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
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