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)
“Pray but one prayer for me ‘twixt thy closed lips,
Think but one thought of me up in the stars.”
—William Morris (1834–1896)
“In my dealing with my child, my Latin and Greek, my accomplishments and my money stead me nothing; but as much soul as I have avails. If I am wilful, he sets his will against mine, one for one, and leaves me, if I please, the degradation of beating him by my superiority of strength. But if I renounce my will, and act for the soul, setting that up as umpire between us two, out of his young eyes looks the same soul; he reveres and loves with me.”
—Ralph Waldo Emerson (1803–1882)