ALMOST GALOIS ω-STABLE CLASSES

Abstract Theorem. Suppose that k = (K, $$\prec_k$$) is an ℵ0-presentable abstract elementary class with Löwenheim–Skolem number ℵ0, satisfying the joint embedding and amalgamation properties in ℵ0. If K has only countably many models in ℵ1, then all are small. If, in addition, k is almost Galois ω-stable then k is Galois ω-stable. Suppose that k = (K, $$\prec_k$$) is an ℵ0-presented almost Galois ω-stable AEC satisfying amalgamation for countable models, and having a model of cardinality ℵ1. The assertion that K is ℵ1-categorical is then absolute.

§1. Introduction. This paper concerns two aspects of pseudoelementary classes in L 1 , , the reducts to a vocabulary ⊆ + of models of an L 1 , ( + )-sentence. In the first two sections we investigate the relationship among the number of countable models of such a class, Scott ranks, and the number of small (i.e., having a countable L 1 , -elementary submodel) models and large (not small) models of the class in ℵ 1 ; this yields some technical information about putative counterexamples to Vaught's conjecture. Building on this material, in the third section, we treat such classes as abstract elementary classes and investigate variations on Galois -stability. In the final section we use the results presented here and in [5] to prove a theorem on the absoluteness of ℵ 1 -categoricity for pseudoelementary classes in L 1 , that are also abstract elementary classes.
We call an abstract elementary class (AEC) almost Galois -stable if for every countable model M , E M (the equivalence relation of 'same Galois type over M ' see Definition 3.1) does not have a perfect set of inequivalent members. An AEC is strictly almost Galois -stable if in addition it is not Galois -stable. The immediate impetus for this paper was [5], which studied what Baldwin and Larson called analytically presented abstract elementary classes. These classes are called by many names: pseudoelementary classes in L 1 , , ℵ 0 -presentable classes, PC(ℵ 0 , ℵ 0 ) or PC ℵ 0 [24], PCΓ(ℵ 0 , ℵ 0 ) [1] or, in the language of Keisler [13], PC in L 1 , . In this paper we will most often refer to them as ℵ 0 -presented. The term 'analytically presented' emphasizes that one can deduce from Burgess's theorem on analytic equivalence relations (see [8], Theorem 9.1.5, for instance) that if such a class is almost Galois -stable then each equivalence relation E M has at most ℵ 1 equivalence classes. This topic first arose in [24] and several of the arguments here just expand ideas Shelah mentioned there; for further background on the context see [1,5,26]. Our main goal, Theorem 3.18, is to prove that an almost Galois -stable ℵ 0 -presentable abstract elementary class with only countably many models in ℵ 1 is Galois -stable. This extends earlier work by Hyttinen-Kesala [11] and Kueker [16] proving the result for sentences of L 1 , with no requirement on the number of uncountable models.
Each class of models in this paper is ℵ 0 -presented. A major tool for this investigation is to expand models of set theory by predicates encoding relevant properties of the models (for some vocabulary ) being studied. This approach appears in Shelah's analysis in [22], Section VII, connecting the Hanf number for omitting families of types with the well-ordering number for classes defined by omitting types. In [21], expanding the vocabulary to describe an analysis of the syntactic types allowed the construction of a 'small' (Definition 2.2) uncountable model in an ℵ 0 -presentable class K from an uncountable model that is small with respect to every countable fragment of L 1 , . In Lemma 2.7, we use this method to show that if, in addition, there are only countably many models in ℵ 1 , then each is small. In Section 3, we combine this technique with constructions using iterated models of set theory to prove Theorem 3. 18. In Section 4, we give sufficient conditions of categoricity in ℵ 1 of an ℵ 0 -presented AEC to be absolute. §2. Small Models. We refer the reader to [1,26] for the definition of abstract elementary class (AEC).
Assumption 2.1. k = (K, ≺ k ) is an AEC which is ℵ 0 -presented. Specifically, K is the class of reducts to of a class defined by a sentence φ ∈ L 1 , ( + ), where + is a countable vocabulary extending .
If is a cardinal, we let K be the class of models in K of cardinality . This section deals with syntactic L 1 , -types in ℵ 0 -presentable classes. As such the arguments are primarily syntactic and are minor variants on arguments Shelah used in [21,23,24]. In particular, no amalgamation assumptions are used in this section.
Definition 2.2. 1. A -structure M is L * -small for L * a countable fragment of L 1 , ( ) if M realizes only countably many L * -types (i.e., only countably many L * -n-types over ∅ for each n < ). 2. A -structure M is called locally -small if for every countable fragment L * of L 1 , ( ), M realizes only countably many L * -types. 3. A -structure M is called small or L 1 , -small if M realizes only countably many L 1 , ( )-types.
Note that 'small' is a much stronger requirement than 'locally small'. If ⊆ and N ∈ , we say that N is locally -small when N is. We emphasize when the ambient larger vocabulary plays a significant role. The following standard fact plays a key role below (see also pages 47, 48 of [1]). We quickly review the proof of this fact, as the details will be important later. For any model M over a countable vocabulary , we can define for each finite tuple a (of size n) from M the n-ary formulas φ a,α (x) (α < |M | + ) as follows.
