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

Abstract

We show that the only degree which cannot co-enumerate a non-trivial $\Pi^0_1$-immune real is $\emptyset$, 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.