Classification theory for abstract elementary classes

Rami Grossberg

Abstract. In this paper some of the basics of classification theory for abstract elementary classes are discussed. Instead of working with types which are sets of formulas (in the first-order case) we deal instead with Galois types which are essentially orbits of automorphism groups acting on the structure.

Some of the most basic results in classification theory for non elementary classes are presented. The motivating point of view is Shelah’s categoricity conjecture for L_{ω_{1},ω}.

Introduction

In recent years the view that stability theory has wider applicability than the originally limited context (i.e. first-order stable theories) is getting increasing recognition among model theorists. The current interest in simple (first-order) theories and beyond signifies a shift in the opinion of many that similar tools and concepts to those of basic stability theory can be developed and are relevant in a wider context. Much of Shelah’s effort in model theory in the last 18-19 years is directed toward development of classification theory for non elementary classes. I feel that the study of classification theory for non elementary classes will not only provide us with a better understanding of classical stability (and simplicity) theory but also will develop new tools and concepts that will be useful in projecting new light on classical problems of “main stream” mathematics.

The purpose of this article is to present some of the basics of classification theory for non elementary classes. For several reasons I will concentrate in what I consider to be the most important and challenging framework: Abstract Elementary Classes, however this is not the only important framework for such a classification theory.

I made an effort to keep most technical details to a minimum, while still having some deep ideas of the subject explained and part of the overall picture described.

1991 Mathematics Subject Classification. Primary: 03C45, 03C52, 03C75. Secondary:

03C05, 03C55 and 03C95.

I would like to express my gratitude to the anonymous referee for A most helpful report and many valuable comments. Also I am grateful to Yi Zhang, Without his endless efforts this paper would not exist.

In the last section I discuss direction for future developments and some open problems.

An interested reader can find more details in Chapter 13 of [Gr].

I thank Oleg Belegradek for an interesting discussion and suggesting several examples, Andrés Villaveces for his comments and Monica VanDieren for comments on a preliminary version.

1. definitions and examples

Definition 1.1. Let K be a class of structures all of the same similarity type L(K). K is an Elementary class if there exists a first-order T in L(K) such that K = Mod(T).

There are very many natural classes that are not elementary:

• Archimedean ordered fields,

• locally-finite groups,

• well-ordered sets,

• Noetherian rings and

• the class of algebraically closed fields with infinite transcendence degree. Extensions of predicate calculus permit a model-theoretic treatment of the

above:

1. The basic infinitary languages: L_{ω_{1},ω} ⊆ L_{λ^{+},ω} ⊆ L_{λ^{+},μ} and L_{∞,ω}. The earliest work on infinitary logic was published in 1931 by Ernst Zermelo [Z].

Later pioneering work was done by: Novikoff (1939, 1943), Bochvar (1940) ([Bo]) and from the late forties to the sixties the main contributors were: Tarski, Hanf, Erdős, Henkin, Chang, Scott, Karp, Lopez-Escobar, Morley, Makkai, Kueker and Keisler.

Recall

Definition 1.2.

• L_{ω_{1},ω} contains all the first-order formulas in the language L and is closed under propositional connectives, first-order quantification and the rule: If {φ_{n}(x) | n < ω} ⊆ L_{ω_{1},ω} then

⋁_{n<ω} φ_{n}(x) ∈ L_{ω_{1},ω}, ℓ(x) < ω.

If {φ_{α}(x) | α < λ} ⊆ L_{λ^{+},ω} then

⋁_{α<λ} φ_{α}(x) ∈ L_{λ^{+},ω}.

• If {φ_{α}(x) | α < λ} ⊆ L_{λ^{+},μ} with ℓ(x) < μ then

⋁_{α<λ} φ_{α}(x) ∈ L_{λ^{+},μ}

and for every sequence of variables ⟨x_{β} | β < α < μ⟩

Qx_{0}Qx_{1} ⋯ Qx_{β} ⋯ φ(⟨x_{β} | β < α⟩) ∈ L_{λ^{+},μ}

for Q ∈ {∀, ∃}.

• For a limit cardinal χ:

L_{χ,μ} := ⋃_{λ<χ} L_{λ^{+},μ}.

• L_{∞,μ} is the proper class obtained by letting λ vary over all cardinals. I.e. it is

L_{∞,μ} := ⋃_{λ∈Card}L_{λ^{+},μ}.

Basic references: For L_{ω_{1},ω} see H. J. Keisler [Ke2] and for L_{λ^{+},μ} see M. A. Dickmann’s [Di] books.

2. Cardinality quantifiers: Andrzej Mostowski in 1957 (see [Mos]) introduced several cardinality quantifiers. The most popular among them was studied extensively in the sixties by Gerhard Fuhrken [Fur ] and Jerry Keisler [Ke1]. It is the ℵ_{1}-interpretation: L(Q) and extensions like L_{ω_{1},ω}(Q) where

M ⊧ Qxφ(x) ⟺ {a ∈ |M| : M ⊧ φ[a]} is uncountable.

3. Other second order quantifiers: Measure-theoretic quantifiers, and topological logics. Barwise, Makkai, Kaufmann, Flum, Ebbinghaus and Ziegler.

4. Other logics: Monadic logics. Rabin, Gurevich, Büchi, Shelah.

There are many more logics, see the volume edited by Jon Barwise and Solomon Feferman [BaFe].

There is a particularly rich model theory for L(Q). In the last 20 years, this theory took on a set-theoretic flavor, see: Fuhrken [Fur ], Keisler [Ke1], [Sh 82],

[Gr2] and [HLSh]. For an overview of some of this I recommend Wilfrid Hodges’s book [Ho1].

A common feature of all the above extensions of first order logic is the failure of the compactness theorem.

An ordered field ⟨F, +, ·, ≤⟩ is archimedean iff

⟨F, +, ·, ≤⟩ ⊧ ∀x[x > 0 → ⋁_{n<ω}[\underbrace{x + … + x}_{n-times} ≥ 1]].

A group G is periodic iff

G ⊧ ∀x ⋁_{n<ω}[x^{n} = 1].

Definition 1.3. Let M and N both be L-structures. Suppose that A ⊆ |M|, and B ⊆ |N|. A function f : A → B is called a partial isomorphism iff it is a bijection and for every relation symbol R(x) and every a ∈ A we have that a ∈ R^{M} ⟺ f(a) ∈ R^{N}, for every function symbol F (x) we have that f(F^{M}(a)) = F^{N}(f(a)) and for every constant symbol c if c^{M} ∈ Dom(f) then f(c^{M}) = c^{N}.

A family F of partial isomorphisms from M into N has the back and forth property iff (forward) for every a ∈ |M| and every g ∈ F there exists h ∈ F such that g ⊆ h and a ∈ Dom(h) and (back) for every b ∈ |N| for every g ∈ F there exists h ∈ F such that g ⊆ h and b ∈ Rang(h).

Denote this by F : M ≅^{p} N . We write M ≅^{p} N for “there exists a non empty F such that F : M ≅^{p} N”.

At first glance the following special case of a theorem of Carol Karp may look a bit surprising:

Fact 1.4 (Karp’s test). Let M and N be L-structures.

M ≅^{p} N ⟺ M ≡_{∞,ω} N.

Shelah’s plenary talk at the International Congress of Mathematicians in Berkeley in 1986 (see [Sh 299] and [Sh tape ]) was dedicated to classification theory for non elementary classes and in particular universal classes.

There are several frameworks for classification theory for non elementary classes. With the exception of the last item they are listed in (more or less) increasing level of generality:

• Homogeneous model theory (this was formerly called Finite diagrams stable in power). In this context we have a fixed (sequentially) homogeneous monster model M and limit the study to elementary submodels of M and its subsets. The subject was started by Shelah in 1970 with [Sh3] and continued by him with [Sh54]. In the last 10-15 years the subject was revived in several publications. See [Gr3], [Gr4],[HySh1], [HySh2], [GrLe1 ], [GrLe3 ], [Le1], [Le2], [BuLe], [Be] and [Kov1].

• Submodels of a given structure. This is a generalization of homogeneous model theory. Start with a given model M (not necessarily homogeneous) and limit the study to submodels of M and its subsets. Started in Grossberg [Gr3], [Gr4]. Further progress is in [GrLe2 ].

• Excellent classes. An excellent class consists of the atomic models of a firstorder countable theory which satisfy a very strong amalgamation property: The (ℵ_{0}, n)-goodness for every n < ω. Shelah introduced them in [Sh87a],

[Sh87b] as a tool to analyze Mod(ψ) when ψ ∈ L_{ω_{1},ω} under the model theoretic assumption that I(ℵ_{n}, ψ) < 2^{ℵ_{n}} holds for every positive integer n. Later in [Sh c] Shelah used excellent classes at a crucial point in his proof of the main gap for first-order theories, (ℵ_{0}, n)-goodness is renamed in [Sh c], page 616 as (ℵ_{0}, n)-existence property. In section 5 of chapter XII Shelah essentially show that for countable and superstable T without the dop, notop is equivalent to the (ℵ_{0}, 2)-existence property which using his resolution technique from [Sh87b] implies excellence. Lately Kolesnikov [Kov2] anounced an n-dimensional analysis to get new results for simple unstable theories. Grossberg and Hart in [GrHa ] developed orthogonality calculus to the level that permitted deriving the main gap for excellent classes. This was the first time that a main gap was proved for non elementary classes.

• Universal classes. The prototypical example is Mod(ψ) when ψ is an L_{ω_{1},ω} sentence of the form ⋀_{n<ω} ψ_{n} where the ψ_{n} are Π_{1}-first-order sentences. This work began in [Sh 300]. Shelah is currently writing a book [Sh h] that, among other things, will include a “main gap”-style of theorem for universal classes.

• Abstract elementary classes. See Definition 1.5 below. This is in my opinion the deepest direction. It is the focal point of this article. Already in the fifties model theorists studied non elementary classes of structures (e.g. Jónsson [Jo1], [Jo2] and Fra¨ issé [Fre]). In [Sh88], Shelah introduced the framework of abstract elementary classes and embarked on the ambitious program of developing a classification theory for abstract elementary classes. This work was continued in many publications of Shelah (totaling more than 700 pages) and members of his school.

• Primal framework. This is a generalization of abstract elementary classes obtained by relaxing the chain axioms (A4 from Definition 1.5 below). See Baldwin and Shelah’s papers [BS1], [BS2], [BS3] and [Gr5].

• Classification theory over a predicate. Unlike the other frameworks this is really an extension of first-order model theory, when the notion of isomorphism is replaced by a stronger one. While in my opinion this framework does not precisely fit into what I call classification theory for non elementary classes, many of the methods are common. The fact that many years ago Shelah announced a solution for the main gap in this context, while for AECs such a theorem is not even on the horizon, indicates to me that this framework is much easier. I suggest to the reader to start with Wilfrid Hodges’s survey article [Ho2]. Pillay and Shelah’s article [PiSh] is the beginning. Further work includes [Sh 234] and [Sh 322]. Shelah has written several hundreds of (unpublished) pages that continue this up to a Main gap. This work is not available to me. I suggest that interested people will contact Shelah directly.

The focus of this article is the framework of abstract elementary classes. This framework, in my opinion, has the best balance of generality, a rich and sophisticated theory. The context of AECs is much more general than that of homogeneous model theory, model theory for L_{ω_{1},ω} or even the framework of submodels of a given structure.

Definition 1.5. Let K be a class of structures all in the same similarity type L(K), and let ≺_{K} be a partial order on K. The ordered pair ⟨K, ≺_{K}⟩ is an abstract elementary class, AEC for short iff

A0 (Closure under isomorphism)

(a) For every M ∈ K and every L(K)-structure N if M ≅ N then N ∈ K.

(b) Let N_{1}, N_{2} ∈ K and M_{1}, M_{2} ∈ K such that there exist f_{l} : N_{l} ≅ M_{l} (for l = 1, 2) satisfying f_{1} ⊆ f_{2} then N_{1} ≺_{K} N_{2} implies that M_{1} ≺_{K}

M_{2}.

A1 For all M, N ∈ K if M ≺_{K} N then M ⊆ N.

A2 Let M, N, M^{∗} be L(K)-structures. If M ⊆ N, M ≺_{K} M^{∗} and N ≺_{K} M^{∗}

then M ≺_{K} N. A3 (Downward Löwenheim-Skolem) There exists a cardinal

LS(K) ≥ ℵ_{0} + |L(K)| such that for every

M ∈ K and for every A ⊆ |M| there exists N ∈ K such that N ≺_{K} M, |N| ⊇ A and ∥N∥ ≤ |A| + LS(K).

