We defined the structure of the Turing degrees as the quotient of the set of functions by the equivalence relation of recursive bi-reducibility, seen as a partial order with the order 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 of any degree . It is strictly above and any r.e. in is T-reducible to .
There are incomparable Turing degrees. They can be built by the finite extension method, an example of forcing in recursion theory.