Abstract We introduce the concept of a locally finite abstract elementary class and develop the theory of disjoint $\left( { \le \lambda ,k} \right)$ -amalgamation) for such classes. From this we find a family of complete ${L_{{\omega _1},\omega }}$ sentences ${\phi _r}$ that a) homogeneously characterizes ${\aleph _r}$ (improving results of Hjorth [11] and Laskowski–Shelah [13] and answering a question of [21]), while b) the ${\phi _r}$ provide the first examples of a class of models of a complete sentence in ${L_{{\omega _1},\omega }}$ where the spectrum of cardinals in which amalgamation holds is other that none or all.

Abstract We introduce the notion of pseudoalgebraicity to study atomic models of first order theories (equivalently models of a complete sentence of ${L_{{\omega _1},\omega }}$ ). Theorem: Let T be any complete first-order theory in a countable language with an atomic model. If the pseudominimal types are not dense, then there are 2ℵ0 pairwise nonisomorphic atomic models of T, each of size ℵ1.

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.

Abstract We study when classes can have the disjoint amalgamation property for a proper initial segment of cardinals. For every natural number k, there is a class Kk, defined by a sentence in Lω1,ω that has no models of cardinality greater than ℶk + 1, but Kk has the disjoint amalgamation property on models of cardinality less than or equal to ℵk − 3 and has models of cardinality ℵk − 1. More strongly, we can have disjoint amalgamation up to ℵ∝ for ∝ < ω1, but have a bound on size of models. For every countable ordinal ∝, there is a class K∝ defined by a sentence in Lω1,ω that has no models of cardinality greater than ℶω1, but K does have the disjoint amalgamation property on models of cardinality less than or equal to ℵ∝. Finally we show that we can extend the ℵ∝ to ℶ∝ in the second theorem consistently with ZFC and while having ℵi ≪ ℶi for 0 < i < ∝. Similar results hold for arbitrary ordinals ∝ with ∣∝∣ = k and Lk + ω.

Abstract We use κ-free but not Whitehead Abelian groups to construct Abstract Elementary Classes (AEC) which satisfy the amalgamation property but fail various conditions on the locality of Galois-types. We introduce the notion that an AEC admits intersections. We conclude that for AEC which admit intersections, the amalgamation property can have no positive effect on locality: there is a transformation of AEC's which preserves non-locality but takes any AEC which admits intersections to one with amalgamation. More specifically we have: Theorem 5.3. There is an AEC with amalgamation which is not (ℵ0, ℵ1)-tame but is ( , ∞)-tame; Theorem 3.3. It is consistent with ZFC that there is an AEC with amalgamation which is not (≤ ℵ2, ≤ ℵ2)-compact.

Baizhanov and Baldwin [1] introduce the notions of benign and weakly benign sets to investigate the preservation of stability by naming arbitrary subsets of a stable structure. They connect the notion with work of Baldwin, Benedikt, Bouscaren, Casanovas, Poizat, and Ziegler. Stimulated by [1], we investigate here the existence of benign or weakly benign sets. Definition 0.1. (1) The set A is benign in M if for every α, β ∊ M if p = tp(α/A) = tp(β/A) then tp*(α/A) = tp*(β/A) where the *-type is the type in the language L* with a new predicate P denoting A. (2) The set A is weakly benign in M if for every α,β ∊ M if p = stp(α/A) = stp(β/A) then tp*(α/A) = tp*(β/A) where the *-type is the type in language with a new predicate P denoting A. Conjecture 0.2 (too optimistic). If M is a model of stable theory T and A ⊆ M then A is benign. Shelah observed, after learning of the Baizhanov-Baldwin reductions of the problem to equivalence relations, the following counterexample. Lemma 0.3. There is an ω-stable rank 2 theory T with ndop which has a model M and set A such that A is not benign in M.

