³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980245797³2ip³2129.97.130.2³2data³2newauthor³31³3minor³30³3text³3A 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.
³3summary³3³2tscreate³2980245797³2keepts³2980497215³2id³21076³2host³2cnts3p02.uwaterloo.ca³2revision³21³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980497215³2ip³2129.97.130.42³2data³2text³3A 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.
³3minor³31³3newauthor³30³3summary³3*³2tscreate³2980245797³2keepts³2980722057³2id³21076³2host³2cnts3p42.uwaterloo.ca³2revision³22³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980722057³2ip³2129.97.182.3³2data³2newauthor³30³3minor³31³3text³3A 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.


³3summary³3*³2tscreate³2980245797³2keepts³2980722159³2id³21076³2host³2cnts2p03.uwaterloo.ca³2revision³23³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980722159³2ip³2129.97.182.3³2data³2text³3A 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

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.
³3minor³31³3newauthor³30³3summary³3*³2tscreate³2980245797³2keepts³2980722225³2id³21076³2host³2cnts2p03.uwaterloo.ca³2revision³24³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980722225³2ip³2129.97.182.3³2data³2newauthor³30³3minor³31³3text³3A 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

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.
³3summary³3*³2tscreate³2980245797³2keepts³2980731902³2id³21076³2host³2cnts2p03.uwaterloo.ca³2revision³25