Open Problems in Mathematical Logic — Problem 29

Definitions and mathematical statement

The supplied record contains two questions referring to a preceding Velickovic--Woodin theorem whose statement is absent. Consequently, the target theorem cannot be reconstructed from the supplied evidence.

Under the customary interpretation of the notation, $\Sigma^1_1(y)$ denotes the class of sets of reals that are effectively analytic relative to the real oracle $y$, and $\omega_1^{CK(z)}$ denotes the least ordinal having no $z$-computable well-order presentation. The source does not explicitly specify the ambient space of reals or its coding.

1. Is there a recursion-theoretic proof of the Velickovic--Woodin theorem referred to in the source as the "above theorem"? The theorem and the intended requirements on such a proof are not supplied.
2. For a parameter $y$ and every $\Sigma^1_1(y)$ set $A$, does \[ \exists x\in A\;\left(\omega_1^{CK(x)}>\omega_1^{CK(y)}\right) \] imply that the Velickovic--Woodin theorem holds for $A$? Here "the theorem holds for $A$" retains the source's unresolved reference; its conclusion and any additional hypotheses are unavailable. The scope and domain of the parameter $y$ are implicit in the source.

These are incomplete formulation targets pending recovery of the referenced theorem, not fully specified mathematical assertions.

Short English statement

The source asks for a recursion-theoretic proof of an unspecified Velickovic--Woodin theorem and asks whether that theorem extends to every effectively analytic set relative to $y$ containing an $x$ with $\omega_1^{CK(x)}>\omega_1^{CK(y)}$. The missing theorem prevents a complete precise statement.

Sources

• Local UnsolvedMath registry: permanent ID $801017$, source code catalogue entry, title "Open Problems in Mathematical Logic --- Problem 29." The registry attributes the record to Barmpalias, Gao et al., with proposed year 2026.
• Supplied source reference: Barmpalias, Gao et al., Open Problems in Mathematical Logic (2026), Problem 29, https://arxiv.org/abs/2608.26628. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not independently verified access or source verification.
• The source names Velickovic--Woodin in connection with the missing preceding theorem but supplies neither that theorem nor a separate bibliographic citation for it. Upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry labels the problem "open"; this is supplied status metadata, not an independently verified mathematical assessment.