Abstract. We study the expansion of stable structures by adding predicates for arbitrary subsets. Generalizing work of Poizat-Bouscaren on the one hand and Baldwin-Benedikt-Casanovas-Ziegler on the other we provide a sufficient condition (Theorem 4.7) for such an expansion to be stable. This generalization weakens the original definitions in two ways: dealing with arbitrary subsets rather than just submodels and removing the ‘small’ or ‘belles paires’ hypothesis. We use this generalization to characterize in terms of pairs, the ‘triviality’ of the geometry on a strongly minimal set (Theorem 2.5). Call a set A benign if any type over A in the expanded language is determined by its restriction to the base language. We characterize the notion of benign as a kind of local homogenity (Theorem 1.7). Answering a question of [8] we characterize the property that M has the finite cover property over A (Theorem 3.9).

Abstract For n < ω, expand the structure (n, S, I, F) (with S the successor relation, I, F as the initial and final element) by forming graphs with edge probability n−α for irrational α, with 0 < α < 1. The sentences in the expanded language, which have limit probability 1, form a complete and stable theory.

Abstract We provide a general framework for studying the expansion of strongly minimal sets by adding additional relations in the style of Hrushovski. We introduce a notion ofseparation of quantifierswhich is a condition on the class of expansions of finitely generated models for the expanded theory to have a countable ω-saturated model. We apply these results to construct for each sufficiently fast growing finite-to-one functionμfrom ‘primitive extensions’ to the natural numbers a theoryTμof an expansion of an algebraically closed field which has Morley rank 2. Finally, we show that ifμis not finite-to-one the theory may not beω-stable.

We combine some tools from stability theory and finite model theory to prove the following results. Theorem. Let T∞ be the almost sure theory for a class K and probability P satisfying the first order 0-1 law. Suppose for some κ, there are infinitely many distinct Lκ-types consistent with T∞. If LFP logic and first order-logic are almost everywhere equivalent with respect to P and T∞ is unstable. Theorem. For appropriate functions f determining the interpretation of the Ramsey quantifier the logic Lω,ω(Qram,f) is almost everywhere equivalent to first-order logic on graphs with respect to edge probability n-α for irrational α.

Hrushovski's generalization of the Fraisse construction has provided a rich source of examples in model theory, model theoretic algebra and random graph theory. The construction assigns to a dimension function δ and a class K of finite (finitely generated) models a countable ‘generic’ structure. We investigate here some of the simplest possible cases of this construction. The class K will be a class of finite graphs; the dimension, δ(A), of a finite graph A will be the cardinality of A minus the number of edges of A. Finally and significantly we restrict to classes which are δ-invariant. A class of finite graphs is δ-invariant if membership of a graph in the class is determined (as specified below) by the dimension and cardinality of the graph, and dimension and cardinality of all its subgraphs. Note that a generic graph constructed as in Hrushovski's example of a new strongly minimal set does not arise from a δ-invariant class. We show there are countably many δ-invariant (strong) amalgamation classes of finite graphs which are closed under subgraph and describe the countable generic models for these classes. This analysis provides ω-stable generic graphs with an array of saturation and model completeness properties which belies the similarity of their construction. In particular, we answer a question of Baizhanov (unpublished) and Baldwin [5] and show that this construction can yield an ω-stable generic which is not saturated. Further, we exhibit some ω-stable generic graphs that are not model complete.

Abstract Saturation is (μ, κ)-transferable in T if and only if there is an expansion T1 of T with |T1| = |T| such that if M is a μ-saturated model of T1 and |M| ≥ κ then the reduct M|L(T) is κ-saturated. We characterize theories which are superstable without f.c.p., or without f.c.p. as, respectively those where saturation is (ℵ0, λ)-transferable or (κ(T), λ)-transferable for all λ. Further if for some μ ≥ |T|,2μ > μ+, stability is equivalent to for all μ ≥ |T|, saturation is (μ, 2μ)-transferable.

