On cardinalities in quotients of inverse limits of groups

Let lambda be aleph_0 or a strong limit of cofinality aleph_0. Suppose that (G_m,p_{m,n}:m =< n<omega) and (H_m,p^t_{m,n}: m=< n < omega) are projective systems of groups of cardinality less than lambda and suppose that for every nG_n such that all the diagrams commute. If for every mu<lambda there exists (f_i in G_omega:i<mu) such that for distinct i,j we have: f_i f_j^{-1} notin h_omega(H_omega), then there exists (f_i in G_omega:i<2^lambda) such that for distinct i,j we have f_i f_j^{-1} notin h_omega(H_omega)

Shelah's Categoricity Conjecture from a successor for Tame Abstract Elementary Classes

Let K be an Abstract Elemenetary Class satisfying the amalgamation and the joint embedding property, let \mu be the Hanf number of K. Suppose K is tame. MAIN COROLLARY: (ZFC) If K is categorical in a successor cardinal bigger than \beth_{(2^\mu)^+} then K is categorical in all cardinals greater than \beth_{(2^\mu)^+}. This is an improvment of a Theorem of Makkai and Shelah ([Sh285] who used a strongly compact cardinal for the same conclusion) and Shelah's downward categoricity theorem for AECs with amalgamation (from [Sh394]).Comment: 19 page

Galois-stability for Tame Abstract Elementary Classes

We introduce tame abstract elementary classes as a generalization of all cases of abstract elementary classes that are known to permit development of stability-like theory. In this paper we explore stability results in this context. We assume that \K is a tame abstract elementary class satisfying the amalgamation property with no maximal model. The main results include: (1) Galois-stability above the Hanf number implies that \kappa(K) is less than the Hanf number. Where \kappa(K) is the parallel of \kapppa(T) for f.o. T. (2) We use (1) to construct Morley sequences (for non-splitting) improving previous results of Shelah (from Sh394) and Grossberg & Lessmann. (3) We obtain a partial stability-spectrum theorem for classes categorical above the Hanf number.Comment: 23 page

On Hanf numbers of the infinitary order property

We study several cardinal, and ordinal--valued functions that are relatives of Hanf numbers. Let kappa be an infinite cardinal, and let T subseteq L_{kappa^+, omega} be a theory of cardinality <= kappa, and let gamma be an ordinal >= kappa^+. For example we look at (1) mu_{T}^*(gamma, kappa):= min {mu^* for all phi in L_{infinity, omega}, with rk(phi)< gamma, if T has the (phi, mu^*)-order property then there exists a formula phi'(x;y) in L_{kappa^+, omega}, such that for every chi >= kappa, T has the (phi', chi)-order property}; and (2) mu^*(gamma, kappa):= sup{mu_T^*(gamma, kappa)| T in L_{kappa^+,omega}}

Excellent Abstract Elementary Classes are tame

The assumption that an AEC is tame is a powerful assumption permitting development of stability theory for AECs with the amalgamation property. Lately several upward categoricity theorems were discovered where tameness replaces strong set-theoretic assumptions. We present in this article two sufficient conditions for tameness, both in form of strong amalgamation properties that occur in nature. One of them was used recently to prove that several Hrushovski classes are tame. This is done by introducing the property of weak $(\mu,n)$-uniqueness which makes sense for all AECs (unlike Shelah's original property) and derive it from the assumption that weak (\LS(\K),n)-uniqueness, (\LS(\K),n)-symmetry and (\LS(\K),n)-existence properties hold for all $n<\omega$. The conjunction of these three properties we call \emph{excellence}, unlike \cite{Sh 87b} we do not require the very strong (\LS(\K),n)-uniqueness, nor we assume that the members of \K are atomic models of a countable first order theory. We also work in a more general context than Shelah's good frames.Comment: 26 page

Classification Theory for Abstract Elementary Classes

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&apos;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