Properties
The following properties hold for the arithmetical hierarchy of sets of natural numbers and the arithmetical hierarchy of subsets of Cantor or Baire space.
- The collections and are closed under finite unions and finite intersections of their respective elements.
- A set is if and only if its complement is . A set is if and only if the set is both and, in which case its complement will also be .
- The inclusions and hold for .
- The inclusions and hold for all and the inclusion holds for . Thus the hierarchy does not collapse.
Read more about this topic: Arithmetical Hierarchy
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