oai-math-revised beta

Browse › Mathematical logic › 240 Shelah's eventual categoricity and the prescribed-threshold obstruction

A CH obstruction to a prescribed categoricity threshold

OpenAI · September 24, 2026

Lean

Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion

Edit NOTES.md · Existing issues · Source on GitHub

Abstract

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.

If the PDF does not display here, open it directly: Open PDF.

Notes

No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.

Automatically collected notes (unreviewed)

work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Formalization

  • OpenAI math release, Shelah's eventual categoricity and the prescribed-threshold obstruction: Lean scope notes (lean/docs/240.md, 2026-10-06). The release's Lean formalization covers Theorem 1.1 of this paper (the CH construction with two nonisomorphic models at $\beth_{\omega_2}$ and categoricity from $\beth_{(2^{\aleph_1})^+}$ on), with comparator statement ComparatorChallenges/CHObstruction.lean; the scope note says the canonical-point (dense-cosieve) obstruction of the final section and the separate eventual-categoricity theorem are not included. [family-level]

Possibly relevant literature (found by the agent, not checked)

  • Saharon Shelah and Andrés Villaveces, The Hart-Shelah example, in stronger logics (arXiv:math/0404258, Ann. Pure Appl. Logic 172 (2021) 102958). Extends the Hart–Shelah examples, which the paper names as the closest predecessor of its finite-support torsor construction, to sentences in $L_{(2^\lambda)^+,\omega}$ that are categorical in $\lambda^+,\dots,\lambda^{+k-1}$ but not in $\beth_{k+1}(\lambda)^+$ or above.
  • Sebastien Vasey, The categoricity spectrum of large abstract elementary classes (arXiv:1805.04068, Selecta Math. 25 (2019)). Assuming cardinal exponentiation is injective, it shows that for AECs with amalgamation and arbitrarily large models the categoricity spectrum is empty, a closed interval of finite successors of LS(K), or an end segment starting below the Hanf number; this is relevant because the paper's CH example has neither amalgamation nor joint embedding.
  • Will Boney and John Baldwin, Hanf Numbers and Presentation Theorems in AECs (arXiv:1511.02935, 2016). Shows that a strongly compact cardinal bounds Hanf numbers for amalgamation and related properties, using a functorial relational presentation theorem; this connects to the paper's use of the Hanf bound $H(K)$ and its discussion of where amalgamation and joint embedding fail.

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026,
  author = {{OpenAI}},
  title = {{A CH obstruction to a prescribed categoricity threshold}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026/paper.pdf}{OAI:A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/240.md. The Lean statements have not been checked by people against the paper's claims.

Family 240: 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.

Other papers in this family: