Sets with negative number of elements?

Boris Leykin

Hi. :)
Look what I've found here http://math.ucr.edu/home/baez/nth_quantization.html
Can anyone say is this nonsense or what, negative cardinality?
I am very curious. :o

Hurkyl

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.

Boris Leykin

Euler Characteristic versus Homotopy Cardinality

Thank you.
All my excitement vanished.

karrerkarrer

isn't that a good methodological abbreviation for anything that is "hyper-nonexistent"?

of similar interest would be considering circles with a negative radius (my favourite object) etc.
any ideas about this??

best
karrerkarrer

dark3lf

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?

gel

Quite simple. A circle of radius r is the solutions to x²+y²=r². So negative radius circle is the same as positive radius.

imaginary radius is probably more interesting. You'd get the hyperbolic plane, depending on how you define it.

CRGreathouse

I found a paper/chapter that may be of interest:
Mathematics of Multisets, pp. 5-6.