This chapter is devoted to the presentation of basic definitions and notation to be used in this monograph. The definitions fall under three general headings; those related to computable partial functionals and computably enumerable sets, those related to the computably enumerable degrees, and those related to trees.

Computably Enumerable Sets

Let ℕ be the natural numbers, i.e., the set of integers {0, 1, 2, …}. We use interval notation on ℕ; thus [k, m] = {n : k ≤ n ≤ m}. Open interval notation and half-open interval notation is used in a similar fashion. The direct sum of two subsets A and B of ℕ is denoted as A ⊕ B and is defined as {2x : x ∈ A} ∪ {2x + 1 : x ∈ B}. |A| will denote the cardinality of the set A.

If A ⊂ ℕ, m ∈ ℕ, and Φ is a partial functional, then we write Φ(A; m) ↓ if m is in the domain of Φ(A), and Φ(A; m) ↑ otherwise. If Φ and Ψ are partial functionals and A and B are sets, then we write Φ(A) ≃ Ψ(B) if Φ(A) and Ψ(B) are compatible, i.e., for all x, if Φ(A; x) ↓ and Ψ(B; x) ↓, then Φ(A; x) = Ψ(B; x); and we write Φ(A) = Ψ(B) if Φ(A) and Ψ(B) are identical, i.e., Φ(A) and Ψ(B) are compatible and for all x, Φ(A) ↓ iff Ψ(B) ↓.