Wikipedia 10K Redux by Reagle from Starling archive. Bugs abound!!!

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TotalOrderedSet

A total-ordered set is a PoSet (T,<) where for any a,b in T, exactly one of the following hold:
* a=b
* ab

The set of all OrdinalNumbers less than a given one form a total-ordered set.  In particular, the finite ordinals (NaturalNumbers) form the unique smallest total-ordered set with no upper bound.  The unique smallest total-ordered set with neither an upper nor a lower bound is the IntegerNumbers.

If the ContinuumHypothesis is true, the any set of CardinalNumbers is total-ordered.  Otherwise things get quite a bit messier.