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:
“Amidst the downward tendency and proneness of things, when every voice is raised for a new road or another statute or a subscription of stock; for an improvement in dress, or in dentistry; for a new house or a larger business; for a political party, or the division of an estate;Mwill you not tolerate one or two solitary voices in the land, speaking for thoughts and principles not marketable or perishable?”
—Ralph Waldo Emerson (1803–1882)
“Don: Why are they closed? They’re all closed, every one of them.
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Pawnbroker: It’s Yom Kippur, a Jewish holiday.
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—Billy Wilder (b. 1906)
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—Jane Mersky Leder (20th century)