Computably inseparable

In computability theory, two disjoint sets of natural numbers are called computably inseparable or recursively inseparable if they cannot be "separated" with a computable set. These sets arise in the study of computability theory itself, particularly in relation to $\Pi^0_1$ classes. Computably inseparable sets also arise in the study of Gödel's incompleteness theorem.

Definition
The natural numbers are the set $$\mathbb{N} = \{0, 1, 2, \dots\}$$. Given disjoint subsets $$ A $$ and $$ B$$ of $$\mathbb{N}$$, a separating set $$ C $$ is a subset of $$\mathbb{N}$$ such that $$A \subseteq C$$ and $$B \cap C = \emptyset$$ (or equivalently,  $$A \subseteq C$$ and $$B \subseteq C'$$, where $$C' = \mathbb{N} \setminus C$$ denotes the complement of $$C$$). For example, $$A$$ itself is a separating set for the pair, as is $$B'$$.

If a pair of disjoint sets $$A$$ and $$B$$ has no computable separating set, then the two sets are computably inseparable.

Examples
If $$A$$ is a non-computable set, then $$A$$ and its complement are computably inseparable. However, there are many examples of sets $$A$$ and $$B $$ that are disjoint, non-complementary, and computably inseparable. Moreover, it is possible for $$A$$ and $$B$$ to be computably inseparable, disjoint, and computably enumerable.
 * Let $$\varphi$$ be the standard indexing of the partial computable functions. Then the sets $$A = \{ e : \varphi_e(0) = 0 \}$$ and $$B = \{ e : \varphi_e(0) = 1 \}$$ are computably inseparable (William Gasarch1998, p. 1047).
 * Let $$\#$$ be a standard Gödel numbering for the formulas of Peano arithmetic. Then the set $$A = \{ \#(\psi) : PA \vdash \psi \}$$ of provable formulas and the set $$B = \{ \#(\psi) : PA \vdash \lnot\psi \}$$ of refutable formulas are computably inseparable. The inseparability of the sets of provable and refutable formulas holds for many other formal theories of arithmetic (Smullyan 1958).