• φ a,0 (x) is the conjunction of all atomic and negated atomic formulas satisfied by a, • φ a,α+1 (x) is the conjunction of the following three formulas: The apparent uncountability of the conjunctions in the previous definition is obviated by identifying formulas φ ac,α and φ a c,α when they are equivalent in M . Working by induction on α, one gets that if M is L * -small for each countable fragment L * of L 1 , ( ), then the set of formulas φ a,α is countable for each α, letting a range over all finite tuples from M . Finally, if M is small there exists an for all finite tuples a. Then is a Scott sentence for M . Fixing the least such α, we say that M has Scott rank α. We will also use the following fundamental result (see [13] or Theorem 5.2.5 of [1]; the notion of fragment is explained in both books). Roughly speaking, the fragment generated by a countable subset X of L 1 , ( ) is the closure of X under first order operations. We preserve Keisler's terminology to emphasize that the theorem deals only with the number of models and does not involve the choice of 'elementary embedding' on the class. Theorem 2.4 (Keisler). If a PC over L 1 , class K has an uncountable model but less than 2 1 models of power ℵ 1 then K is locally -small. That is, for any countable fragment L * of L 1 , ( ), each M ∈ K realizes only countably many L * -types over ∅.
By just changing a few words in the proof of Theorem 6.3.1 of [1], (originally in [21]) one can obtain the following result, which was implicit in [24].
Proof. Let φ be the + -sentence whose reducts to are the members of K. Without loss of generality we may assume the universe of M is 1 . Add to + a binary relation <, interpreted as the usual order on 1 . Using the fact that M realizes only countably many types in any -fragment, define a continuous increasing chain of countable fragments L α for α < ℵ 1 such that • for each quantifier free (first order) n-type over the empty set realized in M , the conjunction of the type is in L 0 , and • the conjunction of each type in L α that is realized in M is a formula in L α+1 .
Extend the similarity type further to by adding new (2n + 1)-ary predicates E n (x, y, z) and (n + 1)-ary functions f n for each n ∈ . Let M satisfy E n (α, a, b) if and only if a and b realize the same L α -type, and let the interpretation of f n map M n+1 into in such a way that E n (α, a, b) if and only if f n (α, a) = f n (α, b) for all suitable α, a, b. Then the following hold.
1. The equivalence relations E n ( , x, y) refines E n (α, x, y) if > α; 2. E n (0, a, b) implies that a and b satisfy the same quantifier free -formulas; 3. If > α and E n ( , a, b), then for every c 1 there exists c 2 such that E n+1 (α, c 1 a, c 2 b), and 4. f n witnesses that for any a ∈ M , each equivalence relation E n (a, x, y) has only countably many classes. All these assertions can be expressed by an L 1 , ( ) sentence . Let L * be the smallest -fragment containing ∧ φ. Now by the Lopez-Escobar bound on L 1 , definable well-orderings, Theorem 5.3.8 of [1], there is a -structure N of cardinality ℵ 1 satisfying ∧ φ such that there is an infinite decreasing sequence d 0 > d 1 > . . . in N (alternately, one could use Lemma 2.5 of [5] for this step). For each n, let E + n (x, y) denote the assertion that for some i, E n (d i , x, y). Using 1), 2), and 3) one can prove by induction on quantifier rank (for all n ∈ simultaneously) that for all n-ary L 1 , ( ) formulas , and all finite tuples a, b . To see this, suppose that this assertion holds for all n and all with quantifier rank at most . Let (z) be an n-ary formula of the form (∃x) (z, x), where has quantifier rank . Let a, b be n-tuples from N for which E + n (a, b) holds and N |= (a). Then for some i, E n (d i , a, b) and for some a, N |= (a, a). By condition 3) above there is a b such that E n+1 (d i+1 , a, a, b, b). By our induction hypothesis we have N |= (b, b) and so N |= (b). Now, for each n, E n (d 0 , x, y) refines E + n (x, y) and by 4) E n (d 0 , x, y) has only countably many classes, so N is small. Definition 2.6. We say a countable structure is extendible if it has an L 1 ,elementary extension to an uncountable model. Lemma 2.7. Suppose that K is the class of reducts to of a class defined by a sentence φ ∈ L 1 , ( + ), where + is a countable vocabulary extending . If some uncountable M ∈ K is locally -small but is not L 1 , ( )-small then 1. There are at least ℵ 1 pairwise-inequivalent complete sentences of L 1 , ( ) which are satisfied by uncountable models in K; 2. K has uncountably many small models in ℵ 1 that satisfy distinct complete sentences of L 1 , ( ); 3. K has uncountably many extendible models in ℵ 0 .
Working by recursion, suppose that we have constructed N α for all α < , for some countable ordinal . This determines each α (α < ) as the Scott sentence of N α and also determines L ( ). Since M is not small, M |= ¬ α for each α < . Apply Theorem 2.5 to M and the restriction of K to models L ( )-elementarily equivalent to M to construct N . Now the N α are pairwise nonisomorphic since each satisfies a distinct complete sentence α of L 1 , ( ), so conclusions 1) and 2) are satisfied. And each N α has a countable elementary submodel with respect to L α+1 ( ), so there are at least ℵ 1 nonisomorphic extendible models in ℵ 0 as well. Proof. If φ ∈ L 1 , is a counterexample to Vaught's conjecture, then every uncountable model of φ is locally small. The result then follows from Theorem 2.9 and Lemma 2.7.
Remark 2.11. Clearly, if K has only countably many models in ℵ 1 then K has at most ℵ 0 nonisomorphic extendible countable models (since each uncountable model is L 1 , -equivalent to at most one model in ℵ 0 ). The three conclusions of Lemma 2.7 are easily seen to be equivalent; we separated them in the statement because both the countable and uncountable models arose naturally in the proof. The converse of Lemma 2.7 asserts that if K has uncountably many extendible countable models and a locally small model in ℵ 1 then it has a nonsmall model in ℵ 1 . Theorem 2.9 shows this is true if the hypothesis is changed to 'uncountably many countable models, but not a perfect set of countable models', without requiring extendibility, and the class of countable models is Borel, as opposed to analytic. In general, the converse is false. The empty theory in a vocabulary with ℵ 0 constants has 2 ℵ 0 models (depending on which constants are identified) in each of ℵ 1 and ℵ 0 ; all are small. But joint embedding and amalgamation fail even under first order elementarity. Example 2.1.1 of [4] is a sentence of L 1 , giving rise to an AEC, with a particular notion of ≺ k (weaker than first order), which satisfies amalgamation and joint embedding and is ℵ 1 -categorical, and for which the model in ℵ 1 is small. In this case there are 2 ℵ 0 countable models, but only one of them is extendible. Definition 2.12. A sentence of L 1 , is large if it has uncountably many countable models. A large sentence is minimal if for every sentence φ either ∧ φ or ∧ ¬φ is not large.
