## Notre Dame Journal of Formal Logic

### A Bounded Jump for the Bounded Turing Degrees

#### Abstract

We define the bounded jump of $A$ by $A^{b}=\{x\in \omega \mid \exists i\leq x[\varphi_{i}(x)\downarrow \wedge\Phi_{x}^{A\upharpoonright \!\!\!\upharpoonright \varphi_{i}(x)}(x)\downarrow ]\}$ and let $A^{nb}$ denote the $n$th bounded jump. We demonstrate several properties of the bounded jump, including the fact 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 $n\geq2$ we have $X\leq_{bT}\emptyset ^{nb}\iff X$ is $\omega^{n}$-c.e. $\iff X\leq_{1}\emptyset ^{nb}$, extending the classical result that $X\leq_{bT}\emptyset '\iff X$ is $\omega$-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 $\emptyset ^{b}\leq_{bT}X\leq_{bT}\emptyset ^{2b}$, there is a $Y\leq_{bT}\emptyset ^{b}$ such that $Y^{b}\equiv_{bT}X$.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 55, Number 2 (2014), 245-264.

Dates
First available in Project Euclid: 24 April 2014

https://projecteuclid.org/euclid.ndjfl/1398345783

Digital Object Identifier
doi:10.1215/00294527-2420660

Mathematical Reviews number (MathSciNet)
MR3201835

Zentralblatt MATH identifier
1307.03024

Subjects
Primary: 03D30: Other degrees and reducibilities

#### Citation

Anderson, Bernard; Csima, Barbara. A Bounded Jump for the Bounded Turing Degrees. Notre Dame J. Formal Logic 55 (2014), no. 2, 245--264. doi:10.1215/00294527-2420660. https://projecteuclid.org/euclid.ndjfl/1398345783

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