Properly $\Sigma _{2}^{0}$ Enumeration Degrees and the High/Low Hierarchy

Journal of Symbolic Logic 71 (4):1125 - 1144 (2006)
We show that there exist downwards properly $\Sigma _{2}^{0}$ (in fact noncuppable) e-degrees that are not high. We also show that every high e-degree bounds a noncuppable e-degree.
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