## Characterizing the continuous degrees

### Speaker : Joseph S. Miller

### Abstract

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
curious 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 (Cai, Ganchev, Lempp, Miller, Soskova), 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. Like them, we identify
our almost total degree as the degree of a point in a computably
regular space with a computable dense sequence, and then we apply the
effective version of Urysohn's metrization theorem (Schröder) to
reveal our space as a computable metric space.

This is joint work with Uri Andrews, Greg Igusa, and Mariya Soskova.