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Journal of Logic and Computation Advance Access published online on August 8, 2007
Journal of Logic and Computation, doi:10.1093/logcom/exm032
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© The Author, 2007. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Original papers

Post's Programme for the Ershov Hierarchy

Bahareh Afshari and George Barmpalias

School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK.
E-mail: bahareh{at}maths.leeds.ac.uk, georgeb{at}maths.leeds.ac.uk

S. Barry Cooper

School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK.
E-mail: pmt6sbc{at}leeds.ac.uk

Frank Stephan

Departments of Mathematics and Computer Science, National University of Singapore, 2 Science Drive 2, Singapore 117543, Republic of Singapore.
E-mail: fstephan{at}comp.nus.edu.sg


  Abstract

This article extends Post's; programme to finite levels of the Ershov hierarchy of {Delta}2 sets. Our initial characterization, in the spirit of Post (1994, Bulletin of the American Mathematical Society, 50, 284–316), of the degrees of the immune and hyperimmune n-enumerable sets leads to a number of results setting other immunity properties in the context of the Turing and wtt-degrees derived from the Ershov hierarchy. For instance, we show that any n-enumerable hyperhyperimmune set must be co-enumerable, for each n ≥ 2. The situation with regard to the wtt-degrees is particularly interesting, as demonstrated by a range of results concerning the wtt-predecessors of hypersimple sets.

Finally, we give a number of results directed at characterizing basic classes of n-enumerable degrees in terms of natural information content. For example, a 2-enumerable degree contains a 2-enumerable dense immune set iff it contains a 2-enumerable r-cohesive set iff it bounds a high enumerable set. This result is extended to a characterization of n-enumerable degrees which bound high enumerable degrees. Furthermore, a characterization for n-enumerable degrees bounding only low2 enumerable degrees is given.

Keywords: Cohesiveness; enumerable sets; Ershov hierarchy; immunity properties; jump classes; Post's programme; Turing degrees


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