Selecta Mathematica

, Volume 23, Issue 2, pp 1469–1506 | Cite as

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

Article

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 \(\psi \) be a universal \(\mathbb {L}_{\omega _1, \omega }\) sentence (in a countable vocabulary). If \(\psi \) is categorical in some\(\lambda \ge \beth _{\beth _{\omega _1}}\), then \(\psi \) is categorical in all\(\lambda ' \ge \beth _{\beth _{\omega _1}}\).

As a byproduct of the proof, we show that a conjecture of Grossberg holds in universal classes:

Corollary 0.2 Let \(\psi \) be a universal \(\mathbb {L}_{\omega _1, \omega }\) sentence (in a countable vocabulary) that is categorical in some \(\lambda \ge \beth _{\beth _{\omega _1}}\), then the class of models of \(\psi \) has the amalgamation property for models of size at least \(\beth _{\beth _{\omega _1}}\).

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 \(M \cup \{a\}\).

Keywords

Abstract elementary classes Universal classes Categoricity Independence Classification theory Smoothness Tameness Prime models 

Mathematics Subject Classification

Primary 03C48 Secondary 03C45 03C52 03C55 03C75 03E55 

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA

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