## 117b – Turing degrees – Lecture 2

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.