Categoricity in multiuniversal classes

Annals of Pure and Applied Logic 170 (11):102712 (2019)
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Abstract

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 result of the second author.

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Citations of this work

An AEC framework for fields with commuting automorphisms.Tapani Hyttinen & Kaisa Kangas - 2023 - Archive for Mathematical Logic 62 (7):1001-1032.
On categoricity in successive cardinals.Sebastien Vasey - 2020 - Journal of Symbolic Logic:1-19.
On categoricity in successive cardinals.Sebastien Vasey - 2022 - Journal of Symbolic Logic 87 (2):545-563.

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References found in this work

Galois-stability for Tame abstract elementary classes.Rami Grossberg & Monica Vandieren - 2006 - Journal of Mathematical Logic 6 (01):25-48.
Shelah's eventual categoricity conjecture in universal classes: Part I.Sebastien Vasey - 2017 - Annals of Pure and Applied Logic 168 (9):1609-1642.
Building independence relations in abstract elementary classes.Sebastien Vasey - 2016 - Annals of Pure and Applied Logic 167 (11):1029-1092.
Tameness from large cardinal axioms.Will Boney - 2014 - Journal of Symbolic Logic 79 (4):1092-1119.
Independence in finitary abstract elementary classes.Tapani Hyttinen & Meeri Kesälä - 2006 - Annals of Pure and Applied Logic 143 (1-3):103-138.

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