Recursively inseparable sets


In computability theory, two disjoint sets of natural numbers are called recursively inseparable if they cannot be "separated" with a recursive set. These sets arise in the study of computability theory itself, particularly in relation to Π classes. Recursively inseparable sets also arise in the study of Gödel's incompleteness theorem.

Definition

The natural numbers are the set ω =. Given disjoint subsets A and B of ω, a separating set C is a subset of ω such that A ⊆ C and B ∩ C = ∅. For example, A itself is a separating set for the pair, as is ω\B.
If a pair of disjoint sets A and B has no recursive separating set, then the two sets are recursively inseparable.

Examples

If A is a non-recursive set then A and its complement are recursively inseparable. However, there are many examples of sets A and B that are disjoint, non-complementary, and recursively inseparable. Moreover, it is possible for A and B to be recursively inseparable, disjoint, and recursively enumerable.