Shelah's eventual categoricity conjecture in universal classes: Part I

Abstract

We prove: Theorem 0.1Let K be a universal class. If K is categorical in cardinals of arbitrarily high cofinality, then K is categorical on a tail of cardinals.The proof stems from ideas of Adi Jarden and Will Boney, and also relies on a deep result of Shelah. As opposed to previous works, the argument is in ZFC and does not use the assumption of categoricity in a successor cardinal. The argument generalizes to abstract elementary classes (AECs) that satisfy a locality property and where certain prime models exist. Moreover assuming amalgamation we can give an explicit bound on the Hanf number and get rid of the cofinality restrictions:Theorem 0.2Let K be an AEC with amalgamation. Assume that K is fullyLS(K)-tame and short and has primes over sets of the formM∪{a}. WriteH2:=ℶ(2ℶ(2LS(K))+)+. If K is categorical in aλ>H2, then K is categorical in allλ′≥H2.