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, 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 smallest connected total-ordered set is the RealNumbers.