Rigidity and biinterpretability in the hyperdegrees

Slaman and Woodin have developed and used set-theoretic methods to prove some remarkable theorems about automorphisms of, and de…nability in, the Turing degrees. Their methods apply to other coarser degree structures as well and, as they point out, give even stronger results for some of them. In particular, their methods can be used to show that the hyperarithmetic degrees are rigid and biinterpretable with second order arithmetic. We give a direct proof using only older coding style arguments to prove these results without any appeal to set-theoretic or metamathematical considerations. Our methods also apply to various coarser