ε : ε < λ + i of M α . withoutloss of generality cf ( δ ) = δ . Case a: δ = λ + . So for some α < δ, N ⊆ M α and we can use item a. Case b: δ < λ + . For each α ∈ δ , let E α be a club of λ + such that for each ε ∈ E α : N F ( N α,ε , N α +1 ,ε , N α,ε +1 , N α +1 ,ε +1 ) and if α is limit then N α,ε = S { N β,ε : β < α } . Let E δ := { α ∈ λ + : N δ,ε ⊆ N δ +1 ,ε , N δ,ε = S { N α,ε : α <δ }} . Denote E := T { E α : α ≤ δ } . By cardinality considerations there is OOD FRAMES WITH A WEAK STABILITY 67 ε ∈ E such that for n < N n ⊆ N δ + n,ε , so by axiom 1.1.1.e N n (cid:22) N δ + n,ε . d ∈ M α id / / M δ id / / M δ +1 N α,ε id / / id O O N δ,ε id / / id O O N δ +1 ,εid O O qN id / / id O O N id O O p Let q ∈ S bs ( N δ +1 ,ε ) be the non-forking extension of p . By Proposition 2.15(the transitivity proposition), q does not fork over N . By Definition 2.1.3.b(monotonicity) q does not fork over N δ,ε , so q ↾ N δ,ε is basic. As ε ∈ E , thesequence h N α,ε : α ≤ δ i is increasing and continuous. So by Definition 2.1.3.c(local character) there is α < δ such that q ↾ N δ,ε does not fork over N α,ε .So by Proposition 2.15 q does not fork over N α,ε . Since M α (cid:22) NF M δ +1 byitem a for some d ∈ M α tp ( d, N δ +1 ,ε , M δ +1 ) = q . So tp ( d, N , M δ +1 ) = p . ⊣ The following proposition is similar to the saturativity = model homo-geneity lemma.
Proposition 8.4.
Suppose(1) M (cid:22) ⊗ M .(2) For n < N n ∈ K λ .(3) N (cid:22) M .(4) N (cid:23) N (cid:22) N (cid:22) M .Then for some N ∗ ∈ K λ and an embedding f : N → M the followinghold:(a) f ↾ N = id N .(b) N F ( N , f [ N ] , N , N ∗ ) .(c) N ∗ (cid:22) M . M id / / M f [ N ] id / / id O O N ∗ id O O N id / / id O O N id O O Proof.
We try to choose N ,ε , N ,ε , N ,ε , f ε by induction on ε < λ + suchthat:(1) For n < h N n,ε : ε < λ + i is an increasing continuous of models in K λ .(2) For n < N n, = N n , f = id N .(3) For ε < λ + , N ,ε (cid:22) M ∧ N ,ε (cid:22) M . (4) h f ε : ε < λ + i is increasing and continuous.(5) f ε : N ,ε → N ,ε is an embedding over N .(6) For every ε ∈ λ + there is a ε such that ( N ,ε , N ,ε +1 , a ε ) is a uniquenesstriple, f ε +1 ( a ε ) ∈ N ,ε and tp ( a ε , N ,ε , N ,ε +1 ) does not fork over N ,ε .(7) N ,ε (cid:22) N ,ε (actually follows by 6). M id / / M N ,ε +1 N ,ε +1 f ε +1 o o id / / id O O N ,ε +1 id O O N ,εid O O N ,εf ε o o id / / id O O N ,εid O O N id d d JJJJJJJJJJ id O O id / / N id O O By clauses 1,4,5 and particularly 6 and Proposition 1.31 we cannot suc-ceed. Where will we get stuck? For ε = 0 or limit we will not get stuck.Suppose we have defined N ,ε , N ,ε , N ,ε , f ε . By clause 5, f ε [ N ,ε ] (cid:22) N ,ε . Case a: f ε [ N ,ε ] = N ,ε . In this case we can find N ,ε +1 , N ,ε +1 , N ,ε +1 ,f ε +1 such that clauses 1-7 above hold [By the existence of the basic types,there is b ∈ N ,ε − f ε [ N ,ε ] such that p := tp ( b, f ε [ N ,ε ] , N ,ε ) is basic. Let q ∈ S bs ( N ,ε ) be the non-forking extension of f − ε ( p ). As M (cid:22) ⊗ M ∧ ( n < ⇒ N n,ε (cid:22) M ∗ n ) ∧ N ,ε (cid:22) N ,ε ∈ K λ , there is a ∈ M which realizes q .So tp ( a, N ,ε , M ) = f − ε ( p ). As s is weakly successful, we can find N ,ε +1 such that ( N ,ε , N ,ε +1 , a ) ∈ K ,uq . As M is saturated in λ + over λ , byLemma 1.32 (the saturation = model homogeneity lemma), without loss ofgenerality N ,ε +1 (cid:22) M . Denote a as a ε . Choose N ,ε +1 (cid:22) M such that N ,ε +1 S N ,ε ⊆ N ,ε +1 . By axiom 1.1.1.e N ,ε +1 (cid:22) N ,ε +1 ∧ N ,ε (cid:22) N ,ε +1 .Now f ε ( tp ( a ε , N ,ε , N ,ε +1 ) = p . So there are N ,ε +1 , f ε +1 such that: N ,ε (cid:22) N ,ε +1 , f ε +1 ( a ε ) = b, f ε ⊆ f ε +1 : N ,ε +1 → N ,ε +1 ]. Case b: f ε [ N ,ε ] = N ,ε . Hence N ,ε , f − ε ↾ N witness that our proposi-tion is true [By 6, Definition 5.7 and Definition 5.6, ζ < ε ⇒ N F ( N ,ζ , N ,ζ +1 , N ,ζ ,N ,ζ +1 ). So by Theorem 5.25 (the long transitivity theorem), N F ( N , N ,ε ,N , N ,ε ). So by the monotonicity of NF, we have N F ( N , f − ε [ N ] , N , N ,ε ).So clause b in the proposition is satisfied. Clauses a,c are satisfied by 5,3respectively].Let ε + 1 be the first ordinal we will get stuck on . In other words, supposewe have defined N ,ε , N ,ε , N ,ε , f ε and we cannot find N ,ε +1 , N ,ε +1 , N ,ε +1 ,f ε +1 such that clauses 1-7 above hold. so case b holds and the propositionis proved. ⊣ OOD FRAMES WITH A WEAK STABILITY 69
