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Characterizing the continuous degrees. (English) Zbl 1442.03023
The paper is devoted to the continuous degrees. It is shown that the almost total degrees coincide with the continuous degrees. Since the total degrees are definable in the partial order of enumeration degrees, it follows that the continuous degrees are also definable. Applying earlier work on the continuous degrees, this shows that the relation “PA above” on the total degrees is definable in the enumeration degrees. The authors also prove that every almost total degree is continuous and that the enumeration degree of \(A\) is continuous if and only if \(A\) is codable, meaning that \(A\) is enumeration above the complement of an infinite tree, every path of which enumerates \(A\).

MSC:
03D35 Undecidability and degrees of sets of sentences
03D25 Recursively (computably) enumerable sets and degrees
03D78 Computation over the reals, computable analysis
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