Superstability and Symmetry

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 Logi

Limit Models in Metric Abstract Elementary Classes: the Categorical case

We study versions of limit models adapted to the context of *metric abstract elementary classes*. Under categoricity and superstability-like assumptions, we generalize some theorems from [GrVaVi]. We prove criteria for existence and uniqueness of limit models in the metric context.Comment: 25 pages, 5 figure

Equivalent definitions of superstability in tame abstract elementary classes

In the context of abstract elementary classes (AECs) with a monster model, several possible definitions of superstability have appeared in the literature. Among them are no long splitting chains, uniqueness of limit models, and solvability. Under the assumption that the class is tame and stable, we show that (asymptotically) no long splitting chains implies solvability and uniqueness of limit models implies no long splitting chains. Using known implications, we can then conclude that all the previously-mentioned definitions (and more) are equivalent: $\mathbf{Corollary}$ Let $K$ be a tame AEC with a monster model. Assume that $K$ is stable in a proper class of cardinals. The following are equivalent: 1) For all high-enough $\lambda$, $K$ has no long splitting chains. 2) For all high-enough $\lambda$, there exists a good $\lambda$-frame on a skeleton of $K_\lambda$. 3) For all high-enough $\lambda$, $K$ has a unique limit model of cardinality $\lambda$. 4) For all high-enough $\lambda$, $K$ has a superlimit model of cardinality $\lambda$. 5) For all high-enough $\lambda$, the union of any increasing chain of $\lambda$-saturated models is $\lambda$-saturated. 6) There exists $\mu$ such that for all high-enough $\lambda$, $K$ is $(\lambda, \mu)$-solvable. This gives evidence that there is a clear notion of superstability in the framework of tame AECs with a monster model.Comment: 24 page

Saturation and solvability in abstract elementary classes with amalgamation

$\mathbf{Theorem.}$ Let $K$ be an abstract elementary class (AEC) with amalgamation and no maximal models. Let $\lambda > \text{LS} (K)$. If $K$ is categorical in $\lambda$, then the model of cardinality $\lambda$ is Galois-saturated. This answers a question asked independently by Baldwin and Shelah. We deduce several corollaries: $K$ has a unique limit model in each cardinal below $\lambda$, (when $\lambda$ is big-enough) $K$ is weakly tame below $\lambda$, and the thresholds of several existing categoricity transfers can be improved. We also prove a downward transfer of solvability (a version of superstability introduced by Shelah): $\mathbf{Corollary.}$ Let $K$ be an AEC with amalgamation and no maximal models. Let $\lambda > \mu > \text{LS} (K)$. If $K$ is solvable in $\lambda$, then $K$ is solvable in $\mu$.Comment: 19 page

Forking and superstability in tame AECs

We prove that any tame abstract elementary class categorical in a suitable cardinal has an eventually global good frame: a forking-like notion defined on all types of single elements. This gives the first known general construction of a good frame in ZFC. We show that we already obtain a well-behaved independence relation assuming only a superstability-like hypothesis instead of categoricity. These methods are applied to obtain an upward stability transfer theorem from categoricity and tameness, as well as new conditions for uniqueness of limit models.Comment: 33 page

Symmetry in abstract elementary classes with amalgamation

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

Symmetry and the Union of Saturated Models in Superstable Abstract Elementary Classes

Our main result (Theorem 1) suggests a possible dividing line ($\mu$-superstable $+$ $\mu$-symmetric) for abstract elementary classes without using extra set-theoretic assumptions or tameness. This theorem illuminates the structural side of such a dividing line. Theoerem 1: Let $\mathcal{K}$ be an abstract elementary class with no maximal models of cardinality $\mu^+$ which satisfies the joint embedding and amalgamation properties. Suppose $\mu\geq LS(\mathcal{K})$. If $\mathcal{K}$ is $\mu$- and $\mu^+$-superstable and satisfies $\mu^+$-symmetry, then for any increasing sequence $\langle M_i\in\mathcal{K}_{\geq\mu^{+}}\mid i<\theta<(\sup\|M_i\|)^+\rangle$ of $\mu^+$-saturated models, $\bigcup_{i<\theta}M_i$ is $\mu^+$-saturated. We also apply results of VanDieren's Superstability and Symmetry paper and use towers to transfer symmetry from $\mu^+$ down to $\mu$ in abstract elementary classes which are both $\mu$- and $\mu^+$-superstable: Theorem 2: Suppose $\mathcal{K}$ is an abstract elementary class satisfying the amalgamation and joint embedding properties and that $\mathcal{K}$ is both $\mu$- and $\mu^+$-superstable. If $\mathcal{K}$ has symmetry for non-$\mu^+$-splitting, then $\mathcal{K}$ has symmetry for non-$\mu$-splitting.Comment: This paper is a synthesis of arXiv:1507.01991 and arXiv:1507.0198

On the structure of categorical abstract elementary classes with amalgamation

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:1508.0325

A survey on tame abstract elementary classes

Tame abstract elementary classes are a broad nonelementary framework for model theory that encompasses several examples of interest. In recent years, progress toward developing a classification theory for them have been made. Abstract independence relations such as Shelah's good frames have been found to be key objects. Several new categoricity transfers have been obtained. We survey these developments using the following result (due to the second author) as our guiding thread: $\mathbf{Theorem}$ If a universal class is categorical in cardinals of arbitrarily high cofinality, then it is categorical on a tail of cardinals.Comment: 84 page

The categoricity spectrum of large abstract elementary classes

The categoricity spectrum of a class of structures is the collection of cardinals in which the class has a single model up to isomorphism. Assuming that cardinal exponentiation is injective (a weakening of the generalized continuum hypothesis, GCH), we give a complete list of the possible categoricity spectrums of an abstract elementary class with amalgamation and arbitrarily large models. Specifically, the categoricity spectrum is either empty, an end segment starting below the Hanf number, or a closed interval consisting of finite successors of the L\"owenheim-Skolem-Tarski number (there are examples of each type). We also prove (assuming a strengthening of the GCH) that the categoricity spectrum of an abstract elementary class with no maximal models is either bounded or contains an end segment. This answers several longstanding questions around Shelah's categoricity conjecture.Comment: 50 page