The Turing Hierarchy of Unsolvability

Abstract

At this point we only know of two degrees of unsolvability: the T-degree shared by all the decidable decision problems, and the T-degree shared by all undecidable decision problems that are T-equivalent to the Halting Problem. In this chapter we will prove that, surprisingly, for every undecidable decision problem there exists a more difficult decision problem. This will in effect mean that there is an infinite hierarchy of degrees of unsolvability and that there is no most difficult decision problem.

A hierarchy is a system of organizing things into different ranks, levels, or positions, depending on how important they are.