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:
“A woman drew her long black hair out tight
And fiddled whisper music on those strings
And bats with baby faces in the violet light
Whistled, and beat their wings
And crawled head downward down a blackened wall....”
—T.S. (Thomas Stearns)
“Since time immemorial, one the dry earth, scraped to the bone, of this immeasurable country, a few men travelled ceaselessly, they owned nothing, but they served no one, free and wretched lords in a strange kingdom. Janine did not know why this idea filled her with a sadness so soft and so vast that she closed her eyes. She only knew that this kingdom, which had always been promised to her would never be her, never again, except at this moment.”
—Albert Camus 1013–1960, French-Algerian novelist, dramatist, philosopher. Janine in Algeria, in The Fall, p. 27, Gallimard (9157)
“And weren’t there special cemetery flowers,
That, once grief sets to growing, grief may rest:
The flowers will go on with grief awhile,
And no one seem neglecting or neglected?
A prudent grief will not despise such aids.”
—Robert Frost (1874–1963)