What is the rationale for all comparisons returning false for IEEE754 NaN values?

Question

Why do comparisons of NaN values behave differently from all other values? That is, all comparisons with the operators ==, <=, >=, <, > where one or both values is NaN

Answer 1

Very short answer:

Because the following: nan / nan = 1 must NOT hold. Otherwise inf/inf would be 1.

(Therefore nan can not be equal to nan. As for > or <, if nan would respect any order relation in a set satisfying the Archimedean property, we would have again nan / nan = 1 at the limit).