Proves Shelah's eventual categoricity conjecture in ZFC: for each bound on the Löwenheim–Skolem number, a uniform threshold makes categoricity of an abstract elementary class in one cardinal above that threshold imply categoricity throughout the same tail. Categoricity means uniqueness up to isomorphism at a given cardinality. Under the continuum hypothesis, a proposed specific Hanf threshold need not suffice. (Lean)
Manuscripts (2)
A CH obstruction to a prescribed categoricity threshold
September 24, 2026 · source · Lean formalization available · cite as \cite{OAI:A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026}
Assuming the continuum hypothesis, we construct an abstract elementary class with Löwenheim–Skolem number ℵ0 that is categorical in every sufficiently large cardinal but has at least two nonisomorphic models of cardinality \beth_{\omega_2}. Thus categoricity does not transfer down to the proposed bound \beth_{(2^{\aleph_0})^+}, which equals \beth_{\omega_2} under CH. Consequently, if ZFC is consistent, the prescribed-threshold form of Shelah's categoricity conjecture is not provable in ZFC.
Eventual categoricity for abstract elementary classes
September 24, 2026 · source · cite as \cite{OAI:Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026}
We prove Shelah's eventual categoricity conjecture for abstract elementary classes in ZFC. For each infinite bound on the Löwenheim–Skolem number there is a uniform threshold such that categoricity in any one cardinal at or above that threshold implies categoricity in every cardinal at or above the same threshold.
Result 240 · Mathematical logic. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md