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

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

#### 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.

First Page:
Primary Subjects: 03D25
Secondary Subjects: 68Q30
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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

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