Aubrey Jaffer <xxxxxx@alum.mit.edu> writes:

>  | From: "Marcin 'Qrczak' Kowalczyk" <xxxxxx@knm.org.pl>
>  | Date: Sat, 22 Oct 2005 20:52:50 +0200
>  |
>  | Aubrey Jaffer <xxxxxx@alum.mit.edu> writes:
>  |
>  | > The total order of the reals is a crucial property for many
>  | > applications.
>  |
>  | It is well known that the default order on the floating point
>  | approximation of reals is not total.
>
>   From Wikipedia, the free encyclopedia.
>   <http://en.wikipedia.org/wiki/Total_order>
>
>   In mathematics, a total order, linear order or simple order on a set
>   X is any binary relation on X that is antisymmetric, transitive, and
>   total.  This means that, if we denote the relation by <=, the
>   following statements hold for all a, b and c in X:
>
>       if a <= b and b <= a then a = b (antisymmetry)
>       if a <= b and b <= c then a <= c (transitivity)
>       a <= b or b <= a (totalness)
>
> Which condition does it violate?

Totalness (as Marcin said).  NaN comparisons (other than not-equals)
always evaluate false.

Thomas