Sets with negative number of elements?

[Boris Leykin]

Hi. :)
Look what I've found here http://math.ucr.edu/home/baez/nth_quantization.html"

something interesting about sets with negative cardinality... but for that, you'll have to read this:
Daniel Loeb, Sets with a negative number of elements, Adv. Math. 91 (1992), 64-74

Can anyone say is this nonsense or what, negative cardinality?
I am very curious.

 

[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]

http://www.math.ucr.edu/home/baez/cardinality/" :yuck:

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]

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

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]

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?

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.