Notre Dame Journal of Formal Logic

On the Degrees of Diagonal Sets and the Failure of the Analogue of a Theorem of Martin

Keng Meng Ng

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Abstract

Semi-hyperhypersimple c.e. sets, also known as diagonals, were introduced by Kummer. He showed that by considering an analogue of hyperhypersimplicity, one could characterize the sets which are the Halting problem relative to arbitrary computable numberings. One could also consider half of splittings of maximal or hyperhypersimple sets and get another variant of maximality and hyperhypersimplicity, which are closely related to the study of automorphisms of the c.e. sets. We investigate the Turing degrees of these classes of c.e. sets. In particular, we show that the analogue of a theorem of Martin fails for these classes.

Article information

Source
Notre Dame J. Formal Logic Volume 50, Number 4 (2009), 469-493.

Dates
First available: 11 February 2010

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1265899126

Digital Object Identifier
doi:10.1215/00294527-2009-022

Mathematical Reviews number (MathSciNet)
MR2598875

Zentralblatt MATH identifier
05778812

Subjects
Primary: 03D25
Secondary: 68Q30

Keywords
maximal hyperhypersimple computable numberings

Citation

Ng , Keng Meng. On the Degrees of Diagonal Sets and the Failure of the Analogue of a Theorem of Martin. Notre Dame Journal of Formal Logic 50 (2009), no. 4, 469--493. doi:10.1215/00294527-2009-022. http://projecteuclid.org/euclid.ndjfl/1265899126.


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