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)
“We are closed in, and the key is turned
On our uncertainty;”
—William Butler Yeats (1865–1939)
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—William Blake (1757–1827)