Recursive and Recursively Enumerable Sets (Continued)

Abstract

In this chapter we continue the study of the recursive and the recursively enumerable subsets of ℕ. First more examples for recursively enumerable sets which are not recursive are given. The non-recursiveness is proved by reduction to the self applicability problem or in one case by a new kind of diagonalization. Several characterizations of the recursive and the recursively enumerable sets are introduced and the corresponding numberings are compared w.r.t. reducibility. In most cases the nonexistence of a computable function can be reduced to the unsolvability of the halting problem. Finally the φ-recursive and the φ-recursively enumerable subsets of P⁽¹⁾ are studied. Rice’s theorem states that no non-trivial property P⁽¹⁾ is φ-recursive. A more general lemma gives two necessary conditions for φ-r.e. sets, by which many subsets of P⁽¹⁾ can be proved to be not φ-r.e.