Shelah's eventual categoricity and the prescribed-threshold obstruction

What OpenAI claims

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)

Papers (2)

1. A CH obstruction to a prescribed categoricity threshold

   24 September 2026 · Preprint, not peer reviewed · PDF

   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 ℶ_ω2. Thus categoricity does not transfer down to the proposed bound ℶ_(2^ℵ0)⁺, which equals ℶ_ω2 under CH. Consequently, if ZFC is consistent, the prescribed-threshold form of Shelah's categoricity conjecture is not provable in ZFC.

2. Eventual categoricity for abstract elementary classes

   24 September 2026 · Preprint, not peer reviewed · PDF

   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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.