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

Abstract

We prove: $\mathbf{Theorem}$ Let $K$ 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: $\mathbf{Theorem}$ Let $K$ be an AEC with amalgamation. Assume that $K$ is fully $\mathrm{LS}(K)$-tame and short and has primes over sets of the form $M\cup \{a\}$. Write $H_2:={\beth }_{{\left(2^{{\beth }_{{\left(2^{\mathrm{LS}(K)}\right)}^{+}}}\right)}^{+}}$. If $K$ is categorical in a $\lambda >H_2$, then $K$ is categorical in all ${\lambda }^{\prime }\ge H_2$.

51 pages

References in corpus (7)

• Building independence relations in abstract elementary classes
• Infinitary stability theory
• Shelah's eventual categoricity conjecture in universal classes. Part II
• Downward categoricity from a successor inside a good frame
• Superstability from categoricity in abstract elementary classes
• Building prime models in fully good abstract elementary classes
• Shelah's eventual categoricity conjecture in tame AECs with primes

Cited by in corpus (18)

• Forking independence from the categorical point of view
• Saturation and solvability in abstract elementary classes with amalgamation
• Superstability from categoricity in abstract elementary classes
• Toward a stability theory of tame abstract elementary classes
• Symmetry in abstract elementary classes with amalgamation
• Universal abstract elementary classes and locally multipresentable categories
• The Kim-Pillay theorem for Abstract Elementary Categories
• Universal classes near ${\aleph }_1$
• Internal sizes in $\mu$-abstract elementary classes
• Quasiminimal abstract elementary classes
• Structural Logic and Abstract Elementary Classes with Intersection
• Shelah's eventual categoricity conjecture in tame AECs with primes
• Building models in small cardinals in local abstract elementary classes
• A Characterization of Uniqueness of Limit Models in Categorical Abstract Elementary Classes
• On categoricity in successive cardinals
• Tameness from two successive good frames
• Non-forking w-good frames
• Notions of amalgamation for AECs and categoricity