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Math-2721 · mathematics · Logic

Downward categoricity transfer to the Hanf threshold under CH

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Assume CH. If an abstract elementary class K in a countable finitary relational language has $LS(K)=\aleph _{0}$ and is categorical in a cardinal above $H(K)=beth_{\omega _{2}}$, must it be categorical in H(K), without amalgamation, tameness or joint-embedding assumptions?

Short English statement

Assume CH. If an abstract elementary class K in a countable finitary relational language has $LS(K)=\aleph _{0}$ and is categorical in a cardinal above $H(K)=beth_{\omega _{2}}$, must it be categorical in H(K), without amalgamation, tameness or joint-embedding assumptions?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2721-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2721

The displayed question has a negative answer under the manuscript's stated construction. Its counterexample meets the conditions in this problem and defeats the universal assertion, so this is a full disproof claim. The supporting result is manuscript 1 (family 240; 2026-09-24).

Downward categoricity transfer to the Hanf threshold under CH

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 240; 2026-09-24), the asserted result is that under CH, an abstract elementary class of countable Löwenheim--Skolem number can be categorical on a tail while failing categoricity at $\beth_{\omega_2}$, refuting the prescribed lower-threshold transfer.

Mathematical statement of the claimed result

Manuscript 1.
With the continuum hypothesis assumed, the claim constructs an abstract elementary class $K$ whose relational vocabulary is countable and whose symbols all have finite arity, satisfying \[ \operatorname{LS}(K)=\aleph_0,\qquad H(K)=\beth_{\omega_2},\qquad I(K,H(K))\ge2,\qquad I(K,\mu)=1\quad(\mu\ge\Lambda), \quad\Lambda=\beth_{(2^{\aleph_1})^+}. \] Here $I(K,\mu)$ counts isomorphism classes of models of size $\mu$, and $\operatorname{LS}(K)$ is the Löwenheim--Skolem number. Thus categoricity at $\Lambda^+>H(K)$ does not force categoricity at $H(K)$; no amalgamation, tameness or joint-embedding hypothesis is added.

Outline of the argument

For manuscript 1 (family 240; 2026-09-24), countable finite-subset diagrams carry color and order data through coherent strong inclusions. Partition bounds forbid an uncountable base pattern and force uniqueness on the tail, while two explicit structures retain different patterns at the proposed endpoint.

Formal verification

For manuscript 1 (family 240; 2026-09-24), OpenAI lists Lean statements for this assertion: under CH, an abstract elementary class of countable Löwenheim--Skolem number can be categorical on a tail while failing categoricity at $\beth_{\omega_2}$, refuting the prescribed lower-threshold transfer.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

The linked family Comparator checks are CHObstruction.lean.

Their named declarations include OAI.CHObstruction.main.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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