No-Least-Join Theorem, Question 1.19
Given a computably enumerable set and sets with , let , , and . Must there exist a nonzero Turing degree such that, for every Turing degree , if and , then ?
Equivalent formulations 2Other wordings
Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.
Cone formulation
For as above, is there necessarily a nonzero Turing cone contained in the collection ?
Common-lower-bound formulation
For as above, does the collection necessarily have a nonzero common lower bound?
References
Primary source
Additional references
- Cone Avoidance and the No-Least-Join Theorem — arXiv — Patrizio Cintioli
Progress summary
An unrefereed preprint claims to settle the question negatively by strengthening the construction so it avoids any chosen noncomputable cone.
Question concerns whether a particular structural phenomenon involving joins and lower bounds occurs in the Turing degrees. Patrizio Cintioli’s construction claims a negative answer.
October 1, 2026 cone-avoidance development
Cintioli’s preprint, Cone Avoidance and the No-Least-Join Theorem, claims a no-least-join construction that avoids an arbitrary noncomputable cone, thereby settling the stated question negatively. The result is presented as an unrefereed preprint and is not independently verified here.
Current status (as of October 2026): The question has a claimed negative answer from Cintioli’s preprint, but that claim remains unverified.
Solutions 0
No solutions have been posted yet.