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Proceedings of the American Mathematical Society

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

 

 

Minimal covers and arithmetical sets


Authors: Carl G. Jockusch and Robert I. Soare
Journal: Proc. Amer. Math. Soc. 25 (1970), 856-859
MSC: Primary 02.70
DOI: https://doi.org/10.1090/S0002-9939-1970-0265154-X
MathSciNet review: 0265154
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Abstract: If $ a$ and $ b$ are degrees of unsolvability, $ a$ is called a minimal cover of $ b$ if $ b < a$ and no degree $ c$ satisfies $ b < c < a$. The degree $ a$ is called a minimal cover if it is a minimal cover of some degree $ b$. We prove by a very simple argument that $ {0^n}$ is not a minimal cover for any $ n$. From this result and the axiom of Borel determinateness (BD) we show that the degrees of arithmetical sets (with their usual ordering) are not elementarily equivalent to all the degrees. We also point out how this latter result can be proved without BD when the jump operation is added to the structures involved.


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DOI: https://doi.org/10.1090/S0002-9939-1970-0265154-X
Keywords: Recursive function, degree of unsolvability, arithmetical hierarchy, axiom of determinateness
Article copyright: © Copyright 1970 American Mathematical Society