Proposition 8.5. If M (cid:22) M , n < ⇒ ( || M n || ) = λ + ∧ A n ⊆ M n ∧ | A n | ≤ λ ) , then there are models N , N ∈ K λ such that: n < ⇒ A n ⊆ N n (cid:22) M n and N T M = N (so of course N (cid:22) N ).Proof. For n < m < ω a model N n,m such that h N n,m : m ≤ ω i is (cid:22) -increasing and continuous of models in K λ , A n ⊆ N n, , N ,m ⊆ N ,m , N ,m T M ⊆ N ,m +1 , N n,m (cid:22) M n . Thisconstruction is possible as LST ( K, (cid:22) ) ≤ λ . Now M T N ,ω = N ,ω [Why?If x ∈ M T N ,ω , then for some m < ω we have x ∈ N ,m T M ⊆ N ,m +1 ⊆ N ,ω and from the other side, if x ∈ N ,ω then for some m < ω we have x ∈ N ,m ⊆ N ,m , so x ∈ M T N ,ω ]. ⊣ Proposition 8.6. If M ∗ (cid:22) ⊗ M ∗ then there is an increasing continuoussequence of models in K sat , h M ε : ε ≤ λ + + 1 i such that:(a) M λ + = M ∗ , M λ + +1 = M ∗ .(b) ε < λ + ⇒ M ε ≺ + M ε +1 .(c) ε < λ + ⇒ M ε (cid:22) NF M ∗ .Proof. By Proposition 7.12.c, there is a winning strategy for player 2 in the ≺ + -game. Let F be such a winning strategy. Enumerate M ∗ by { a ε : ε <λ + } . We construct h N α,ε : ε ≤ α i , N α by induction on α such that thefollowing hold:(1) For each ε ≤ α , N α,ε ∈ K λ and N α,ε (cid:22) M ∗ .(2) h N α,ε : ε ≤ α < λ + i is increasing continuous in the variables α, ε .(3) h N α : α < λ + i is an increasing continuous of models in K λ .(4) N α,α (cid:22) N α (cid:22) M ∗ .(5) If α + 1 is odd then for each ε ≤ α , N α +1 ,ε +1 is isomorphic to F ( h N β,ε : ε + 1 ≤ β ≤ α + 1 i , h N β,ε +1 : ε + 1 ≤ β ≤ α i ) over N α,ε +1 S N α +1 ,ε .(6) If α + 1 is odd then N F ( N α,α , N α , N α +1 ,α +1 , N α +1 )(7) a α ∈ N α +2 .(8) N α T M ∗ ⊆ N α, α .(9) If α + 1 is odd then N α +1 ,α +1 = N α +1 ,α .(10) If α + 1 is odd then N α +1 , T N α = N α, , N α +1 , = N α, .(11) If α + 1 is even then for each ε ≤ α N α +1 ,ε = N α,ε . M ε id / / M ε +1 id / / M α id / / M λ + = M ∗ id / / M λ + +1 = M ∗ N α,ε id / / id O O N α,ε +1 id / / id O O N α,α id / / id O O N αid O O N ε +1 ,ε id / / id O O N ε +1 ,ε +1 id / / id O O N ε +1 id O O N ε,ε id / / id O O N εid O O [Explanation: N α,α , N α are approximations for M ∗ , M ∗ respectively. N α,ε is an approximation for M ε . When α + 1 is even, we increase the ap-proximations of M ∗ , M ∗ such that in the end we will have M ∗ ⊆ S { N α : α < λ + } , M ∗ = S { N α,α : α < λ + } by 7,8 respectively. when α + 1 is odd,we increase the approximations of M ε (mainly by clause 10). Clause 11 saysthat in even step the approximations to M ε do not increase. Clause 5 insurethat in the end we will have M ε ≺ + M ε +1 . Clause 6 insure that in the endrequirement c will be satisfied. The point of the proof is, that we could notdemand 6 for every α , (as otherwise we prove M ∗ (cid:22) NF M ∗ , which mightbe wrong). But we succeed to prove that N F ( N α,ε , N α , N α +1 ,ε , N α +1 ) so M ε (cid:22) NF M ∗ ]. Why can we carry out the construction?
We construct by induction on α . For limit α , by clauses 2,3 there is no freedom. Clauses 1,4 are satisfiedby the smoothness, clauses 5,6,7,9,10,11 are not relevant and clause 8 issatisfied. For α = 0 we choose N , N , by Proposition 8.5. Suppose wehave defined h N α,ε : ε ≤ α i , N α . what will we do in step α + 1? Case a: α + 1 is even. For ε ≤ α define N α +1 ,ε := N α,ε . By Proposition 8.5there are N α +1 , N α +1 ,α +1 as required, especially clauses 7,8 are satisfied. Case b: α + 1 is odd. Define N tempα +1 ,ε by induction on ε ≤ α such that:(1) h N tempα +1 ,ε : ε ≤ α i is an (cid:22) -increasing continuous sequence.(2) N tempα +1 ,ε +1 = F ( h N β,ε : ε + 1 ≤ β ≤ α i ⌢ h N tempα +1 ,ε i , h N β,ε +1 : ε + 1 ≤ β < α i ).(3) N α, (cid:22) N tempα +1 , .Now by Proposition 8.4, there are N α +1 and an embedding g : N tempα +1 ,α → M ∗ over N α,α such that we have N F ( N α,α , N α , g [ N tempα +1 ,α ] , N α +1 ). For every ε ≤ α define N α +1 ,ε := g [ N tempα +1 ,ε ]. Now define N α +1 ,α +1 := N α +1 ,α . So wecan carry out the construction. OOD FRAMES WITH A WEAK STABILITY 71
Why is it sufficient?
