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.

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. 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. 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).

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