OpenAI · 240-01 · 2026-09-24 · Mathematical logic
A CH obstruction to a prescribed categoricity threshold
preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026
Family 240. Shelah's eventual categoricity conjecture.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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.
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 . Thus categoricity does not transfer down to the proposed bound , which equals under CH. Consequently, if ZFC is consistent, the prescribed-threshold form of Shelah's categoricity conjecture is not provable in ZFC.
Preprint source and Lean scope
The abstract above is the catalogue abstract. The sections below are the full preprint source, 0 sections. This page is not only the abstract. The manuscript file is loading.
This preprint is named in the Lean scope note for 240: Shelah's eventual categoricity and the prescribed-threshold obstruction. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.