Open Problems in Mathematical Logic — Problem 23

Definitions and mathematical statement

The supplied record concerns a map $A\mapsto I(A)$, but supplies neither its definition nor its domain or codomain. It also uses $H^f$ without defining the admissible functions $f$ or the associated set function. Consequently, the following four questions retain unresolved notation and require the missing source context before they constitute fully specified mathematical problems.

1. Determine whether \[ \left|\operatorname{ran}(I)\right|=\left|\mathcal P({\mathbb{R}})\right|, \] where $\mathcal P({\mathbb{R}})$ is the power set of ${\mathbb{R}}$, interpreting the source's $2^{\mathbb R}$ in its usual power-set sense. Here $\operatorname{ran}(I)=\{I(A):A\in\operatorname{dom}(I)\}$, with $\operatorname{dom}(I)$ unspecified. The source adds "Seems yes under GCH"; this is a supplied tentative qualification, not an established implication asserted here. GCH denotes the generalized continuum hypothesis.

2. The source states that for a closed set $C$, $I(C)$ is arithmetic, of complexity $\mathbf{\Sigma}^{0}_{2}$. It asks for the complexity of \[ \mathcal S=\left\{\varphi:\varphi\text{ is a }\mathbf{\Sigma}^{0}_{2}\text{ formula and defines }I(C)\text{ for some closed }C\right\}, \] and specifically asks whether $\mathcal S$ is $\mathbf{\Sigma}^{1}_{2}$-complete. The source further states that $\mathcal S$ is $\mathbf{\Pi}^{1}_{1}$-hard. These complexity assertions are supplied claims. The ambient space in which $C$ is closed, the objects defined by the formulas, permitted parameters, coding of formulas, and reductions used for hardness and completeness are not specified. Thus $\mathcal S$ is presently a schematic description, not a fixed coded set.

3. Determine whether the assertion described in the source as "Borel hierarchy is proper w.r.t. the range of $I$" is independent of ZFC, the Zermelo--Fraenkel axioms with choice. The source does not specify the underlying topological space, how the range of $I$ is compared with Borel classes, or what precise strictness assertion is intended. A definite sentence whose independence is being asked about therefore remains to be supplied.

4. Determine whether it is consistent with ZFC that, for every admissible $f$ and every $A\subseteq{\mathbb{R}}$, positivity of $H^f(A)$ implies the existence of a subset $A_0$ with finite positive $H^f$-value. If "a subset $A_0$" means a subset of $A$, the intended assertion has the conditional form \[ \forall f\;\forall A\subseteq{\mathbb{R}}\;\left(H^f(A)>0\ \Longrightarrow\ \exists A_0\subseteq A\;\left(0<H^f(A_0)<\infty\right)\right). \] The quantifier over $f$ must be restricted to the intended, presently unspecified class. The displayed formulation explicitly records the natural interpretation $A_0\subseteq A$; the supplied wording does not state that inclusion explicitly. No definition of $H^f$, including any conventions governing its values, is supplied.

Short English statement

Problem 23 asks four questions about an unspecified assignment $A\mapsto I(A)$: the size of its range, the definability complexity of outputs arising from closed sets, independence of a Borel-hierarchy strictness assertion, and consistency of a finite-positive-subset property for $H^f$. Missing definitions and coding conventions prevent a complete formal statement. The registry labels the problem open; that label and the accompanying complexity and GCH comments are supplied information requiring review.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $801011$, exact source problem code catalogue entry, title Open Problems in Mathematical Logic --- Problem 23. The supplied proposer attribution is Barmpalias, Gao et al., with proposed year 2026. The registry status is open; it is not an independently verified literature assessment.
• Supplied source attribution: Barmpalias, Gao et al., Open Problems in Mathematical Logic (2026), Problem 23. Supplied source URL: https://arxiv.org/abs/2608.26628. The record reports access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance details; the source has not been independently visited or verified for this draft. The supplied excerpt omits the definitions needed to resolve the notation.