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ON CATEGORICITY IN SUCCESSIVE CARDINALS
Published online by Cambridge University Press: 20 July 2020
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.
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- © The Author(s), 2020. Published by Cambridge University Press on behalf of The Association for Symbolic Logic