ON CATEGORICITY IN SUCCESSIVE CARDINALS

We investigate, in ZFC, the behavior of abstract elementary classes (AECs) categorical in many successive small cardinals. We prove for example that a universal L-omega 1,L-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.