# Chapter 12. Degrees of Unsolvability

Degrees of unsolvability study how undecidable problems can be compared according to their relative computational difficulty, rather than treating all undecidable problems as having the same level of complexity.

The chapter begins with reducibility, where one problem is transformed into another in a computable way, so that a solution to the second problem would also give a solution to the first.

Turing degrees are then introduced as equivalence classes of problems that can compute one another, giving a formal measure of relative unsolvability.

Recursively enumerable sets are studied as sets whose elements can be listed by an algorithm, even when membership in the set may not be decidable.

Post's problem is then presented as a central question about whether there exist recursively enumerable degrees strictly between decidable sets and the halting problem.

The final part of the chapter introduces the structure of degrees, where the collection of Turing degrees is studied as an ordered mathematical object with rich and subtle internal behavior.