Quote by Canute
That makes sense to me, and it leaves the axiom ambiguous rather than contradictory. (It is or is not ambiguous/paradoxical regardlessw of whether it is an axiom or a theorem). If the dividing point is in one of the classes then that would be a solution to my problem. However, someone stated that the dividing point is unique and that threw me off the track, since this is untrue if the point is in one the classes. If the dividing point is in one of the classes then it cannot be the dividing point between the classes, and if it is one of the classes it is no different to its complement in the other class (sup x = inf x).

Huh? Why can't the dividing point be unique if it's in one of the classes. It's the point that "produces the division" it is not the point that is between both classes. Like I said (I think), "produces the division" is not a technical term, but it's meaning is obvious. The other thing to note that while sup((infinity, 0)) = 0 = inf([0, infinity)), 0 is an element of [0, infinity) while not an element of (infinity, 0). That is, the supremum, if it exists, of an open interval of the reals is not in the interval. This is how the supremum of one class is the infimum of the other, without this unique point being in both classes.