C Notations and symbols

–A real x is a member of 2ω or ωω. A string is a member of either 2<ω or ω<ω. Strings are denoted σ, τ etc.

–If x is a real, then x ↾ n is a finite string σ of length n such that for all i < n, σ(i)= x(i).

–If σ is a finite string, then |σ| is its length.

–x ≻ σ means x ↾|σ|= σ.

–[σ]={x | x ∈ ωω ∧ x ≻ σ}.

–If f is a function, then Dom(f) denotes the domain of f, and Rang(f) its range. Dom(σ) and Rang(σ) carry the obvious meaning where σ is considered as a finite function.

–If T is a tree, then [T]={x | x ∈ ωω ∧(∀n)(x ↾ n ∈ T)}.

–Given a set A, (A)={B | B ⊆ A} is the power set of A.

–≤T denotes Turing reduction.

–x′ is the Turing jump of x.

–x ⊕ y is the real z such that z(2n)= x(n) and z(2n + 1)= y(n).

–〈⋅, ⋅〉 is a recursion bijection from ω2 to ω.

–For any n and x ∈ 2ω, x[n] ={m | 〈n, m〉 ∈ x}.

–If A is a set, then (A) denotes the power set of A.

–A ⊆ 2ω or ωω is or (resp. or ) if it is or (resp. or definable with real number parameters.

–Given a partial ordering 〈P, ≤〉, p|q means p and q are incomparable, i.e. p q and q p.