For ε < λ + define M ε := S { N α,ε : ε ≤ α < λ + } .Define M λ + := S { M ε : ε < λ + } , M λ + +1 := S { N α : α < λ + } . We will provethat the sequence h M ε : 0 < ε < λ + + 1 i satisfies requirements a,b,c:(a) By 3,4,7 M λ + +1 = M ∗ . Why is M λ + = M ∗ ? By 1 M λ + ⊆ M ∗ . Let x ∈ M ∗ . Then x ∈ M ∗ = M λ + +1 . So by the definition of M λ + +1 and 3,there is α such that x ∈ N α . So by 8 x ∈ N α, α . But by the definitions of M ε , M λ + , N α, α ⊆ M α ⊆ M λ + .(b) By 2,10 | M | = λ + . By 2 and the smoothness, the sequence h M ε : ε <λ + i is (cid:22) -increasing and continuous. So | M ε | = λ + . Does ε < λ + ⇒ M ε ∈ K sat ? Not exactly, but we can prove by induction on ε that 0 < ε < λ + ⇒ ( M ε ∈ K sat ∧ M ε ≺ + M ε +1 ): For ε = 0 by 10. For limit ε by Theorem7.18.a. For ε successor by 5 and Proposition 7.10.b. So requirement b issatisfied.(c) The sequences h N α,ε : ε ≤ α < λ + i , h N α : ε ≤ α < λ + i are rep-resentations of M ε , M λ + +1 respectively. Let α ∈ λ + . We will prove N F ( N α,ε , N α , N α +1 ,ε , N α +1 ). If α + 1 is even, this is satisfied by clause11. So let α + 1 be odd. By 6 we have: (*) N F ( N α,α , N α , N α +1 ,α +1 , N α +1 ).By 5 and Theorem 5.25 (the transitivity of NF), N F ( N α,ε , N α,α , N α +1 ,ε ,N α +1 ,α ) [Why? By 5 (and Proposition 7.12.c), ∀ ζ ∈ [ ε, α ) N F ( N α,ζ , N α,ζ +1 ,N α +1 ,ζ , N α +1 ,ζ +1 ). The sequences h N α,ζ : ζ ∈ [ ε, α ) i , h N α +1 ,ζ : ζ ∈ [ ε, α ) i are increasing and continuous. So by Theorem 5.25 (the long transitiv-ity theorem), N F ( N α,ε , N α,α , N α +1 ,ε , N α +1 ,α ). So by the monotonicity ofNF, we have: (**) N F ( N α,ε , N α,α , N α +1 ,ε , N α +1 ,α +1 )]. Now by (*),(**) andTheorem 5.25 N F ( N α,ε , N α +1 ,ε , N α , N α +1 ). Note that we use here freelyTheorem 5.22 (the symmetry theorem of NF). ⊣ Non-smoothness implies non-structure
Hypothesis . s is a weakly successful semi-good λ -frame with conjugation. Definition 9.2.
Let ¯ M = h M α : α < α ∗ i be an increasing sequence ofmodels in K λ + . We say that ¯ M is (cid:22) NF -increasing in the successor ordinals if β < γ < α ∗ ⇒ M β +1 (cid:22) NF M γ +1 . Definition 9.3.
Let α ≤ λ +2 and let ¯ M = h M α : α < λ +2 i be an (cid:22) NF -increasing in the successor ordinals and continuous sequence with union M . Define S ( ¯ M ) =: { δ ∈ λ +2 : ∃ α ∈ ( δ, λ +2 ) M δ (cid:14) NF M α } . Define S ( M ) =: S ( ¯ M ) /D λ +2 where D λ +2 is the clubs filter on λ +2 . (By Proposition9.5 S ( M ) does not depend on the representation ¯ M ). Proposition 9.4.
Let ¯ M = h M α : α < λ +2 i be a (cid:22) NF -increasing in thesuccessor ordinals and continuous sequence. Then:(a) For each α, β with α < β < λ +2 , M α (cid:22) NF M α +1 ⇔ M α (cid:22) NF M β .(b) S ( ¯ M ) = { δ ∈ λ +2 : ∀ α ∈ ( δ, λ +2 ) M δ (cid:14) NF M α } .Proof. (a) Easy (by Proposition 6.5.c). (b) By item a. ⊣ Proposition 9.5.
Suppose:(1) The sequences ¯ M := h M α, : α < λ +2 i , ¯ M := h M α, : α < λ +2 i are (cid:22) NF -increasing in the successor ordinals and continuous.(2) M = S { M α, : α < λ +2 } and M = S { M α, : α < λ +2 } .(3) M , M are isomorphic.Then S ( ¯ M ) /D λ +2 = S ( ¯ M ) /D λ +2 .Proof. Let f : M → M be an isomorphism. Define E := { α ∈ λ +2 : f [ M ,α ] = M ,α } . So S ( h M α, : α ∈ E i ) = S ( h f [ M α, ] : α ∈ E i ) = S ( h M α, : α ∈ E i ). By Proposition 9.4.b S ( h M α, : α ∈ E i ) = S ( ¯ M ) T E and S ( h M α, : α ∈ E i ) = S ( ¯ M ) T E . Hence S ( ¯ M ) T E = S ( ¯ M ) T E . ⊣ Proposition 9.6.