We extend bounds on the expressive power of first-order logic over finite structures and over ordered finite structures, by generalizing to the situation where the finite structures are embedded in an infinite structure M, where M satisfies some simple combinatorial properties studied in model-theoretic stability theory. We first consider first-order logic over finite structures embedded in a stable structure, and show that it has the same generic expressive power as first-order logic on unordered finite structures. It follows from this that having the additional structure of, for example, an abelian group or an equivalence relation, does not allow one to define any new generic queries. We also consider first-order logic over finite structures living within any model M that lacks the independence property and show that its expressive power is bounded by first-order logic over finite ordered structures. This latter result gives an enormous class of structures in which the expressive power of first-order logic is sharply limited; it shows that common queries such as parity and connectivity cannot be defined for finite structures living within structures from this huge class. It also gives a pure combinatorial property of an interpreted structure M that is sufficient to extend results on first-order logic on ordered structures to first-order logic on finite structures embedded in M.

We work throughout in a finite relational language L. This paper is built on [2] and [3]. We repeat some of the basic notions and results from these papers for the convenience of the reader but familiarity with the setup in the first few sections of [3] is needed to read this paper. Spencer and Shelah [6] constructed for each irrational α between 0 and 1 the theory Tα as the almost sure theory of random graphs with edge probability n−α. In [2] we proved that this was the same theory as the theory Tα built by constructing a generic model in [3]. In this paper we explore some of the more subtle model theoretic properties of this theory. We show that Tα has the dimensional order property and does not have the finite cover property. We work in the framework of [3] so probability theory is not needed in this paper. This choice allows us to consider a wider class of theories than just the Tα. The basic facts cited from [3] were due to Hrushovski [4]; a full bibliography is in [3]. For general background in stability theory see [1] or [5]. We work at three levels of generality. The first is given by an axiomatic framework in Context 1.10. Section 2 is carried out in this generality. The main family of examples for this context is described in Example 1.3. Sections 3 and 4 depend on a function δ assigning a real number to each finite L-structure as in these examples. Some of the constructions in Section 3 (labeled at the time) use heavily the restriction of the class of examples to graphs. The first author acknowledges useful discussions on this paper with Sergei Starchenko.

This paper is concerned with a class K of models and an abstract notion of submodel ≤. Experience in first order model theory has shown the desirability of finding a ‘monster model’ to serve as a universal domain for K. In the original constructions of Jónsson and Fraïssé, K was a universal class and ordinary substructure played the role of ≤. Working with a cardinal λ satisfying λ<λ = λ guarantees appropriate downward Löwenheim-Skolem theorems; the existence and uniqueness of a homogeneous-universal model appears to depend centrally on the amalgamation property. We make this apparent dependence more precise in this paper. The major innovation of this paper is the introduction of a weaker notion (chain homogeneous-universal) to replace the natural notion of (K, <)-homogeneous-universal model. Modulo a weak extension of ZFC (provable if V = L), we show (Corollary 5.24) that a class K obeying certain minimal restrictions satisfies a fundamental dichotomy. For arbitrarily large λ, either K has the maximal number of models in power λ or K has a unique chain homogeneous-universal model of power λ. We show (5.25) in a class with amalgamation this dichotomy holds for the notion of K-homogeneous-universal model in the more normal sense. The methods here allow us to improve our earlier results [5] in two other ways: certain requirements on all chains of a given length are replaced by requiring winning strategies in certain games; the notion of a canonically prime model is avoided. A full understanding of these extensions requires consideration of the earlier papers but we summarize them quickly here.

