Comparing the degrees of enumerability and the closed Medvedev degrees

Paul Shafer, University of Leeds. Part of the proofs, constructions and computations seminar series.

Both the Turing degrees and the enumeration degrees embed into the Medvedev degrees, with the embedding of the Turing degrees being particularly nice. 

Every Turing degree is mapped to the Medvedev degree of a closed set (in particular, a singleton), and the range of the embedding is definable (a theorem of Dyment and Medvedev).  On the other hand, little is known about the embedding of the enumeration degrees in the Medvedev degrees.

Call a Medvedev degree 'closed' if it is the degree of a closed subset of Baire space, and call a Medvedev degree a 'degree of enumerability' if it is in the range of the embedding of the enumeration degrees into the Medvedev degrees. 

We explore the distribution of the degrees of enumerability with respect to the closed degrees and find that many situations occur. There are nonzero closed degrees that do not bound nonzero degrees of enumerability, there are nonzero degrees of enumerability that do not bound nonzero closed degrees, and there are degrees that are nontrivially both degrees of enumerability and closed degrees. We also show that the compact degrees of enumerability exactly correspond to the cototal enumeration degrees.

This work is joint with Andrea Sorbi (University of Siena).