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:
“Both the man of science and the man of art live always at the edge of mystery, surrounded by it. Both, as a measure of their creation, have always had to do with the harmonization of what is new with what is familiar, with the balance between novelty and synthesis, with the struggle to make partial order in total chaos.... This cannot be an easy life.”
—J. Robert Oppenheimer (1904–1967)
“I’ve got orders to obey, thank God.”
—Robert Bolt (1924–1995)