we-solve (beta)

Math-2243 · mathematics · Logic

Shelah's eventual categoricity conjecture

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
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 $\mu(\lambda)$.

Formulation references: [S1], [S2].

Short English statement

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 $\mu(\lambda)$.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2243-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2243

Theorem 1.2, labelled main:uniform, states the uniform eventual-categoricity transfer for arbitrary abstract elementary classes. Its threshold depends only on the Löwenheim--Skolem bound, as required by this question; no extra amalgamation or tameness condition is imposed.

Shelah's eventual categoricity conjecture

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 in ZFC, each bound on Löwenheim--Skolem number has a uniform threshold above which categoricity in one cardinal implies categoricity throughout the same tail.

Mathematical statement of the claimed result

Manuscript 1.
For every infinite cardinal $\lambda$, the claim gives a uniform cardinal threshold $\mu(\lambda)\ge\lambda$ for all abstract elementary classes $K$ of Löwenheim--Skolem number at most $\lambda$: \[ (\exists\kappa\ge\mu(\lambda):I(K,\kappa)=1) \Longrightarrow(\forall\kappa'\ge\mu(\lambda):I(K,\kappa')=1). \] Here $I(K,\kappa)$ is the number of model isomorphism classes of cardinality $\kappa$. The same threshold occurs in the premise and conclusion. A second formulation asserts that categoricity in unboundedly many cardinals forces categoricity on a tail, without adding amalgamation or tameness assumptions.

Outline of the argument

For manuscript 1 (family 240; 2026-09-24), fixed order-indexed presentations encode short model orbits. A direct reversal theorem for strong maps preserves bounded tuple records; elimination and stabilization of strong-map classes then create the stable diagrams and closed-level amalgamation needed for tail transfer.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 240; 2026-09-24).

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

Go to contribution →