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)
“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 (18191892)
“Until, accustomed to disappointments, you can let yourself rule and be ruled by these strings or emanations that connect everything together, you havent fully exorcised the demon of doubt that sets you in motion like a rocking horse that cannot stop rocking.”
—John Ashbery (b. 1927)