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)
“Alas for the cripple Practice when it seeks to come up with the bird Theory, which flies before it. Try your design on the best school. The scholars are of all ages and temperaments and capacities. It is difficult to class them, some are too young, some are slow, some perverse. Each requires so much consideration, that the morning hope of the teacher, of a day of love and progress, is often closed at evening by despair.”
—Ralph Waldo Emerson (1803–1882)
“The world can doubtless never be well known by theory: practice is absolutely necessary; but surely it is of great use to a young man, before he sets out for that country, full of mazes, windings, and turnings, to have at least a general map of it, made by some experienced traveller.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)