Upper bounds on ideals in the Turing degrees
G. Barmpalias and A. Nies
Summary
We study ideals in the computably enumerable Turing degrees, and their upper bounds.
Every proper $\Sigma^0_4$ ideal in the c.e. Turing degrees has an incomplete upper bound.
It follows that there is no $\Sigma^0_4$ prime ideal in the c.e. Turing degrees. This answers a question of Calhoun.
Every proper $\Sigma^0_3$ ideal in the c.e. Turing degrees has a $low_2$ upper bound.
Furthermore, the partial order of $\Sigma^0_3$ ideals under inclusion is dense.