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 $\Pi _{1}^{0}$ classes. Computably inseparable sets also arise in the study of Gödel's incompleteness theorem.

Definition

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.

Examples

If $A$ is a non-computable set, then $A$ and its complement are computably inseparable. However, there are many examples of sets $A$ and $B$ that are disjoint, non-complementary, and computably inseparable. Moreover, it is possible for $A$ and $B$ to be computably inseparable, disjoint, and computably enumerable.

• 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).

References

• Cenzer, Douglas (1999), "Π⁰
  ₁ 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