Effectively separable

Effectively separable

In computability theory, two sets of natural numbers are effectively separable if it is possible to separate the sets with a computable set, and effectively inseparable otherwise.

Formal definition

Let "A" and "B" be disjoint sets of natural numbers. These sets are effectively separable if there is a computable set "C" of natural numbers such that A subseteq C and B cap C is empty.

If "A" and "B" are not effectively separable then they are effectively inseparable.

Examples

#Let "A" be the set of Godel numbers of Turing machines "M" such that on input "0" the machine "M" halts and outputs "0". Let "B" be the set of Godel numbers of Turing machines "M" such that on input "0" the machine "M" halts and outputs "1". Then "A" and "B" are effectively inseparable recursively enumerable sets.
#Let "T" be the set of (Godel numbers of) theorems provable from the axioms of Peano arithmetic, which is a recursively enumerable set, and let "R" be the set of negations of theorems of the axioms of Peano arithmetic, which is also recursively enumerable. Then "T" and "R" are effectively inseparable sets. A similar result would hold with any sufficiently strong axiom system in place of Peano arithmetic.

References

Soare, R. "Recursively enumerable sets and degrees." Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1987. ISBN 3-540-15299-7


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