216 research outputs found

    Toward categoricity for classes with no maximal models

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    We provide here the first steps toward Classification Theory of Abstract Elementary Classes with no maximal models, plus some mild set theoretical assumptions, when the class is categorical in some lambda greater than its Lowenheim-Skolem number. We study the degree to which amalgamation may be recovered, the behaviour of non mu-splitting types. Most importantly, the existence of saturated models in a strong enough sense is proved, as a first step toward a complete solution to the Los Conjecture for these classes

    On categoricity in successive cardinals

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    We investigate, in ZFC, the behavior of abstract elementary classes (AECs) categorical in many successive small cardinals. We prove for example that a universal Lω1,ω\mathbb{L}_{\omega_1, \omega} sentence categorical on an end segment of cardinals below ω\beth_\omega must be categorical also everywhere above ω\beth_\omega. This is done without any additional model-theoretic hypotheses (such as amalgamation or arbitrarily large models) and generalizes to the much broader framework of tame AECs with weak amalgamation and coherent sequences.Comment: 19 page

    Saturation and solvability in abstract elementary classes with amalgamation

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    Theorem.\mathbf{Theorem.} Let KK be an abstract elementary class (AEC) with amalgamation and no maximal models. Let λ>LS(K)\lambda > \text{LS} (K). If KK is categorical in λ\lambda, then the model of cardinality λ\lambda is Galois-saturated. This answers a question asked independently by Baldwin and Shelah. We deduce several corollaries: KK has a unique limit model in each cardinal below λ\lambda, (when λ\lambda is big-enough) KK is weakly tame below λ\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): Corollary.\mathbf{Corollary.} Let KK be an AEC with amalgamation and no maximal models. Let λ>μ>LS(K)\lambda > \mu > \text{LS} (K). If KK is solvable in λ\lambda, then KK is solvable in μ\mu.Comment: 19 page

    On the structure of categorical abstract elementary classes with amalgamation

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    For KK 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. Theorem\mathbf{Theorem} Let μLS(K)\mu \ge \text{LS} (K). If KK is categorical in a λ(2μ)+\lambda \ge \beth_{\left(2^{\mu}\right)^+}, then: 1) Whenever M0,M1,M2KμM_0, M_1, M_2 \in K_\mu are such that M1M_1 and M2M_2 are limit over M0M_0, we have M1M0M2M_1 \cong_{M_0} M_2. 2) If μ>LS(K)\mu > \text{LS} (K), the model of size λ\lambda is μ\mu-saturated. 3) If μ(2LS(K))+\mu \ge \beth_{(2^{\text{LS} (K)})^+} and λ(2μ+)+\lambda \ge \beth_{\left(2^{\mu^+}\right)^+}, then there exists a type-full good μ\mu-frame with underlying class the saturated models in KμK_\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

    Shelah's eventual categoricity conjecture in tame AECs with primes

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    A new case of Shelah's eventual categoricity conjecture is established: Theorem\mathbf{Theorem} Let KK be an AEC with amalgamation. Write H2:=(2(2LS(K))+)+H_2 := \beth_{\left(2^{\beth_{\left(2^{\text{LS} (K)}\right)^+}}\right)^+}. Assume that KK is H2H_2-tame and KH2K_{\ge H_2} has primes over sets of the form M{a}M \cup \{a\}. If KK is categorical in some λ>H2\lambda > H_2, then KK is categorical in all λH2\lambda' \ge H_2. The result had previously been established when the stronger locality assumptions of full tameness and shortness are also required. An application of the method of proof of the theorem is that Shelah's categoricity conjecture holds in the context of homogeneous model theory (this was known, but our proof gives new cases): Theorem\mathbf{Theorem} Let DD be a homogeneous diagram in a first-order theory TT. If DD is categorical in a λ>T\lambda > |T|, then DD is categorical in all λmin(λ,(2T)+)\lambda' \ge \min (\lambda, \beth_{(2^{|T|})^+}).Comment: 16 pages. Generalizes arXiv:1506.0702

    Shelah's eventual categoricity conjecture in universal classes: part I

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    We prove: Theorem\mathbf{Theorem} Let KK be a universal class. If KK is categorical in cardinals of arbitrarily high cofinality, then KK is categorical on a tail of cardinals. The proof stems from ideas of Adi Jarden and Will Boney, and also relies on a deep result of Shelah. As opposed to previous works, the argument is in ZFC and does not use the assumption of categoricity in a successor cardinal. The argument generalizes to abstract elementary classes (AECs) that satisfy a locality property and where certain prime models exist. Moreover assuming amalgamation we can give an explicit bound on the Hanf number and get rid of the cofinality restrictions: Theorem\mathbf{Theorem} Let KK be an AEC with amalgamation. Assume that KK is fully LS(K)\operatorname{LS} (K)-tame and short and has primes over sets of the form M{a}M \cup \{a\}. Write H2:=(2(2LS(K))+)+H_2 := \beth_{\left(2^{\beth_{\left(2^{\operatorname{LS} (K)}\right)^+}}\right)^+}. If KK is categorical in a λ>H2\lambda > H_2, then KK is categorical in all λH2\lambda' \ge H_2.Comment: 51 page

    Equivalent definitions of superstability in tame abstract elementary classes

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    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: Corollary\mathbf{Corollary} Let KK be a tame AEC with a monster model. Assume that KK is stable in a proper class of cardinals. The following are equivalent: 1) For all high-enough λ\lambda, KK has no long splitting chains. 2) For all high-enough λ\lambda, there exists a good λ\lambda-frame on a skeleton of KλK_\lambda. 3) For all high-enough λ\lambda, KK has a unique limit model of cardinality λ\lambda. 4) For all high-enough λ\lambda, KK has a superlimit model of cardinality λ\lambda. 5) For all high-enough λ\lambda, the union of any increasing chain of λ\lambda-saturated models is λ\lambda-saturated. 6) There exists μ\mu such that for all high-enough λ\lambda, KK is (λ,μ)(\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

    Categoricity may fail late

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    We build an example that generalizes math.LO/9201240 to uncountable cases. In particular, our example yields a sentence psi in L_{(2^lambda)^+, omega} that is categorical in lambda, lambda^+, ..., lambda^{+k} but not in beth_{k+1}(lambda)^+. This is connected with the Los Conjecture and with Shelah's own conjecture and construction of excellent classes for the psi in L_{omega_1,omega} case

    Forking and superstability in tame AECs

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    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

    The categoricity spectrum of large abstract elementary classes

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    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
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