A4 (Tarski-Vaught Chain)

(a) For every regular cardinal μ and every N ∈ K if {M_{i} ≺_{K} N : i < μ} ⊆ K is ≺_{K}-increasing (i.e. i < j ⟹ M_{i} ≺_{K} M_{j}) then ⋃_{i<μ} M_{i} ∈ K and ⋃_{i<μ} M_{i} ≺_{K} N.

(b) For every regular μ, if {M_{i} : i < μ} ⊆ K is ≺_{K}-increasing then ⋃_{i<μ} M_{i} ∈ K and M_{0} ≺_{K} ⋃_{i<μ} M_{i}.

Remark 1.6. I invite the reader to verify the following important basic observation: Given M, N_{1}, N_{2} ∈ K such that N_{1} ≺_{K} N_{2} and f : N_{1} ≅ M then there are N ∈ K and g : N_{2} ≅ N such that M ≺_{K} N and g extends f.

Example 1.7 (elementary classes). Let T be a first-order theory, K = Mod(T) and ≺_{K} the usual notion of elementary submodel. Then ⟨K ≺_{K}⟩ is an AEC with LS(K) = |L(T)| + ℵ_{0}.

Example 1.8 (ℵ_{1}-saturated models of a f.o. theory). Let T be a complete countable superstable and not ℵ_{0}-stable, ≺_{K} be the elementary submodel relation and LS(K) = 2^{ℵ_{0}}. K := {M ⊧ T : M is ℵ_{1}-saturated}. By Theorem III.3.12 of [Sh c] K is closed under unions (recall that superstability implies κ(T) = ℵ_{0}).

Recall (Keisler [Ke2]): for ψ ∈ L_{ω_{1},ω} a subset L_{A} of L_{ω_{1},ω} is a fragment containing ψ iff ψ ∈ L_{A}, L_{A} is closed under: Taking subformulas, first-order connectives and quantifiers.

Definition 1.9. Let M and N be L-structures. Suppose L_{A} is an L-fragment. M ⊆_{T V,L_{A}} N iff

1. M ⊆ N and

2. for every a ∈ |M| and every φ(y; x) ∈ L_{A} if N ⊧ ∃yφ(y; a) then there

exists b ∈ |M| such that N ⊧ φ[b; a].

When L_{A} consists of only the first-order formulas from the language of the structure we omit it.

It is the contents of the Tarski-Vaught test that M is an elementary submodel of N iff M ⊆_{T V} N.

Example 1.10 (L_{ω_{1},ω}). Let ψ ∈ L_{ω_{1},ω} be a sentence in a countable language and suppose that L_{A} ⊆ L_{ω_{1},ω} is a countable fragment containing ψ. Take ≺_{K} to be defined by M ≺_{K} N ⟺ M ⊆_{T V,L_{A}} N.

Let K := Mod(T). Clearly ⟨K, ≺_{K}⟩ is an abstract elementary class with LS(K) = ℵ_{0}.

Example 1.11 (L_{ω_{1},ω}(Q)). T ⊆ L_{ω_{1},ω}(Q) be a countable theory in a countable language and suppose that L_{A} ⊆ L_{ω_{1},ω}(Q) is a countable fragment containing T. ⟨K, ≺_{K}⟩ is an abstract elementary class, when K = Mod(T) and M ≺_{K} N iff M ⊆_{T V,L_{A}} N

1. if a ∈ |M| and M ⊧ Qxφ(x; a) then there exists b ∈ |N| − |M| such that N ⊧ φ[b; a] and

2. if a ∈ |M| and M ⊧ ¬Qxφ(x; a) then φ(M; a) = φ(N; a).

In fact AECs are more general than L_{ω_{1},ω}(Q). The contents of the following Theorem is that in the Chain Axioms (A4) it is

possible to replace the regular cardinal μ by an arbitrary directed set.

Theorem 1.12 (Theorem 21.4 of [Gra ]). Let ⟨K, ≺_{K}⟩ be an AEC and ⟨M_{s} : s ∈ I⟩ be a directed system. Then ⋃_{s∈I} M_{s} ∈ K. Moreover

(a) If M_{s} ≺_{K} N for every s ∈ I, then ⋃_{s∈I} M_{s} ≺_{K} N.

(b) M_{s} ≺_{K} ⋃_{t∈I} M_{t}, for every s ∈ I.

Proof. By induction on |I|.

An early version of 1.12 can be found in Tsurane Iwamura’s paper from 1944 ([Iw]).

Notation 1.13. Denote by K_{<λ} the class {M ∈ K : ∥M∥ < λ} and by K_{λ} the class {M ∈ K : ∥M∥ = λ}. I(λ, K) is the cardinality of K_{λ} / ≅.

Some examples from “main stream mathematics”:

Example 1.14 (Normed fields). Let

K := {⟨F, +, ·, | |⟩ | F is an algebraically closed field }

⟨F_{1}, +, ·, | |_{1}⟩ ≺_{K} ⟨F_{2}, +, ·, | |_{2}⟩ ⟺ F_{1} ≤ F_{2}, (∀a ∈ F_{1})[|a|_{1} = |a|_{2}] and the value groups are the equal.

Example 1.15 (Local fields). Let ⟨F, +, ·, | |⟩ be a non archimedean normed field (i.e. |a + b| ≤ Max{|a|, |b|} for all a, b ∈ F). It is well known that ([Cass]) R := {a ∈ F : |a| ≤ 1_{ℝ}} is a subring of F and I := {a ∈ F : |a| < 1_{ℝ}} is a maximal ideal of R. The field R/I is called the residue field of F. A field is local iff its residue field is finite.

The class K := {⟨F, +, ·, | |⟩ : F is local} is an AEC when ⟨F_{1}, +, ·, | |_{1}⟩ ≺_{K} ⟨F_{2}, +, ·, | |_{2}⟩ is as in the previous example. Another relation K is defined by dropping the requirement of equal value groups.

Example 1.16 (Noetherian rings). Define R ≺_{K} S iff R is a subring of S and R ≡_{∞,ω} S. Notice that by Karp’s test when R is Noetherian then also S is.

Example 1.17 (Rings of finite dimension). Let K be a class of rings. Define R ≺_{K} S iff R is a subring of S and if I ⊊ J are ideals of R and I′ ⊊ J′ are ideals of S such that

1. I′ ∩ R = I,

2. J′ ∩ R = J and

3. there is no ideal of R strictly between I and J then there is no ideal of S strictly between I′ and J′.

Example 1.18 (coordinate rings). Let F be a finite field or the rationals. Suppose p ∈ F [x_{1}, …, x_{n}] is such that \{a ∈ \overline{F}^{n} | p(a) = 0\} is an irreducible variety. For an algebraically closed field K ≥ F let K_{p} be the coordinate ring of {a ∈ K^{n} | p(a) = 0}. The following is an AEC:

K_{p} := {K_{p} | K is an algebraically closed field extending F }.

The relation ≺_{K} is the subring relation.

2. what is the purpose of this?

Develop a classification theory for non elementary classes for K = Mod(T) when T is a complete L_{ω_{1},ω}-theory (or maybe some other non first-order theory).

We try to imitate the rich and productive theory for elementary classes. Eventually this will have applications in “mainstream” mathematics. What is classification theory?

We want to be able to answer every question about K_{λ} / ≅.

Basic test questions:

1. Given λ ≥ LS(K) is K_{λ} ≠ ∅?

2. Does K_{λ} ≠ ∅ imply K_{λ^{+}} ≠ ∅?

3. Does I(λ^{+}, K) = 1 imply that I(λ, K) = 1?

4. What are the possible functions λ ↦ I(λ, K)?

5. Under what condition on K is it possible to find a nice dependence relation on subsets of every M ∈ K?

In pure model theory the �oś Conjecture (see [Lo]) was a major driving force:

Conjecture 2.1 (�oś conjecture 1954 and Morley 1965). Let T be a firstorder theory. If there exists λ > |T | + ℵ_{0} such that I(λ, T) = 1 then I(μ, T) = 1 holds for every μ > |T | + ℵ_{0}.

In 1965 Morley [Mo1] confirmed the conjecture for theories in a countable language. For his proof Morley discovered the notions of Morley rank, prime model over a set and implicitly strongly minimal formulas. Important progress was made earlier by Ehrenfeucht and Mostowski as well as by Vaught. At the end of his article [Mo1] Morley raised the question whether his categoricity theorem holds for theories in uncountable languages. Several people recognized it as an important problem. Fredrick Rowbottom [Ro] and J. P. Ressayre [Re] made important progress toward a complete solution. In 1970, in addition to building on earlier work, Shelah invented superstable theories, weakly minimal formulas and local rank to prove the conjecture for all first-order theories.

There is a need for good test-questions to measure progress in classification theory for non-elementary classes. Around 1977 Shelah proposed a conjecture that would serve as a benchmark for progress of the theory and may serve as a guide for future developments:

Conjecture 2.2 (Shelah’s conjecture). Let T be a countable theory in L_{ω_{1},ω}. If there exists λ ≥ ℶ_{ω_{1}} such that I(λ, T) = 1 then I(μ, T) = 1 holds for every μ ≥ ℶ_{ω_{1}}.

There is a similar conjecture for AEC generalizing the above conjecture. See Conjecture 3.6 in the next section.

Based on experience with the first-order version it is likely that any attempt to prove 2.2 will produce a rich and powerful machinery. Indeed the partial results obtained so far indicate that this is the case.

A word about examples: Essentially all known examples for categorical classes are derived from first-order ones.

Generic example: Let T be a countable first-order theory categorical in ℵ_{1}. Suppose M ⊧ T of cardinality ℵ_{1}, pick a countable A subset of |M|. Let L′ := L(T) ∪ {P } when P is a new unary predicate.

ψ_{T,A} := ⋀ T ∧ ∀x[P (x) → ⋁_{a∈A} x = a]

By the proof of Morley’s theorem, ℵ_{1}-categoricity of T implies that also T h(⟨M, a⟩_{a∈A}) is categorical in every λ ≥ ℵ_{1}. This is an example of an L_{ω_{1},ω}-theory categorical in all uncountable cardinals.

The only exception seems to be the example of coordinate rings (Example 1.18) which is categorical in every uncountable cardinal.

I think it is too early at this stage of the theory to formulate a Zilber-like conjecture of the nature of all categorical AECs or even the categorical L_{ω_{1},ω}-theories, but perhaps there is a classifcation of the classes categorical classes above the Hanf number.

Conjecture 2.2 is very open. There are more than 500 published pages dedicated for partial results. Among them are:

1. Keisler (1971), using a two cardinal theorem that improves Vaught’s theorem, has shown that under the additional assumption of the existence of sequentially homogeneous model the conjecture is true. Unfortunately, Shelah observed that using a construction of Leo Marcus [Ma], Keisler’s additional assumption does not follow from categoricity. Thus Keisler’s strategy, while being very elegant, is a dead end.

2. Shelah (1978) in [Sh87a] and [Sh87b] building on [Sh48] proved a form of the conjecture under the additional assumption of I(ℵ_{n+1}, T) < 2^{ℵ_{n+1}} for every n < ω. Such a class of structures is called an excellent class. Grossberg and Hart [GrHa ] proved the main gap for excellent classes.

3. Lessmann ([Le1]) proved the conjecture for countable finite diagrams, using a Baldwin-Lachlan style argument by introducing the necessary pregeometries via a new rank function.

4. Makkai and Shelah [Sh285] proved a downward version of the conjecture under the additional assumption that both λ and μ are above a strongly compact cardinal, and λ = χ^{+}. It is a major open problem of [Sh 702] to get rid of the assumption that λ is a successor. There are nicely behaved forking and orthogonality calculi for this.

5. Kolman-Shelah [KoSh] and Shelah [Sh472] contain partial going down results for λ above a measurable cardinal with the additional assumptions that K has the amalgamation property and λ is a successor cardinal.

6. [Sh 394] deals with classes that satisfy the amalgamation property. Several important concepts are introduced and a downward categoricity is concluded without using a large cardinal assumption as in [KoSh].

7. In [ShVi] Shelah and Andrés Villaveces embarked on an even more ambitious program: Deal with the categoricity conjecture for classes with slightly weaker model and set-theoretic assumptions. Namely, they assume GCH and work with classes that have no maximal models (this is a weakening of the amalgamation property). Recently more progress on this direction was made by Monica VanDieren [Va].

