Editing Total order (section)

===Finite total orders===

A simple [[counting]] argument will verify that any non-empty finite totally ordered set (and hence any non-empty subset thereof) has a least element. Thus every finite total order is in fact a [[well order]].  Either by direct proof or by observing that every well order is [[order isomorphic]] to an [[Ordinal number|ordinal]] one may show that every finite total order is [[order isomorphic]] to an [[initial segment]] of the natural numbers ordered by <. In other words, a total order on a set with ''k'' elements induces a bijection with the first ''k'' natural numbers. Hence it is common to index finite total orders or well orders with [[order type]] &omega; by natural numbers in a fashion which respects the ordering (either starting with zero or with one).