Open Problems in Mathematical Logic — Problem 3

Definitions and mathematical statement

Let $\emptyset'$ denote the halting set, and let $\mathrm{ML}(\emptyset')$ denote the class of reals that are Martin--Löf random relative to $\emptyset'$, equivalently the $2$-random reals. In the usual binary-sequence presentation, randomness relative to $\emptyset'$ means avoiding $\bigcap_{n\in{\mathbb{N}}}U_n$ for every sequence $(U_n)_{n\in{\mathbb{N}}}$ of uniformly $\emptyset'$-effectively open subsets of $2^{{\mathbb{N}}}$ with $\mu(U_n)\leq 2^{-n}$, where $\mu$ is the fair-coin product measure. This explains the standard randomness terminology; the supplied excerpt does not specify its representation of reals.

For a specified comparison relation $\preccurlyeq$, a subset $B$ is a chain if \[ \forall x,y\in B\quad (x\preccurlyeq y\ \lor\ y\preccurlyeq x). \] The supplied question is whether there exists a chain $B$ such that \[ B\subseteq\mathrm{ML}(\emptyset') \quad\text{and}\quad \forall\alpha\in\mathrm{ML}(\emptyset')\quad [B]\neq[\alpha]. \] Here $\preccurlyeq$ is only a placeholder for the intended comparison relation. The supplied record does not define that relation, the construction $[B]$, or the construction $[\alpha]$. In particular, it does not specify the objects being compared by the displayed inequality. Consequently, this display preserves the source question but is not a fully determined mathematical assertion until those definitions are recovered. No nonemptiness, cardinality, or effectiveness requirement on $B$ is supplied. The introductory description of this class as a special case refers to surrounding material absent from the record.

Short English statement

The source asks whether some chain of $2$-random reals has an associated object $[B]$ different from $[\alpha]$ for every individual $2$-random real $\alpha$. The ordering and bracket constructions are missing, so the intended statement requires clarification from the source context.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800991$, exact source problem code catalogue entry, title "Open Problems in Mathematical Logic --- Problem 3." Upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The record attributes the problem to Barmpalias, Gao et al., with proposed year $2026$, and identifies Problem 3 of Open Problems in Mathematical Logic (2026): https://arxiv.org/abs/2608.26628. The supplied extraction method is arxiv_latex; the supplied access date is 2026-10-16. These are supplied provenance claims, not independent source verification.
• The local record labels the problem open and records its update at 2026-10-16T14:00:00Z. This status is supplied registry curation, not an independently verified mathematical or literature assessment.