Total Orders
Countable sets can be totally ordered in various ways, e.g.:
- Well orders (see also ordinal number):
- The usual order of natural numbers (0, 1, 2, 3, 4, 5, ...)
- The integers in the order (0, 1, 2, 3, ...; −1, −2, −3, ...)
- Other (not well orders):
- The usual order of integers (..., -3, -2, -1, 0, 1, 2, 3, ...)
- The usual order of rational numbers (Cannot be explicitly written as a list!)
Note that in both examples of well orders here, any subset has a least element; and in both examples of non-well orders, some subsets do not have a least element. This is the key definition that determines whether a total order is also a well order.
Read more about this topic: Countable Set
Famous quotes containing the words total and/or orders:
“For, the expectation of gratitude is mean, and is continually punished by the total insensibility of the obliged person. It is a great happiness to get off without injury and heart-burning, from one who has had the ill luck to be served by you. It is a very onerous business, this being served, and the debtor naturally wishes to give you a slap.”
—Ralph Waldo Emerson (1803–1882)
“What is all wisdom save a collection of platitudes? Take fifty of our current proverbial sayings—they are so trite, so threadbare, that we can hardly bring our lips to utter them. None the less they embody the concentrated experience of the race and the man who orders his life according to their teaching cannot go far wrong.”
—Norman Douglas (1868–1952)