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.^[1] These sets arise in the study of computability theory itself, particularly in relation to ${\displaystyle {\mathrm{\Pi }}^{0}1}$ classes. Computably inseparable sets also arise in the study of Gödel's incompleteness theorem.

The natural numbers are the set ${\displaystyle \mathbb{\mathbb{N}}=\{0,1,2,\dots \}}$. Given disjoint subsets ${\displaystyle A}$ and ${\displaystyle B}$ of ${\displaystyle \mathbb{\mathbb{N}}}$, a separating set ${\displaystyle C}$ is a subset of ${\displaystyle \mathbb{\mathbb{N}}}$ such that ${\displaystyle A\subseteq C}$ and ${\displaystyle B\cap C=\mathrm{\varnothing }}$ (or equivalently, ${\displaystyle A\subseteq C}$ and ${\displaystyle B\subseteq C^{\prime }}$, where ${\displaystyle C^{\prime }=\mathbb{\mathbb{N}}\setminus C}$ denotes the complement of ${\displaystyle C}$). For example, ${\displaystyle A}$ itself is a separating set for the pair, as is ${\displaystyle B^{\prime }}$.

If a pair of disjoint sets ${\displaystyle A}$ and ${\displaystyle B}$ has no computable separating set, then the two sets are computably inseparable.

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

• Let ${\displaystyle \phi }$ be the standard indexing of the partial computable functions. Then the sets ${\displaystyle A=\{e:{\phi }_e(0)=0\}}$ and ${\displaystyle B=\{e:{\phi }_e(0)=1\}}$ are computably inseparable (William Gasarch1998, p. 1047).
• Let ${\displaystyle \mathrm{\#}}$ be a standard Gödel numbering for the formulas of Peano arithmetic. Then the set ${\displaystyle A=\{\mathrm{\#}(\psi ):PA⊢\psi \}}$ of provable formulas and the set ${\displaystyle B=\{\mathrm{\#}(\psi ):PA⊢\mathrm{\neg }\psi \}}$ 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., 140, Amsterdam: North-Holland, pp. 37–85, doi:10.1016/S0049-237X(99)80018-4 
• Gasarch, William (1998), "A survey of recursive combinatorics", Handbook of recursive mathematics, Vol. 2, Stud. Logic Found. Math., 139, Amsterdam: North-Holland, pp. 1041–1176, doi:10.1016/S0049-237X(98)80049-9 
• Monk, J. Donald (1976), Mathematical Logic, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90170-1, https://archive.org/details/mathematicallogi00jdon 
• 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 

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