{"id":"openai-math/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold","pdf":"https://github.com/openai/math/blob/main/preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026","tags":["set-theory","model-theory","categoricity"],"score":0,"title":"A CH obstruction to a prescribed categoricity threshold","domain":"Mathematical logic","family":"240","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.LO","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-24","reactions":{},"formalized":true,"family_title":"Shelah's eventual categoricity and the prescribed-threshold obstruction"}