8. Inspired by a question of mine, whether it is possible to generalize the results of [Sh48] and [Sh88] for uncountable cardinals (or for P C_{κ} classes for uncountable κ). E.g. generalize [Sh48] from ℵ_{1} to arbitrary λ. the problem is that both in [Sh48] and [Sh88] the assumption that the categoricity is the successor of ℵ_{0} was used heavily in a form of applying a weak compactness phenomenon (in the form of undefinability of a well ordering in L_{ω_{1},ω}). I suspected that this attempt must produce new model-theoretic machinery. It turns out that I was right. In several massive papers Shelah answered that question and more. This appears in [Sh 576] (102 pages), [Sh 600] (82 pages), and [Sh 603] (20 pages). In particular he has shown (under weak GCH and no large cardinals) that if K is categorical in both λ and λ^{+} then I(λ^{++}, K) < 2^{λ^{++}} ⟹ K_{λ^{+++}} ≠ ∅.

In 1986 I proposed

Conjecture 2.3 (Intermediate �oś conjectures). Let ψ ∈ L_{ω_{1},ω}

1. If ψ is categorical in some λ ≥ ℶ_{ω_{1}}, then Mod(ψ) has the amalgamation property in every μ ≥ ℶ_{ω_{1}}.

2. If Mod(ψ) has the amalgamation property for all μ ≥ ℶ_{ω_{1}}, then ψ categorical in some λ ≥ ℶ_{ω_{1}} implies that ψ categorical in χ for every χ ≥ ℶ_{ω_{1}}.

In section 6 it will be shown that the amalgamation property permits a nice theory of types.

In [KoSh] Oren Kolman and Shelah derive the amalgamation property from the assumption that ψ is categorical above a measurable cardinal. Lately Shelah and Villaveces in [ShVi] have shown that if K has no maximal models (i.e. every M ∈ K has a proper ≺_{K}-extension in K) the weak GCH implies that every model can be extended to an amalgamation base.

[Sh 394] is dedicated to progress toward #2. It contains a proof of a downward version of the categoricity conjecture under the assumption that the class has the amalgamation property.

Why?

What could be the benefits of such a theory? Looking at the first-order example the notions of independence, several model-theoretic rank functions, forking, orthogonality calculus, regular types, pre-weight, prime models etc. all have found concrete applications in algebra, generalizing Krull’s dimension theory from commutative algebra.

1. Clarify the above notions:

“Studying only the model theory of first-order logic would be analogous to the study of real analysis never knowing of any but the polynomial functions; core concepts like continuity, differentiability, analyticity, and their relations would remain at best vaguely perceived. It is only the study of more general functions that one sees the importance of these notions, and their different roles, even for the simple case.” -Jon Barwise, page 15–16 of

[BaFe].

2. It is beautiful and difficult mathematics.

3. Effect on model theory for first-order theories. Already [Sh87a] and [Sh87b] had a profound effect on the proof of main gap for first-order theories. Especially good sets and stable systems. See the last 5 sections of Chapter XII in [Sh c] and what Shelah named the book’s main theorem-Theorem XII.6.1. I expect that a similar n-dimensional analysis will be used to better our understanding of simple unstable first-order theories. Also classification theory over a predicate benefited much from this.

4. Many interesting concepts of classical mathematics are not first-order (Archimedean, Noetherian and any chain conditions. etc).

5. Potential applications in classification theory of finite models. See Baldwin and Lessmann [BaLe] and Lessmann [Le3].

3. basic facts

The following is the notion of pseudo-elementary (or projective) class.

Definition 3.1. Let L_{1} ⊇ L and let T_{1} be a first-order theory in L_{1}, suppose that Γ is a set of T_{1}-types without parameters. We denote by

PC(T_{1}, Γ, L) = { M ↾ L : M ⊧ T_{1} and M omits all types fromΓ }.

Definition 3.2. A class K of structures is called a PC-class if there exists an expansion L_{1} of L(K), a first-order theory T_{1} in L_{1} and a set of T_{1}-types Γ such that K = PC(T_{1}, Γ, L(K)). When |T_{1}| + |Γ| + ℵ_{0} = μ we say that the class is PC_{μ}.

In the special case when L_{1} = L(K) we write EC(T, Γ) for P C(T, Γ, L(K)) and say that the class K is an EC-class.

Theorem 3.3 (C.C. Chang 1968). If T is a theory in L_{μ^{+},ω} of cardinality ≤ μ then Mod(T) is PC_{μ}.

Similar to Birkhoff’s presentation theorem for varieties/equational classes, there is a syntactic presentation theorem which generalizes Theorem 3.3:

Theorem 3.4 (Shelah’s presentation theorem). If (K, ≺_{K}) is an AEC, then there exists μ ≤ 2^{LS(K)} such that K is a PC_{μ} class. Also K^{≺_{K}} := {(N, M) | M ≺_{K} N} is a PC_{μ} class, for L(K^{≺_{K}}) consisting of a single unary predicate symbol.

A proof uses Theorem 1.12, details can be found in [Sh88] or in Chapter 13 of [Gr].

Using Morley’s result on the bound on computing Hanf numbers this immediately gives a corollary that is surprising and difficult to prove directly from the definitions, but is a trivial consequence of Shelah’s presentation theorem.

Corollary 3.5. Let (K, ≺_{K}) be an AEC. If K_{λ} ≠ ∅ for some λ ≥ ℶ_{(2^{2^{LS(K)}})^{+}} then K_{χ} ≠ ∅ holds for all χ ≥ LS(K).

A generalization of Conjecture 2.2 to AECs appears in [Sh 702]:

Conjecture 3.6 (Shelah’s conjecture for AEC). Let K be an AEC. If there exists λ ≥ ℶ_{(2^{LS(K)})^{+}} such that I(λ, K) = 1 then I(μ, K) = 1 holds for every μ ≥ ℶ_{(2^{LS(K)})^{+}}.

Now we introduce the appropriate generalization of elementary embedding.

Definition 3.7. For M, N ∈ K a monomorphism f : M → N is called a

K-embedding iff f[M] ≺_{K} N. Denote this by writing f : M ↪^{K} N. When the identity of K is clear it is omitted and we write f : M ↪ N.

Definition 3.8. Let (K, ≺_{K}) be an AEC, and let μ, κ ≥ λ ≥ LS(K) be cardinals.

1. We say that a model M ∈ K_{λ} is a (μ, κ)-amalgamation base iff for every

M_{1} ∈ K_{κ}, M_{2} ∈ K_{μ} and f_{ℓ} : M ↪^{K} M_{ℓ} for ℓ = 1, 2, there exists N ∈ K so

that there are g_{1} : M_{1} ↪^{K} N and g_{2} : M_{2} ↪^{K} N satisfying g_{1} ∘f_{1} = g_{2} ∘f_{2}. Namely, the following diagram is commutative:

\xymatrix{M_{1} \ar[r]^{g_{1}} & N \\ M \ar[u]^{f_{1}} \ar[r]_{f_{2}} & M_{2} \ar[u]_{g_{2}}}

N is called an amalgam of M, M_{1}, M_{2}, f_{1}, f_{2}.

When κ = μ = λ we say that M is an amalgamation base.

2. M ∈ K_{λ} is an (< μ, ≤ κ)-amalgamation base iff for every λ ≤ μ_{1} < μ,

λ ≤ κ_{1} ≤ κ, M_{1} ∈ K_{κ_{1}}, M_{2} ∈ K_{μ_{1}} and f_{ℓ} : M ↪^{K} M_{ℓ} for ℓ = 1, 2, there

exists an amalgam N of M, N_{1}, N_{2}, f_{1} and f_{2}.

3. K satisfies the (λ, μ, κ)-amalgamation property iff every M ∈ K_{λ} is an (μ, κ)-amalgamation base.

4. K satisfies the λ-amalgamation property iff every M ∈ K_{λ} is an amalgamation base.

It is a corollary of the Robinson consistency theorem that if K = Mod(T) for some complete first-order theory then K has the λ-amalgamation property for all λ ≥ ℵ_{0} + |L(T)|.

K_{λ} has the JEP iff for every M_{1}, M_{2} ∈ K_{λ} there are N ∈ K and K-embeddings

f_{ℓ} : M_{ℓ} ↪^{K} N.

Remark 3.9. Note that for first-order theories an even stronger amalgamation property holds. Every model is an amalgamation base for arbitrarily large models. I.e. remove the requirements that the models N_{ℓ} have the same cardinality as M. The stronger amalgamation property is used in [Va], but we will work here with that from the previous definition (when both extensions are of the same cardinality as the base.)

Already in 1956 B. Jónsson studied abstract elementary classes with the amalgamation and JEP properties. This influenced Morley and Vaught to introduce saturated models ([MoV]). Shelah was the first in [Sh88] to consider AEC in the context of classification theory. Implicitly indicating that the methods of stability/classification theory are not limited to elementary classes only and potentially have a broader applicability than first order.

In the proof of Morley’s categoricity theorem (as well as in Shelah’s generalization to uncountable theories) saturated models play a central role. At first one shows that saturated models exists (at least at the categoricity cardinal λ) and using the uniqueness of them it suffices to show that having an uncountable non-saturated model implies the existence of a non-saturated model in λ, contradicting categoricity. Non elementary classes in their very nature are connected to omitting types, so working with saturated models is not reasonable.

It turns out that model homogeneity is a good replacement for saturation. In that it generalizes saturation (for elementary classes) and we have the analog existence and uniqueness theorems.

Definition 3.10. Let K be an AEC.

1. Let λ > LS(K). We say that M is λ-model homogeneous iff for all N ≺_{K}

N′ ∈ K_{<λ} such that N ≺_{K} M there exists f : N′ ↪^{K} M such that f ↾ |N| = id_{|N|}.

2. M is said to be model homogeneous iff M is ∥M∥-model homogeneous.

It is an exercise (using the compactness theorem) to show:

Proposition 3.11. Let K be an elementary class. For λ > LS(K) and M ∈ K_{λ}. M is saturated iff M is model homogeneous.

By imitating the argument of the proof of existence of saturated models one can show:

Theorem 3.12 (existence). Let K is an AEC and λ ≥ LS(K). Suppose that 2^{λ} = λ^{+}. Further assume that K_{λ^{+}} is not empty. If K has the λ-amalgamation property, then there exists a model homogeneous M ∈ K_{λ^{+}}.

Theorem 3.13 (better existence). Suppose that K be an AEC such that μ > LS(K) satisfies μ = μ^{<μ} and K_{μ} is not empty. If K has the (< μ, μ)-amalgamation property, then for all N ∈ K_{μ} there exists M ≻_{K} N, M ∈ K_{μ} which is model homogeneous.

Theorem 3.14 (uniqueness). Let K be an AEC. Suppose that K_{λ} is categorical for some λ ≥ LS(K). If M and N are model homogeneous members of K_{λ^{+}}, then M ≅ N.

There are uniqueness theorems that follow from weaker assumptions.

While model homogeneity is very nice, model homogeneous models are natural to consider only when K has the amalgamation property. I.e. Suppose that K is categorical in λ and K has no maximal models. The model of cardinality λ is model homogeneous iff K has the (< μ, μ)-amalgamation property for every μ < λ.

A substitute called (μ, σ)-limit model was introduced in [KoSh] and where used in a substantial way to obtain the amalgamation property (from categoricity above a measurable cardinal), limit models reappeared also in [Sh 394] under the assumption that K has the amalgamation property. Further study of limit models (without requiring the amalgamation property) is in Shelah and Villaveces in [ShVi]. The uniqueness: Any two (μ, σ)-limit models are isomorphic for different σ’s was proved only lately by Monica VanDieren who has introduced and offered a characterization of the correct notion of model homogeneity for classes not requiring amalgamation in [Va]. Since most of this article deals with classes that have the amalgamation property I will not discuss (μ, σ)-limit models here.

4. the beginning of classification theory for AEC

In his JSL list of open problems from 1975 Harvey Friedman reproduced a question that started classification theory for non elementary classes.

Question 4.1 (Baldwin’s problem 1975). Does there exists a countable similarity type and a countable T ⊆ L(Q) (in the ℵ_{1} interpretation) such that T has a unique uncountable model (up to isomorphism)?

Since the Downward Löwenheim Skolem theorem holds for L(Q), Baldwin’s question is equivalent to “Does there exists a countable similarity type and a countable T ⊆ L(Q) such that T is categorical in ℵ_{1} but does not have a model of cardinality ℵ_{2}?”

The question is important since it suggested for the first time a connection between categoricity in a cardinal and existence of models in its successor.