As part of their proof of Theorem 2.9, Harnik and Makkai showed that any counterexample to Vaught's conjecture can be strengthened to a minimal counterexample. We call a model of cardinality ℵ 1 large if it is not L 1 , -small in the sense of Definition 2.2. Lemma 2.7 implies that if φ has a large model in ℵ 1 then φ is large.
Corollary 2.13. If φ is a minimal counterexample to Vaught's conjecture then φ has a large model in ℵ 1 , and all large models of φ in ℵ 1 are L 1 , -elementarily equivalent.
Proof. Theorem 2.9 says that φ has a large model N . Suppose that ∈ L 1 , holds in N . The fact that φ ∧ has a large model implies by Lemma 2.7 that φ ∧ has uncountably many models in ℵ 0 . By minimality, φ ∧ ¬ has only countably many models in ℵ 0 and so by Lemma 2.7 again, all uncountable models of φ ∧ ¬ are small.
Harrington 1 showed that any counterexample to Vaught's conjecture has models in ℵ 1 with Scott ranks (using sentences in L 2 , ) cofinal in ℵ 2 .
Question 2.14. Can one say anything about the embedability relation on the large models of a counterexample to Vaught's conjecture?

Connections with the Morley Analysis.
We pause to connect this analysis in Section 2.2 with a related but subtly distinct procedure. (a) Let L K 0 be the set of first order -sentences. (b) Let L K α+1 be the smallest fragment generated by L K α and the sentences of the form (∃x) p(x) where p is an L K α -type realized in a model in K. (c) For limit , L K = α< L K α . 2. K is scattered if and only if for each α < 1 , L K α is countable. Recall Morley's theorem, which is key to his approach to Vaught's conjecture.
Theorem 2.16 (Morley). If K is the class of models of a sentence in L 1 , that has less than 2 ℵ 0 models of power ℵ 0 then K is scattered.
Remark 2.17. We cannot conclude that K is scattered from just counting models in ℵ 1 , even from the hypothesis that K is ℵ 1 -categorical. Again, Example 2.1.1 of [4] (Remark 2.11) is ℵ 1 -categorical and has joint embedding for ≺ k . But there are 2 ℵ 0 first order types that give models that are not even first order mutually embeddible and the class K is not scattered. Clearly if K is scattered, every model of K is locally small. So from Theorem 2.16 and Theorem 2.4, we conclude. If K has either less than 2 ℵ 0 models in ℵ 0 or less than 2 ℵ 1 models in ℵ 1 , then every uncountable model of K is locally small.
Remark 2.19. The arguments of Morley and Shelah have different goals. Being scattered is a condition on all models of (in the interesting case for the Vaught conjecture) an incomplete sentence in L 1 , . The Shelah argument contracts K to a smaller class where every model is small and thus finds a K ⊂ K that is small and is axiomatized by a complete sentence. The hard part is to make sure K has an uncountable model. In the most used case, K and a fortiori K is ℵ 1 -categorical.

Alternate proofs using Scott sentences.
In this subsection we prove alternate versions of Theorem 2.5 and part of Lemma 2.7. Theorem 2.20 has essentially the same proof as Theorem 2.5 and was proved (by the third author, modulo the fragment of ZFC used) at essentially the same time. It can be used in place of Theorem 2.5 in all of our applications of Theorem 2.5. An ill-founded model as in the statement of the theorem can be obtained either by Lopez-Escobar or by iterated ultrapowers of models of set theory. For convenience we use the theory ZFC • from [5]. Any theory strong enough to carry out the construction of Scott sentences should be sufficient.
Theorem 2.20. Let be a countable vocabulary, let M be a -structure, and let N be an -model of ZFC • with N 1 ill-founded. Let be the ordinal isomorphic to the longest well-founded initial segment of N 1 .

Suppose that, in N , M is locally -small and either large or small with Scott rank in the ill-founded part of N . Then M is small, and the Scott rank of M is exactly .
Proof. Let t be the Scott rank of M in N if N thinks that M is small, and N 1 otherwise. Let φ a,s : a ∈ M < , s < t be the set of formulas defined in N in the first t many steps of the search for a Scott sentence for M . Then φ a,α : a ∈ M < , α < is also the set of formulas defined in V in the first many steps of the search for a Scott sentence for M . Since the Scott rank of M in N is in the ill-founded part of N if it exists, the Scott rank of M in V is at least . We claim that for any n ∈ and any pair a, b of n-tuples from M , if φ a,s = φ b,s for any ill-founded s < t, then a and b satisfy all the same L 1 , ( )-formulas in M (from the point of view of V ). To see this, suppose that this assertion holds for all n and all formulas with quantifier rank at most . Let (z) be an n-ary formula of the form (∃x) (z, x), where has quantifier rank . Let a, b be n-tuples from N , let s < t be an ill-founded ordinal of N such that φ a,s = φ b,s , and suppose that M |= (a). Then there is an ill-founded r < s, and for any such r, φ a,r = φ b,r . Since M |= (a), there is a c ∈ M such that M |= (a, c). Since r < s and φ a,s = φ b,s , φ a,r+1 = φ b,r+1 , which means that there is some d ∈ M such that φ bd,r = φ ac,r . Thus by our induction hypothesis, M |= (b, d ) and thus M |= (b).
