22 found
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  1.  18
    Shelah's eventual categoricity conjecture in universal classes: Part I.Sebastien Vasey - 2017 - Annals of Pure and Applied Logic 168 (9):1609-1642.
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  2.  20
    Forking and superstability in Tame aecs.Sebastien Vasey - 2016 - Journal of Symbolic Logic 81 (1):357-383.
  3.  23
    Canonical forking in AECs.Will Boney, Rami Grossberg, Alexei Kolesnikov & Sebastien Vasey - 2016 - Annals of Pure and Applied Logic 167 (7):590-613.
  4.  23
    Building independence relations in abstract elementary classes.Sebastien Vasey - 2016 - Annals of Pure and Applied Logic 167 (11):1029-1092.
  5.  19
    Toward a stability theory of tame abstract elementary classes.Sebastien Vasey - 2018 - Journal of Mathematical Logic 18 (2):1850009.
    We initiate a systematic investigation of the abstract elementary classes that have amalgamation, satisfy tameness, and are stable in some cardinal. Assuming the singular cardinal hypothesis, we prove a full characterization of the stability cardinals, and connect the stability spectrum with the behavior of saturated models.We deduce that if a class is stable on a tail of cardinals, then it has no long splitting chains. This indicates that there is a clear notion of superstability in this framework.We also present an (...)
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  6.  17
    Superstability from categoricity in abstract elementary classes.Will Boney, Rami Grossberg, Monica M. VanDieren & Sebastien Vasey - 2017 - Annals of Pure and Applied Logic 168 (7):1383-1395.
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  7.  24
    Downward categoricity from a successor inside a good frame.Sebastien Vasey - 2017 - Annals of Pure and Applied Logic 168 (3):651-692.
  8.  37
    Categoricity in multiuniversal classes.Nathanael Ackerman, Will Boney & Sebastien Vasey - 2019 - Annals of Pure and Applied Logic 170 (11):102712.
    The third author has shown that Shelah's eventual categoricity conjecture holds in universal classes: class of structures closed under isomorphisms, substructures, and unions of chains. We extend this result to the framework of multiuniversal classes. Roughly speaking, these are classes with a closure operator that is essentially algebraic closure (instead of, in the universal case, being essentially definable closure). Along the way, we prove in particular that Galois (orbital) types in multiuniversal classes are determined by their finite restrictions, generalizing a (...)
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  9.  15
    Universal classes near ${\aleph _1}$.Marcos Mazari-Armida & Sebastien Vasey - 2018 - Journal of Symbolic Logic 83 (4):1633-1643.
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  10.  25
    Saturation and solvability in abstract elementary classes with amalgamation.Sebastien Vasey - 2017 - Archive for Mathematical Logic 56 (5-6):671-690.
    Theorem 0.1LetK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {K}$$\end{document}be an abstract elementary class with amalgamation and no maximal models. Letλ>LS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda > {LS}$$\end{document}. IfK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {K}$$\end{document}is categorical inλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}, then the model of cardinalityλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}is Galois-saturated.This answers a question asked independently by Baldwin and (...)
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  11.  21
    Abstract elementary classes stable in ℵ0.Saharon Shelah & Sebastien Vasey - 2018 - Annals of Pure and Applied Logic 169 (7):565-587.
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  12.  13
    Induced and higher-dimensional stable independence.Michael Lieberman, Jiří Rosický & Sebastien Vasey - 2022 - Annals of Pure and Applied Logic 173 (7):103124.
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  13.  28
    Good frames in the Hart–Shelah example.Will Boney & Sebastien Vasey - 2018 - Archive for Mathematical Logic 57 (5-6):687-712.
    For a fixed natural number \, the Hart–Shelah example is an abstract elementary class with amalgamation that is categorical exactly in the infinite cardinals less than or equal to \. We investigate recently-isolated properties of AECs in the setting of this example. We isolate the exact amount of type-shortness holding in the example and show that it has a type-full good \-frame which fails the existence property for uniqueness triples. This gives the first example of such a frame. Along the (...)
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  14.  21
    Quasiminimal abstract elementary classes.Sebastien Vasey - 2018 - Archive for Mathematical Logic 57 (3-4):299-315.
