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Abstract. We show that the expressive power of first-order logic over finite models embedded in a model M is determined by stability-theoretic properties of M. In particular, we show that if M is stable, then every class of finite structures that can be defined by embedding the structures in M, can be defined in pure first-order logic. We also show that if M does not have the independence property, then any class of finite structures that can be defined by embedding the structures in M, can be defined in first-order logic over a dense linear order. This extends known results on the definability of classes of finite structures and ordered finite structures in the setting of embedded finite models. These results depend on several results in infinite model theory. Let I be a set of indiscernibles in a model M and suppose (M,I) is elementarily equivalent to (M1,I1) whereM1 is |I1 | +-saturated. If M is stable and (M,I) is saturated, then every permutation of I extends to an automorphism of M and the theory of (M,I) isstable. LetI be a sequence of <-indiscernibles in a model M, which does not have the independence property, and suppose (M,I) is elementarily equivalent to (M1,I1) where(I1,<) is a complete dense linear order and M1 is |I1 | +-saturated. Then (M, I)-types over I are order-definable and if (M, I) isℵ1-saturated, every order preserving permutation of I can be extended to a back-and-forth system. 1.

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Abstract. This article surveys the model theory of differentially closed fields, an interesting setting where one can use model-theoretic methods to obtain algebraic information. The article concludes with one example showing how this information can be used in diophantine applications. A differential field is a field K equipped with a derivation δ: K → K; recall that this means that, for x, y ∈ K, we have δ(x + y) = δ(x) + δ(y) and δ(xy) = x δ(y) + yδ(x). Roughly speaking, such a field is called differentially closed when it contains enough solutions of ordinary differential equations. This setting allows one to use model-theoretic methods, and particularly dimensiontheoretic ideas, to obtain interesting algebraic information. In this lecture I give a survey of the model theory of differentially closed fields, concluding with an example — Hrushovski’s proof of the Mordell–Lang conjecture in characteristic zero — showing how model-theoretic methods in this area can be used in diophantine applications. I will not give the proofs of the main theorems. Most of the material in Sections 1–3 can be found in [Marker

Shelah’s bibliography. This the first of a series of articles dealing with abstract classification theory. The apparatus to assign systems of cardinal invariants to models of a first order theory (or determine its impossibility) is developed in [5]. It is natural to try to extend this theory to classes of models which are described in other ways. Work on the classification theory for nonelementary classes [8] and for universal classes [9] led to the conclusion that an axiomatic approach provided the best setting for developing a theory of wider application. This approach is reminiscent of the early work of Fraissé and Jónson on the existence of homogeneous-universal models. As this will be a long project it seems appropriate to report our progress as we go along. In large part this series of articles will parallel the development in [9]. A survey of that paper which could serve as an introduction to this one is [1]. The first chapter of this article corresponds to Section 2 of [9]. In it we describe the axioms on which the remainder of the article depends

ABSTRACT. The study of classes of models of a finite diagram was initiated by S. Shelah in 1969. A diagram D is a set of types over the empty set, and the class of models of the diagram D consists of the models of T which omit all the types not in D. In this work, we introduce a natural dependence relation on the subsets of the models for the ℵ0-stable case which share many of the formal properties of forking. This is achieved by considering a rank for this framework which is bounded when the diagram D is ℵ0-stable. We can also obtain pregeometries with respect to this dependence relation. The dependence relation is the natural one induced by the rank, and the pregeometries exist on the set of realizations of types of minimal rank. Finally, these concepts are used to generalize many of the classical results for models of a totally transcendental first-order theory. In fact, strong analogies arise: models are determined by their pregeometries or their relationship with their pregeometries; however the proofs are different, as we do not have compactness. This is illustrated with positive results (categoricity) as well as negative results (construction of nonisomorphic models).

Abstract. We present a self-contained exposition of the basic aspects of simple theories while developing the fundamentals of forking calculus. We expound also the deeper aspects of S. Shelah’s 1980 paper Simple unstable theories. The concept of weak dividing has been replaced with that of forking. The exposition is from a contemporary perspective and takes into account contributions due to S. Buechler, E. Hrushovski, B. Kim, O. Lessmann, S. Shelah and A. Pillay.

Zilber’s proposes [60] to prove ‘canonicity results for pseudo-analytic ’ structures. Informally, ‘canonical means the theory of the structure in a suitable possibly infinitary language (see Section 2) has one model in each uncountable power ’ while ‘pseudoanalytic means the model of power 2 ℵ0 can be taken as a reduct of an expansion of the complex numbers by analytic functions’. This program interacts with two other lines of research. First is the general study of categoricity theorems in infinitary languages. After initial results by Keisler, reported in [31], this line was taken up in a long series of works by Shelah. We place Zilber’s work in this context. The second direction stems from Hrushovski’s construction of a counterexample to Zilber’s conjecture that every strongly minimal set is ‘trivial’, ‘vector space-like’, or ‘field-like’. This construction turns out to be very concrete example of the Abstract Elementary Classes which arose in Shelah’s analysis. This paper examines the intertwining of these three themes. The study of (C, +, ·, exp) leads one immediately to some extension of first order logic; the integers with all their arithmetic are first order definable in (C, +, ·, exp). Thus, the first order theory of complex exponentiation is horribly complicated; it is certainly unstable and so can’t be first order categorical. One solution is to use infinitary logic to pin down the pathology. Insist that the kernel of the exponential map is fixed as a single copy of the integers while allowing the rest of the structure to grow. We describe in Section 5 Zilber’s program to

For the specification of abstract data types, quite a number of logical systems have been developed. In this work, we will try to give an overview over this variety. As a prerequisite, we first study notions of {\em representation} and embedding between logical systems, which are formalized as {\em institutions} here. Different kinds of representations will lead to a looser or tighter connection of the institutions, with more or less good possibilities of faithfully embedding the semantics and of re-using proof support. In the second part, we then perform a detailed ``empirical'' study of the relations among various well-known institutions of total, order-sorted and partial algebras and first-order structures (all with Horn style, i.e.\ universally quantified conditional, axioms). We thus obtain a {\em graph} of institutions, with different kinds of edges according to the different kinds of representations between institutions studied in the first part. We also prove some separation results, leading to a {\em hierarchy} of institutions, which in turn naturally leads to five subgraphs of the above graph of institutions. They correspond to five different levels of expressiveness in the hierarchy, which can be characterized by different kinds of conditional generation principles. We introduce a systematic notation for institutions of total, order-sorted and partial algebras and first-order structures. The notation closely follows the combination of features that are present in the respective institution. This raises the question whether these combinations of features can be made mathematically precise in some way. In the third part, we therefore study the combination of institutions with the help of so-called parchments (which are certain algebraic presentations of institutions) and parchment morphisms. The present book is a revised version of the author's thesis, where a number of mathematical problems (pointed out by Andrzej Tarlecki) and a number of misuses of the English language (pointed out by Bernd Krieg-Br\"uckner) have been corrected. Also, the syntax of specifications has been adopted to that of the recently developed Common Algebraic Specification Language {\sc Casl} \cite{CASL/Summary,Mosses97TAPSOFT}.