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How enumeration reductibility yields extended Harrington non-splitting

Published online by Cambridge University Press:  12 March 2014

Mariya I. Soskova  and
S. Barry Cooper
Mariya I. Soskova
Affiliation:
University of Leeds, Department of Pure Mathematics, Leeds, LS2 9JT., UK, E-mail: s.b.cooper@leeds.ac.uk
S. Barry Cooper
Affiliation:
University of Leeds, Department of Pure Mathematics, Leeds, LS2 9JT., UK, E-mail: mariya@maths.leeds.ac.uk
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§1. Introduction. Sacks [16] showed that every computably enumerable (c.e.) degree > 0 has a c.e. splitting. Hence, relativising, every c.e. degree has a Δ2 splitting above each proper predecessor (by ‘splitting’ we understand ‘nontrivial splitting’). Arslanov [1] showed that 0′ has a d.c.e. splitting above each c.e. a < 0′. On the other hand, Lachlan [11] proved the existence of a c.e. a < 0 which has no c.e. splitting above some proper c.e. predecessor, and Harrington [10] showed that one could take a = 0′. Splitting and nonsplitting techniques have had a number of consequences for definability and elementary equivalence in the degrees below 0′.

Heterogeneous splittings are best considered in the context of cupping and non-cupping. Posner and Robinson [15] showed that every nonzero Δ2 degree can be nontrivially cupped to 0′, and Arslanov [1] showed that every c.e. degree > 0 can be d.c.e. cupped to 0′ (and hence since every d.c.e., or even n-c.e., degree has a nonzero c.e. predecessor, every n-c.e. degree > 0 is d.c.e. cuppable). Cooper [4] and Yates (see Miller [13]) showed the existence of degrees noncuppable in the c.e. degrees. Moreover, the search for relative cupping results was drastically limited by Cooper [5], and Slaman and Steel [17] (see also Downey [9]), who showed that there is a nonzero c.e. degree a below which even Δ2 cupping of c.e. degrees fails.

We prove below what appears to be the strongest possible of such nonsplitting and noncupping results.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 2 , June 2008 , pp. 634 - 655
Copyright
Copyright © Association for Symbolic Logic 2008

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References

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