Diagonalization in degree constructions

Extract

We present two theorems whose applications are to eliminate diagonalization arguments from a variety of constructions of degrees of unsolvability.

All definitions and notations come from [1, Chapter 1]. We give a brief resumé of them here.

We identify a set with its characteristic function. (A(x)= 1 if x ∈ A and A (x) = 0 if x ∉ A.) A string σ is the restriction of a characteristic function to a finite initial segment of natural numbers, lh(σ) = length of σ = n + 1 if σ = A[n] for some set A. (A [n] is the restriction of A to {m: m ≤ n}.) If i = 0 or 1, σ * i is defined as the string of length lh(σ) + 1 such that σ * i ⊇ σ and σ * i(lh(σ)) = i. We write σ ∣ τ if σ ⊉ τ and τ ⊉ σ.

{Φ_n} is a listing of the partial recursive functionals. We write “A ≤_TB” (“A is Turing reducible to B”) if ∃n∀xΦ_n(B)(x) = A(x).

A partial function, T, from strings to strings is a tree if T is order preserving and for all strings, σ, if one of T(σ * 0), T(σ * 1) is defined then T(σ), T(σ * 0), T(σ * 1) are all defined and T(σ * 0)∣(σ * 1).