Assume that we can assign to each S ∈ S λ +2 λ + := { S : S isa stationary subset of λ +2 and ( ∀ α ∈ S ) cf ( α ) = λ + } , a model M S ∈ K λ +2 with S ( M S ) = S/D λ +2 (especially it is defined).Then there are λ +2 non-isomorphic models in K λ +2 .Proof. Since | S λ +2 λ + | = 2 λ +2 it follows by Proposition 9.5. ⊣ The following theorem says that there is a kind of a witness for non- (cid:22) NF -smoothness, such that if it holds, then there are 2 λ +2 non-isomorphic modelsin K λ +2 . Theorem 9.7.
Suppose that there is an increasing continuous sequence h M ∗ α : α ≤ λ + +1 i of models in K sat such that for each α, β with α < β < λ + we have M ∗ α ≺ + M ∗ β (cid:22) NF M ∗ λ + +1 but M ∗ λ + (cid:14) NF M ∗ λ + +1 .Then there are λ +2 pairwise non-isomorphic models in K λ +2 .Proof. By Proposition 9.6, it is enough to assign to each S ∈ S λ +2 λ + a model M S ∈ K λ +2 with S ( M S ) = S/D λ +2 . Let S be a stationary subset of λ +2 such that α ∈ S ⇒ cf ( α ) = λ + . We will choose a model M β by inductionon β < λ +2 such that:(1) M β ∈ K sat .(2) The sequence h M β : β < λ +2 i is continuous.(3) β ∈ λ +2 − S ⇒ M β ≺ + M β +1 .(4) If β ∈ S then ( M β , M β +1 ) ∼ = ( M ∗ λ + , M ∗ λ + +1 ).(5) For each β < λ +2 M β (cid:22) NF M β +1 ⇔ β / ∈ S .Note that clause 5 is the crucial point and it actually follows by clauses3,4.[Why is it possible to choose M β ? For β = 0 we choose a model M ∈ K sat . For limit ordinal β , define M β = S { M γ : γ < β } . What will we doin the β + 1 step? Clause 5 follows by clauses 3,4. So it is enough to find M β +1 which satisfies clauses 3,4. OOD FRAMES WITH A WEAK STABILITY 73 case a: β / ∈ S . In this case we choose M β +1 such that M β ≺ + M β +1 (seeProposition 7.12.a). case b: β ∈ S . Since M β , M ∗ λ + are saturated in λ + over λ , they areisomorphic. Hence we can find M β +1 with clause 4]Define M S := S { M α : α < λ +2 } . It remains to prove that S ( M S ) = S/D λ +2 (especially S ( M S ) is defined). But if S ( h M α : α < λ +2 i ) is definedthen by clause 5 S ( M S ) = S ( h M α : α < λ +2 i ) /D λ +2 = S/D λ +2 . So it isenough to prove that it is defined, namely to prove that for each α, β with α < β < λ +2 we have M α +1 (cid:22) NF M β +1 . But it is easier to prove more: Claim 9.8.
For every β ≤ λ + ( ∗ ) β : For each α with α < β the followinghold:(1) M α +1 (cid:22) NF M β +1 .(2) If β / ∈ S then M α +1 ≺ + M β +1 .Proof. ( ∗ ) is vacuous.Why does ( ∗ ) β ⇒ ( ∗ ) β +1 hold? Fix α < β + 1. We prove that M α +1 ≺ + M β +2 . By clause 3 M β +1 ≺ + M β +2 . So if α = β then M α +1 ≺ + M β +2 .So without loss of generality α < β . By ( ∗ ) β M α +1 (cid:22) NF M β +1 . But M β +1 ≺ + M β +2 . So by Proposition 7.10.c M α +1 ≺ + M β +2 . This establishes( ∗ ) β +1 .Assume that δ is a limit ordinal and ( ∗ ) β holds for each β with β < δ .We have to prove ( ∗ ) δ . Let h γ ( ε ) : ε < cf ( δ ) i be an increasing continuousof ordinals with limit δ , such that for every ε, γ ( ε + 1) is a successor of asuccessor ordinal. Note that for every ε < cf( δ ) γ ε / ∈ S , because cf ( γ ε ) < cf( δ ) ≤ λ + . Consider the sequence h M γ ε : ε < cf ( δ ) i . Claim 9.9. M γ ε ≺ + M γ ε +1 for each ε < cf ( δ ) .Proof. Since γ ε / ∈ S , by clause 3 M γ ε ≺ + M γ ε +1 . If γ ε +1 = γ ε + 1 then theclaim is proved. Assume γ ε +1 > γ ε + 1. γ ε +1 = ζ + 1 for some successor ζ . ζ / ∈ S . So by ( ∗ ) ζ . M γ ε +1 ≺ + M ζ +1 = M γ ε +1 . So M γ ε ≺ + M γ ε +1 ≺ + M γ ε +1 . Hence by Proposition 7.10.d M γ ε ≺ + M γ ε +1 . ⊣ Claim 9.10.
The sequence h M γ ε : ε < cf ( δ ) i ⌢ h M δ i is continuous.Proof. Take δ ′ ∈ { γ ε : ε < cf( δ ) } S { δ } and take x ∈ M δ ′ . We have tofind ε < cf ( δ ) such that γ ε < δ ′ and x ∈ M γ ε . By clause 2 the sequence h M β : β < λ +2 i is continuous, so for some β < δ ′ x ∈ M β . The ordinalssequence h γ ε : ε < cf ( δ ) i ⌢ h δ i is increasing and continuous. Hence for some ε < cf ( δ ) with β < γ ε < δ ′ . Since M β ⊆ M γ ε , x ∈ M γ ε . ⊣ Claim 9.11. M γ ε (cid:22) NF M δ for each ε < cf( δ ) .Proof. By Proposition 6.5.d (and Claim 9.9, Claim 9.10 and Proposition7.10.a). ⊣ Now we return to the proof of ( ∗ ) δ . Fix α < δ . Claim 9.12. M α +1 (cid:22) NF M γ ε +1 for some ε < cf ( δ ) . Proof.
