Simple set

In computability theory a simple set is an example of a set which is recursively enumerable but not recursive.

Definition

A subset S of the natural numbers N is called simple if it satisfies the following properties

1. N\S is infinite
2. S is recursively enumerable
3. S ∩ X ≠ ø for any infinite recursively enumerable set X

Properties

• The set of simple sets and the set of creative sets are disjoint. A simple set is never creative and a creative set is never simple.
• The collection of sets that are simple or cofinite forms a filter in the lattice of recursively enumerable sets.

Reference

• Robert I. Soare, Recursively Enumerable Sets and Degrees, Springer-Verlag, 1987. ISBN 0-387-15299-7