A natural extension (generalizing L(Q) by an AEC and more importantly replacing ℵ_{0} by an arbitrary λ):

Question 4.2. Let K be an AEC and λ ≥ LS(K). Does categoricity in λ^{+} of K imply existence of a model of cardinality λ^{++}?

The following is a relatively simple example of a family of deep results that was motivated by Baldwin’s question.

Theorem 4.3. Suppose 2^{λ} < 2^{λ^{+}}. For an AEC K which fails to have the λ-amalgamation property. If I(λ, K) = 1 and λ ≥ LS(K) then I(λ^{+}, K) = 2^{λ^{+}}.

Section 8 of this paper is dedicated to the a proof of Theorem 4.3.

The set-theoretic assumption 2^{λ} < 2^{λ^{+}} is known as the weak continuum hypothesis since it follows from 2^{λ} = λ^{+}. However instead of using cardinal arithmetic we use a “diamond-like” combinatorial principle known as the Devlin-Shelah’s weak diamond:

Fact 4.4 (Devlin-Shelah). If 2^{λ} < 2^{λ^{+}} then there are λ^{+} pairwise disjoint stationary subsets of λ^{+} such that for any of these sets S the principle Φ_{λ^{+}}^{2}(S) holds, where Φ_{λ^{+}}^{2}(S) is: For all F : ^{λ^{+}>}2 → 2 there exists g : λ^{+} → 2 so that for every f : λ^{+} → 2 the set {δ ∈ S | F (f ↾ δ) = g(δ)} is stationary.

For more details see [DS] or section 13.5 of [Gr] or Chapter 13 of [Sh f]. The use of the weak diamond is essential, the statement of Theorem 4.3 is false

under MA_{ℵ_{1}} + 2^{ℵ_{0}} > ℵ_{1}:

Fact 4.5. There exists an AEC, K with LS(K) = ℵ_{0} such that

1. K is categorical in ℵ_{0},

2. K_{ℵ_{1}} ≠ ∅ and

3. The amalgamation property fails in K_{ℵ_{0}}. We have that

MA_{ℵ_{1}} + 2^{ℵ_{0}} > ℵ_{1} ⟹ I(ℵ_{1}, K) = 1.

This class is obtained by essentially considering the countable substructures of the random bipartite graph whose left side is ω and right side is ω_{1}. The categoricity proof is similar to Baumgartner’s ([Bau]) proof of the uniqueness of ℵ_{1}-dense orders.

I will conclude this section with a typical application of the weak diamond to AEC, this is not a particular case of Theorem 4.3 but rather a different theorem. The following theorem is a simple prototype of several more sophisticated results (e.g. [ShVi] and [Va]). A structure M ∈ K_{μ} is called universal model iff for every N ∈ K_{μ} there exists a K-embedding from N into M.

Theorem 4.6. Suppose 2^{λ} < 2^{λ^{+}}. For an AEC K which fails to have the λ-amalgamation property. If I(λ, K) = 1 and λ ≥ LS(K) then K_{λ^{+}} does not have a universal model.

In this section, as well as in sections 5 and 8, I will use some elementary facts about stationary sets of ordinals (all can be found in Kunen’s book [Ku] or in section 1.8 of [Gr]): Let λ be an uncountable regular cardinal. A set C of ordinals all less than λ is called a closed unbounded set (club) iff for every α < λ there exists β ∈ C such that β > α and for every bounded A ⊆ C we have that ⋃ A ∈ C. A set S ⊆ λ is stationary iff S ∩ C ≠ ∅ for every club C ⊆ λ.

Fact 4.7. Let λ be an uncountable regular cardinal. Let M be a structure of cardinality λ in a countable language.

1. If {M_{i} ≺ M | i < λ} and {N_{i} ≺ M | i < λ} are elementary chains which are increasing and continuous such that M = ⋃_{i<λ} M_{i} = ⋃_{i<λ} N_{i} and ∥M_{i}∥ + ∥N_{i}∥ < λ, for all i < λ, then the set {δ < λ | N_{δ} = M_{δ}} contains a closed unbounded subset of λ.

2. Suppose that L(M) contains a unary predicate P such that P^{M} is the set of ordinals less than λ. Then for every continuous increasing elementary chain {M_{i} ≺ M | i < λ} such that M = ⋃_{i<λ} M_{i} and ∥M_{i}∥ < λ for all i < λ the set {δ < λ | P^{M_{δ}} = δ} contains a club of λ.

Instead of using the principle Φ_{λ^{+}}^{2} directly we use another combinatorial principle:

Definition 4.8. Θ_{λ^{+}} is said to hold if and only if for all {f_{η} : η ∈ ^{λ^{+}}2 where f_{η} : λ^{+} → λ^{+}} and for every club C ⊆ λ^{+}, there exists η ≠ ν ∈ ^{λ^{+}}2 and there exists a δ ∈ C such that η ↾ δ = ν ↾ δ, f_{η} ↾ δ = f_{ν} ↾ δ and η[δ] ≠ ν[δ].

In [DS] Devlin and Shelah have shown that Θ_{λ^{+}} follows from Φ_{λ^{+}}^{2} (see also Chapter 13 of [Gr]).

Now to the proof of Theorem 4.6:

Proof. By assumption, we may take N_{0}, N_{1}, N_{2} ∈ K_{λ}, that can not be amalgamated.

For ρ ∈ ^{λ^{+}>}2 define a family of M_{ρ} ∈ K_{λ} so that the following hold:

1. |M_{ρ}| = λ(1 + ℓ(ρ)),

2. ν ⋖ η → M_{ν} ≺_{K} M_{η},

3. when ℓ(ρ) is a limit ordinal, M_{ρ} = ⋃_{α<ℓ(ρ)} M_{ρ↾α},

4. M_{ρˆ 0} and M_{ρˆ 1} cannot be amalgamated over M_{ρ}. Using λ-categoricity and the triple N_{0}, N_{1}, N_{2} the construction is possible. For η ∈ ^{λ^{+}>}2 let M_{η} := ⋃_{α<λ^{+}} M_{η↾α}. Now suppose that M ∈ K_{λ^{+}} is universal. Without loss of generality we may

assume that |M| = λ^{+}. By universality for every η ∈ ^{λ^{+}}2 there is a K-embedding f_{η} : M_{η} → M.

Now consider C := {δ < λ^{+} | δ = λ(1 + δ)}, it contains a club. Using Θ_{λ^{+}} there are η ≠ ν ∈ ^{λ^{+}}2 and δ ∈ C as in Θ_{λ^{+}}. Denote by ρ the largest common initial segment of η and ν (it is η ↾ δ). Since η[δ] ≠ ν[δ] we assume that η[δ] = 0 and ν[δ] = 1. Pick M^{∗} ≺_{K} M of cardinality λ containing the set f_{η}[M_{ρˆ 0}] ∪ f_{ν}[M_{ρˆ 1}].

Note that the diagram

\xymatrix{M_{ρˆ 0} \ar[r]_{f_{η}} & M^{∗} \\ M_{ρ} \ar[u]^{id} \ar[r]_{id} & M_{ρˆ 1} \ar[u]_{f_{ν}}}

is commutative in contradiction to requirement 4 in the construction.

The argument used in the proof of Theorem 4.6 can be used to prove the following useful:

Corollary 4.9. Suppose 2^{λ} < 2^{λ^{+}} and K is an AEC with LS(K) ≤ λ. If I(λ, K) = 1 and there exists μ > λ such that I(μ, K) = 1 then K has the λ-amalgamation property.

5. Solution to Baldwin’s question and K_{λ}^{3}

Suppose T ⊆ L(Q). Given M ⊧ T of cardinality ℵ_{1} pick a ψ ∈ L_{ω_{1},ω} a Scott sentence of a countable substructure of M. Now let L_{A} ⊆ L_{ω_{1},ω}(Q) be a countable fragment containing T and ψ. Let K := Mod(⋀ T ∧ ψ) and let N_{1} ≺_{K} N_{2} iff N_{1} ⊆_{T V,L_{A}} N_{2}. It is enough to prove:

Theorem 5.1. Suppose ⟨K, ≺_{K}⟩ is an AEC which is PC_{ℵ_{0}}. If K is categorical both in ℵ_{0} and ℵ_{1} then K_{ℵ_{2}} ≠ ∅.

Proof. Since K is categorical in ℵ_{1} and closed under union it is enough to show that

(∗) There are N_{0} ≠ N_{1} ∈ K_{ℵ_{1}} such that
N_{0} ≺_{K} N_{1}.

The following concept is central to the theory:

Definition 5.2.

K_{λ}^{3} := {(M, N, a) : M, N ∈ K_{λ}, M ≺_{K} N,
a ∈ |N| − |M|}.

On K_{λ}^{3} define a partial ordering by

(M, N, a) < (M′,N′, a′) ⟺^{def} M ≺_{K} M′ ∧
M ≠ M′ ∧ N ≺_{K} N′ ∧ a = a′.

When K is categorical in λ then the assumption K_{λ^{+}} ≠ ∅ implies that K_{λ}^{3} ≠ ∅.

Remark 5.3. In the next section we will show that the element (M, N, a) of K_{λ}^{3} plays a similar role to that of tp(a/M, N) in first-order logic.

Using (∗) and the assumption that I(ℵ_{1}, K) = 1 once more, Theorem 5.1 follows from:

Theorem 5.4. Suppose ⟨K, ≺_{K}⟩ is an AEC which is PC_{ℵ_{0}}. If K is categorical in ℵ_{0} and the poset ⟨K_{ℵ_{0}}^{3}, <⟩ is not empty and has a maximal element then I(ℵ_{1}, K) = 2^{ℵ_{1}}.

Why enough? If every element of K_{ℵ_{0}} has an extension define {(M_{α}, N_{α}, a) | α < ω_{1}} ⊆ K_{ℵ_{0}}, strictly increasing and continuous. Then the following is a witness for (∗):

⋃_{α<ω_{1}} M_{α} ≺ ⋃_{α<ω_{1}} N_{α}, since a ∉ ⋃_{α<ω_{1}} M_{α}.

It is time to recall a fact from the mid sixties that was discovered independently by Lopez-Escobar (using proof-theoretic methods) and Morley (using model-theoretic techniques):

Fact 5.5 (undefinability of well-ordering). Let K^{∗} be a P C_{ℵ_{0}}-class such that L(K^{∗}) contains a binary relation < and a unary predicate P. If there exists 𝔄 ∈ K^{∗} with ⟨P^{𝔄}, <^{𝔄}⟩ = ⟨ω_{1}, ∈⟩, then there exists 𝔅 ∈ K^{∗} such that there is {α_{n} ∈ P^{𝔅} | n < ω} satisfying

𝔅 ⊧ α_{n+1} < α_{n} for all n < ω.

The main step is the following:

Lemma 5.6 (technical lemma). Suppose ⟨K, ≺_{K}⟩ is an AEC which is PC_{ℵ_{0}}, and K is categorical both in ℵ_{0} and ℵ_{1}. Then for every M ∈ K_{ℵ_{0}} there are {M_{n} ∈ K_{ℵ_{0}} n < ω} such that for every n < ω we have that M_{n+1} ≺_{K} M_{n} and M = ⋂_{n<ω} M_{n}.

Proof. By categoricity in ℵ_{0} it is sufficient to show that there exists a model M as in the statement.

Let T_{1}, Γ_{1}, T_{2}, Γ_{2} be at most countable such that K = PC(T_{1}, Γ_{1}, L(K)) and {⟨|N|, |M|⟩ : M ≺_{K} N} = PC(T_{2}, Γ_{2}, {P (x)}). Since ⟨K, ≺_{K}⟩ is an abstract elementary class and K_{ω_{1}} ≠ ∅ we can fix an ≺_{K}- increasing continuous chain of models {M_{α} ∈ K_{ℵ_{0}} : α < ω_{1}}. Denote by f : ω_{1} → K_{ℵ_{0}} the mapping α ↦ M_{α}. Let M := ⋃_{α<ω_{1}} M_{α}. For every a ∈ |M| let h(a) := Min{ξ < ω_{1} : a ∈ |M_{ξ}|}.

