This paper is devoted to the proof of the following upward categoricity theorem: Let K be a tame abstract elementary class with amalgamation, arbitrarily large models, and countable Löwenheim-Skolem number. If K is categorical in ‮א‬₁ then K is categorical in every uncountable cardinal. More generally, we prove that if K is categorical in a successor cardinal λ⁺ then K is categorical everywhere above λ⁺.

Let K be the class of atomic models of a countable first order theory. We prove that if K is excellent and categorical in some uncountable cardinal, then each model is prime and minimal over the basis of a definable pregeometry given by a quasiminimal set. This implies that K is categorical in all uncountable cardinals. We also introduce a U-rank to measure the complexity of complete types over models. We prove that the U-rank has the usual additivity properties, that (...) quasiminimal types have U-rank 1, and that the U-rank of any type is finite in the uncountably categorical, excellent case. However, in contrast to the first order case, the supremum of the U-rank over all types may be ω (and is not achieved). We illustrate the theory with the example of free groups, and Zilber's pseudo analytic structures. (shrink)

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 0-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 0-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 as well as negative results . We also give a proof of a Two Cardinal Theorem for this context. (shrink)

We present Saharon Shelah's Stability Spectrum and Homogeneity Spectrum theorems, as well as the equivalence between the order property and instability in the framework of Finite Diagrams. Finite Diagrams is a context which generalizes the first order case. Localized versions of these theorems are presented. Our presentation is based on several papers; the point of view is contemporary and some of the proofs are new. The treatment of local stability in Finite Diagrams is new.

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.

Djordjević [Dj 1] proved that under natural technical assumptions, if a complete L n -theory is stable and has amalgamation over sets, then it has arbitrarily large finite models. We extend his study and prove the existence of arbitrarily large finite models for classes of models of L n -theories (maybe omitting types) under weaker amalgamation properties. In particular our analysis covers the case of vector spaces.

We study the class of elementary submodels of a large superstable homogeneous model. We introduce a rank which is bounded in the superstable case, and use it to define a dependence relation which shares many (but not all) of the properties of forking in the first order case. The main difference is that we do not have extension over all sets. We also present an example of Shelah showing that extension over all sets may not hold for any dependence relation (...) for superstable homogeneous models. (shrink)

This paper is concerned with extensions of geometric stability theory to some nonelementary classes. We prove the following theorem:Theorem. Let [Formula: see text] be a large homogeneous model of a stable diagram D. Let p, q ∈ SD(A), where p is quasiminimal and q unbounded. Let [Formula: see text] and [Formula: see text]. Suppose that there exists an integer n < ω such that [Formula: see text] for any independent a1, …, an∈ P and finite subset C ⊆ Q, but (...) [Formula: see text] for some independent a1, …, an, an+1∈ P and some finite subset C ⊆ Q.Then [Formula: see text] interprets a group G which acts on the geometry P′ obtained from P. Furthermore, either [Formula: see text] interprets a non-classical group, or n = 1,2,3 and•If n = 1 then G is abelian and acts regularly on P′.•If n = 2 the action of G on P′ is isomorphic to the affine action of K ⋊ K* on the algebraically closed field K.•If n = 3 the action of G on P′ is isomorphic to the action of PGL2(K) on the projective line ℙ1(K) of the algebraically closed field K.We prove a similar result for excellent classes. (shrink)

. We study a local version of the order property in several frameworks, with an emphasis on frameworks where the compactness theorem fails: (1) Inside a fixed model, (2) for classes of models where the compactness theorem fails and (3) for the first order case. Appropriate localizations of the order property, the independence property, and the strict order property are introduced. We are able to generalize some of the results that were known in the case of local stability for the (...) first order theories, and for stability for nonelementary classes (existence of indiscernibles, existence of averages, stability spectrum, equivalence between order and instability). In the first order case, we also prove the local version of Shelah's Trichotomy Theorem. Finally, as an application, we give a new characterization of stable types when the ambient first order theory is simple. (shrink)