For each n ∈ and each pair a, b of n-tuples from M , if φ a,α = φ b,α for all α < , then φ a,s = φ b,s for some ill-founded s < t, since if φ a,r = φ b,r for any r < t, then N thinks that there is a least such r, and there is no least ill-founded ordinal of N . It follows then that the Scott rank of M (in V ) is exactly . Lemma 2.21 will make up part of the proof of our main theorem (Theorem 3.18). The proof is in fact a simplified part of the main argument in the proof of that theorem.
Lemma 2.21. Suppose that K is the class of reducts to of a class defined by a sentence φ ∈ L 1 , ( + ), where + is a countable vocabulary extending . If K has a model in K ℵ 1 that is locally -small, but is not L 1 , ( )-small then K has small models in ℵ 1 of club many distinct Scott ranks.
Proof. Suppose that M is a model in K with cardinality ℵ 1 that is is locally -small but is not L 1 , ( )-small. Fix a regular cardinal > 2 2 ℵ 1 . It suffices to show that for every countable elementary submodel X of H ( ) with , φ, and M in X , there exists a small model in K of cardinality ℵ 1 whose Scott rank is X ∩ 1 . Fix such an X . Let P be the transitive collapse of X , and let : X → P be the corresponding collapsing map. Then ( 1 ) = P 1 is the ordinal X ∩ 1 . By iterating the construction in [14], one can find an elementary extension P of P with corresponding elementary embedding : P → N , with critical point P 1 , such that N 1 is ill-founded and uncountable, and such that the well-founded ordinals of N are exactly the members of P 1 . Since N 1 is ill-founded, Theorem 2.20 implies that ( (M )) is L 1 , ( )-small, with Scott rank equal to the longest well-founded initial segment of N 1 , which is X ∩ 1 . Theorem 2.20 gives part of the proof of the following fact, which is used in the proof of Theorem 4.2.
having an uncountable model. The statement that every uncountable model in K satisfies the same Scott sentence in L 1 , can be expressed as both a Σ 1 2 sentence and a Π 1 2 sentence, each in a given real parameter for K. Proof. First consider the statement that there is a complete sentence in L 1 , such that whenever M is a countable model in K and N is a countable -model of ZFC • with ∈ N and M uncountable in N , M |= . By Theorem 8.9 in Marker's appendix to [2], being a complete sentence in L 1 , is Π 1 1 , so this sentence is Σ 1 2 . If there is a nonsmall uncountable model in K, or if there are uncountable small models with distinct Scott sentences in L 1 , , then this Σ 1 2 statement can be shown to be false by taking the transitive collapses of a suitable countable elementary submodels (note that a nonsmall model satisfies the negation of each complete sentence in L 1 , ). On the other hand, for any sentence of L 1 , , if there exist a countable model M in K and a countable -model N of ZFC • with ∈ N , M uncountable in N and M |= ¬ , then one can find an uncountable model in K satisfying ¬ , by taking an iterated generic elementary embedding of length 1 (as in the proof of Theorem 2.1 of [5]). Now consider the statement that whenever M and N are countable models in K and P and Q are countable -models of ZFC • with M an uncountable model in P and N an uncountable model in Q, then M and N are isomorphic. This statement is easily seen to be Π 1 2 in a code for K. As above, if there is a nonsmall uncountable model in K, or if there are uncountable small models with distinct Scott sentences in L 1 , , then this Π 1 2 statement can be shown to be false by taking the transitive collapses of suitable countable elementary submodels of H (κ), for any regular κ greater than 2 2 ℵ 1 .
In the other direction, suppose first that there exist a countable model P of ZFC • with P 1 wellfounded, and a model M ∈ K in P which P thinks has uncountable Scott rank. Then we can produce two uncountable models in K of distinct Scott ranks by taking elementary extensions of P. We start by finding two elementary embeddings, k 1 : P → R 1 and k 2 : P → R 2 , each with critical point P 1 , where the well-founded part of R 1 is exactly P 1 , and the well-founded part of R 2 is at least P 2 (for the first of these, use the construction in [14]; for the second use [10] or [5]). We can then iteratively extend R 1 and R 2 each 1 many times (iterating either the construction in [14] or the one in [5]), producing elementary embeddings j 1 : R 1 → R * 1 and j 2 : 1 are uncountable. By Theorem 2.20, j 1 (k 1 (M )) will have Scott rank P 1 . By the elementarity of j 2 • k 2 , the Scott rank of j 2 (k 2 (M )) will be at least P 2 (and uncountable if R * 2 1 = 1 ). Supposing now that there exists no pair (M, P) as in the previous paragraph, suppose that we have two -models P and Q of ZFC • , containing countable models M and N (respectively) in K which they think are uncountable, and suppose that M and N satisfy different Scott sentences in V . Then either P thinks that M is small, or P 1 is ill-founded, and the same holds for Q and N . Using the constructions from either [14] or [5], iterate P and Q each 1 times, producing models M and N of cardinality ℵ 1 . Then M and M have the same Scott sentence, as do N and N . To see this, note that the Scott rank of M as computed in P is either in the well-founded part of P (in which case it must be countable in P, since we are not in the case of the previous paragraph) or not. In the first case, M and M have the same Scott sentence by elementarity. In the second case, they have the same Scott sentence by Theorem 2.20. Since this applies to Q and N also, M and N are uncountable models in K satisfying distinct Scott sentences in L 1 , , and we are done. §3. Almost Galois Stability. The section is concerned about stability and almost stability with respect to Galois types. Note that if K satisfies the amalgamation property then ∼ = ∼ 0 . This identity is used crucially in proving the equivalence of model-homogeneity and Galoissaturation. When we use this equivalence we will assume amalgamation.
