Shelah's eventual categoricity conjecture: For every cardinal lambda there exists a cardinal mu(lambda) such that if an AEC K with LS(K) ≤ lambda is categorical in a cardinal above mu(lambda) then it is categorical in all cardinals above mu(lambda).

Shelah's eventual categoricity conjecture

About 49 years old · traced to 1977

Shelah's eventual categoricity conjecture: For every cardinal $\lambda$ there exists a cardinal $\mu(\lambda)$ such that if an AEC K with LS(K) ${} \le \lambda$ is categorical in a cardinal above $\mu(\lambda)$ then it is categorical in all cardinals above $\mu(\lambda)$ .

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Model Theory problems from Wikipedia

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InAMR Open Problem Lists

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Claimed by OpenAI.

Shelah eventual categoricity for abstract elementary classes in ZFC, with a uniform threshold for each infinite bound on the Lowenheim–Skolem number, above which categoricity in one cardinal implies categoricity in every cardinal.

Repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/paper.pdf

OpenAI-240-02-Eventual-categoricity-for-abstract-elementary-classes.pdf975,455 bytesOpen

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Claimed by OpenAI.

Related threshold result under the continuum hypothesis: claims an abstract elementary class with countable Lowenheim–Skolem number that is categorical in every sufficiently large cardinal but has at least two nonisomorphic models at beth_(omega_2). This obstructs a specific prescribed categoricity threshold. The example is compatible with eventual categoricity at a higher threshold and does not refute the existential threshold statement of this problem.

Repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026/paper.pdf

OpenAI-240-01-A-CH-obstruction-to-a-prescribed-categoricity-threshold.pdf549,483 bytesOpen