Let $A\subseteq\mathbb{N}$ be the set of Gödel codes of theorems of Peano arithmetic, and $B\subseteq\mathbb{N}$ be the set of codes of antitheorems (i.e, refutable statements, statements whose negation is a theorem). It is a standard fact (see, e.g., Cooper, *Computability Theory* (1994), exercise 9.2.13) that $A$ and $B$ are computably inseparable: any $D\subseteq\mathbb{N}$ such that $A\subseteq D$ and $B\cap D=\varnothing$ (“separating $A$ and $B$”) is of Turing degree $>\mathbf{0}$. Clearly, we can find such a $D$ having degree $\mathbf{0'}$, since $A$ itself (or, if we prefer, the complement of $B$) is such.

**Main question:** Does there exist $D$ separating $A$ and $B$ having degree $<\mathbf{0'}$?

**Bonus questions:** What can be said about the set of possible degrees of $D$ separating $A$ and $B$? Does it have a lower bound? Also, if we take $D$ at random w.r.t. the obvious probability measure (take $A$ and add teach element of the complement of $B$ with probability $\frac{1}{2}$ independently), what is the probability that $D$ has degree $\not\geq\mathbf{0'}$?