The Capability Approach is in the first place an approach to the evaluation of individual well-being and social welfare. Many disciplines refer to the CA, first and foremost welfare economics, development studies and political philosophy. Educational theory was not among the first disciplines that took notice of the CA, but has a rising interest in it. This paper argues that the CA would also profit from looking into educational theory. The first part of the paper shows why and where educational (...) theory—or more precisely: a theory of learning—is missing in the CA. This is done in three steps: the first section gives a brief overview of the core concepts of Sen’s CA. Section “Capability and Choosing” focuses on the role of choosing in the CA. It states the views of Sen and Nussbaum on choosing and shows the shortcomings in their appreciation of choosing. In consequence, the third section derives some demands on a theory of learning in the CA. The second part of the paper presents Dewey’s educational theory on experience as a possible starting point when looking for a learning theory that lends itself to the integration in the CA. Section “Opportunity of Choosing, Experience and Education” introduces Dewey’s conception of experience, freedom of the learner, conditions of experience and education. Section “Capability and Experience” discusses how Dewey’s concepts fit into the CA. On the first glance, there are three points in which the CA and Dewey’s concepts match: the importance of freedom for human life, the role of participation in education and the need to take internal and external factors as well as their interaction into account in assessing choice situations. This establishes a basis for linking both theories. Yet, more research is needed to explore the issue further. Section “Conclusion and Outlook” concludes and sketches the lines for future research. (shrink)

We introduce Lascar strong types in excellent classes and prove that they coincide with the orbits of the group generated by automorphisms fixing a model. We define a new independence relation using Lascar strong types and show that it is well-behaved over models, as well as over finite sets. We then develop simplicity and show that, under simplicity, the independence relation satisfies all the properties of nonforking in a stable first order theory. Further, simplicity for an excellent class, as well (...) as the independence relation itself, is uniquely determined. Finally, we show that an excellent class is simple if and only if it has extensible U-rank . We deduce that any excellent class of finite U-rank is simple, and that any uncountably categorical excellent class has an expansion with countably many constants which is simple. (shrink)

We study the class of elementary submodels of a large superstable homogeneous model. We introduce a rank which is bounded in the superstable case, and use it to define a dependence relation which shares many (but not all) of the properties of forking in the first order case. The main difference is that we do not have extension over all sets. We also present an example of Shelah showing that extension over all sets may not hold for any dependence relation (...) for superstable homogeneous models. (shrink)

We give a complete and elementary proof of the following upward categoricity theorem: let be a local abstract elementary class with amalgamation and joint embedding, arbitrarily large models, and countable Löwenheim–Skolem number. If is categorical in 1 then is categorical in every uncountable cardinal. In particular, this provides a new proof of the upward part of Morley’s theorem in first order logic without any use of prime models or heavy stability theoretic machinery.

We show that any (atomic) excellent class K can be expanded with hyperimaginaries to form an (atomic) excellent class Keq which has canonical bases. When K is, in addition, of finite U-rank, then Keq is also simple and has a full canonical bases theorem. This positive situation contrasts starkly with homogeneous model theory for example, where the eq-expansion may fail to be homogeneous. However, this paper shows that expanding an ω-stable, homogeneous class K gives rise to an excellent class, which (...) is simple if K is of finite U-rank. (shrink)

We study simplicity and stability in some large strongly homogeneous expansions of Hilbert spaces. Our approach to simplicity is that of Buechler and Lessmann 69). All structures we consider are shown to have built-in canonical bases.

We present a new proof of the existence of Morley sequences in simple theories. We avoid using the Erdős–Rado theorem and instead use only Ramsey’s theorem and compactness. The proof shows that the basic theory of forking in simple theories can be developed using only principles from “ordinary mathematics,” answering a question of Grossberg, Iovino, and Lessmann, as well as a question of Baldwin.