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OP-11831Status unassessedLogic and foundations

A Davies example provable in ZFC — Problem 7.1

A Davies example provable in ZFC — Problem 7.1. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today.

MSC 03 · Mathematical logic and foundations

Editorial MSC2020 mapping of A Davies example provable in ZFC — Problem 7.1. https://arxiv.org/html/2608.26628v2#S7.Thmproblem1 Not an author-supplied code.

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Problem statement

Problem 7.1.

Is there a Davies example provably in ZFC?

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Background

  1. Theodore Slaman

In 1954, [1] shows that for E⊆R2E\subseteq\mathbb{R}^{2} an analytic set with Hausdorff dimension 1. Then, for almost all θ\theta, the projection of EE onto a line with slope θ\theta has Hausdorff dimension 1. Later in 1969, [2] shows that CH could imply there is some E⊆R2E\subseteq\mathbb{R}^{2} with Hausdorff dimension 1 and for all θ\theta, projection of EE onto a line of slope θ\theta has Hausdorff dimension 0.

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Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today.

References
  1. [1]Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; A Davies example provable in ZFC — Problem 7.1 linkhttp://creativecommons.org/licenses/by/4.0/; attributed adaptation; HTML converted to Markdown with original mathematical notation and source links.

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