117b – Turing degrees – Lecture 2

We define the structure D of the Turing degrees as the quotient of the set of functions f:N®N by the equivalence relation ºT 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 a¢ of any degree a. It is strictly above a and any A r.e. in a is T-reducible to a¢.

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

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