Fixing a coding of hereditarily countable sets by subsets of , the notion of Galois types naturally induces an equivalence relation on P( ). 2 For each M ∈ K ℵ 0 we let E M denote the corresponding equivalence relation for Galois types over M (This notation was used in [5] 3 .). The domain of E M is then the set of subsets of coding triples of the form (M, a, N ), where N ∈ K ℵ 0 , M ≺ K N and a ∈ N \ M . If k is ℵ 0 -presented, then each E M is an analytic equivalence relation, and by Burgess's trichotomy for analytic equivalence relations, E M has either countably many equivalence classes, ℵ 1 many, or a perfect set of inequivalent reals. Because there are two notions of weak-stability in the literature of AEC ( [11,26], we call the following notion almost Galois -stability.  If an AEC has no maximal models 4 then it has arbitrarily large models. In general the converse fails; but the converse holds under joint embedding with one exception: an AEC with a unique maximal model may satisfy joint embedding (See part (1) of Corollary 3.6). The class of wellorders of order type ≤ 1 , with ≺ k as end extension is a standard example of part 1 of Theorem 3.7: an AEC with a unique maximal model in ℵ 1 but amalgamation in ℵ 0 .
Assuming amalgamation, the relation 'M and N can be ≺ k -embedded into a common model' is an equivalence relation and each equivalence class is an AEC with joint embedding. Often we will assume amalgamation and joint embedding to avoid assuming only amalgamation and then having to restrict to one joint embedding class. Failure to make this assumption yields trivial counterexamples. There are no universal models for the class of algebraically closed fields (because of characteristic) but fixing the characteristic (that is the joint embedding class) yields a family of classes each with the joint embedding property. The technique of restricting to an equivalence class is illustrated by the generalization of Theorem 3.18 to Corollary 3.22.
We record some additional observations about amalgamation, joint embedding and maximal models. Lemma 3.4 is Lemma 2.11 on page 134 of [26], in the case κ = ℵ 0 (the relevant case for this paper). It shows that joint embedding and amalgamation in κ implies members of K with cardinality κ + can be amalgamated over submodels of cardinality κ. It is an open question (usually conjectured to be false) whether amalgamation in κ implies amalgamation in κ + . Note that (1) of Lemma 3.4 is an easy consequence of part (2). 3 It might be natural to write E 1 M instead, as we are referring to the Galois 1-types. Recent work of Boney [6] shows that for an AEC satisfying amalgamation for countable models, the set of Galois n-types over M has the same cardinality for each n ∈ . 4 Maximal means there is no proper ≺ k -extension, even one isomorphic to itself.
1. If M, N ∈ K ℵ 0 and P ∈ K ℵ 1 , with M ≺ k P and M ≺ k N , then there exist Q ∈ K ℵ 1 with N ≺ k Q, and a ≺ k -embedding f : P → Q such that f is the identity function on M . 2. If M ∈ K ℵ 0 , P, Q ∈ K ℵ 1 with M ≺ k P and M ≺ k Q, then there exist R ∈ K ℵ 1 with P ≺ k R and a ≺ k -embedding f : Q → R such that f is the identity function on M . In Part (1) of Lemma 3.4, we have not asserted that Q is a proper extension of P; the first example in Remark 3.3 shows that is too strong. Lemma 3.4 implies that for AEC's with Löwenheim-Skolem number ℵ 0 satisfying amalgamation for countable models, maximal models of cardinality ℵ 1 are strongly ℵ 1 -model homogeneous. By Theorem 8.3 of [1], ℵ 1 -model homogeneous models are isomorphic for AEC's with Löwenheim-Skolem number ℵ 0 satisfying joint embedding for countable models. We note two additional consequences of Lemma 3.4 for maximal models.  Amalgamation with some form of joint embedding easily allows one to show the following (see Corollary 8.23 of [1]); we give two variants. Note that in the second case the Galois-saturated model may not be unique. Furthermore, there may be countable models that are not extendible, even when there is a unique Galoissaturated model in ℵ 1 .
Theorem 3.7. Let k = (K, ≺ k ) be an AEC which is almost Galois -stable and satisfies amalgamation in ℵ 0 .
1. If K satisfies joint embedding in ℵ 0 then there is a unique Galois-saturated model M in K ℵ 1 .

If N ∈ K ℵ 0 has an uncountable extension in K, then there is a Galois
Proof. For the first part, carefully construct an interweaving enumeration the Galois types over an increasing chain of countable models in order type 1 so that each Galois type over each model in the chain is realized. For uniqueness, suppose that M and M are Galois-saturated models in K ℵ 1 . Choose countable M 0 ≺ k M and M 0 ≺ k M . By joint embedding there is a countable M 1 that ≺ k -extends both M 0 and M 0 . Applying Galois saturation, a countable recursive construction shows that M 1 is ≺ k -embeddable into both M and M . Then a recursive construction of length 1 using Galois saturation shows M and M are isomorphic (over M 1 ).
For the second part, let K N be the equivalence class under joint embedding of the models that are jointly embeddable with N . Apply the first argument to this class.
For any AEC k = (K , ≺ k ), if M, N ∈ K and M ≺ K N , then M is a substructure of N , but the definition of AEC does not require even that M be a first-order elementary submodel of N . Before proving the main result of this section, Theorem 3.18, we prove a lemma which reduces the proof to the case where M ≺ K N implies L 1 , ( )-elementarity. A similar reduction appears in Theorem 3.6E of [24] and Lemma 2.5 of [21].
