Recursively inseparable sets

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In computability theory, recursively inseparable sets are pairs of sets of natural numbers that cannot be "separated" with a recursive set (Monk 1976, p. 100). These sets arise in the study of computability theory itself, particularly in relation to Π01 classes. Recursively inseparable sets also arise in the study of Gödel's incompleteness theorem.

Definition[edit]

The natural numbers are the set ω = {0, 1, 2, ...}. Given disjoint subsets A and B of ω, a separating set C is a subset of ω such that AC and BC = ∅ (or equivalently, AC and BC). For example, A itself is a separating set for the pair, as is ω\B.

If a pair of disjoint sets A and B has no recursive separating set, then the two sets are recursively inseparable.

Examples[edit]

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

  • Let φ be the standard indexing of the partial computable functions. Then the sets A = {e : φe(0) = 0} and B = {e : φe(0) = 1} are recursively inseparable (Gasarch 1998, p. 1047).
  • Let # be a standard Gödel numbering for the formulas of Peano arithmetic. Then the set A = { #(ψ) : PA ⊢ ψ} of provable formulas and the set B = { #(ψ) : PA ⊢ ¬ψ} of refutable formulas are recursively inseparable. The inseparability of the sets of provable and refutable formulas holds for many other formal theories of arithmetic (Smullyan 1958).

References[edit]

  • Cenzer, Douglas (1999), "\Pi^0_1 classes in computability theory", Handbook of computability theory, Stud. Logic Found. Math. 140, Amsterdam: North-Holland, pp. 37–85, MR 1720779 
  • 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, 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: 143–147, doi:10.1002/malq.19580040705, ISSN 0044-3050, MR 0099293