Recursively inseparable sets

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.

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 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.

• Let φ be the standard indexing of the partial computable functions. Then the sets A = {e : φ_e(0) = 0} and B = {e : φ_e(0) = 1} are recursively inseparable (William Gasarch1998, p. 1047).
• Let # be a standard Gödel numbering for the formulas of Peano arithmetic. Then the set A = { #(ψ) : PA ⊢ ψ} of provable formulas and the set B = { #(ψ) : PA ⊢ ¬ψ} of refutable formulas are recursively inseparable. The inseparability of the sets of provable and refutable formulas holds for many other formal theories of arithmetic (Smullyan 1958).