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.[1] These sets arise in the study of computability theory itself, particularly in relation to
classes. Computably inseparable sets also arise in the study of
Gödel's incompleteness theorem.
Definition
The natural numbers are the set . Given disjoint subsets and of , a separating set is a subset of such that and (or equivalently, and , where denotes the
complement of ). For example, itself is a separating set for the pair, as is .
If a pair of disjoint sets and has no
computable separating set, then the two sets are computably inseparable.
Examples
If is a non-computable set, then and its complement are computably inseparable. However, there are many examples of sets and that are disjoint, non-complementary, and computably inseparable. Moreover, it is possible for and to be computably inseparable, disjoint, and
computably enumerable.
Let be a standard
Gödel numbering for the formulas of
Peano arithmetic. Then the set of provable formulas and the set 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).