Definition 3.8. Let K be an AEC in a countable similarity type , with Löwenheim-Skolem number ℵ 0 , such that K has a unique Galois-saturated model M in ℵ 1 , which is small.
Let K * be the set of N ∈ K ℵ 0 which satisfy the Scott sentence of M . 3. Let (K , ≺ k ) be the closure of (K * , ≺ k * ) under isomorphism and direct limits of arbitrary length.
To discuss the relationship between (almost) Galois stability of K and K , we introduce some notation. We first give a standard equivalent for the definition of Galois type, but parameterized for the comparisons we need here. The class K 0 below will be K or K in our applications. This construction is implicit in [24] and in the extension of those arguments towards the construction of examples of a good frame in [27] and chapter III of [26]. The next lemma shows the properties of the induced class K . We describe a slightly more general situation from [26] in Remark 3.12.
We need to require the joint embedding property to guarantee that (K, ℵ 1 )homogeneous-universal 5 is equivalent to Galois saturated. Most of the argument for the next lemma would work if we just assume there is a unique Galois saturated model (which is small); but it might not be universal (in either K or K ). (See Chapter 16 of [1] or Remark 1 of [25] for more detailed remarks.) Suppose further that unique Galois-saturated model M in ℵ 1 is small. Then the following hold. be the vocabularies which witness that K is ℵ 0presented. Let 1 be the sentence whose reducts are the models in K; let 2 be the sentence whose reducts are pairs (N, M ) with N ≺ k M . Further suppose that φ is the Scott sentence of M . The following sentences witness that

(K , ≺ k ) is an AEC with Löwenheim-Skolem number
7) This is evident since N is small.

Conclusion 5 immediately yields.
Corollary 3.11. Under the hypotheses of Lemma 3.10, Moreover, the hypothesis of joint embedding is in some ways only a convenience; see Corollary 3.22. If K has the amalgamation property then joint embedability is an equivalence relation and each of the equivalence classes is an AEC with joint embedding preserving the other properties defining AEC's. At least one class fails Galois -stability if K does. But some classes may not have any uncountable models.
where ≺ k is defined as before. It is again straightforward to see that each K M is an ℵ 0 categorical AEC. If there are less than 2 ℵ 1 models in ℵ 1 of K and a fortiori of each K M then under 2 ℵ 0 < 2 ℵ 1 , K M has the amalgamation property and since all models are extension of a single one, the joint embedding property. Then Shelah argues that by way of the notion of 'materialization of types' (Chapter 1 of [26]) one can deduce almost Galois stability.
The following variants on an example of Jarden and Shelah [12] will illustrate the situation and also provide some context for Theorem II.3.4 of [26]. That theorem aims to construct a good frame from an ℵ 0 -presentable class that has few models in ℵ 1 , is ℵ 0 -categorical, has amalgamation in ℵ 0 and is -Galois stable or at least -almost Galois stable. We show several of these conditions are necessary. In particular, these examples are not ℵ 0 -categorical. Note that one use of Lemma 3.10 is to extract an ℵ 0 -categorical AEC from a given AEC with few models in ℵ 1 . Recall that there are only ℵ 1 countable linear orders that are one transitive (any two points are automorphic) [20].
Here is the basic example Example 3.13. Let contain equality and a binary symbol <. Let (K, ≺ k ) be the class of structures such that each M ∈ K is a partially ordered set such that each component is a countable 1-transitive linear order.
K is an ℵ 0 -presentable AEC. It has exactly ℵ 1 countable models and 2 ℵ 1 in ℵ 1 . It is almost Galois -stable but not Galois -stable. K ℵ 0 satisfies the amalgamation property and the joint embedding property. Thus there is a unique Galois saturated model in ℵ 1 . Now we vary the example so there are ℵ 1 nonisomorphic Galois-saturated models in ℵ 1 .
Example 3.14. Let consist of a binary symbol < and another binary relation symbol E. Let (K, ≺ k ) be the class of -structures such that each M ∈ K is a partially ordered set such that each component (maximal connected component) is a countable 1-transitive linear order. Further E is an equivalence relation; each class intersects each component in exactly one point. Moreover E induces an orderisomorphism between each pair of components. M ≺ k N means M ⊆ N but each element of N − M is incomparable with all elements of M .
K is an ℵ 0 -presentable AEC as it is describable in L(Q) using only assertions of the form 'φ(x) is countable'. It has exactly ℵ 1 models in each infinite cardinality. It is almost Galois -stable but not Galois -stable. K ℵ 0 satisfies the amalgamation property but does not satisfy the joint embedding property. There are in fact ℵ 1 , pairwise nonisomorphic Galois saturated models in ℵ 1 ; each model is ℵ 1 copies of a particular 1-transitive order.
There is no countable fragment L * such that syntactic type in L * is the same as the Galois type in K.
Because the joint embedding property fails, Lemma 3.10 does not apply to this example. Applying the construction in Definition 3.8 gives rise to ℵ 1 distinct ℵ 0 -presentable AEC; each is categorical in every infinite cardinality; each is -stable. In each derived AEC, Galois type is equivalent to syntactic type.
The refined AEC, where all components have the same order type, are indexed by φ α for α < 1 , which list the Scott sentences of countable transitive linear orders.
Neither Lemma 3.10 nor II.3.4 (page 285) of [26] applies to either of these examples because there are too many models in ℵ 1 in the first case and the Galois saturated model is not locally small in the second. Nevertheless there are ℵ 1 restrictions of K to AEC K α , where models in K α contain only components satisfying φ α . Each of them is Galois -stable. In each K α , Galois types are equivalent to syntactic types in an appropriate fragment L α .
Question 3.15. Find an example of an ℵ 0 -presented AEC with the joint embedding and amalgamation properties that has fewer than 2 ℵ 1 many models in ℵ 1 and is strictly almost Galois -stable.
