Upper bounds on ideals in the computably enumerable Turing degrees

We study ideals in the computably enumerable Turing degrees, and their upper bounds. Every proper Σ0 4 ideal in the c.e. Turing degrees has an incomplete upper bound. It follows that there is no Σ0 4 prime ideal in the c.e. Turing degrees. This answers a question of Calhoun [Cal93]. Every proper Σ0 3 ideal in the c.e. Turing degrees has a low2 upper bound. Furthermore, the partial order of Σ0 3 ideals under inclusion is dense.

• Type: Journal
• Year: 2011
• Where: APAL
• Authors: George Barmpalias, André Nies
• Keywords: APAL 2011, Proper Σ0, Software Engineering, Turing Degrees, Upper Bound