By the reflection principle there exists a regular cardinal χ sufficiently large so that

H(χ) ⊇ {λ^{+}, f, h, M, T_{1}, Γ_{1}, T_{2}, Γ_{2}, L(K), ...} and ⟨H(χ), ∈⟩ reflects all relevant information. Namely

\begin{matrix}⟨H(χ), ∈⟩ ⊧ & ∀a ∈ |M|[h(a) ∈ ω_{1}] ∧ ∀α < γ ∈ ω_{1} \\ [f(α) ∈ PC(T_{1}, Γ_{1}, L(K)) ∧ \\ ⟨f(γ), f(α)⟩ ∈ PC(T_{2}, Γ_{2}, \{P (x)\})] \end{matrix} (1)

⟨H(χ), ∈⟩ ⊧ ∀α < γ ∈ ω_{1}[α < γ → f(α) ≺_{K} f(γ)] (2)

\begin{matrix}⟨H(χ), ∈⟩ ⊧ & ∀α ∈ ω_{1}[∀γ[γ < α → γ + 1 < α]] → \\ ∀M ∈ Rang(f)[M ≺_{K} f(α) ∧ M ≠ f(α)] → \\ ∃i < α[M ≺_{K} f(i)]] \end{matrix} (3)

⟨H(χ), ∈⟩ ⊧ “⟨K, ≺_{K}⟩ is an AEC” (4)

(5) ⟨H(χ), ∈⟩ ⊧ “⟨K, ≺_{K}⟩ satisfies Theorem 1.12”

\begin{matrix}⟨H(χ), ∈⟩ & ⊧ M = ⋃_{α<ω_{1}} f(α) \\ ⟨H(χ), ∈⟩ ⊧ ∀a ∈ |M|[a ∈ f(h(a))] \\ ∧ (∀α ∈ ω_{1})[α < h(a) → a ∉ f(α)]. \end{matrix} (6)

Let

𝔄 := ⟨H(χ), ∈, |M|, ω, f, h, Q, T̄_{1}, T̄_{2}Γ̄_{1}, Γ̄_{2}, φ, p, ζ⟩_{ζ<λ,φ∈T_{1}∪T_{2},p∈Γ_{1}∪Γ_{2}}.

Where Γ̄_{1}, Γ̄_{2} are unary predicates interpreted by the corresponding sets of types, similarly T̄_{l} are unary predicates interpreting T_{l}, Q is a unary predicate interpreted by the set of ordinals ω_{1}, ω is a unary predicate interpreted by the set of natural numbers and f and h are unary function symbols interpreted by the corresponding functions. ζ, φ and p are individual constants interpreted by the corresponding elements. Now let p_{1}(y) := {Γ̄_{1}(y) ∧ y ≠ q : q ∈ Γ_{1}}, p_{2}(y) := {Γ̄_{2}(y) ∧ y ≠ q : q ∈ Γ_{2}} and p_{3}(φ) := {T̄_{1}(φ) ∧ φ ≠ ψ : ψ ∈ T_{1}}, p_{4}(φ) := {T̄_{2}(φ) ∧ φ ≠ ψ : ψ ∈ T_{2}}, finally let p_{5}(j) := {λ(j) ∧ j ≠ ζ : ζ < λ}.

Denote by T_{3} the theory of 𝔄. Clearly 𝔄 ∈ EC(T_{3}, {p_{1}, p_{2}, p_{3}, p_{4}, p_{5}}) and for all 𝔅 ∈ EC(T_{3}, {p_{1}, p_{2}, p_{3}, p_{4}, p_{5}}) since T_{l}^{𝔅} = T_{l}, Γ_{l}^{𝔅} = Γ_{l} and we have (using (∗)_{0} and (∗)_{1}) that

For all α < γ ∈ Q^{𝔅} [f(α) ≺_{K} f(γ)] ∧ [f(α) ∈ PC(T_{1}, Γ_{1}, L(K))].

Since 𝔅 omits p_{5} we get that λ^{𝔅} = λ, namely we have that K_{λ}^{𝔅} / ≅ = K_{λ} / ≅.

Since ⟨Q^{𝔄}, ∈⟩ has order type λ^{+} and from the assumption on λ an application of Fact 5.5 to EC(T_{3}, {p_{1}, p_{2}, p_{3}, p_{4}, p_{5}}) produces a model 𝔅 ∈ EC(T_{3}, {p_{1}, p_{2}, p_{3}, p_{4}, p_{5}}) such that there exists {α_{n} : n < ω} ⊆ Q^{𝔅} such that for every n < ω we have that 𝔅 ⊧ α_{n+1} < α_{n}. For n < ω let M_{n} := f(α_{n}). We conclude with showing that

Claim 5.7. There exists N ∈ K_{λ}^{𝔅} such that N = ⋂_{n<ω} M_{n}.

Proof. Since ω ⊆ Q^{𝔅} and for all k < ω and for every n < ω we have

𝔅 ⊧ [k < α_{n}],

the set I := {β ∈ Q^{𝔅} : ∀n < ω[β < α_{n}]} is nonempty and directed. Since ⟨K, ≺_{K}⟩^{𝔅} is an abstract elementary class (by (∗)_{3}) we have that there exists N := ⋃_{s∈I} f(s). By the definition of N we have that for all n < ω N ≺_{K} M_{n}. So clearly N ⊆ ⋂_{n<ω} M_{n}. Using the function h we show that the last two sets are equal. Suppose that there exists a ∈ ⋂_{n<ω} M_{n} − |N|. By (∗)_{5} there exists a first γ ∈ Q such that a ∈ f(γ). Since a ∈ M_{n} for every n by minimality we get that γ ≤ α_{n} for all n < ω. Since f is order-preserving we get that f(γ) ≺_{K} f(α_{n}), namely f(γ) ⊆ M_{n} for all n which is a contradiction to the choice of the element a.

Suppose (M, N, a) ∈ K_{ℵ_{0}}^{3} is maximal. Given any S ⊆ ω_{1}, define {M_{α}^{S} : α < ω_{1}} ⊆ K_{ℵ_{0}} as follows:

1. |M_{α}^{S}| = ω(1 + α),

2. For α limit, M_{α}^{S} = ⋃_{β<α} M_{β}^{S},

3. for α = β + 1, there are two cases:

(a) if β ∈ S, using ℵ_{0}-categoricity take M_{β+1}^{S} ⪶_{K} M_{β}^{S} and a_{β}^{S} ∈ |M_{β+1}^{S}| − |M_{β}^{S}| so that (M_{β}^{S}, M_{β+1}^{S}, a_{β}^{S}) ≅ (M, N, a)

(b) if β ∉ S, then take a descending decomposition. I. e. apply Lemma 5.6 to the model M_{β}^{S} to get a descending {M_{n} | n < ω} ⊆ K_{ℵ_{0}} such that M_{β}^{S} = ⋂_{n<ω} M_{n}. Now define M_{β+1}^{S} to be M_{0}.

Notice that α < β ⟹ M_{α}^{S} ⪵_{K} M_{β}^{S}.
For each set S let M_{S} := ⋃_{α<ω_{1}} M_{α}^{S}.
Claim 5.8. If S_{1} ≢ S_{2} mod D_{ω_{1}} then M^{S_{1}} ≅ M̸^{S_{2}}.

Proof. Let S_{1} ≢ S_{2} mod D_{ω_{1}} and f : M^{S_{1}} ≅ M^{S_{2}} be given. By requirement (1) of the construction |M^{S_{1}}| = |M^{S_{2}}| = ω_{1}. Observe that

C_{1} := {δ < ω_{1} : (∀α < δ)[f(α) < δ]} is a club.

Using continuity of the chains we get that

C_{2} := {δ < ω_{1} : f ↾ δ : M_{δ}^{S_{1}} ≅ M_{δ}^{S_{2}}} is also a club.

Take C := C_{1} ∩ C_{2}. Since C is a club we may assume without loss of generality that there exists δ ∈ C ∩ (S_{1} − S_{2}). Since δ ∈ S_{1} by the construction we have that (M_{δ}^{S_{1}}, M_{δ+1}^{S_{1}}, a_{δ}^{S_{1}}) is a maximal element of K_{ℵ_{0}}^{3}. Since

f(a_{δ}^{S_{1}}) ∉ M_{δ}^{S_{2}},

by the assumption δ ∉ S_{2} there exists n < ω such that

f(a_{δ}^{S_{1}}) ∉ M_{n}.

Let N := f^{−1}[M_{n}]. Since M_{δ}^{S_{2}} is a proper substructure of M_{n} we have that M_{δ}^{S_{1}} ⊊ N. Since |N| is a countable subset of |M^{S_{1}}| there exists γ < ω_{1} such that N ≺ M_{γ}^{S_{1}}. Since C ∩ (S_{1} − S_{2}) is unbounded there exists ξ ∈ C ∩ (S_{1} − S_{2}) greater than max{δ, γ}. Thus we have (N, M_{ξ}^{S_{1}}, a_{δ}^{S_{1}}) is a proper K_{ℵ_{0}}^{3}-extension of (M_{δ}^{S_{1}}, M_{δ+1}^{S_{1}}, a_{δ}^{S_{1}}) contradicting its maximality.

By Ulam’s theorem the claim gives I(ℵ_{1}, K) = 2^{ℵ_{1}}.

6. Galois types

Recall the following basic result of elementary model theory:

Fact 6.1. Let ℭ be an uncountable saturated model of cardinality greater than |L(ℭ)|. For a, b ∈ | ℭ | and A ⊆ | ℭ | such that |A| < ∥ ℭ ∥.

tp(a/A) = tp(b/A) ⟺ ∃f ∈ Aut_{A}(ℭ) such that f(a) = b.

Namely the orbit of the element a under the group action of Aut_{A}(ℭ) on ℭ can be identified with tp(a/A). The set of L(K)-formulas satisfied by a does not have the corresponding property for abstract elementary classes. Thus we need a replacement. A replacement introduced by Shelah in [Sh 300] and since takes a prominent role in model theory of AECs is to work directly with orbits instead of set of formulas! This is the notion of Galois type to be defined below. Unfortunately not having formulas (in any logic) creates many technical difficulties.

Definition 6.2. Let M ∈ K_{λ}, We say that (M, N_{1}, a_{1}) and (M, N_{2}, a_{2}) ∈ K_{λ}^{3}, are ∼-related, written

(M, N_{1}, a_{1}) ∼ (M, N_{2}, a_{2}),

if there exists N ∈ K_{λ} and K-embeddings

h_{1} : N_{1} → N and h_{2} : N_{2} → N,

such that

h_{1}(a_{1}) = h_{2}(a_{2})

and the following diagram commutes:

\xymatrix{N_{1} \ar[r]^{h_{1}} & N \\ M \ar[u]^{id} \ar[r]_{id} & N_{2} \ar[u]_{h_{2}}}

Lemma 6.3. Suppose that K_{λ} has the amalgamation property. For M ∈ K_{λ} the relation ∼ is an equivalence relation on K_{λ}^{3}.

Proof. Exercise, notice that we use that M and N_{ℓ} are amalgamation bases.

Definition 6.4 (Galois types).

1. For (M, N, a) ∈ K_{λ}^{3}, we let

ga-tp(a/M, N) = (M, N, a)/ ∼ .

This is the type of a over M in N.

2. For M ∈ K_{λ}, we let

ga-S(M) = { ga-tp(a′/M, N′) : (M, N′, a′) ∈ K_{λ}^{3}}.

3. Given p ∈ ga-S(M) and N ∈ K_{≥λ}, we say that p is realized by a ∈ N, if there exists N′ ≺_{K} N of cardinality λ containing |M| ∪ a such that

(M, N′, a) ∈ K_{λ}^{3} and p = ga-tp(a/M, N′).

Assuming that K_{ℵ_{0}} has the amalgamation property, the content of Theorem 5.4 is a weak replacement of the compactness theorem:

Theorem 6.5 (extension property of types). Let K be an AEC which is PC_{ℵ_{0}}. Then the assumption I(ℵ_{1}, K) = 1 implies that for every M ≺_{K} N ∈ K_{ℵ_{0}} and every p ∈ ga-S(M) there exists q ∈ ga-S(N) extending p.

Thus the assumptions of categoricity in ℵ_{1} together with K_{ℵ_{0}} has the amalgamation property are a replacement of an easy fact in first-order logic which is a corollary of the compactness theorem. This is a typical example of “compactness regained” which appears also in much more complicated results.

Definition 6.6. Let ⟨K, ≺_{K}⟩ be an abstract elementary class and suppose that λ > LS(K). For N ∈ K_{≥λ} the model N is λ-Galois saturated iff for every M ≺_{K} N of cardinality less than λ and every p ∈ ga-S(M) is realized in N.

The contents of the following theorem that for AEC classes that have the amalgamation property, model homogeneity and saturation are equivalent properties. It is further evidence that the notion of Galois type makes sense.

