Shelah’s eventual categoricity conjecture in universal classes: part II

Abstract

We prove that a universal class categorical in a high-enough cardinal is categorical on a tail of cardinals. As opposed to other results in the literature, we work in ZFC, do not require the categoricity cardinal to be a successor, do not assume amalgamation, and do not use large cardinals. Moreover we give an explicit bound on the “high-enough” threshold: Theorem 0.1 Let ψ be a universal Lω1,ω sentence (in a countable vocabulary). If ψ is categorical in someλ≥ℶℶω1, then ψ is categorical in allλ′≥ℶℶω1. As a byproduct of the proof, we show that a conjecture of Grossberg holds in universal classes: Corollary 0.2 Let ψ be a universal Lω1,ω sentence (in a countable vocabulary) that is categorical in some λ≥ℶℶω1, then the class of models of ψ has the amalgamation property for models of size at least ℶℶω1. We also establish generalizations of these two results to uncountable languages. As part of the argument, we develop machinery to transfer model-theoretic properties between two different classes satisfying a compatibility condition (agreeing on any sufficiently large cardinals in which either is categorical). This is used as a bridge between Shelah’s milestone study of universal classes (which we use extensively) and a categoricity transfer theorem of the author for abstract elementary classes that have amalgamation, are tame, and have primes over sets of the form M∪ { a}.