[einhverfr]

Sets may be unordered by definition but that doesn't mean you can't define something interesting as an ordered set.

Consider the Pythagorean attempt to prove that all numbers were rational by trying to prove that the square root of two was rational. That they were able to prove that it was not rational meant that we ended up with a new category of numbers. Similarly once you get into the square root of -1 you get into yet another category of numbers designed to address that problem.

Our numeric model isn't complete with just rational numbers, or just rational and irrational numbers. Today we have to add imaginary and complex numbers as well. Why shouldn't we be expanding relational math in the same way?