A Bounded Jump for the Bounded Turing Degrees

We define the bounded jump of A by A = {x ∈ ω | ∃i ≤ x[φi(x) ↓ ∧ Φ φi(x) x (x)↓]} and let A denote the n-th bounded jump. We demonstrate several properties of the bounded jump, including that it is strictly increasing and order preserving on the bounded Turing (bT ) degrees (also known as the weak truth-table degrees). We show that the bounded jump is related to the Ershov hierarchy. Indeed, for n ≥ 2 we have X ≤bT ∅ ⇐⇒ X is ω-c.e. ⇐⇒ X ≤1 ∅, extending the classical result that X ≤bT ∅′ ⇐⇒ X is ω-c.e.. Finally, we prove that the analogue of Shoenfield inversion holds for the bounded jump on the bounded Turing degrees. That is, for every X such that ∅ ≤bT X ≤bT ∅, there is a Y ≤bT ∅ such that Y b ≡bT X.