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ON CATEGORICITY IN SUCCESSIVE CARDINALS
Part of:
Model theory
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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