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 Π 1 0 {\displaystyle \Pi _{1}^{0}} classes. Computably inseparable sets also arise in the study of Gödel's incompleteness theorem.
Definition
The natural numbers are the set N = { 0 , 1 , 2 , … } {\displaystyle \mathbb {N} =\{0,1,2,\dots \}} . Given disjoint subsets A {\displaystyle A} and B {\displaystyle B} of N {\displaystyle \mathbb {N} } , a separating set C {\displaystyle C} is a subset of N {\displaystyle \mathbb {N} } such that A ⊆ C {\displaystyle A\subseteq C} and B ∩ C = ∅ {\displaystyle B\cap C=\emptyset } (or equivalently, A ⊆ C {\displaystyle A\subseteq C} and B ⊆ C ′ {\displaystyle B\subseteq C'} , where C ′ = N ∖ C {\displaystyle C'=\mathbb {N} \setminus C} denotes the complement of C {\displaystyle C} ). For example, A {\displaystyle A} itself is a separating set for the pair, as is B ′ {\displaystyle B'} .
If a pair of disjoint sets A {\displaystyle A} and B {\displaystyle B} has no computable separating set, then the two sets are computably inseparable.
Examples
If A {\displaystyle A} is a non-computable set, then A {\displaystyle A} and its complement are computably inseparable. However, there are many examples of sets A {\displaystyle A} and B {\displaystyle B} that are disjoint, non-complementary, and computably inseparable. Moreover, it is possible for A {\displaystyle A} and B {\displaystyle B} to be computably inseparable, disjoint, and computably enumerable.
- Let φ {\displaystyle \varphi } be the standard indexing of the partial computable functions. Then the sets A = { e : φ e ( 0 ) = 0 } {\displaystyle A=\{e:\varphi _{e}(0)=0\}} and B = { e : φ e ( 0 ) = 1 } {\displaystyle B=\{e:\varphi _{e}(0)=1\}} are computably inseparable (William Gasarch1998, p. 1047).
- Let # {\displaystyle \#} be a standard Gödel numbering for the formulas of Peano arithmetic. Then the set A = { # ( ψ ) : P A ⊢ ψ } {\displaystyle A=\{\#(\psi ):PA\vdash \psi \}} of provable formulas and the set B = { # ( ψ ) : P A ⊢ ¬ ψ } {\displaystyle 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).
- Cenzer, Douglas (1999), "Π01 classes in computability theory", Handbook of computability theory, Stud. Logic Found. Math., vol. 140, Amsterdam: North-Holland, pp. 37–85, doi:10.1016/S0049-237X(99)80018-4, MR 1720779
- Gasarch, William (1998), "A survey of recursive combinatorics", Handbook of recursive mathematics, Vol. 2, Stud. Logic Found. Math., vol. 139, Amsterdam: North-Holland, pp. 1041–1176, doi:10.1016/S0049-237X(98)80049-9, MR 1673598
- Monk, J. Donald (1976), Mathematical Logic, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90170-1
- Smullyan, Raymond M. (1958), "Undecidability and recursive inseparability", Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 4 (7–11): 143–147, doi:10.1002/malq.19580040705, ISSN 0044-3050, MR 0099293
References
Monk 1976, p. 100 ↩