We finish the first topic of the course with a survey of results about the theory of ** D**.

In particular, we state the Slaman-Woodin theorems on coding countable relations inside ** D **and on the structure of Aut(

**).**

*D*The coding theorem and the *arithmetic definability *of <_{T} give a new proof of Simpson’s theorem on the complexity of Th(** D**): Not only it is undecidable, but it is

*recursively isomorphic*to the theory of

*second-order arithmetic.*That <

_{T }is definable in first order arithmetic will be shown as part of the second topic.

The results on Aut(** D**) give that there are only countably many automorphisms of

**, that they are all arithmetically definable, and coincide with the identity above**

*D***0**². This was then used by Slaman and Shore to prove that the relation R(

*x*,

*y*) iff

*y=x*¢ is definable in

*.*

**D**We then state Martin’s theorem on *Turing Borel determinacy* that any degree invariant property which is Borel as a subset of 2^{N }and <_{T}-cofinal holds on a *cone*.