Theorem 7.3.1 may be iterated into the transfinite in several different ways. Here is the first version.
Theorem 7.3.2. (Slaman and Steel ). Let 0 < and let z be a real such that (γ) <T z for each γ<β. Then there is a g such that g ⊕ z ≡T g(β).
The rest of the section is devoted to a proof of Theorem 7.3.2.
Let a ∈ . The first step is to introduce an enumeration of arithmetical predicates indexed by ...