Recursively enumerable sets and degrees
Recursively enumerable sets and degrees
Degrees with almost universal cupping property
CiE'10 Proceedings of the Programs, proofs, process and 6th international conference on Computability in Europe
Cupping and diamond embeddings: a unifying approach
CiE'11 Proceedings of the 7th conference on Models of computation in context: computability in Europe
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Cupping nonzero computably enumerable (c.e. for short) degrees to 0′ in various structures has been one of the most important topics in the development of classical computability theory. An incomplete c.e. degree a is cuppable if there is an incomplete c.e. degree b such that a∪b=0′, and noncuppable if there is no such degree b. Sacks splitting theorem shows the existence of cuppable degrees. However, Yates(unpublished) and Cooper [3] proved that there are noncomputable noncuppable degrees. After that, Harrington and Shelah were able to employ the cupping/noncupping properties to show that the theory of the c.e. degrees under relation ≤ is undecidable. Cuppable and noncuppable degrees were further studied later. See Harrington [7], Miller [10], Fejer and Soare [6], Ambos-Spies, Lachlan and Soare [1], etc..