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Shelah's eventual categoricity conjecture for abstract elementary classes

Morley proved that a countable first-order theory categorical in one uncountable cardinal is categorical in all of them. Shelah's abstract elementary classes (AECs) generalise elementary classes to nonelementary settings such as Lω1,ωL_{\omega_1,\omega} sentences, without compactness. Shelah's eventual categoricity conjecture asks for an analogue: for each infinite cardinal λ\lambda there should be a threshold μ(λ)\mu(\lambda) such that every AEC KK with LS(K)≤λLS(K)\le\lambda that is categorical in some cardinal ≥μ(λ)\ge\mu(\lambda) is categorical in every cardinal ≥μ(λ)\ge\mu(\lambda). Prior proofs needed amalgamation, tameness, a successor starting cardinal, large cardinals or weak GCH. Is eventual categoricity provable in ZFC for all AECs?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Model theory; abstract elementary classes
Posed by
Saharon Shelah
Year posed
2009
Years open
17y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
46 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.2: for every infinite cardinal λ\lambda there is μ(λ)≥λ\mu(\lambda)\ge\lambda such that every AEC KK with LS(K)≤λLS(K)\le\lambda categorical in some κ≥μ(λ)\kappa\ge\mu(\lambda) is categorical in every κ′≥μ(λ)\kappa'\ge\mu(\lambda); equivalently (Theorem 1.3) an AEC categorical in unboundedly many cardinals is categorical on a tail. It applies to Lω1,ωL_{\omega_1,\omega} sentences in countable languages. It gives no explicit value for μ(λ)\mu(\lambda); the prescribed ℶω1\beth_{\omega_1} threshold for Lω1,ωL_{\omega_1,\omega} and the Hanf threshold for AECs are left separate (the companion refutes the latter under CH).

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has two manuscripts dated September 24, 2026: the ZFC eventual categoricity theorem (this entry) and a CH counterexample to the prescribed Hanf threshold (its own entry).

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.2 was read against the conjecture. It gives, for every infinite λ\lambda, a cardinal μ(λ)\mu(\lambda) such that every AEC with LS(K)≤λLS(K)\le\lambda categorical in one cardinal ≥μ(λ)\ge\mu(\lambda) is categorical in all cardinals ≥μ(λ)\ge\mu(\lambda), in ZFC, with no amalgamation, joint embedding, tameness or no-maximal-models hypothesis and any starting cardinal. This is the qualitative conjecture. It does not give a numerical threshold. The long proof (order presentations, working schemes, geometry built from unbounded categoricity) was not refereed and is not formalised; the family's Lean formalization covers only the CH companion.

Sources

Changelog1 change

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