Theorem 6.7. Let ⟨K, ≺_{K}⟩ be an abstract elementary class and suppose that λ > LS(K). Suppose that K has the (< λ, < λ)-amalgamation property then for M ∈ K_{≥λ} we have that M is λ-Galois saturated iff M is λ-model homogeneous.

Proof. It is easy to show that homogeneity implies saturation.

Let μ ≥ LS(K) be such that μ < λ.

Let N_{1} ≺ N_{2} ∈ K_{≤μ} be given such that N_{1} ≺ M.

We may assume that μ is the cardinality of N_{2}. Fix ⟨a_{i} | i < μ⟩ an enumeration of N_{2}. Now by induction on i < μ define two increasing continuous chains of models ⟨N_{l}^{i}, f_{i} | i ≤ μ, l ∈ {1, 2}⟩ and mappings ⟨f_{i} | i < μ⟩ satisfying:

1. N_{l}^{0} = N_{l}, f_{0} = id_{N_{1}}, ∥N_{2}^{i}∥ = μ,

2. N_{1}^{i} ≺_{K} N_{2}^{i},

3. f_{i} : N_{1}^{i} ↪ M and

4. a_{i} ∈ N_{1}^{i+1} for every i < μ.

Since the chains are continuous we only have to define two models and an embedding for i = j + 1:

Let M_{1}^{j} := f_{j}[N_{1}^{j}]. Using Remark 1.6 let M_{2}^{j} be an amalgam of N_{2}^{j} and M_{1}^{j} over N_{1}^{j} such that M_{2}^{j} ≻ M_{1}^{j} and let g_{j} : N_{2}^{j} ≅ M_{2}^{j} be an extension of f_{j}. Namely the diagram

\xymatrix{N_{2}^{j} \ar[r]^{g_{j}} & M_{2}^{j} \\ N_{1}^{j} \ar[u]^{id} \ar[r]_{f_{j}} & M_{1}^{j} \ar[u]_{id} \ar[r]_{id} & M}

commutes.

If g_{j}(a_{j}) ∈ M_{1}^{j} then do nothing. Otherwise consider p := ga-tp(g_{j}(a_{j}), M_{1}^{j}, M_{2}^{j}) and use the hypothesis that M is μ_{2}^{+}-saturated to get M^{∗} ∈ K_{μ_{2}} and to find b ∈ M^{∗} such that M_{1}^{j} ≺ M^{∗} ≺ M and (M_{1}^{j}, M_{2}^{j}, g_{j}(a_{j})) ∼ (M_{1}^{j}, M^{∗}, b).

Unwinding the definition of ∼ gives: There exists N^{∗∗} ∈ K_{μ} and mappings h_{1}, h_{2} such that the diagram

\xymatrix{M_{2}^{j} \ar[r]^{h_{2}} & N^{∗∗} \\ M_{1}^{j} \ar[u]^{id} \ar[r]_{id} & M^{∗} \ar[u]_{h_{1}}}

commutes and in addition we have h_{2}(g_{j}(a_{j})) = h_{1}(b). By renaming the elements of N^{∗∗} and changing the mapping h_{1} accordingly we may assume that h_{2} is the identity. Thus by gluing the last two diagrams together we get that the diagram

\xymatrix{N_{2}^{j} \ar[r]^{g_{j}} & M_{2}^{j} \ar[r]^{id} & N^{∗∗} \\ N_{1}^{j} \ar[u]^{id} \ar[r]_{f_{j}} & M_{1}^{j} \ar[u]_{id} \ar[r]_{id} & M^{∗} \ar[u]_{h_{1}} \ar[r]_{id} & M}

commute. Now pick N_{2}^{i} ≻ N_{2}^{j} and h_{j} ⊇ g_{j} such that h_{j} : N_{2}^{i} ≅ N^{∗∗}.

So we have that

\xymatrix{N_{2}^{i} \ar[dr]^{h_{j}} \\ N_{2}^{j} \ar[u]^{id} \ar[r]_{g_{j}} & N^{∗∗} \\ N_{1}^{j} \ar[u]^{id} \ar[r]_{f_{j}} & M^{∗} \ar[u]_{h_{1}}}

Commutes. Let N_{1}^{i} := h_{j}^{−1}[h_{1}[M^{∗}]]. Since g_{j}(a_{j}) = h_{1}(b) (using also that b ∈ M^{∗}) we get that a_{j} ∈ N_{1}^{j+1} and from h_{j} ⊇ g_{j} and the fact that

\xymatrix{N_{2}^{i} \ar[dr]^{h_{j}} \\  & N^{∗∗} \\ N_{1}^{j} \ar[uu]^{id} \ar[r]_{f_{j}} & M^{∗} \ar[u]_{h_{1}}}

commutes we get that N_{1}^{i} ⊇ N_{1}^{j}. Now we are ready to define the mapping f_{i}, let f_{i} := h_{1}^{−1} ∘ (h_{j} ↾ N_{1}^{j}). It is a K-embedding that extends f_{j} as required. Verify using Axiom A2 that ⋃_{i<μ} f_{i} is an embedding of N_{2} over N_{1} into M.

7. Stability-like properties

Remark 7.1. Notice that K_{λ}^{3} has no maximal element corresponds to “every type over M can be extended to a type over M′ for all M ⊊ M′ ∈ K_{λ}”. The contents of Theorem 5.4 is that if K is an P C_{ℵ_{0}} class categorical in ℵ_{0} such that I(ℵ_{1}, K) < 2^{ℵ_{1}} every type over a countable model has a proper extension.

Theorem 7.2 ([Sh 576]). Suppose K is a PC_{μ}-class. Let λ ≥ μ. If I(λ, K) = I(λ^{+}, K) = 1 and K_{λ^{++}} ≠ ∅, then K_{λ}^{3} has no maximal element.

Definition 7.3. (M_{0}, M_{1}, a) ∈ K_{λ}^{3} is minimal iff for every
(M′_{0}, M′_{ℓ}, a_{ℓ}) ≥ _{h_{ℓ}}(M_{0}, M_{1}, a) ℓ = 1, 2

we have that ga-tp(a_{1}, M′_{0}, M_{1}) = ga-tp(a_{2}, M′_{0}, M_{2}). (M′_{0}, M′_{1}, a′) ≥ _{h}(M_{0}, M_{1}, a) stands for “h is an embedding of M_{1} into M′_{1} taking M_{0} into M′_{0} and h(a) = a′”.

The above notion of minimal triple is a generalization of strongly minimal non algebraic type. We are actually generalizing the stationarity property of strongnly minimal types. I.e. if a_{1}, a_{2} realize extensions (to |M′_{0}|) of the minimal type of a then they types of a_{1} and a_{2} are equal.

There are several basic existence results of strongly minimal types. I.e.

Fact 7.4. (W. Marsh) If T is an ℵ_{0}-stable first-order then there exists a strongly minimal formula.

It is natural at this stage to introduce the following “obvious” concept:

Definition 7.5. K is stable in λ iff | ga-S(M)| ≤ λ for every M ∈ K_{λ} .

An analog to Marsh’e theorem is the following result:

Theorem 7.6. Let λ ≥ LS(K). Suppose K has the λ-amalgamation property and K_{λ}^{3} does not have a maximal triple. If there is no minimal (M_{0}, M_{1}, a) ∈ K_{λ}^{3}, let p := ga-tp(a/M_{0}, M_{1}) then the following holds:

1. There is M′_{0} ∈ K_{λ} such that M_{0} ⪯ M′_{0} and
|{p′ ∈ ga-S(M_{0}) : p′ ≥ p}| ≥ 2^{log(λ)};
2. There exists N ∈ K_{λ^{+}} such that M′_{0} ⪯ N and
|{p ∈ ga-S(M′_{0}) : p is realized in N}| = λ^{+}.

Namely,

|{c ∈ N : ∃N′ ⪰ N, M′_{0} ⪯ N′, N′ ∈ K_{λ}, such that
ga-tp(c/M′_{0}, N′) ≥ ga-tp(a/M_{0}, M_{1})}| = λ^{+}.

The previous theorem is further evidence that the notion of Galois types is useful.

The notion of minimal element of K_{λ}^{3} plays a central role in proving categoricity results. A key to categoricity: Under strong assumptions on K we have that if p ∈ ga-S(M) is minimal then for every M_{2} ≻ M_{1} ≻ M if M_{1} ⊊ M_{2} then there exists a ∈ |M_{2}| − |M_{1}| realizing p.

At the current state of affairs there is no nice forking-like relation for AECs (even under the assumption that they are categorical above the hanf number).

However there are several approximations. Since not even the parallel to Morley’s theorem is available for AECs one can investigate one of the coarser notions from the days of stability theory before forking. Below it is shown that a notion parallel to splitting of types is moderately nicely behaved. For this we make the following

Hypothesis 7.7. K has the (λ, < κ, μ)-amalgamation property for every λ, κ and μ, we may assume that there exists a monster model ℭ. By monster model we mean a very large model homogeneous model.

Notice that existence of large Galois-saturated models is also desireable, but we need more than just amalgamation to prove their existence.

Definition 7.8. Suppose N ≺_{K} M ∈ K, λ ≥ ∥N∥ a cardinal number. The type p ∈ ga-S(M) λ-splits over N iff there exists N_{1}, N_{2}, h of cardinality λ such that N ≺ N_{ℓ} ≺ M, h : N_{1} ≅_{N} N_{2} and the types p ↾ N_{2} and h(p ↾ N_{1}) are

contradictory (= there if N^{∗} is an extension of M then there is no a ∈ |N^{∗}| such that a ⊧ p ↾ N_{2} ∪ h(p ↾ N_{1})).

It is tempting to call the previoous notion Galois-splitting. However I feel that doing so will make certain passages unreadable. It is important to recognize the similarity of Galois-splitting to the ususal first-order splitting as well as the differences. The key difference that here we don’t have a formula wittnessing the splititng, moreover the splitting is evidenced by models rather than a finite sequence of parameters.

Theorem 7.9. Let λ ≥ LS(K). Suppose K is stable in λ. For every M ∈ K_{≥λ} and every p ∈ ga-S(M) there exists M_{0} ≺ M of cardinality ≤ λ such that p does not split over M_{0}.

Proof. Suppose N ≻_{K} M, a ∈ N such that p = ga-tp(a/M, N) and p splits over N_{0}, for every N_{0} ≺_{K} M of cardinality λ.

Let χ := min{χ | 2^{χ} > λ}. Notice that χ ≤ λ and 2^{<χ} ≤ λ.

We’ll define {M_{α} ≺ M | α < χ} ⊆ K_{λ} increasing and continuous ≺_{K}-chain which will be used to construct M_{χ}^{∗} ∈ K_{λ} such that

| ga-S(M_{χ}^{∗})| ≥ 2^{χ} > λ obtaining a contradiction to λ-stability.

Pick M_{0} ≺ M any model of cardinality λ.

For α = β + 1; since p splits over M_{β} there are N_{β,ℓ} ≺_{K} M of cardinality λ for ℓ = 1, 2 and there is h_{β} : N_{β,1} ≅_{M_{β}} N_{β,2} such that h_{β}(p ↾ N_{β,1}) ≠ p ↾ N_{β,2}. Pick M_{β} ≺_{K} M of cardinality λ containing the set |N_{β,1}| ∪ |N_{β,2}|.

Now for α < χ define M_{α}^{∗} ∈ K_{λ} and for η ∈ ^{α}2 define a K-embedding h_{η} such that

1. β < α ⟹ M_{β}^{∗} ≺_{K} M_{α}^{∗},

2. for α limit let M_{α}^{∗} = ⋃_{β<α} M_{β}^{∗},

3. β < α ∧ η ∈ ^{α}2 ⟹ h_{η↾β} ⊆ h_{η},

4. η ∈ ^{α}2 ⟹ h_{η} : M_{α} ↪^{K} M_{α}^{∗} and

5. α = β + 1 ∧ η ∈ ^{α}2 ⟹ h_{ηˆ 0}(N_{β,1}) = h_{ηˆ 1}(N_{β,2}).

