Abstract not found

Abstract. This is a non-standard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried to solve them, and sometimes with failed tries, anecdote and opinion. So the discussion is quite personal, in other words, egocentric and somewhat accidental. As we discuss many problems, history and side references are erratic, usually kept at a minimum (“see... ” means: see the references there and possibly the paper itself). The base were lectures in Rutgers Fall ’97 and reflect my knowledge then. The other half, [122], concentrating on model theory, will subsequently appear. I thank Andreas Blass and Andrzej Ros̷lanowski for many helpful comments. (666) revision:2001-11-12 modified:2003-11-18

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.

Consider the classical universal cover of the one dimensional complex torus C ∗ , which gives us the exact sequence 0 − → Z i

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# 1 ,# . While only very basic theorems are proved, an effort is made to present number of different technologies: Flavors of weak diamond, models of weak set theories, and commutative diagrams. We focus in issues involving existence of Galois types, extensions of types and Galois-stability.

elementary classes. Theorem 0.1. Suppose that K is a χ-tame abstract elementary class and satisfies the amalgamation and joint embedding properties and has arbitrarily large models. Let λ ≥ Max{χ, LS(K) +}. If K is categorical in λ and λ +, then K is categorical in λ ++. Combining this theorem with some results from [Sh 394], we derive a form of Shelah’s Categoricity Conjecture for tame abstract elementary classes: Corollary 0.2. Suppose K is a χ-tame abstract elementary class satisfying the amalgamation and joint embedding properties. Let µ0:= Hanf(K). If χ ≤ ℶ (2 µ 0) + and K is categorical in some λ +> ℶ (2 µ 0) +, then K is categorical in µ for all µ> ℶ (2 µ 0) +.

Abstract. We prove that from categoricity in λ + we can get categoricity in all cardinals ≥ λ + in a χ-tame abstract elementary classes which has arbitrarily large models and satisfies the amalgamation and joint embedding properties, provided λ> LS(K) and λ ≥ χ. For the missing case when λ = LS(K), we prove that K is totally categorical provided that K is categorical in LS(K) and LS(K) +. 1. introduction The benchmark of progress in the development of a model theory for abstract elementary classes (AECs) is Shelah’s Categoricity Conjecture. Conjecture 1.1. Let K be an abstract elementary class. If K is categorical in some λ> Hanf(K) 1, then for every µ ≥ Hanf(K), K is categorical in µ. With the exception of [MaSh], [KoSh], [Sh 576], [ShVi] and [Va] in which extra set theoretic assumptions are made, all work towards Shelah’s Categoricity Conjecture has taken place under the assumption of the amalgamation property. An AEC satisfies the amalgamation property if for every triple of models M0,M1,M2 in which M0 ≺K M1 and M0 ≺K M2 there exist K-mappings g1 and g2 and an amalgam N ∈ K such that the diagram below commutes.

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 continue the work of [2] and prove that for λ successor, a λ-categorical theory T in Lκ ∗,ω is µ-categorical for every µ,µ ≤ λ which is above the (2 LS(T) ) +-beth cardinal. (472) revision:2001-11-12 modified:2001-11-12 We deal here with the categoricity spectrum of theories T in the logic: Lκ ∗,ω with κ ∗ measurable and more generally, continued the attempts develop classification theory of non elementary classes in particular non forking. Makkai and Shelah

Abstract. The results in this paper are in a context of abstract elementary classes identified by Shelah and Villaveces in which the amalgamation property is not assumed. The long-term goal is to solve Shelah’s Categoricity Conjecture in this context. Here we tackle a problem of Shelah and Villaveces by proving that in their context, the uniqueness of limit models follows from categoricity under the assumption that the subclass of amalgamation bases is closed under unions of bounded, ≺Kincreasing chains.