Woodin's question answered: GCH below a strongly compact cardinal implies GCH everywhere (Zhixing You, 4-page proof with GPT-6 Astra)
On Oct 6, 2026 Zhixing You (Hebrew University of Jerusalem) posted arXiv 2610.07537, a four-page paper answering a long-standing question of Hugh Woodin: if κ is a strongly compact cardinal and the generalized continuum hypothesis holds below κ, then GCH holds everywhere. The analogous facts for supercompact cardinals, and Solovay's theorem for singular strong limits, were classical; Apter had shown the answer can be negative without the axiom of choice. The acknowledgement says the work "used AI assistance from GPT-6-Astra with the ultra reasoning setting, with GPT-6.1-Sol and GPT-6-Astra invoked dynamically". Unrefereed.
Key facts
- Paper: 'The generalized continuum hypothesis above a strongly compact cardinal', Zhixing You, arXiv 2610.07537 (math.LO), 6 Oct 2026, 4 pages; preprint dated 5 Oct
- Theorem A: if κ is strongly compact and GCH holds below κ, then GCH holds (in ZFC). Remark 3.3: the proof does not use crit(j) = κ, so it also works for ω₁-strongly compact κ
- Background (paper): Solovay — above a strongly compact, 2^λ = λ⁺ for singular strong limit λ; Scott — measurable κ with GCH below gives 2^κ = κ⁺; supercompact κ with GCH below gives full GCH (Kanamori, Prop. 22.2). Woodin asked whether strong compactness suffices (Apter, 'On a problem of Woodin', Arch. Math. Logic 2000), and Apter gave a negative answer without choice
- AI statement (quote): 'This work used AI assistance from GPT-6-Astra with the ultra reasoning setting, with GPT-6.1-Sol and GPT-6-Astra invoked dynamically during the investigation. The author takes full responsibility for the paper.' The paper does not say which steps came from the models
- Author: Zhixing You, Einstein Institute of Mathematics, Hebrew University of Jerusalem (a set theorist with earlier work on strong compactness, e.g. arXiv 2302.13171)
- Not in OpenAI's 6 Oct github.com/openai/math catalogue (no strongly-compact or GCH family)
- Verification: unrefereed preprint; no expert reaction found as of 7 Oct 2026. A four-page answer to a 25-year-old large-cardinal question will need checking by set theorists
Science result
- Field
- mathematics / set theory / large cardinals
- Problem
- Woodin's question: if κ is strongly compact and GCH holds below κ, does GCH hold everywhere? (open since 2000)
- Result
- Yes, in ZFC; the argument also works for ω₁-strongly compact cardinals.
- AI system
- GPT-6 Astra (ultra reasoning), GPT-6.1 Sol
- Human role
- AI-assisted; the author acknowledges GPT-6 Astra and GPT-6.1 Sol 'during the investigation' without detailing their contribution and takes responsibility
- Verification
- Unrefereed preprint
- Status
- pending
- Why surprising
- A question in cardinal arithmetic that resisted large-cardinal specialists for about 25 years is answered in four pages.
What happened
In ZFC, the continuum function λ ↦ 2^λ is almost unconstrained at regular cardinals (Easton's theorem), but large cardinals force regularity higher up. Solovay showed that above a strongly compact cardinal κ, 2^λ = λ⁺ at every singular strong limit λ. It was also known that if κ is supercompact and GCH holds below κ, then GCH holds everywhere. Hugh Woodin asked whether the weaker hypothesis of strong compactness is enough. In 2000 Arthur Apter showed that without the axiom of choice it is not.
On 6 October 2026 Zhixing You (Hebrew University of Jerusalem) posted a four-page paper answering Woodin's question positively in ZFC: if κ is strongly compact and GCH holds below κ, then GCH holds. The argument works with elementary embeddings witnessing λ-strong compactness and does not use that the critical point is κ, so it also applies to ω₁-strongly compact cardinals.
The acknowledgement reads: "This work used AI assistance from GPT-6-Astra with the ultra reasoning setting, with GPT-6.1-Sol and GPT-6-Astra invoked dynamically during the investigation. The author takes full responsibility for the paper." The paper does not say which ideas came from the models.
Why it matters
Woodin's question is a well-known open problem about large cardinals and the continuum function, and a short positive answer would be a notable result in set theory. It also adds mathematical logic to the fields where frontier-model help is now disclosed, after the Medvedev-logic undecidability result. A short proof of a long-open question is also exactly the kind of claim experts will want to check before accepting.
Changelog
- 2026-10-07: created (sweep 2026-10-07, arXiv AI-disclosure section; PDF read)
Related events
- Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help (July–September catalogue) ★★★★
- OpenAI releases GPT-6 Astra, its first GPT-6 model ★★★★★
- OpenAI releases GPT-6.1 Sol: near-Astra performance at one-fifth of Astra's price ★★★★
- Medvedev's logic of finite problems (1962) shown undecidable; key idea from ChatGPT Sol 5.6, central argument checked in Lean by Claude Opus 5 ★★★
Sources (2)
- paperarXiv 2610.07537: The generalized continuum hypothesis above a strongly compact cardinal
- paperApter (2000): On a problem of Woodin, Archive for Mathematical Logic 39
id: 2026-10-06-gch-above-strongly-compact-woodin-question · updated 2026-10-07 · open in the interactive timeline