The construction is possible by using the λ-amalgamation property at α = β + 1 several times. Given η ∈ ^{β}2 let N^{∗} be of cardinality λ and f_{0} be such that the diagram

\xymatrix{M_{β+1} \ar[r]^{f_{0}} & N^{∗} \\ M_{β} \ar[u]^{id} \ar[r]_{h_{η}} & M_{β}^{∗} \ar[u]_{id}}

commutes. Denote by N_{2} the model f_{0}(N_{β,2}). Since h_{β} : N_{β,1} ≅_{M_{β}} N_{β,2} there is a K-mapping g fixing M_{β} such that g(N_{β,1}) = N_{2}. Using the amalgamation property now pick N^{∗∗} ∈ K_{λ} and a mapping f_{1} such that the diagram

\xymatrix{M_{β+1} \ar[r]^{f_{1}} & N^{∗∗} \\ N_{β,1} \ar[u]^{id} \ar[r]_{g} & N_{2} \ar[u]^{id} \\ M_{β} \ar[u]^{id} \ar[r]_{h_{η}} & M_{β}^{∗} \ar[u]_{id}}

Finally apply the amalgamation property to find M_{β+1}^{∗} ∈ K_{λ} and mappings e_{0}, e_{1} such that

\xymatrix{N^{∗∗} \ar[r]^{e_{1}} & M_{β+1}^{∗} \\ M_{β}^{∗} \ar[u]^{id} \ar[r]_{id} & N^{∗} \ar[u]_{e_{0}}}

commutes. After renaming some of the elements of M_{β+1}^{∗} and changing e_{1} we may assume that e_{0} = id_{N^{∗}}.

Let h_{ηˆ 0} := f_{0} and h_{ηˆ 1} := e_{1} ∘ f_{1}.

Now for η ∈ ^{χ}2 let

M_{χ}^{∗} := ⋃_{α<χ} M_{α}^{∗} and H_{η} := ⋃_{α<χ} h_{η↾α}.

Take N_{η}^{∗} ≻_{K} M_{χ}^{∗} from K_{λ}, an amalgam of N and M_{χ}^{∗} over M_{χ} such that

\xymatrix{N \ar[r]^{H_{η}} & N_{η}^{∗} \\ M_{χ} \ar[u]^{id} \ar[r]_{h_{η}} & M_{χ}^{∗} \ar[u]_{id}}

commutes.

Notice that

η ≠ ν ∈ ^{χ}2 ⟹ ga-tp(H_{η}(a)/M_{χ}^{∗}, N_{η}^{∗}) ≠ ga-tp(H_{ν}(a)/M_{χ}^{∗}, N_{ν}^{∗}).

Thus | ga-S(M_{χ}^{∗})| ≥ 2^{χ} > λ.

Question 7.10. Is it possible to replace ℵ_{0} in Remark 7.1 with an uncountable λ?

This is important since a positive answer will require developing a replacement for the undefinability of well ordering in L_{ω_{1},ω}. The fact that undefinability of well ordering does not have a natural generalization for uncountable cardinals follows from an independence result of Jon Barwise and Ken Kunen [BaKu].

Here is a very simple example of a result from [Sh 576] in this direction:

Theorem 7.11. Let λ ≥ μ. Suppose K is an AEC such that K is P C_{μ}, I(λ, K) = I(λ^{+}, K) = 1 and K_{λ^{++}} ≠ ∅. Then K_{λ}^{3} does not have a maximal triple.

Proof. (K_{λ}^{3} has no max element). Since I(λ, K) = 1 and K_{λ^{+}} ≠ 0, it is enough to show that if (M_{0}, M_{1}, a) ∈ K_{λ}^{3}, then there exists

(M′_{0}, M′_{1}, a′) ≥ (M_{0}, M_{1}, a)

such that M_{0} ≠ M′_{0}.

Suppose for contradiction that (M_{0}, M_{1}, a) ∈ K_{λ}^{3} is a maximal triple. Define {N_{i} : i < λ^{+}} ⊆ K_{λ} increasing continuous and {h_{i} : M_{1} ≅ N_{i+1} : i < λ^{+}} such that N_{i} = h_{i}[M_{0}]. Since we want these sets to be continuous, it is enough to define them at successor stages. Given N_{i}, where i < λ^{+}, by I(λ, K) = 1 there exists an isomorphism h̄ : M_{0} ≅ N_{i}. Since M_{0} ≺_{K} M_{1}, we can find N_{i+1} and h_{i} such that h̄ ⊆ h_{i} and h : M_{1} ≅ N_{i+1}. Note that h_{i}(a) ∈ N_{i+1} − N_{i}. From the construction we get that that (N_{i}, N_{i+1}, h_{i}(a)) ∈ K_{λ}^{3} is a maximal triple, for i < λ^{+}.

Now let N = ⋃_{i<λ^{+}} N_{i}. Then, since h_{i}(a) ∈ N_{i+1} − N_{i} for all i < λ^{+} and since {N_{i} : i < λ^{+}} is a chain, N ∈ K_{λ^{+}}. Since I(λ^{+}, K) = 1 and K_{λ^{++}} ≠ 0, there is N^{1} ≠ N in K_{λ^{+}} such that N ≺_{K} N^{1}.

Pick {N_{β}^{1} ≺_{K} N^{1} : β < λ^{+}} ⊆ K_{λ} increasing and continuous such that for all β < λ^{+}, N_{β} ⊂ N_{β}^{1} but N_{β} ≠ N_{β}^{1}. (This is possible since |N^{1}| ≠ |N|).

Define a function g : |N^{1}| → λ^{+} as follows:

g(b) := \begin{cases}i & if h_{i}(a) = b \\ 0 & Otherwise. \end{cases}

Notice that the relation g(b) = i is a function since we have seen above that i ≠ j ⟹ h_{i}(a) ≠ h_{j}(a).

Applying the reflection theorem from set theory, let α be large enough such that V_{α} contains the set

{N, N^{1}, M_{0}, M_{1}, a, ⟨N_{i}, N_{i}^{1}, h_{i} : i < λ^{+}⟩, g, λ^{+},

“the PC_{μ} definition of K ”}

and such that the model

B^{∗} = ⟨V_{α}, ϵ, N, N^{1}, M_{0}, M_{1}, Q, K, g, i ↦ (N_{i}, N_{i}^{1}, h_{i}), a⟩,

where Q = λ^{+}, reflects the following sentences:

(i) ”{N_{i} : i < λ} is increasing and continuous in K_{λ+}” (ii) (∀i ∈ Q)[N_{i} ⊂ N_{i}^{1} ∧ N_{i} ≠ N_{i}^{1}] (iii) ”for all i ∈ Q, (N_{i}, N_{i+1}, h_{i}(a)) is a maximal element in K_{λ}^{3}” (iv) g : |N^{1}| → Q

(v) (∀b ∈ N^{1})(∃i ∈ Q)((h_{i}(a) = b) → g(b) = i) (vi) N = ⋃_{i∈Q} N_{i} (vii) N^{1} = ⋃_{i∈Q} N_{i}^{1}.

Let B ≺_{K} B^{∗} of cardinality λ such that Q^{B} = δ, a limit ordinal. (For example, take {B_{i} ≺_{K} B^{∗} : i < λ^{∗}} increasing continuous such that ||B_{i}|| = λ and Q^{B_{i}} ⊇ i. Then use the fact that {δ < λ^{+} : δ = Q^{B_{δ}}} is a club.)

Now, N^{B} = ⋃_{α<δ} N_{α} = ⋃_{α<δ} N_{α} = N_{δ}. Similarly, N^{1}^{B} = N_{δ}^{1}.

Denote a_{δ} := h_{δ}(a).
Claim N_{δ+1} ∩ (N^{1})^{B} = N_{δ}. In particular, a_{δ} ∉ N_{δ}^{1}.

Proof. If a_{δ} ∈ N_{δ}^{1} then a_{δ} ∈ B. Since B is closed under g, g(a_{δ}) ∈ B. But recall that

B ⊧(g : |N^{1}| → Q) ∧ (∀b ∈ N^{1})
((∃i ∈ Q)(h_{i} = b) → (g(b) = i)).
So g(a_{δ}) ∈ Q. Compute g(a_{δ}) = δ.

Since B ⊧ ZF^{−}, also δ + 1 ∈ Q^{B}, which contradicts the fact that δ = Q^{B}.

If N_{δ+1} ∩ N_{δ}^{1} properly contains N_{δ}, then since a_{δ} ∈ N_{δ+1} ⊆ N_{δ+1}^{1}, by the fact that a ∉ N_{δ}^{1} we get that (N_{δ}^{1}, N_{δ+1}^{1}, a_{δ}) ∈ K_{λ}^{3}. And since then

(N_{δ}, N_{δ+1}, a_{δ}) < (N_{δ}^{1}, N_{δ+1}^{1}, a_{δ}),

we get a contradiction to the assumption that (N_{δ}, N_{δ+1}, a_{δ}) is a maximal triple.

In [GrVa ] Grossberg and VanDieren have shown that for categorical AEC Morley-sequences exist when the dependence relation is non splitting.

Denote by M ≺_{K}^{univ} N the statement N is universal over M i.e. for every M′ ≻ M of cardinality ∥M∥ there exists a K-embedding from M′ into N over M.

Definition 7.12 (from [GrVa ]). Let K be an AEC. The class is called μsuperstable iff there exists μ > LS(K) satisfying

1. for every M ∈ K_{μ}, there exists M ≺_{K}^{univ} M′ ∈ K_{μ} and

2. for every κ = cf(κ) < μ^{+} whenever ⟨M_{i} ∈ K_{μ} | i ≤ κ⟩ is ≺_{K}^{univ}- increasing and continuous and p ∈ ga-S(M_{κ}), there exists i < κ such that p does not μ-split over M_{i}.

Theorem 7.13 (from [GrVa ]). Suppose K is μ-superstable for some μ ≥ LS(K) and K has the amalgamation property. Let M ∈ K_{>μ}, A, I ⊂ M be given such that |I| ≥ μ^{+} > |A|. Then there exists J ⊂ I of cardinality μ^{+}, indiscernible over A. Moreover J can be chosen to be a Morley sequence over A.

Where

Definition 7.14 ([GrVa ]). {a_{i} ∈ μ^{+}} is a Morley sequence over M_{0} iff there exists an ≺_{K}-chain {M_{i} | i < μ^{+}} such that p_{i} := ga-tp(a_{i}/M_{i}, M_{i+1}) does not μ-split over M_{0} and ⟨a_{i} | i ∈ λ⟩ is an indiscernible sequence over M_{0}.

8. proof of Theorem 4.3

Before starting the proof let me point out that we will be using eth Devlin-Shelah weak diamond. Instead of using the principle Φ_{λ^{+}} we use a principle that may look little stronger but using a pairing function together with with Fact 4.7 one can show that it follows from Φ_{λ^{+}}:

We will be using:

There exists a stationary subset S of λ^{+} such that for every

F : ^{<λ^{+}}λ^{+} ×^{<λ^{+}}λ^{+} ×^{<λ^{+}}λ^{+} → 2,

there exists a guess g : λ^{+} → 2, such that for every η, ν, h : λ^{+} → λ^{+}, the set

{δ ∈ S | F (η ↾ δ, ν ↾ δ, h ↾ δ) = g(δ)} is stationary.

Proof of Theorm 4.3. Recall that I(λ, K) = 1 and K fails to have the λamalgamation property implies that K_{λ^{+}} is nonempty.

By assumption, we may take N_{0}, N_{1}, N_{2} ∈ K_{λ}, that can not be amalgamated. For η ∈ ^{λ^{+}>}2 define a family of M_{η} ∈ K_{λ} so that the following hold:

1. |M_{η}| = λ(1 + ℓ(η)),

2. ν ⋖ η → M_{ν} ≺_{K} M_{η},

3. when ℓ(η) is a limit ordinal, M_{η} = ⋃_{α<ℓ(η)} M_{η↾α},

4. M_{ηˆ 0} and M_{ηˆ 1} cannot be amalgamated over M_{η}. Using λ-categoricity and the triple N_{0}, N_{1}, N_{2} the construction is possible. Divide the proof into two cases, in the first case assume a stronger failure of

the amalgamation and the second is the negation of the first.

\underline{Case A:} Suppose that there exist N ≺ M ∈ K_{λ} so that for every M′ extending M in K_{λ}, there is a pair M^{0} and M^{1} extending M′ so that M^{0} and M^{1} cannot be amalgamated over N.

To requirements 1-4 we add M_{⟨⟩} = N and replace (4) by

(4)′ M_{ηˆ 0} and M_{ηˆ 1} cannot be amalgamated over N. For η ∈ ^{λ^{+}}2, let M_{η} := ⋃_{α<λ^{+}} M_{η↾α}.

Claim 8.1. η ≠ ν ⟹ ⟨M_{η}, a⟩_{a∈|N|} ≅ ⟨M_{ν}, a⟩_{a∈|N|}.