Take ε < cf ( δ ) with α + 1 < γ ε +1 . γ ε +1 = ζ + 1 for some ζ . So by( ∗ ) ζ . M α +1 (cid:22) NF M ζ +1 = M γ ε +1 . ⊣ Case a: δ / ∈ S . In this case by clause 4 M δ ≺ + M δ +1 . So by Proposition7.10.c it is enough to prove that M α +1 (cid:22) NF M δ . By Claim 9.12 M α +1 (cid:22) NF M γ ε +1 for some ε . By Claim 9.11 M γ ε +1 (cid:22) NF M δ . So by Proposition 6.5.b M α +1 (cid:22) NF M δ . Case b: δ ∈ S . In this case we have to prove that M α +1 (cid:22) NF M δ +1 . Wechoose f α by induction on α ≤ λ + such that:(1) For every α ≤ λ + , f α : M ∗ α → M γ α is an isomorphism.(2) h f α : α ≤ λ + i is an increasing continuous sequence of isomorphisms.There is no problem to carry out this induction [Why? We can choose f by Theorem 1.33, (the uniqueness of the saturated model in λ + over λ ). M ∗ α ≺ + M ∗ α +1 . By Claim 9.8 M γ α ≺ + M γ α +1 . So by Theorem 7.13.a, forevery α , we can find f α +1 . For α limit take union].Now by clause 4, ( M δ , M δ +1 ) ∼ = ( M ∗ λ + , M ∗ λ + +1 ). So we can find anisomorphism f : M λ + +1 → M δ +1 that extends f λ + . For every ε < λ + M ∗ ε (cid:22) NF M ∗ λ + +1 , so M γ ε = f [ M ∗ ε ] (cid:22) NF f [ M ∗ λ + +1 ] = M δ +1 . So M γ ε (cid:22) M δ +1 for each ε < cf ( δ ). Hence M γ ε +1 (cid:22) NF M δ +1 for each ε < cf ( δ ). But byClaim 9.12 for some ε < cf ( δ ) M α +1 (cid:22) NF M γ ε +1 . Therefore by Proposition6.5.b M α +1 (cid:22) NF M δ +1 . ⊣⊣ Theorem 9.13.
The following conditions are equivalent:(a) ( K sat , (cid:22) NF ↾ K sat ) does not satisfy smoothness.(b) There are M ∗ , M ∗ ∈ K sat such that M ∗ (cid:22) ⊗ M ∗ but M ∗ ⊀ NF M ∗ .(c) There is a sequence h M ε : ε ≤ λ + + 1 i of models in K sat such that foreach ε, ζ with ε < ζ ≤ λ + + 1 we have ε = λ + ⇔ M ε ≺ + M ζ ⇔ M ε (cid:22) NF M ζ .Proof. c ⇒ a is clear. b ⇒ c holds by Proposition 8.6. a ⇒ b holds byProposition 8.3.b. ⊣ Now we can prove Theorem 7.3, but first we remind it: If ( K sat , (cid:22) NF ↾ K sat ) does not satisfy smoothness, then there are 2 λ +2 pairwise non-isomorphicmodels in K λ +2 . Proof.
Condition a of Theorem 9.13 is satisfied, so condition c is satisfiedtoo. Hence by Theorem 9.7 we have the conclusion of the theorem. ⊣ a good λ + -frame Discussion:
In Definitions 2.18, 2.20 and 2.22 we expanded the definitionof the non-forking relation and basic types to models in K >λ . In Theorem2.26 we proved some axioms of a good frame for this expansions. Here weare going to prove the other axioms. So why are sections 3-9 needed? In OOD FRAMES WITH A WEAK STABILITY 75 other words, what are the difficulties in proving that S + (defined below)is a good λ + -frame? The main problem is that amalgamation may nothold in ( K λ + , (cid:22) ↾ K λ + ). Now we can overcome this problem by restrictingthe relation (cid:22) K λ + to the relation (cid:22) NF . But then there is a problem withsmoothness. We overcome this problem by showing that non-smoothnessis a non-structure property, see section 9. For the non-structure theorem,we had to restrict to the class of saturated models in λ + over λ . Now therelation ≺ + and the locality enable use to prove the remaining axioms. Definition 10.1.
Let s be a semi good frame. We say that s is successful when:(1) s is weakly successful (i.e. we have existence for K ,uq s ).(2) ( K sat , (cid:22) NF ↾ K sat ) satisfies smoothness. Hypothesis . s is a successful semi-good λ -frame with conjugation.We remind that the types in this paper are classes of triples under someequivalence relation. But this relation depends on the partial order, wedefine on the class of models. For M , M ∈ K λ + when we write tp ( a, M, N )we mean to the partial order (cid:22) . But when we want to consider the partialorder (cid:22) NF we have to write it explicitly. Definition 10.3.
For M , M ∈ K sat and a ∈ M − M we define tp + ( a, M , M ) := tp (( K sat ) up , ( (cid:22) NF ↾ K sat ) up ) ( a, M , M ) . (About ‘ sat ’ see Definition 7.2 (page 54) and about ‘ up ’ see Definition1.13 (page 5)). Proposition 10.4.