If two models of a first-order theory are isomorphic, then they remain isomorphic in any forcing extension of the universe of sets. In general however, such a forcing extension may create new isomorphisms. For example, any forcing that collapses cardinals may easily make formerly nonisomorphic models isomorphic. However, if we place restrictions on the partially-ordered set to ensure that the forcing extension preserves certain invariants, then the ability to force nonisomorphic models of some theory T to be isomorphic implies that the invariants are not sufficient to characterize the models of T. A countable first-order theory is said to be classifiable if it is superstable and does not have either the dimensional order property (DOP) or the omitting types order property (OTOP). If T is not classifiable, Shelah has shown in [5] that sentences in L∞,λ do not characterize models of T of power λ. By contrast, in [8] Shelah showed that if a theory T is classifiable, then each model of cardinality λ is described by a sentence of L∞,λ. In fact, this sentence can be chosen in the . ( is the result of enriching the language by adding for each μ < λ a quantifier saying the dimension of a dependence structure is greater than μ) Further work ([3], [2]) shows that ⊐+ can be replaced by ℵ1.

Abstract Let T be a complete countable first order theory and λ an uncountable cardinal. Theorem 1. If T is not superstable, T has 2λ resplendent models of power λ. Theorem 2. If T is strictly superstable, then T has at least min(2λ, ℶ2) resplendent models of power λ. Theorem 3. If T is not superstable or is small and strictly superstable, then every resplendent homogeneous model of T is saturated. Theorem 4 (with Knight). For each μ ∈ ω ∪ {ω, 2ω} there is a recursive theory in a finite language which has μ resplendent models of power κ for every infinite κ.

Abstract Let I(μ, Κ) denote the number of nonisomorphic models of power μ and IE(μ, Κ) the number of nonmutually embeddable models. We define in this paper the notion of a diverse class and use it to prove a number of results. The major result is Theorem B: For any diverse class Κ and μ greater than the cardinality of the language of Κ, From it we deduce both an old result of Shelah, Theorem C: If T is countable and λ0 > ℵ0 then for every μ > ℵ0, IE(μ, T) ≥ min(2μ, ⊐2), and an extension of that result to uncountable languages, Theorem D: If ∣T∣ < 2ωλ0 > ∣T∣, and ∣D(T)∣ = ∣T∣ then for μ > ∣T∣,

One of the first results in model theory [12] asserts that a first-order sentence is preserved in extensions if and only if it is equivalent to an existential sentence. In the first section of this paper, we analyze a natural program for extending this result to a class of languages extending first-order logic, notably including L(Q) and L(aa), respectively the languages with the quantifiers “there exist un-countably many” and “for almost all countable subsets”. In the second section we answer a question of Bruce [3] by showing that this program cannot resolve the question for L(Q). We also consider whether the natural class of “generalized Σ-sentences” in L(Q) characterizes the class of sentences preserved in extensions, refuting the relativized version but leaving the unrestricted question open. In the third section we show that the analogous class of L(aa)-sentences preserved in extensions does not include (up to elementary equivalence) all such sentences. This particular candidate class was nominated, rather tentatively, by Bruce [3]. In the fourth section we show that under rather general conditions, if L is a countably compact extension of first-order logic and T is an ℵ1-categorical first-order theory, then L is trivial relative to T.

There are two origins for first-order theories. One type of theory arises by generalizing the common features of a number of different structures, e.g. the theory of groups, and formulating a set of axioms to encode these common features. Here the set of axioms is well understood, frequently it is finite or at least recursive, but there usually is no clear understanding of all the logical consequences of these axioms. The second type of theory arises by considering the set, T = Th(A), of all sentences true in a fixed structure A,3 e.g. the theory of arithmetic (N, +, 0) or the theory of the field of complex numbers (alias: the theory of algebraically closed fields of characteristic zero). The second case gives little more insight as to the truth in A (i.e. membership in T) of a given sentence ∅. But it does guarantee that for a given sentence ∅, either ∅ or ¬∅ is in T, that is, that T is a complete theory. When does a theory T of the first type, i.e. with well-understood axioms, posses this completeness property? An obvious sufficient condition is that T be secretly of the second type, that it have only one model, or, in jargon, T is categorical. Unfortunately (or fortunately depending on your point of view) for any theory with an infinite model, the Löwenheim-Skolem theorem shows this to be impossible: The theory has a model in every infinite power. In the mid-50's Łoś and Vaught discovered that if a theory T with no finite models is categorical in some infinite power α (all models with cardinality α are isomorphic) then T is complete. We will be dealing below with countable complete theories and will assume, unless stated to the contrary, that each theory has no finite models.

