Countable Set - Total Orders

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:

    Parenting is the one area of my life where I can feel incompetent, out of control and like a total failure all of the time.
    —Attorney Father. As quoted in Reviving Ophelia, by Mary Pipher, ch. 4 (1994)

    Our own physical body possesses a wisdom which we who inhabit the body lack. We give it orders which make no sense.
    Henry Miller (1891–1980)