Shelah's eventual categoricity and the prescribed-threshold obstruction
OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
OpenAI's result summary
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.
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MANUSCRIPT 01
A CH obstruction to a prescribed categoricity threshold
Assuming the continuum hypothesis, we construct an abstract elementary class with Löwenheim–Skolem number ℵ_(0) 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…
Full source abstract
Assuming the continuum hypothesis, we construct an abstract elementary class with Löwenheim–Skolem number ℵ_(0) 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.
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Eventual categoricity for abstract elementary classes
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.
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