For every M , M , M with M (cid:22) NF M ∧ M (cid:22) NF M and every a , a with a ∈ M − M ∧ a ∈ M − M : tp + ( a , M , M ) = tp + ( a , M , M ) ⇔ tp ( a , M , M ) = tp ( a , M , M ) . Proof.
The first direction: Suppose tp + ( a , M , M ) = tp + ( a , M , M ). ByTheorem 7.18.c (page 65) ( K sat , (cid:22) NF ↾ K sat ) has amalgamation. So thereare f , f , M such that: M (cid:22) NF M , f n : M n → M is a (cid:22) NF -embeddingover M and f ( a ) = f ( a ). But K sat ⊆ K , and the relation (cid:22) NF isincluded in the relation (cid:22) so the amalgamation ( f , f , M ) witnesses that tp ( a , M , M ) = tp ( a , M , M ).The second direction: Suppose tp ( a , M , M ) = tp ( a , M , M ). Take anamalgamation ( f , f , M ) of M , M over M with f ( a ) = f ( a ). Foreach N ∈ K λ with N (cid:22) M tp ( f ( a ) , N, f [ M ]) = tp ( f ( a ) , N, f [ M ]).So by Theorem 7.13.b tp + ( a , M , M ) = tp + ( a , M , M ). ⊣ Although we defined restriction of types in Definition 1.21.3 (on page 7),the following definition is needed:
Definition 10.5.
For p = tp + ( a, M , M ) and N ∈ K λ with N (cid:22) M wedefine p ↾ N := tp ( a, N, M ). The following definition is based on Definition 2.18 (page 18).
Definition 10.6. s + := (( K sat ) up , ( (cid:22) NF ↾ K sat ) up , s bs, + , + S ), where:(1) For each M ∈ K sat we define S bs, + ( M ) := { tp + ( a, M, N ) : { M, N } ⊆ K sat , M (cid:22) NF N, tp ( a, M, N ) ∈ S bs>λ } (2) + S is defined by: tp + ( a, M , M ) does not fork over M if { M , M , M }⊆ K sat , M (cid:22) NF M (cid:22) NF M and tp ( a, M , M ) does not fork over M . Proposition 10.7. (a) S bs is well defined: It does not depend on the triple ( M , M , a ) thatrepresents the type.(b) + S is well defined: It does not depend on the triple ( M , M , a ) thatrepresents the type.Proof. By Proposition 10.4. ⊣ Proposition 10.8.
Let s be a successful semi-good λ -frame with conjuga-tion.(1) ( K sat , (cid:22) NF ↾ K sat ) satisfies axiom c of a.e.c. in λ + (i.e. Definition1.1.2.c).(2) ( K sat , (cid:22) NF ↾ K sat ) is an a.e.c. in λ + .(3) ( K sat , (cid:22) NF ↾ K sat ) satisfies the amalgamation property.Proof. By Theorem 7.18 and hypothesis 10.2. ⊣ Theorem 10.9.
Let s be a successful semi-good λ -frame with conjugation.Then s + is a good λ + -frame.Proof. By Proposition 10.8 ( K sat , (cid:22) NF ↾ K sat ) is an a.e.c. in λ + with amal-gamation. So by Fact 1.15 (page 5) (( K sat ) up , ( (cid:22) NF ↾ K sat ) up ) is an a.e.c.with LST number λ + . By Theorem 1.33 (page 9) K sat is categorical. So( K sat , (cid:22) NF ↾ K sat ) has joint embedding. By Proposition 7.12.a (page 58)and Proposition 7.10.a there is no (cid:22) NF -maximal model in K sat . Whatabout the axioms of the basic types and the non-forking relation? By The-orem 2.26 the following axioms are satisfied: Density, monotonicity, localcharacter and continuity. Proposition 10.10. s + satisfies basic stability.Proof. Let M ∈ K sat . M ∈ K λ + , so it has a representation h N α : α ∈ λ + i .For p ∈ S bs, + ( M ) define ( α p , q p ) by: α p is the minimal ordinal in λ + suchthat p does not fork over N α . q p =: p ↾ N α p . For every α ∈ λ + we have | S bs ( N α ) | ≤ λ + , so | ( α p , q p ) : p ∈ S bs, + ( M ) | ≤ λ + × λ + = λ + . So it issufficient to prove that the function p → ( α p , q p ) is an injection. For every p , p ∈ S bs, + ( M ) if α p = α p ∧ q p = q p Then by Corollary 7.17.b (locality,page 65) p = p . ⊣ OOD FRAMES WITH A WEAK STABILITY 77
Proposition 10.11. (1) If(a) N ∈ K λ and M ∈ K λ + .(b) For n = 1 , p n ∈ S bs, + ( M ) and does not fork over N .(c) p ↾ N = p ↾ N .Then p = p .(2) s + satisfies uniqueness.Proof.
