We finished the first topic of the course with a survey of results about the theory of .

In particular, we stated the Slaman-Woodin theorems on coding countable relations inside and on the structure of .

The coding theorem and the *arithmetic definability *of give a new proof of Simpson’s theorem on the complexity of : Not only it is undecidable, but it is *recursively isomorphic* to the theory of *second-order *arithmetic.* *That _{ }is definable in first order arithmetic will be shown as part of the second topic.

The results on imply that there are only countably many automorphisms of , that they are all arithmetically definable, and coincide with the identity above . This was then used by Slaman and Shore to prove that the relation iff is definable in .

We then stated Martin’s theorem on *Turing Borel determinacy* that any degree invariant property which is Borel as a subset of and -cofinal holds on a *cone*.