You could arbitrarily make NaN sort as if it was a certain value, and that would be useful when you want to sort a big array, but it would have unpleasant side effects when you're doing math. IEEE decided "always false" was less likely to cause problems, but to be clear you get problems no matter what you choose.
Defining just error, undefined, positive infinity, and negative infinity is far from an exhaustive list but they obviously should be treated differently in many contexts.
0/0, ∞/∞, ∞%n, n%0, ∞-∞, results with imaginary components
The first five have no meaningful approximation or way to interact with anything. And there's no good way to pretend a single float is a complex number.
So those results get the "this doesn't exist" treatment. Coder's choice if NaN triggers errors or not.
Would you try to define any more behavior for any of those NaNs?
Mathematically different infinities are not equivalent because infinity is not a number. How you represent that is dependent on the specific system involved.
It is true that different infinities exists[0][1] and there are whole areas of logic examining them.
It is also true that _number_ without any extra qualifier generally means the Real numbers (R) or Complex numbers (C) and those domain do not define infinity as a number, but even then there are only a few good ways to add infinity into each number system:
In R generally you either add a projective point of infinity ∞ [2] that makes geometry sometimes nices or two signed infinities (-∞ and +∞) that make calculus nicer (especially limits and integration)
In C it is often simpler as typically you want to treat them as a sphere and so add an extra point so that the inversion f(x) = 1/x is a well-behaved function. In this domain you often end up working with holomorphic functions[3] and then there is not really an intrinsic difference between a function like f(x) = 1/x and g(x) = x they simply have both a _pole_ f at 0 and g at infinity.
If you want to get trippy even integers can have unusual definitions [4] and then there is always one of my favorite topic in math: surreal numbers [5] (for which I recommend both [6] and [7]) a field where √∞ < ∞/2 < ∞ - 1 < ∞ < ∞ + 1 and is perfectly well defined (but still 0/0 doesn't have any meaning in any of these theories, that is a though nut to crack)
[0]https://en.wikipedia.org/wiki/Ordinal_number [1]https://en.wikipedia.org/wiki/Cardinal_number [2]https://en.wikipedia.org/wiki/Projective_geometry [3]https://en.wikipedia.org/wiki/Holomorphic_function [4]https://en.wikipedia.org/wiki/Algebraic_integer [5]https://en.wikipedia.org/wiki/Surreal_number [6]https://www-cs-faculty.stanford.edu/~knuth/sn.html [7]https://books.google.it/books?id=tXiVo8qA5PQC&redir_esc=y
