Such generalizations are easy enough to construct. I imagine you have no trouble with the notion of a multiset: a set that's allowed to contain multiple copies of something. e.g. <1, 1, 2> would be different from <1, 2>.

It's easy to see that a multiset can be described as a function that tells you how many copies of an object there are. e.g. if S = <1, 1, 2>, then S(1) = 2, S(2) = 1, and S(x) = 0 for anything else.

From there, it's a small step to allow functions to have negative values. Then *voila*, you have a generalization of the notion of a set that permits a set to have a negative number of elements.

I don't know exactly what sort of generalization that article is planning on discussing, though. It might be this one, or it might be something entirely different.

This would imply that the circle's negative radius causes the circle to "fold in on itself" so-to-speak into a negative dimension below the circle's two. This raises the question of negative dimensions... Theories?