Submission date:

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
Let ψ be a universal Lω1,ω sentence. If ψ is categorical in some λ ≥ ℶℶω1, then ψ is categorical in all λ' ≥ ℶℶω1.
As a byproduct of the proof, we show that a conjecture of Grossberg holds in universal classes:
Corollary
Let ψ be a universal Lω1,ω sentence 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. 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}.

Mathematics Subject Classification: 03C48 (Primary), 03C45, 03C52, 03C55 (Secondary)

Keywords and phrases: Abstract elementary classes; Universal classes; Categoricity; Independence; Classification theory; Smoothness; Tameness; Prime models

Full text arXiv 1602.02633: pdf, ps.