a=b a<b a>b
If the ContinuumHypothesis? is true, then any set of CardinalNumbers? is total-ordered...otherwise things get quite a bit messier. The set of all OrdinalNumbers? less than any given one definitely form a total-ordered set, though, and 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 lower bound is the IntegerNumbers. The unique smallest unbounded total-ordered set which also happens to be dense, that is have non-empty (a,b) for every a<b, is the RationalNumbers. The unique smallest unbounded connected total-ordered set is the RealNumbers?.