Shelah's eventual categoricity and the prescribed-threshold obstruction. 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.
released 2026-09-24 | 4 theorems · 21 lemmas · 29 proofs · 14,153 words |
PLAY LEVEL 1 »(pdf)
Assuming the continuum hypothesis, we construct an abstract elementary class with Löwenheim–Skolem number ℵ0 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.
released 2026-09-24 | 21 theorems · 47 lemmas · 99 proofs · 69,471 words |
PLAY LEVEL 2 »(pdf)
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.