In computability theory, recursively inseparable sets are pairs of sets of natural numbers that cannot be "separated" with a recursive set (Monk 1976, p. 100). These sets arise in the study of computability theory itself, particularly in relation to Π0
1 classes. Recursively inseparable sets also arise in the study of Gödel's incompleteness theorem.
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Definition
The natural numbers are the set ω = {0, 1, 2, ...}. Given disjoint subsets A and B of ω, a separating set C is a subset of ω such that A ⊆ C and B ∩ C = ∅ (or equivalently, A ⊆ C and B ⊆ C). For example, A itself is a separating set for the pair, as is ωB.
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.