TotalOrderedSet

A total-ordered set is a LatticE (T,v,^) where for any a,b in T, either avb=a and a^b=b, or avb=b and a^b=a. A PartialOrder <= on a set T defines a total order if and only for every a,b in T, exactly one of the following hold:

a=b

a<b

a>b

The set of all OrdinalNumbers less than any 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.