Structural Properties and $\Sigma^0_2$ Enumeration Degrees

Abstract

We prove that each $\Sigma^0_2$ set which is hypersimple relative to $\emptyset$' is noncuppable in the structure of the $\Sigma^0_2$ enumeration degrees. This gives a connection between properties of $\Sigma^0_2$ sets under inclusion and and the $\Sigma^0_2$ enumeration degrees. We also prove that some low non-computably enumerable enumeration degree contains no set which is simple relative to $\emptyset$'.