117b – Turing degrees – Lecture 2

January 9, 2007

We defined the structure {\mathcal D} of the Turing degrees as the quotient of the set of functions f:{\mathbb N}\to{\mathbb N} by the equivalence relation \equiv_T of recursive bi-reducibility, seen as a partial order with the order <_T induced by Turing reducibility. This structure has a least degree and any degree has only countably many predecessors. It is an upper semilattice, and any countably many degrees have a common upper bound.

Generalizing the halting problem, we can define the Turing jump {\bf a}' of any degree {\bf a}. It is strictly above {\bf a} and any A r.e. in {\bf a} is T-reducible to {\bf a}'.

There are incomparable Turing degrees. They can be built by the finite extension method, an example of forcing in recursion theory.