By Theorem 3.18 below, K must fail joint embedding or have at least ℵ 1 models in ℵ 1 .
Given a -structure M and a fragment L of L 1 , ( ), we say that M is L-atomic if for each finite sequence a from M there exists an |a|-ary formula a (x) ∈ L such that M |= a (a), and, for each |a|-ary formula (  We turn to the main result. Corollary 3.22 derives a slightly weaker conclusion than Theorem 3.18 in the absence of the joint embedding property. By Corollary 2.8, the hypotheses of Theorem 3.18 imply that all models in K are small. By Theorem 3.7, K 1 contains a unique Galois-saturated model. Theorem 3.18. Suppose that K is an ℵ 0 -presented AEC (over a countable vocabulary ) which satisfies amalgamation, and JEP for countable models, such that that K is almost Galois -stable, and |K ℵ 1 | ≤ ℵ 0 . Let M be the unique Galois-saturated model in K ℵ 1 , and let k = (K , ≺ k ) be as in Definition 3.8. Letˆ be formed by adding many new constant symbols to . Then We will show that case 1) contradicts the assumption of almost Galois -stability of k (which by Corollary 3.11 is equivalent to that of k), and that case 2a) contradicts the assumption that |K ℵ 1 | ≤ ℵ 0 . We are reduced to case 2b) and Lemma 3.17 gives that k is Galois -stable.
For case 1, we use the following fact 6 . By Fact 3.19, in Case 1 there exists a perfect set of syntactic 1-types in L * (ˆ ) that are realized in countableˆ -structures whose -reducts are in K ℵ 0 and for which the interpretation of the c i 's enumerates M 0 in the same manner that M does. Since ≺ k implies L 1 , ( )-elementarity, this implies the existence of a perfect set of Galois 1-types over M 0 , contradicting the almost Galois -stability of k .
The bulk of the proof derives a contradiction from Case 2a. Let φ be the Scott sentence for M . Let M = M α : α < 1 be such that (as above) M 0 is the model introduced in the statement of the theorem, M = α< 1 M α and the following hold for each α < 1 : • M α is a countable element of K; • M α is a proper subset of M α+1 ; • if α is a limit ordinal, then M α = <α M . The models M α are all isomorphic, as they satisfy the same Scott sentence. As M is Galois saturated, there is a set F = {F α : α < 1 } such that each F α is an automorphism of M mapping M 0 setwise to M α . For each pair α, < 1 , let F α, denote F • F −1 α . Let + be the expansion of our vocabulary to the of Theorem 2.5 (i.e., add the symbols E n , f n (n ∈ ), and a binary relation ordering the domain of M in order type 1 ; alternately, using Theorem 2.20 we could skip this step). Fix a regular cardinal large enough so that M , + , M , and F are elements of H ( ) (to apply the methods of [5], we need to be larger than 2 2 ℵ 1 ).
Let X α : α < 1 be a ⊆-increasing continuous chain of countable elementary submodels of H ( ) such that M , + , M , and F are elements of X 0 , and such that for each α < 1 there is a countable ordinal ∈ X α+1 − X α . For each α < 1 , let P α be the transitive collapse of X α , and let α : X α → P α be the corresponding collapsing map. Then α ( 1 ) = Pα 1 is the ordinal X α ∩ 1 . The following is a paraphrase of Theorem 2.1 of [10] (Hutchinson built on work of Keisler and Morley [14]; Enayat provides a useful source on this work in [7]). It can be proved via iterated ultrapowers as in [5]. Section 4 of [10] describes the fragment of ZFC needed for Fact 3.20; this fragment is easily seen to follow from the theory ZFC • of [5]. Construct a family {P α : α < 1 } of uncountable models of set theory so that, for each α < 1 , there is an elementary extension of P α to P α (with corresponding elementary embedding α : P α → P α ) such that 1. the critical point of α is Pα 1 , so Pα 1 is an initial segment of P α 1 ; 2. P α 1 is ill-founded; 3. in V , there is a continuous increasing 1 -sequence t α : < 1 consisting of elements of P α 1 . Item 3 above implies in particular that each P α 1 is uncountable. Each P α can be realized as the union of an increasing elementary chain of models P α : < 1 , where P α 0 = P α , P α = < 1 P α for limit α, and each P α +1 can be obtained by applying Fact 3.20 to P α . Then each t α (the c of Fact 3.20) can be taken to be P α

.
Recall that M is the union of the continuous ⊆-increasing chain M α : α < 1 . It follows then for each α < 1 , that M Pα 1 = α (M ) ⊂ P α , and that M Pα 1 has cardinality ℵ 1 in P α . For each α < 1 , let N α = α (M Pα 1 ) and let N α = α ( α (M )). Then each N α is an expansion of N α via the given enumeration of M 0 by the constants c i , and it has cardinality ℵ 1 in P α .
In the argument for Theorem 2.5 replace the appeal to Lopez-Escobar (Theorem 5.3.8 of [1]) with the observation that the induced ordering on N α is not well-founded by construction. The rest of the argument for Theorem 2.5 (or Theorem 2.20) shows that, in V , each N α is small for L 1 , ( ). Nevertheless, by the elementarity of α • α , each P α thinks that N α is not L 1 , ( )-small.
Since M is a sequence indexed by 1 in V (or in X α ), α ( α (M )) is a sequence indexed by P α 1 in P α . So, in P α , for each element t of its 1 , there is a t-th element of the sequence, which we denote by M α t . Furthermore, in P α , α ( α (F )) is a set {F α t : t ∈ P α 1 } consisting of automorphisms of N α , such that each F α t ∈ P α is an automorphism of N α sending M 0 setwise to M α t . Each F α t is then an automorphism of N α in V also.
