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: $\mathbf{Theorem}$ Let $\psi$ be a universal ${\mathbb{L}}_{{\omega }_1,\omega }$ sentence. If $\psi$ is categorical in some $\lambda \ge {\beth }_{{\beth }_{{\omega }_1}}$, then $\psi$ is categorical in all ${\lambda }^{\prime }\ge {\beth }_{{\beth }_{{\omega }_1}}$. As a byproduct of the proof, we show that a conjecture of Grossberg holds in universal classes: $\mathbf{Corollary}$ Let $\psi$ be a universal ${\mathbb{L}}_{{\omega }_1,\omega }$ sentence 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. 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 $M\cup \{a\}$.

49 pages

References in corpus (2)

• Building independence relations in abstract elementary classes
• Infinitary stability theory

Cited by in corpus (10)

• Shelah's eventual categoricity conjecture in universal classes: part I
• Forking independence from the categorical point of view
• Superstability from categoricity in abstract elementary classes
• Saturation and solvability in abstract elementary classes with amalgamation
• Toward a stability theory of tame abstract elementary classes
• Universal classes near ${\aleph }_1$
• Shelah's eventual categoricity conjecture in tame AECs with primes
• Structural Logic and Abstract Elementary Classes with Intersection
• On categoricity in successive cardinals
• Tameness from two successive good frames