TotalOrderedSet

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

• a=b
• a<b
• a>b

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.