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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:


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?.

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Last edited January 29, 2001 9:31 am by RoseParks (diff)