The continuous degrees measure the computability-theoretic content of elements of computable metric spaces. They properly extend the Turing degrees and naturally embed into the enumeration degrees. Although nontotal (i.e., non-Turing) continuous degrees exist, they are all very close to total: joining a continuous degree with a total degree that is not below it always results in a total degree. We call this property almost totality.
We prove that the almost total degrees coincide with the continuous degrees. Since the total degrees are definable in the partial order of enumeration degrees , we see 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.
In order to prove that every almost total degree is continuous, we pass through another characterization of the continuous degrees that slightly simplifies one of Kihara and Pauly . We prove 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.
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