1) By the proof of Corollary 7.17.b (locality, page 65). Remember that if N (cid:22) NF N ≺ + N then N (cid:22) N (By Proposition 7.10.c) .2) Suppose n < ⇒ M n ∈ K sat , M (cid:22) M , p, q ∈ S bs, + ( M ) , p ↾ M = q ↾ M and p, q does not fork over M . By the definition of + S , there are N p , N q ∈ K λ , such that N p (cid:22) M , N q (cid:22) M , p does not fork over N p and q does not fork over N q . As LST ( K, (cid:22) ) ≤ λ , there is a model N ∈ K λ with N p S N q ⊆ N (cid:22) M . By axiom 1.1.1.e N p (cid:22) N and N q (cid:22) N . ByTheorem 2.26(2) (monotonicity, page 19), p, q does not fork over N . By theassumption p ↾ M = q ↾ M , so p ↾ N = q ↾ N . Hence by item 1, p = q . ⊣ Proposition 10.12. s + satisfies symmetry.Proof. M id / / M M id = = {{{{{{{{ M id F F (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) id / / M id = = {{{{{{{{ N id E E (cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10) id / / id O O N id O O N id O O id = = {{{{{{{{ N id O O id F F (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) id / / N id O O id = = {{{{{{{{ Suppose 1-5 where:(1) { M , M , M } ⊆ K sat .(2) M (cid:22) NF M (cid:22) NF M .(3) tp ( a , M , M ) ∈ S bs, + ( M ). (4) a ∈ M .(5) tp ( a , M , M ) does not fork over M . Step a:
We choose models N , N , N ∈ K λ which satisfies 6-12 where:(6) n ∈ { , , } ⇒ N n (cid:22) M n and N (cid:22) N (cid:22) N .(7) tp ( a , M , M ) does not fork over N .(8) tp ( a , M , M ) does not fork over N .(9) a ∈ N .(10) a ∈ N .(11) d N F ( N , N , M , M ).(12) d N F ( N , N , M , M ).(Why is it possible? By 2, there are representations h N ,α : α < λ + i , h N ,α : α < λ + i , h N ∗ ,α : α < λ + i , h N ,α : α < λ + i of M , M , M , M respectively,such that: α < λ + ⇒ N F ( N ,α , N ,α , N ,α +1 , N ,α +1 ) , N F ( N ∗ ,α , N ,α ,N ∗ ,α +1 , N ,α +1 ). Let E be a club of λ + such that α ∈ E ⇒ N ,α = N ∗ ,α .Choose α ∈ E big enough such that 7,8,9,10 will satisfied for N = N ,α N = N ,α , N = N ,α ) Step b: [We use the symmetry axiom] By 6,8 we have:(13) tp ( a , N , N ) ∈ S bs ( N ).by 6,7 we have:(14) tp ( a , N , N ) does not fork over N .Now by Definition 2.1.3.e (symmetry) there are N ∗ , N ∗ ∈ K λ which satisfies15-18:(15) N (cid:22) N ∗ (cid:22) N ∗ .(16) N (cid:22) N ∗ .(17) a ∈ N ∗ .(18) tp ( a , N ∗ , N ∗ ) does not fork over N . Step c: [Move everything to K sat ]We can choose f which satisfies 19,20:(19) f is an injection, dom ( f ) = N ∗ and f ↾ N is the identity.(20) f [ N ∗ ] T M = N .Define N := f [ N ∗ ] , N := f [ N ∗ ]. By the existence proposition of the ≺ + -extensions (Proposition 7.12.c), there is M ∈ K λ which satisfies 21,22:(21) d N F ( N , N , M , M ).(22) M ≺ + M .By 20 (mainly) we know:(23) N T M = N .(Why? By 15 and the definitions of f, N , we have N (cid:22) N . By 6 N (cid:22) M .Let x ∈ N T M . By 2,15 x ∈ N T M . So by 20 x ∈ N . So x ∈ N T M .Hence by 12, x ∈ N . So x ∈ N T M . Hence by 11, we have x ∈ N ). Soby the existence proposition of d N F (Proposition 6.3.c), there is M ∈ K sat such that: OOD FRAMES WITH A WEAK STABILITY 79 (24) d N F ( N , N , M , M ).Without loss of generality N T M = N as M T N = N . By Proposition7.12.b there is M ∈ K sat which satisfies 25,26:(25) M ≺ + M .(26) d N F ( N , N , M , M ). Step d:
We will prove 27,28:(27) tp ( a , M , M ) does not fork over N .(28) There is an isomorphism g : M → M over M S N .Then we will conclude:(29) tp ( a , g [ M ] , M ) does not fork over M . By 25, Proposition 7.12.c=7.10and 24 we have 30:(30) M ≺ + M .By 24,25 and Theorem 6.3.b (monotonicity, on page 50): (31) N F ( N , N , M , M ).By 24,26,28 and the transitivity of the relation d N F we have:(32)
N F ( N , N , M , M ).By 2,22 and Proposition 7.10.c:(33) M ≺ + M .By 30-33 and Theorem 7.13.c, we know 28. By 26, and Theorem 6.3.e(respecting the frame, page 50):(34) tp ( a , M , M ) does not fork over N . By 18 (and 12,9,19):(35) tp ( a , N , N ) does not fork over N . By 26 N (cid:22) M , so by Theorem2.26(3) (the transitivity of the non-forking relation), we have:(27) tp ( a , M , M ) does not fork over N . Step e:
It remains to prove(36) a ∈ g [ M ]. By 28 , g is an isomorphism over N , so it is sufficient toprove a ∈ N . By 17 a ∈ N ∗ . So by 10,19 a ∈ N . ⊣ By the following proposition, s + satisfies extension. Proposition 10.13. (1) If N (cid:22) M ∈ K sat , p ∈ S bs ( N ) , N ∈ K λ , then there is q ∈ S bs, + ( M ) such that q ↾ N = p and q does not fork over N .(2) If { M , M } ⊆ K sat , M (cid:22) NF M , p ∈ S bs, + ( M ) than there is anextension of p to S bs, + ( M ) .Proof. (1) Let a, N be such that tp ( a, N, N ) = p . By Theorem 6.3.c (page 50)without loss of generality there is a model M such that d N F ( N, N ,M, M ). By Theorem 6.3.e q := tp ( a, M, M ) does not fork over N .(2) By the definition of S bs, + , there is a model N ∈ K λ such that N (cid:22) M and p does not fork over N. By item (1), there is q ∈ S bs, + ( M )which does not fork over N , and q ↾ N = p ↾ N . q does not fork over M as it does not fork over N . So it is sufficient to prove that q := q ↾ M = p . By Theorem 2.26.2 (monotonicity), q does notfork over N . q ↾ N = q ↾ N = p ↾ N . Hence by Corollary 7.17.b(locality) p = q . ⊣ This ends the proof of Theorem 10.9. ⊣ Conclusions
Theorem 11.1.
