Notre Dame Journal of Formal Logic

A Bounded Jump for the Bounded Turing Degrees

Bernard Anderson and Barbara Csima

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Abstract

We define the bounded jump of A by Ab={xωix[φi(x)ΦxAφi(x)(x)]} and let Anb denote the nth 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 n2 we have XbTnbX is ωn-c.e. X1nb, extending the classical result that XbT'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 bbTXbT2b, there is a YbTb such that YbbTX.

Article information

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

Dates
First available in Project Euclid: 24 April 2014

Permanent link to this document
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

Keywords
bounded jump jump bounded Turing degrees $bT$-degrees wtt degrees

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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