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.