A Note on Lowness for ${\Pi }_1^0$-Immunity

Abstract

We show that the only degree which cannot co-enumerate a non-trivial ${\Pi }_1^0$-immune real is $\varnothing$, resolving a conjecture of the author. The main ingredient is that if a real $A$ has that all $A$-maximal sets are c.e., then $A$ has c.e. degree, which can be proved via a coding mechanism and two classical results of Lachlan.

arXiv commentInformal note; an expanded paper version containing related results is in preparation