    We propose the notion of a quasiminimal abstract elementary class. This is an AEC satisfying four semantic conditions: countable Löwenheim–Skolem–Tarski number, existence of a prime model, closure under intersections, and uniqueness of the generic orbital type over every countable model. We exhibit a correspondence between Zilber’s quasiminimal pregeometry classes and quasiminimal AECs: any quasiminimal pregeometry class induces a quasiminimal AEC, and for any quasiminimal AEC there is a natural functorial expansion that induces a quasiminimal pregeometry class. We show in particular (...)
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  15.  31
    Shelah's eventual categoricity conjecture in tame abstract elementary classes with primes.Sebastien Vasey - 2018 - Mathematical Logic Quarterly 64 (1-2):25-36.
    A new case of Shelah's eventual categoricity conjecture is established: Let be an abstract elementary class with amalgamation. Write and. Assume that is H2‐tame and has primes over sets of the form. If is categorical in some, then is categorical in all. 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 mentioned result is that Shelah's categoricity conjecture holds in the context (...)
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  16.  16
    Building prime models in fully good abstract elementary classes.Sebastien Vasey - 2017 - Mathematical Logic Quarterly 63 (3-4):193-201.
    We show how to build prime models in classes of saturated models of abstract elementary classes (AECs) having a well‐behaved independence relation: Let be an almost fully good AEC that is categorical in and has the ‐existence property for domination triples. For any, the class of Galois saturated models of of size λ has prime models over every set of the form. This generalizes an argument of Shelah, who proved the result when λ is a successor cardinal.
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  17.  36
    Indiscernible Extraction and Morley Sequences.Sebastien Vasey - 2017 - Notre Dame Journal of Formal Logic 58 (1):127-132.
    We present a new proof of the existence of Morley sequences in simple theories. We avoid using the Erdős–Rado theorem and instead use only Ramsey’s theorem and compactness. The proof shows that the basic theory of forking in simple theories can be developed using only principles from “ordinary mathematics,” answering a question of Grossberg, Iovino, and Lessmann, as well as a question of Baldwin.
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  18.  19
    On the uniqueness property of forking in abstract elementary classes.Sebastien Vasey - 2017 - Mathematical Logic Quarterly 63 (6):598-604.
    In the setup of abstract elementary classes satisfying a local version of superstability, we prove the uniqueness property for μ‐forking, a certain independence notion arising from splitting. This had been a longstanding technical difficulty when constructing forking‐like notions in this setup. As an application, we show that the two versions of forking symmetry appearing in the literature (the one defined by Shelah for good frames and the one defined by VanDieren for splitting) are equivalent.
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  19.  15
    Cellular Categories and Stable Independence.Michael Lieberman, Jiří Rosický & Sebastien Vasey - forthcoming - Journal of Symbolic Logic:1-24.
    We exhibit a bridge between the theory of cellular categories, used in algebraic topology and homological algebra, and the model-theoretic notion of stable independence. Roughly speaking, we show that the combinatorial cellular categories (those where, in a precise sense, the cellular morphisms are generated by a set) are exactly those that give rise to stable independence notions. We give two applications: on the one hand, we show that the abstract elementary classes of roots of Ext studied by Baldwin–Eklof–Trlifaj are stable (...)
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  20.  11
    Superstability and Categoricity in Abstract Elementary Classes, Carnegie Mellon University, USA, 2017. Supervised by Rami Grossberg.Christian Rosendal & Sebastien Vasey - 2018 - Bulletin of Symbolic Logic 24 (2):192-194.
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  21.  11
    On categoricity in successive cardinals.Sebastien Vasey - 2020 - Journal of Symbolic Logic:1-19.
    We investigate, in ZFC, the behavior of abstract elementary classes categorical in many successive small cardinals. We prove for example that a universal $\mathbb {L}_{\omega _1, \omega }$ sentence categorical on an end segment of cardinals below $\beth _\omega $ must be categorical also everywhere above $\beth _\omega $. This is done without any additional model-theoretic hypotheses and generalizes to the much broader framework of tame AECs with weak amalgamation and coherent sequences.
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  22.  8
    On categoricity in successive cardinals.Sebastien Vasey - 2022 - Journal of Symbolic Logic 87 (2):545-563.
    We investigate, in ZFC, the behavior of abstract elementary classes categorical in many successive small cardinals. We prove for example that a universal $\mathbb {L}_{\omega _1, \omega }$ sentence categorical on an end segment of cardinals below $\beth _\omega $ must be categorical also everywhere above $\beth _\omega $. This is done without any additional model-theoretic hypotheses and generalizes to the much broader framework of tame AECs with weak amalgamation and coherent sequences.
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