Random Axioms in Computability and Kolmogorov Complexity

Definitions and mathematical statement

The supplied record asks:

Are random axioms useful in computability theory and Kolmogorov complexity?

This is a research question whose terms are insufficiently specified to determine a precise mathematical proposition. The record does not define "random axioms," a probability space or selection procedure for such axioms, a background formal theory, or a criterion for "useful." It also does not specify a particular notion of Kolmogorov complexity, a target computability-theoretic result, or any quantifiers governing the intended assertion.

Consequently, no formal existence, universal, or probabilistic assertion can be extracted without adding hypotheses or choosing an interpretation absent from the supplied evidence. The question is retained with these qualifications rather than replaced by an invented formal statement.

Short English statement

The question asks whether random axioms can be useful in computability theory and Kolmogorov complexity. The supplied record leaves both random axioms and usefulness undefined, so a precise mathematical target remains unresolved. Its open status is a supplied curation label.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30001981$, source problem code catalogue entry, title "Random Axioms in Computability and Kolmogorov Complexity," in the Oberwolfach Reports Open Problems collection. The proposed year is recorded as 2012; no individual proposer is supplied.
• Supplied report citation: Computability Theory (2012), Oberwolfach Reports, pp. 397--455, DOI 10.4171/owr/2012/08. Source URL and supplied literature reference labelled "original Oberwolfach report": https://doi.org/10.4171/owr/2012/08.
• The registry labels the statement "exact" and reports that its mathematical content was checked against the official report, with notation, context, and question form normalized without changing the source-level claim. This is supplied verification metadata, not an independent verification performed for this draft.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the question open and states that the extraction contains no precise formal statement about the usefulness of random axioms, leaving it unresolved as a research question. This is supplied dated curation, not independently verified. The supplied background calls for independent verification of time-sensitive status and cited claims.