Proof. Let ρ be the meet of η and ν. If there was an isomorphism between ⟨M_{η}, a⟩_{a∈|N|} and ⟨M_{ν}, a⟩_{a∈|N|}, we would have that M_{ν} is an amalgam of M_{ρˆ 0} and M_{ρˆ 1} over N, a contradiction to requirement (4).

Suppose for the sake of contradiction that

μ := I(λ^{+}, K) < 2^{λ^{+}}.

Take {M_{i} | i < μ} to be a complete set of representatives for K_{λ^{+}}.

Clearly,

|{⟨M_{η}, a⟩_{a∈|N|}/ ≅ : η ∈ ^{λ^{+}}2}|
≤ |{⟨M_{i}, a⟩_{a∈|N|} : i < μ}|
≤ μ∥M_{i}∥^{λ} = μ(λ^{+})^{λ} = μλ^{+}2^{λ} =
μ2^{λ} < 2^{λ^{+}}.

But this is a contradiction to the fact we have many pairwise non-embeddable models. Notice how we used the assumption 2^{λ} < 2^{λ^{+}} in the final step.

\underline{Case B:} Suppose for all N and every M ∈ K_{λ} there is an M′ extending M so that for any extensions M_{0} and M_{1} of M′, M_{0} and M_{1} can be amalgamated over N.

Again, we tweak requirement (4) and replace it by:

(4)′ if M^{0} and M^{1} are extensions of M_{ηˆ 0} and M_{ηˆ 1}, then M_{η}, M^{0} and M^{1} can be amalgamated.

To perform the construction at successors, we do the following, to define M_{ηˆ 0} and M_{ηˆ 1} from M_{η}:

Apply λ categoricity to fix an isomorphism f : M_{η} ≅ N_{0} (the unamalgamable triple we chose at the beginning).

Using N_{1}, N_{2} and their preimages pick M^{∗} and M^{∗∗} extensions of M_{η} which cannot be amalgamated over M_{η}.

By the assumption of case B, we can take M_{ηˆ 0} to be an extension of M^{∗} so that any extensions of M_{ηˆ 0} can be amalgamated over M_{η}. Take M_{ηˆ 1} similarly for M^{∗∗}.

Let C := {δ < λ : δ = λ(1 + δ)}, notice that it is a club. By Ulam’s theorem there are {S_{γ} ⊆ C : γ < λ^{+}} stationary sets such that γ_{1} ≠ γ_{2} ⟹ S_{γ_{1}} ∩ S_{γ_{2}} = ∅ and for all γ < λ^{+} we have that Φ_{λ^{+}}(S_{γ}) holds.

For every δ < λ^{+} such that δ = λ(1 + δ), h : δ → δ and η, ν ∈ ^{δ}2 let

F (η, ν, h) := \begin{cases}1 if h : M_{η} & ↪ M_{ν} and \\ M_{ηˆ 0} & M_{η} & id & [figure] & h & [figure] & M_{νˆ 0} \\ & can be amalgamated. \\ 0 & Otherwise. \end{cases}

By Φ_{λ^{+}}(S_{γ}) pick g_{γ} : λ^{+} → 2 such that

for all η, ν ∈ ^{λ^{+}}2 and every h : λ^{+} → λ^{+} we have that

S′_{γ} := {δ ∈ S_{γ} : F (η ↾ δ, ν ↾ δ, h ↾ δ) = g_{γ}(δ)}

is stationary. For X ⊆ λ^{+} and every δ < λ^{+} let

η_{X}[δ] := \begin{cases}g_{γ}(δ) & if δ ∈ S_{γ} and δ ∈ X \\ 0 & Otherwise. \end{cases}

Notice that since {S_{γ} : γ < λ^{+}} are pairwise disjoint for any δ ∈ X there is at most one γ such that δ ∈ S_{γ} (maybe none), so η_{X} is well defined. We finish by showing:

Claim 8.2. For every X ≠ Y ⊆ λ^{+} we have that M_{η_{X}} ≅ M̸_{η_{Y}}.

Proof. Suppose X ≠ Y ⊆ λ^{+}, h : M_{η_{X}} ↪ M_{η_{Y}}. An application of Φ_{λ^{+}}(S_{γ}) to η_{X}, η_{Y} and h yields S′_{γ} as above. Let D := {δ < λ^{+} | (h ↾ δ) : δ → δ} be a club. For γ < λ^{+} let S′′_{γ} := S′_{γ} ∩ D. Without loss of generality there exists γ ∈ X − Y. Pick δ ∈ S′′_{γ}.

Denote by η the sequence η_{X} ↾ δ and by ν the sequence η_{Y} ↾ δ. Since γ ∉ Y by the definition of the sequence η_{Y} we have that η_{Y} [δ] = 0, namely ν ⋖ νˆ 0 ⋖ η_{Y}.

Now consider η_{X}[δ]. There are two possibilities (according to the value of η_{X}[δ]):

1. If η_{X}[δ] = 1, by the definition of η_{X}[δ] necessarily η_{X}[δ] = g_{γ}(δ), since δ ∈ S′_{γ} we have that F (η, ν, h) = 1. From the definition of F we get that M_{ηˆ 0} and M_{νˆ 0} can be amalgamated over M_{η}. Denote by M^{1} the amalgam and let f and g be such that the following diagram commutes:

(∗)_{1} \xymatrix{M_{ηˆ 0} \ar[r]_{g} & M^{1} \\ M_{η} \ar[u]^{id} \ar[r]_{h} & M_{νˆ 0} \ar[u]^{f}}

Since ηˆ 1 ⋖ η_{X}, νˆ 0 ⋖ η_{Y} and from the assumption that h : M_{η_{X}} ↪ M_{η_{Y}}, the following diagram

\xymatrix{M_{ηˆ 1} \ar[r]_{h} & M_{η_{Y}} \\ M_{η} \ar[u] \ar[r]_{h} & M_{νˆ 0} \ar[u]^{id}}

must commute. By axiom A4 there exists M^{2} ≺_{K} M_{η_{Y}} of cardinality λ such that h(M_{ηˆ 1}) ≺_{K} M^{2} and M_{νˆ 0} ≺_{K} M^{2}. Namely the following diagram commutes:

(∗)_{2} \xymatrix{M_{ηˆ 1} \ar[r]_{h} & M^{2} \\ M_{η} \ar[u] \ar[r]_{h} & M_{νˆ 0} \ar[u]^{id}}

Since

\xymatrix{M^{1} \\ M_{νˆ 0} \ar[u]_{f} \ar[r]_{id} & M^{2}}

by the second half of requirement (4)′ there are M^{3} ∈ K_{λ} and K-mappings e_{l} : M^{l} ↪ M^{3} (for l = 1, 2) such that

(∗)_{3} \xymatrix{M^{1} \ar[r]_{e_{1}} & M^{3} \\ M_{ν} \ar[u]_{f} \ar[r]_{id} & M^{2} \ar[u]^{e_{2}}}

is commutative. Combining (∗)_{1}, (∗)_{2} and (∗)_{3} together we get

(∗)_{4} \xymatrix{M_{ηˆ 0} \ar[r]_{g} & M^{1} \ar[r]_{e_{1}} & M^{3} \\ M_{η} \ar[u]^{id} \ar[d]_{id} \ar[r]_{h} & M_{ν} \ar[u]_{f} \ar[r]_{id} & M^{2} \ar[u]^{e_{2}} \\ M_{ηˆ 1} \ar[urr]_{h}}

Thus we have that

\xymatrix{M_{ηˆ 1} \ar[r]^{h∘e_{2}} & M^{3} \\ M_{η} \ar[u]_{id} \ar[r]_{id} & M_{ηˆ 0} \ar[u]_{g∘e_{1}}}

is commutative, which is a contradiction to the first half of requirement (4)′.

2. If η_{X}[δ] = 0 then we have that ηˆ 0 ⋖ η_{X}. Since h : M_{η_{X}} ↪ M_{η_{Y}} we get that M_{η_{Y}} is an amalgam of M_{ηˆ 0} and M_{νˆ 0} over M_{η} so by the definition of F we get that F (η, ν, h ↾ δ) = 1. Since δ ∈ S′′_{γ} we have that g_{γ}(δ) = 1 which by the definition of η_{X} gives η_{X}[δ] = 1 and this contradicts the assumption of this case.

9. The main problems

In my opinion most of the future results will come from two separate lines of research, that many years from now will merge.

1. Stability theory for AECs. The most ambitious form of this program can be stated as follows: Let MT stand for Morley’s categoricity theorem, f.o. stability for the contents of Shelah’s book ([Sh c]) and let n.e.categ stand for the partial results toward Shelah’s categoricity conjecture (the contents of [Ke2],[Sh48], [Sh87a], [Sh87b], [MakSh ], [KoSh], [Sh 394], [Sh472], [Le1]).

By developing stability theory for AECs I mean solving the equation

(∗) \frac{f.o. stability}{MT} = \frac{x}{n.e.categ}.

Examples of progress toward this direction can be found in [Sh3], [Sh88], [Sh 300], [GrHa ], [Gr3],[Gr4], [GrHa ],[GrLe1 ],[GrLe2 ],[GrLe4 ],[GrSh1 ], [GrSh2 ], [GrSh3 ], [GrSh4 ],[Le1], [Le2],[Sh 576],[Sh 600],[ShVi], [GrVa ],[Kov1] and [Va].

2. Geometric model theory for AECs. This program can be described as an attempt to solve the following equation:

(#) \frac{HZ}{f.o. stability} = \frac{?}{x}

where HZ stands for Hrushovski’s extension of Zilber’s geometric ideas, and x is a (partial) solution for (∗). There very few results in this direction. Among them are [Le1], [GrLe3 ] and [Le3].

I expect that several interactions of (∗) and (#) will eventually yield among other things a solution for Shelah’s categoricity conjecture.

While I am convinced that eventually the theory will have more applications to main stream mathematics via commutative algebra, algebraic geometry or analytic structures than model theory of first-order logic, so far there are no applications in sight.

The greater potential is due to the ability to axiomatize local finiteness and structures satisfying various chain conditions. It is too early to predict what exactly these applications will be. It is natural to expect that studying AECs of some concrete structures (rings and groups) may produce valuable results.

There is one conjecture that may eventually be solved using non elementary methods. This is Zilber’s conjecture (stronger than Schanuel’s conjecture) concerning analytic structure from [Zi]. Namely that ℂ_{exp} is the canonical structure of cardinality 2^{ℵ_{0}} in the class H(ex/st).

Zilber points out that this class fits the framework of EC classes. The following are major concrete problems in AEC:

1. In Makkai-Shelah [MakSh ] as well as in [Sh 394] and [Sh472] partial results toward a categoricity theorem are presented. In both cases the hypothesis is that the class is categorical in a successor cardinal. It is not clear at all if this is just a technical limitation or a central problem. I suspect it is central. Probably replacing the assumption that the class is categorical in λ^{+} with the assumption of categoricity in λ is of similar difficulty.

2. The categoricity theorem in [Sh 394] is a going down theorem. Is there a going up theorem? The simplest instance of this is:

Conjecture 9.1. Let K be an AEC. If there exists λ ≥ Hanf(K) such that K is categorical in λ then K is categorical in λ^{+}.

3. It is a major open problem to find a nice (forking-like) dependence notion for AEC. In fact even under the assumption that K has the amalgamation property and the class is categorical in a cardinal above Hanf(K) this is open (see Remark 4.10(1) in [Sh 394]).

4. One of the technical problems of working in AEC without the amalgamation property is the inexistence of monster models and therefore types over models can not always be extended to global types. To deal with this Shelah and Andres Villaveces have introduced in ([ShVi]) an interpolant, which is the framework of AEC without maximal models, under GCH. They have managed to show that categoricity implies that every small model (below the categoricity cardinal) can be extended to an amalgamation base and several other basic facts. Some extensions of this work can be found in [Va]. As of today there are no categoricity results in this context. It is natural to expect the following:

Conjecture 9.2. Let K be an AEC without maximal models and suppose that λ ≥ Hanf(K). If K is categorical in λ^{+} then K is categorical in every μ ≤ λ.

5. Probably the appropriate name for this is existence of Hanf number for amalgamation:

Conjecture 9.3. Let K be an AEC. Suppose (for simplicity?) that K does not have maximal models. There exists a cardinal number μ(K) such that if K has the μ(K)-amalgamation property then K has the λamalgamation property for all λ ≥ μ(K).

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    E-mail address, Rami Grossberg: rami@cmu.edu, URL:  www.math.cmu.edu/~rami

Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213