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:
“Go on, high ship, since now, upon the shore,
The snake has left its skin upon the floor.
Key West sank downward under massive clouds
And silvers and greens spread over the sea. The moon
Is at the mast-head and the past is dead.”
—Wallace Stevens (1879–1955)
“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 (1819–1892)
“The vain man does not wish so much to be prominent as to feel himself prominent; he therefore disdains none of the expedients for self-deception and self-outwitting. It is not the opinion of others that he sets his heart on, but his opinion of their opinion.”
—Friedrich Nietzsche (1844–1900)