Since each N α is small, each N α is as well. Since we are assuming that there are only countably many models in K of cardinality ℵ 1 , there exists an uncountable set S ⊆ 1 such that N α 0 and N α 1 are isomorphic (in V ) for all α 0 , α 1 in S. Fix for a moment a pair of elements α 0 , α 1 of S and an isomorphism : N α 0 → N α 1 . Applying item 3 above and the continuity (in the sense of P αj , for j = 0, 1) of the sequences M α 0 t : t ∈ such that maps M α 0 t 0 setwise to M α 1 t 1 . To see this, start with 0 = 0 and, for each n ∈ , let n+1 be large enough so that Then let s 0 = t α 0 sup n∈ n and let s 1 = t α 1 sup n∈ n . By the continuity in item 3, the s j 's are in the respective P αj , for j ∈ {0, 1}. So, for each j, by the continuity in P αj of M Then (F α 1 s 1 ) −1 • • F α 0 s 0 is an isomorphism of N α 0 and N α 1 fixing M 0 setwise, though not necessarily pointwise.
Finally, we show that for each α 0 < 1 such an isomorphism is impossible for sufficiently large α 1 < 1 .
Each model P α thinks that N α is small for every countable fragment of L 1 , ( ) but not L 1 , ( )-small. Thus, from the point of view of P α , there is no ordinal t such that φ a,t (x) ≡ φ a,t+1 (x) (in the terms of the Scott construction) for all finite tuples a of N α . For each well-founded ordinal of P α (this includes the members of P α 1 = 1 ∩ X α , by item 1 above), and each finite tuple a of N α , P α sees the same formula φ a, (x) that the true universe V does, which means that the Scott sentence for N α has rank at least 1 ∩ X α (and slightly more than this, in fact, in the approach from [5]) 7 . Now choose α 0 , α 1 ∈ S such that 1 ∩ X α 1 is greater than the Scott rank (in V) of N α 0 . Since permuting the constants c i in terms of their enumeration of M 0 has no 7 Alternatively, Lemma 2.21 implies that the Scott rank of N α is exactly the well-founded part of effect on the rank of the Scott sentence for N α 1 , there cannot be then an isomorphism of N α 0 and N α 1 fixing M 0 setwise, since this would imply that N α 0 and N α 1 have the same Scott rank (Indeed, their Scott sentences would differ only by a permutation of the c i 's). Thus we have a contradiction in case 2a.
We have ruled out cases 1) and 2a) and are left with case 2b). Again, Lemma 3.17 gives the second conclusion of the theorem.
Remark 3.21. Note that argument ruling out case 2a) uses the set theoretic argument to find ℵ 1 -small models in ℵ 1 with distinct -Scott rank. By the automorphism argument, this contradicts the assumption that there are only ℵ 0 -models in ℵ 1 .
We return to the slightly more complicated situation where joint embedding is not assumed. Proof. Since there are only countably many models in ℵ 1 , the equivalence relation of common extension has at most countably many classes. Each satisfies the hypothesis and therefore the conclusion of Theorem 3.18. §4. Absoluteness of Categoricity. In this section we show that ℵ 1 -categoricity is absolute for an ℵ 0 -presented almost Galois -stable AEC with an uncountable model that satisfies amalgamation for countable models. While we rely on Theorem 6.1 of [5], there are two key differences in the argument here; we do not assume any form of joint embedding and we give a different proof than in [5] that such a class is Galois -stable using Lemma 3.18 Remark 4.1. Suppose that k = (K , ≺ k ) is an analytically presented AEC, and that M ∈ K is countable. Then there exists an uncountable N ∈ K with M ≺ k N if and only if there exists a countable -model P of ZFC • (containing a real parameter for K ) such that M ∈ P and P thinks there exists an uncountable N ∈ K with M ≺ N . This follows from the same argument as for Theorem 2.1 of [5] (one direction consists of taking the transitive collapse of a countable elementary submodel; the other consists of building an iterated generic elementary embedding of length 1 ). Note that the latter clause is Σ 1 1 in the given parameter for K, and therefore absolute.
Similarly, the statement that K is almost Galois -stable is Π 1 1 in a real coding K (this is Theorem 6.6. of [5]), and therefore absolute. Amalgamation of countable models for such a K is easily seen to be Π 1 2 in a code for K. In this section we apply Theorem 3.18 to prove the following theorem. 8 Theorem 4.2. Suppose that K is an ℵ 0 -presented almost Galois -stable AEC with Löwenheim-Skolem number ℵ 0 , satisfying amalgamation for countable models and having an uncountable model. The assertion that K is ℵ 1 -categorical is then absolute, as it is equivalent to a statement of the form φ 1 ∧ φ 2 , where φ 1 and φ 2 are Π 1 2 and Σ 1 2 , respectively, in a code for K.
We rely on the following fact from [5].
Fact 4.3. Suppose that k = (K, ≺ k ) is an ℵ 0 -presented AEC. The following assertion is equivalent to a Σ 1 2 statement in a parameter for k: There exist M ∈ K ℵ 0 and N ∈ K ℵ 1 such that By the first part of Lemma 3.18, we have clause (4) for all countable N ∈ K . The second part of Lemma 3.18 implies Galois -stability for K . Now the ℵ 1categoricity of K (hence K ) and the Galois -stability of K imply clause (4).
For the other direction, since K and K have the same uncountable models, it suffices to show that K is ℵ 1 -categorical. From clause (4) and the first part of Theorem 3.7, we get that K has a unique small uncountable Galois saturated model. Applying Lemma 3.17 to K , clause (4) implies that K is Galois -stable. Then clause (4) implies that the Galois saturated model is the only model in K (a fortiori K) of cardinality ℵ 1 .