A varietyV(equational class of algebras) satisfies a strong Malcev condition ∃f1,…, ∃fnθ(f1, …,fn,x1, …,xm) where θ is a conjunction of equations in the function variablesf1, …,fnand the individual variablesx1, …,xm, if there are polynomial symbolsp1, …,pnin the language ofVsuch that ∀x1, …,xmθ(p1…,pn,x1, …,xm) is a law ofV. Thus a strong Malcev condition involves restricted second order quantification of a strange sort. The quantification is restricted to functions which are “polynomially definable”. This notion was introduced by Malcev [6] who used it to describe those varieties all of whose members have permutable congruence relations. The general formal definition of Malcev conditions is due to Grätzer [1]. Since then and especially since Jónsson's [3] characterization of varieties with distributive congruences there has been extensive study of strong Malcev conditions and the related concepts: Malcev conditions and weak Malcev conditions. In [9], Taylor gives necessary and sufficient semantic conditions for a class of varieties to be defined by a (strong) Malcev condition. A key to the proof is the translation of the restricted second order concepts into first order concepts in a certain many sorted language. In this paper we show that, given this translation, Taylor's theorem is an easy consequence of a result of Tarski [8] and the standard preservation theorems of first order logic.

In [1] the notions of strongly minimal formula and algebraic closure were applied to the study of ℵ1-categorical theories. In this paper we study a particularly simple class of ℵ1-categorical theories. We characterize this class in terms of the analysis of the Stone space of models of T given by Morley [3]. We assume familiarity with [1] and [3], but for convenience we list the principal results and definitions from those papers which are used here. Our notation is the same as in [1] with the following exceptions. We deal with a countable first order language L. We may extend the language L in several ways. If is an L-structure, there is a natural extension of L obtained by adjoining to L a constant a for each (the universe of ). For each sentence A(a1, …, an) ∈ L(A) we say satisfies A(a1, …, an) and write if in Shoenfield's notation If is an L-structure and X is a subset of , then L(X) is the language obtained by adjoining to L a name x for each is the natural expansion of to an L(X)-structure. A structure is an inessential expansion [4, p. 141] of an L-structure if for some .

The notion of an almost strongly minimal theory was introduced in [1]. Such a theory is a particularly simple sort of ℵ1-categorical theory. In [1] we characterized this simplicity in terms of the Stone space of models of T. Here, we characterize almost strongly minimal theories which are not ℵ0-categorical in terms of D. M. R. Park's notion [4] of a theory with the strong elementary intersection property. In addition we prove a useful sufficient condition for an elementary theory to be an almost strongly minimal theory. Our notation is from [1] but this paper is independent of the results proved there. We do assume familiarity with §1 and §2 of [2]. In [4], Park defines a theory T to have the strong elementary intersection property (s.e.i.p.) if for each model C of T and each pair of elementary submodels of C either is an elementary submodel of C. T has the nontrivial strong elementary intersection property (n.s.e.i.p.) if for each triple C, as above Park proves the following two statements equivalent:

The purpose of this paper is twofold. In §1 and §2 which are largely expository we develop the known theory of ℵ1-categoricity in terms of strongly minimal sets. In §3 we settle affirmatively Vaught's conjecture that a complete ℵ1-categorical theory has either just one or just ℵ0 countable models, and in §4 we present an example which serves to illustrate the ideas of §3. As far as we know the only work published on strongly minimal sets is that of Marsh [3]. The present exposition goes beyond [3] in showing that any ℵ-categorical theory has a principal extension in which some formula is strongly minimal.