Categoricity in multiuniversal classes

Boney, Will; Vasey, Sebastien

doi:10.1016/j.apal.2019.06.001

Cited by 8 publications

(11 citation statements)

References 18 publications

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“…Let be a universal L 1 , sentence (i.e., is of the form ∀xφ(x), where φ is quantifier-free). If is categorical in ℵ 0 and 1 ≤ I( , ℵ 1 ) < 2 ℵ 1 , then: (1) has arbitrarily large models. (2) If is categorical in some uncountable cardinal then is categorical in all uncountable cardinals.…”

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confidence: 99%

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UNIVERSAL CLASSES NEAR ${\aleph _1}$

Mazari-Armida¹,

Vasey²

2018

J. symb. log.

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Shelah has provided sufficient conditions for an Lω 1 ,ω -sentence ψ to have arbitrarily large models and for a Morley-like theorem to hold of ψ. These conditions involve structural and set-theoretic assumptions on all the ℵn's. Using tools of Boney, Shelah, and the second author, we give assumptions on ℵ 0 and ℵ 1 which suffice when ψ is restricted to be universal:(1) If ψ is categorical in ℵ 0 and 1 ≤ I(ψ, ℵ 1 ) < 2 ℵ 1 , then ψ has arbitrarily large models and categoricity of ψ in some uncountable cardinal implies categoricity of ψ in all uncountable cardinals.(2) If ψ is categorical in ℵ 1 , then ψ is categorical in all uncountable cardinals.The theorem generalizes to the framework of Lω 1 ,ω -definable tame abstract elementary classes with primes.

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confidence: 99%

“…Similarly, Fact 1.4 can be generalized. See for example the recent result of Ackerman, Boney, and the second author on multiuniversal classes[1].…”

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confidence: 99%

UNIVERSAL CLASSES NEAR ${\aleph _1}$

Mazari-Armida¹,

Vasey²

2018

J. symb. log.

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“…It is easy to see that an FCA-class actually is an abstract elementary class (AEC) and multiuniversal in the sense of [1] (see Definition 2.6 in the present paper). Example 2.9 in [1] lists several examples of multiuniversal classes, and FCA-classes serve as additional examples of a multiuniversal class that is not universal (as an AEC) and not axiomatizable by a first order theory.…”

Section: The Aec Frameworkmentioning

confidence: 84%

“…In [1] (Theorem 3.3), it is proved that in multiuniversal classes, Galois types of infinite sequences are determined by the Galois types of finite subsequences, and this implies a multiuniversal class with AP and JEP is homogeneous. However, our result (Lemma 2.15) will imply Theorem 3.3. in [1], as we will point out in Remark 2. 16.…”

Section: A Remark On Existential Types In Multiuniversal Classesmentioning

confidence: 99%

An AEC framework for fields with commuting automorphisms

Hyttinen¹,

Kangas²

2019

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In this paper, we introduce an AEC framework for studying fields with commuting automorphisms. Fields with commuting automorphisms generalise difference fields. Whereas in a difference field, there is one distinguished automorphism, a field with commuting automorphisms can have several of them, and they are required to commute. Z. Chatzidakis and E. Hrushovski have studied in depth the model theory of ACFA, the model companion of difference fields. Hrushovski has proved that in the case of fields with two or more commuting automorphisms, the existentially closed models do not necessarily form a first order model class. In the present paper, we introduce FCA-classes, an AEC framework for studying the existentially closed models of the theory of fields with commuting automorphisms. We prove that an FCA-class has AP and JEP and thus a monster model, that Galois types coincide with existential types in existentially closed models, that the class is homogeneous, and that there is a version of type amalgamation theorem that allows to combine three types under certain conditions. Finally, we use these results to show that our monster model is a simple homogeneous structure in the sense of S. Buechler and O. Lessman (this is a non-elementary analogue for the classification theoretic notion of a simple first order theory). Contents 1. Introduction 1 2. A remark on existential types in multiuniversal classes 5 3. The AEC framework 13 4. Independence 18 5. Simplicity 24 References 28

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“…+ . This holds more generally for multiuniversal classes [ABV19] Other approximations to categoricity (for example from tameness, a locality property of types that has not been discussed here) are in [She99,GV06b,GV06a]. Note that given any finitely accessible category K, we have that K mono is an AEC and the embedding K mono → K preserves and reflects presentability ranks, hence categoricity.…”

Section: The Following Are Classical Examples Of the Occurrence Of Ca...mentioning

confidence: 99%

Accessible categories, set theory, and model theory: an invitation

Vasey¹

2019

Preprint

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We give a self-contained introduction to accessible categories and how they shed light on both model-and set-theoretic questions. We survey for example recent developments on the study of presentability ranks, a notion of cardinality localized to a given category, as well as stable independence, a generalization of pushouts and model-theoretic forking that may interest mathematicians at large. We give many examples, including recently discovered connections with homotopy theory and homological algebra. We also discuss concrete versions of accessible categories (such as abstract elementary classes), and how they allow nontrivial "element by element" constructions. We conclude with a new proof of the equivalence between saturated and homogeneous which does not use the coherence axiom of abstract elementary classes.

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Categoricity in multiuniversal classes

Cited by 8 publications

References 18 publications

UNIVERSAL CLASSES NEAR ${\aleph _1}$

UNIVERSAL CLASSES NEAR ${\aleph _1}$

An AEC framework for fields with commuting automorphisms

Accessible categories, set theory, and model theory: an invitation

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