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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Defining jump classes in the degrees below $ {\bf0}'$


Author: Richard A. Shore
Journal: Proc. Amer. Math. Soc. 104 (1988), 287-292
MSC: Primary 03D30; Secondary 03D20
MathSciNet review: 958085
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Abstract: We prove that, for each degree $ {\mathbf{c}}$ r.e. in and above $ {{\mathbf{0}}^{(3)}}$, the class of degrees $ {\mathbf{x}} \leq {\mathbf{0}}'$ with $ {{\mathbf{x}}^{(3)}} = {\mathbf{c}}$ is definable without parameters in $ \mathcal{D}( \leq 0')$, the degrees below $ {\mathbf{0'}}$. Indeed the same definitions work below any r.e. degree $ {\mathbf{r}}$ in place of $ {\mathbf{0'}}$. Thus for each r.e. degree $ {\mathbf{r}}$, $ \operatorname{Th} (\mathcal{D}( \leq {\mathbf{r}}))$ uniquely determines $ {{\mathbf{r}}^{(3)}}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0958085-4
Keywords: Degrees below $ {\mathbf{0'}}$, jump classes, definability
Article copyright: © Copyright 1988 American Mathematical Society