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ON CATEGORICITY IN SUCCESSIVE CARDINALS

Part of: Model theory

Published online by Cambridge University Press:  20 July 2020

SEBASTIEN VASEY
SEBASTIEN VASEY*
Affiliation:
RADIX TRADING LLC. CHICAGO, IL 60654, USA E-mail: vasey.sebastien@gmail.com URL: svasey.github.io
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Abstract

We investigate, in ZFC, the behavior of abstract elementary classes (AECs) 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 (such as amalgamation or arbitrarily large models) and generalizes to the much broader framework of tame AECs with weak amalgamation and coherent sequences.

Keywords

abstract elementary classescategoricityamalgamationno maximal modelsgood frames

MSC classification

Primary: 03C48: Abstract elementary classes and related topics
Secondary: 03C45: Classification theory, stability and related concepts 03C52: Properties of classes of models 03C55: Set-theoretic model theory 03C75: Other infinitary logic

Information

Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 2 , June 2022 , pp. 545 - 563
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Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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