13 found
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  1.  24
    Canonical forking in AECs.Will Boney, Rami Grossberg, Alexei Kolesnikov & Sebastien Vasey - 2016 - Annals of Pure and Applied Logic 167 (7):590-613.
  2.  32
    Tameness from large cardinal axioms.Will Boney - 2014 - Journal of Symbolic Logic 79 (4):1092-1119.
    We show that Shelah’s Eventual Categoricity Conjecture for successors follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC withLS below a strongly compact cardinalκis <κ-tame and applying the categoricity transfer of Grossberg and VanDieren [11]. These techniques also apply to measurable and weakly compact cardinals and we prove similar tameness results under those hypotheses. We isolate a dual property (...)
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  3.  18
    Tameness and extending frames.Will Boney - 2014 - Journal of Mathematical Logic 14 (2):1450007.
    We combine two notions in AECs, tameness and good λ-frames, and show that they together give a very well-behaved nonforking notion in all cardinalities. This helps to fill a longstanding gap in classification theory of tame AECs and increases the applicability of frames. Along the way, we prove a complete stability transfer theorem and uniqueness of limit models in these AECs.
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  4.  18
    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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  5.  38
    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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  6.  17
    Computing the Number of Types of Infinite Length.Will Boney - 2017 - Notre Dame Journal of Formal Logic 58 (1):133-154.
    We show that the number of types of sequences of tuples of a fixed length can be calculated from the number of 1-types and the length of the sequences. Specifically, if κ≤λ, then sup ‖M‖=λ|Sκ|=|)κ. We show that this holds for any abstract elementary class with λ-amalgamation. No such calculation is possible for nonalgebraic types. However, we introduce a subclass of nonalgebraic types for which the same upper bound holds.
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  7.  22
    Forking in short and tame abstract elementary classes.Will Boney & Rami Grossberg - 2017 - Annals of Pure and Applied Logic 168 (8):1517-1551.
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  8.  29
    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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  9.  26
    Advances in Classification Theory for Abstract Elementary Classes.Will Boney - 2018 - Bulletin of Symbolic Logic 24 (4):454-455.
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  10.  31
    A presentation theorem for continuous logic and metric abstract elementary classes.Will Boney - 2017 - Mathematical Logic Quarterly 63 (5):397-414.
    In recent years, model theory has widened its scope to include metric structures by considering real-valued models whose underlying set is a complete metric space. We show that it is possible to carry out this work by giving presentation theorems that translate the two main frameworks into discrete settings. We also translate various notions of classification theory.
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  11.  7
    Cofinality Quantifiers in Abstract Elementary Classes and Beyond.Will Boney - forthcoming - Journal of Symbolic Logic:1-15.
    The cofinality quantifiers were introduced by Shelah as an example of a compact logic stronger than first-order logic. We show that the classes of models axiomatized by these quantifiers can be turned into an Abstract Elementary Class by restricting to positive and deliberate uses. Rather than using an ad hoc proof, we give a general framework of abstract Skolemizations. This method gives a uniform proof that a wide rang of classes are Abstract Elementary Classes.
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  12.  20
    Tameness, powerful images, and large cardinals.Will Boney & Michael Lieberman - 2020 - Journal of Mathematical Logic 21 (1):2050024.
    We provide comprehensive, level-by-level characterizations of large cardinals, in the range from weakly compact to strongly compact, by closure properties of powerful images of accessible functors. In the process, we show that these properties are also equivalent to various forms of tameness for abstract elementary classes. This systematizes and extends results of [W. Boney and S. Unger, Large cardinal axioms from tameness in AECs, Proc. Amer. Math. Soc.145(10) (2017) 4517–4532; A. Brooke-Taylor and J. Rosický, Accessible images revisited, Proc. AMS145(3) (2016) (...)
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  13.  7
    Which Classes of Structures Are Both Pseudo-Elementary and Definable by an Infinitary Sentence?Will Boney, Barbara F. Csima, D. A. Y. Nancy A. & Matthew Harrison-Trainor - 2023 - Bulletin of Symbolic Logic 29 (1):1-18.
    When classes of structures are not first-order definable, we might still try to find a nice description. There are two common ways for doing this. One is to expand the language, leading to notions of pseudo-elementary classes, and the other is to allow infinite conjuncts and disjuncts. In this paper we examine the intersection. Namely, we address the question: Which classes of structures are both pseudo-elementary and ${\mathcal {L}}_{\omega _1, \omega }$ -elementary? We find that these are exactly the classes (...)
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