Suppose:(1) s = ( K, (cid:22) , S bs , S ) is a semi-good λ -frame with conjugation.(2) I ( λ +2 , K ) < µ unif ( λ +2 , λ + ) .(3) λ < λ + < λ +2 , and W dmId ( λ + ) is not saturated in λ +2 .Then(1) There is a good λ + -frame s + = (( K sat , (cid:22) NF ↾ K sat ) up , S bs, + , + S ) ,such that K sat ⊆ K λ + and the relation (cid:22) NF ↾ K sat is included inthe relation (cid:22) ↾ K sat .(2) s + has the conjugation property.(3) There is a model in K of cardinality λ +2 .(4) There is a model in K of cardinality λ +3 .Proof. (1) By Corollary 4.23 (page 37) s is weakly successful in the densitysense. s has conjugation, so by Proposition 4.7 (page 31), s is weakly suc-cessful. By clause 2 of our assumption, I ( λ +2 , K ) < µ unif ( λ +2 , λ + ). Butby Proposition 4.15 µ unif ( λ +2 , λ + ) ≤ λ +2 . So I ( λ +2 , K ) < λ +2 . Henceby Theorem 7.3 (page 54), ( K sat , (cid:22) NF ↾ K sat ) satisfies smoothness, i.e. s issuccessful (Definition 10.1). So Hypothesis 10.2 is satisfied. Therefore byTheorem 10.9, s + is a good λ + -frame. Obviously K sat ⊆ K λ + and (cid:22) NF isincluded in the relation (cid:22) ↾ K λ + .(2) Why does s + have conjugation? Suppose M (cid:22) NF M , { M , M } ⊆ K sat and p ∈ S bs, + ( M ) does not fork over M . By the definition of + S , thereis N ∈ K λ such that N (cid:22) M and p does not fork over N . p ↾ M f ( p ↾ M ) = pM idf / / M N id O O By Theorem 1.33.a (the uniqueness of the saturated model), there is anisomorphism f : M → M over N . By Theorem 2.26(2) (monotonicity), p ↾ M does not fork over N . So f ( p ↾ M ) does not fork over N . But also p does OOD FRAMES WITH A WEAK STABILITY 81 not fork over N and f ( p ↾ M ) ↾ N = ( p ↾ M ) ↾ N = p ↾ N [Why do we havethe first equality? There are M +0 , f + , a such that p ↾ M = tp ( a, M , M +0 )and f ⊆ f + , dom ( f + ) = M +0 . So ( p ↾ M ) ↾ N = tp ( a, N, M +0 ) = tp ( f + ( a ) , N, f + [ M +0 ]) = tp ( f + ( a ) , M , f + [ M +0 ]) ↾ N = f ( p ↾ M ) ↾ N ].So by 10.11(1), f ( p ↾ M ) = p .(3) By Proposition 3.4.3 (page 23).(4) Substitute s + instead of s in Proposition 3.4.3. ⊣ Corollary 11.2.
Suppose:(1) n < ω .(2) s = ( K, (cid:22) , S bs , S ) is a semi-good λ -frame with conjugation.(3) m < n ⇒ I ( λ +(2+ m ) , K ) < µ unif ( λ +(2+ m ) , λ +(1+ m ) ) .(4) For every m < n , λ < λ + < λ +2 < ... λ +(1+ n ) and W dmId ( λ +1+ m ) is not saturated in λ +(2+ m ) . then there is a good λ + n -frame s n =: (( K n , ≤ n ) , S bs, + n , + n S ) , such that:(1) K nλ + n ⊆ K λ + n , ≤ n ⊆(cid:22) k ↾ K n .(2) s n has conjugation.(3) There is a model in K n of cardinality λ +(2+ n ) .Proof. By induction on n , using Theorem 11.1. ⊣ Corollary 11.3.
Suppose:(1) s = ( K, (cid:22) , S bs , S ) is a semi-good λ -frame with conjugation.(2) m < ω ⇒ I ( λ +(2+ m ) , K ) < µ unif ( λ +(2+ m ) , λ +(1+ m ) ) .(3) λ + m < λ + m +1 and for every m < ω , W dmId ( λ +1+ m ) is not satu-rated in λ +(2+ m ) . Then there is a model in K n of cardinality λ + n for every n < ω .Proof. By Corollary 11.2. ⊣ Comparison to [Sh 600]A reader who knows [Sh 600], might ask about the main problems inwriting our paper. As in [Sh 600], there is a wide use of brimmed extensions(i.e. using stability), we had to find alternatives.First the relation
N F is defined in [Sh 600] using brimness, so we found anatural definition (maybe an easier one) which is equivalent to the definitionin [Sh 600], but not using brimness.Another problem was proving conjugation (see Definition 2.12, page 15).But in the main examples there is conjugation, so it is reasonable to assumeconjugation.Another problem was to find a relation ≺ + on k nice which satisfies therequired properties (see the discussion before Definition 7.4, page 55). In[Sh 600] it uses essentially brimness. But as the needed relation is on modelsof cardinality λ + , We can find such a relation, using just weak stability. Acknowledgment.
We thank Boaz Tsaban and Alon Sitton for their use-ful suggestions and comments. We thank the referee for doing an outstand-ing and extremely conscientious job in improving the paper, for his usefulsuggestions and for an example he pointed out.
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Department of Mathematics and computer science., Bar-Ilan University,Ramatgan 52900, Israel
E-mail address , Saharon Shelah: [email protected]