Arithmetical Hierarchy - Properties

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

Famous quotes containing the word properties:

    A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.
    Ralph Waldo Emerson (1803–1882)

    The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.
    John Locke (1632–1704)