Search

We study uniqueness of limit models in abstract elementary classes (AECs) with no maximal models. We prove (assuming instances of diamonds) that categoricity in a cardinal of the form $\mu^{+(n + 1)}$ implies the uniqueness of limit models of cardinality $\mu^{+}, \mu^{++}, \ldots, \mu^{+n}$. This sheds light on a paper of Shelah and Villaveces, who were the first to consider uniqueness of limit models in this context. We also prove that (again assuming instances of diamonds) in an AEC with no maximal models, tameness (a locality property for types) together with categoricity in a proper class of cardinals imply categoricity on a tail of cardinals. This is the first categoricity transfer theorem in that setup and answers a question of Baldwin., Comment: Paper withdrawn due to a fatal mistake

Starting from an abstract elementary class with no maximal models, Shelah and Villaveces have shown (assuming instances of diamond) that categoricity implies a superstability-like property for a certain independence relation called nonsplitting. We generalize their result as follows: given an abstract notion of independence for Galois (orbital) types over models, we derive that the notion satisfies a superstability property provided that the class is categorical and satisfies a weakening of amalgamation. This extends the Shelah-Villaveces result (the independence notion there was splitting) as well as a result of the first and second author where the independence notion was coheir. The argument is in ZFC and fills a gap in the Shelah-Villaveces proof., Comment: 14 pages

In this paper we examine the task set forth by Shelah and Villaveces in \cite{ShVi} of proving the uniqueness of limit models of cardinality $\mu$ in $\lambda$-categorical abstract elementary classes with no maximal models, where $\lambda$ is some cardinal larger than $\mu$. In \cite{Va} and \cite{Va-errata} we identified several gaps in the approach outlined in \cite{ShVi}, and we added the assumption that the union of an increasing chain of limit models is a limit model. Here we replace this assumption with the seemingly weaker statement that the union of an increasing and continuous chain of limit models is an amalgamation base. Moreover, we prove that this assumption is not only sufficient but is necessary to settle the uniqueness of limit models problem attempted in \cite{ShVi} for $\lambda=\mu^{+n}$ when $0

For $K$ an abstract elementary class with amalgamation and no maximal models, we show that categoricity in a high-enough cardinal implies structural properties such as the uniqueness of limit models and the existence of good frames. This improves several classical results of Shelah. $\mathbf{Theorem}$ Let $\mu \ge \text{LS} (K)$. If $K$ is categorical in a $\lambda \ge \beth_{\left(2^{\mu}\right)^+}$, then: 1) Whenever $M_0, M_1, M_2 \in K_\mu$ are such that $M_1$ and $M_2$ are limit over $M_0$, we have $M_1 \cong_{M_0} M_2$. 2) If $\mu > \text{LS} (K)$, the model of size $\lambda$ is $\mu$-saturated. 3) If $\mu \ge \beth_{(2^{\text{LS} (K)})^+}$ and $\lambda \ge \beth_{\left(2^{\mu^+}\right)^+}$, then there exists a type-full good $\mu$-frame with underlying class the saturated models in $K_\mu$. Our main tool is the symmetry property of splitting (previously isolated by the first author). The key lemma deduces symmetry from failure of the order property., Comment: 19 pages. This has since been merged with arXiv:1508.03252

In this paper, we examine the locality condition for non-splitting and determine the level of uniqueness of limit models that can be recovered in some stable, but not superstable, abstract elementary classes. In particular we prove (note that no tameness is assumed): Suppose that $\mathcal{K}$ is an abstract elementary class satisfying 1. the joint embedding and amalgamation properties with no maximal model of cardinality $\mu$. 2. stability in $\mu$. 3. $\kappa^*_\mu(\mathcal{K})<\mu^+$. 4. continuity for non-$\mu$-splitting (i.e. if $p\in\text{ga-S}(M)$ and $M$ is a limit model witnessed by $\langle M_i\mid i<\alpha\rangle$ for some limit ordinal $\alpha<\mu^+$ and there exists $N \prec M_0$ so that $p\restriction M_i$ does not $\mu$-split over $N$ for all $i<\alpha$, then $p$ does not $\mu$-split over $N$). For $\theta$ and $\delta$ limit ordinals $<\mu^+$ both with cofinality $\geq\kappa^*_\mu(\mathcal{K})$, if $\mathcal{K}$ satisfies symmetry for non-$\mu$-splitting (or just $(\mu,\delta)$-symmetry), then, for any $M_1$ and $M_2$ that are $(\mu,\theta)$ and $(\mu,\delta)$-limit models over $M_0$, respectively, we have that $M_1$ and $M_2$ are isomorphic over $M_0$., Comment: This article generalizes some results from arXiv:1507.01990

This paper is part of a program initiated by Saharon Shelah to extend the model theory of first order logic to the non-elementary setting of abstract elementary classes (AECs). An abstract elementary class is a semantic generalization of the class of models of a complete first order theory with the elementary substructure relation. We examine the symmetry property of splitting (previously isolated by the first author) in AECs with amalgamation that satisfy a local definition of superstability. The key results are a downward transfer of symmetry and a deduction of symmetry from failure of the order property. These results are then used to prove several structural properties in categorical AECs, improving classical results of Shelah who focused on the special case of categoricity in a successor cardinal. We also study the interaction of symmetry with tameness, a locality property for Galois (orbital) types. We show that superstability and tameness together imply symmetry. This sharpens previous work of Boney and the second author., Comment: 37 pages. This merges with arXiv:1509.01488 . Was previously titled "Transferring symmetry downward and applications"

This paper continues the study of superstability in abstract elementary classes (AECs) satisfying the amalgamation property. In particular, we consider the definition of $\mu$-superstability which is based on the local character characterization of superstability from first order logic. Not only is $\mu$-superstability a potential dividing line in the classification theory for AECs, but it is also a tool in proving instances of Shelah's Categoricity Conjecture. In this paper, we introduce a formulation, involving towers, of symmetry over limit models for $\mu$-superstable abstract elementary classes. We use this formulation to gain insight into the problem of the uniqueness of limit models for categorical AECs., Comment: Accepted for publication by Annals of Pure and Applied Logic

In this paper we prove: Theorem 1. Let $\mathcal{K}$ be an abstract elementary class which satisfies the joint embedding and amalgamation properties. Suppose $\lambda>\mu\geq LS(\mathcal{K})$ and $\theta$ is a limit ordinal $<\lambda^+$. If $\mathcal{K}$ is $\mu$ superstable and $\mu^+$-superstable and satisfies $\mu^+$-symmetry, then for any increasing sequence $\langle M_i\mid i<\theta\rangle$ of $\mu^+$-saturated models of cardinality $\lambda$, the model $\bigcup_{i<\theta}M_i$ is $\mu^+$-saturated., Comment: This paper has been combined with another paper. The content appears in arXiv:1512.01786