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 [142]). 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 ...

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