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)
“No other creative field is as closed to those who are not white and male as is the visual arts. After I decided to be an artist, the first thing that I had to believe was that I, a black woman, could penetrate the art scene, and that, further, I could do so without sacrificing one iota of my blackness or my femaleness or my humanity.”
—Faith Ringgold (b. 1934)
“Nothing sets a person up more than having something turn out just the way it’s supposed to be, like falling into a Swiss snowdrift and seeing a big dog come up with a little cask of brandy round its neck.”
—Claud Cockburn (1904–1981)