Ordinal Number - Downward Closed Sets of Ordinals

Downward Closed Sets of Ordinals

A set is downward closed if anything less than an element of the set is also in the set. If a set of ordinals is downward closed, then that set is an ordinal—the least ordinal not in the set.

Examples:

  • The set of ordinals less than 3 is 3 = { 0, 1, 2 }, the smallest ordinal not less than 3.
  • The set of finite ordinals is infinite, the smallest infinite ordinal: ω.
  • The set of countable ordinals is uncountable, the smallest uncountable ordinal: ω1.

Read more about this topic:  Ordinal Number

Famous quotes containing the words downward, closed and/or sets:

    All places where women are excluded tend downward to barbarism; but the moment she is introduced, there come in with her courtesy, cleanliness, sobriety, and order.
    Harriet Beecher Stowe (1811–1896)

    On a flat road runs the well-trained runner,
    He is lean and sinewy with muscular legs,
    He is thinly clothed, he leans forward as he runs,
    With lightly closed fists and arms partially raised.
    Walt Whitman (1819–1892)

    There is the name and the thing; the name is a sound which sets a mark on and denotes the thing. The name is no part of the thing nor of the substance; it is an extraneous piece added to the thing, and outside of it.
    Michel de Montaigne (1533–1592)