Open Problems in Mathematical Logic — Problem 2 (Levin)

Definitions and mathematical statement

Let $\omega=\{0,1,2,\ldots\}$ and let $2^\omega=\{\alpha:\omega\to\{0,1\}\}$ be Cantor space. The supplied record asks whether the following assertion holds:

For every $X\subseteq 2^\omega$ with $\mu(X)=1$, there exists $Y\subseteq 2^\omega$ with $\mu(Y)=1$ such that, for every chain $B\subseteq Y$, there is some $\alpha\in X$ satisfying \[ [B]=[\alpha]. \]

The supplied record does not define $\mu$, the relation with respect to which $B$ is a chain, or the meanings of $[B]$ and $[\alpha]$. In particular, it does not specify whether the measure conditions require measurable sets or refer to outer measure, nor whether empty chains are included. These unresolved qualifications are retained: the assertion above reproduces the source question but cannot yet be interpreted as a fully specified mathematical statement.

Short English statement

The question attributed to Levin asks whether every set $X\subseteq 2^\omega$ of measure one admits a set $Y\subseteq 2^\omega$ of measure one such that each chain in $Y$ has its bracketed value represented by an element of $X$. The measure, chain relation, and bracket notation require clarification from the source.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800990$, source problem code catalogue entry, title Open Problems in Mathematical Logic --- Problem 2 (Levin). The registry labels the problem open; this is a supplied status, not an independently verified assessment.
• Barmpalias, Gao et al., Open Problems in Mathematical Logic (2026), Problem 2, https://arxiv.org/abs/2608.26628. The supplied background attributes the problem/context to Levin; the registry lists Barmpalias, Gao et al. in its proposer field and 2026 in its proposed-year field. These entries do not establish an original proposal date for Levin.
• The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance claims; the cited source has not been independently accessed or verified for this draft. Upstream record ID: $800990$; upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.