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I'm not a set theorist. Perhaps the answer to my question is so well known that it is not appropriate to answer here or my question is absurd. Nevertheless I'll try: I have heard the rumor that, similar to the extension of positive integers to negative integers, recently(?) also attempts have been made to investigate negative cardinalities.

My question: What would be the intuitive picture behind negative cardinality and can also in that domain countable and uncountable cardinals exist?

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Negative numbers have been around for thousands of years. Perhaps you mean something more specific? Googling "negative cardinality" brings up essentially nothing. –  Ryan Budney Jul 11 '13 at 8:00
    
The question is what do you want cardinals to represent. If so, what does a negative cardinal represents? –  Asaf Karagila Jul 11 '13 at 9:27
    
Get me thinking of Mandelbrots ideas about "degrees of emptiness of a set", see worldscientific.com/doi/abs/10.1142/… –  Kjetil B Halvorsen Jul 11 '13 at 13:29
    
One naive guess does not work: Isomorphism classes of sets (aka cardinals) under disjoint union form a commutative monoid - for size issues one should first choose a Grothendieck universe (or something similar) and consider only sets with smaller cardinality. A commutative monoid can be group completed. But by a naive variant of the Eilenberg swindle, every cardinal becomes 0 in the group construction: For example $1+\aleph_0 = \aleph_0$, so $1 = 0$ in the group completion. Note that this problem does not occur for finite cardinals and here we get just $\mathbb{Z}$. –  Lennart Meier Jul 11 '13 at 15:58
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closed as unclear what you're asking by Ryan Budney, Asaf Karagila, Ramiro de la Vega, Chris Godsil, Benjamin Steinberg Jul 11 '13 at 14:58

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3 Answers

Consider reading Jim Propp's curious paper "Euler measure as generalized cardinality" (arXiv:math/0203289).

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First you need to ask yourself what is cardinality? It's a mathematical notion which is designed to give us a way to measure the size of a set.

When you want to extend cardinals to negative integers you need to ask yourself, what is the purpose of this extension? Can we measure "more sets" now? In the context of $\sf ZFC$ or similar theories the answer is no. You don't have any sets which can be measured by negative cardinals because you wish to extend the current system, where every set is already assigned a positive (or infinite) cardinal.

What about other set theories? Well, maybe there you could extend this context. But then you have a question of usability. Cardinality comes with a wonderful set of arithmetical operations. We know that $|A\times B|=|A|\cdot|B|$ and $|A+ B|=|A|+|B|$ (where $A+B$ is the disjoint union of the sets).

Can we extend these notions of arithmetics as well? If $|A|=-1$ then $|A\times A|=1$, and therefore $|A+A\times A|=0$. So we have a disjoint union of two non-empty sets is empty, or that cardinality zero is no longer the unique property of the empty set. Either case is troubling.

This means that cardinality, when extended to negative integers, cannot extend the usual arithmetics as well.

The last resort, if so, would be to measure a whole other object. Polytopes, for example. But this sort of misses the point that cardinality is the size of sets. So you can talk about the concept of "generalized cardinality" but that won't be a measure of size of sets anymore. What would characterize it exactly? I don't know. I just feel that the term "cardinality" wouldn't be very fitting here.

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this joke seems a propos. –  Willie Wong Jul 11 '13 at 10:55
    
@Willie: I bet that the mathematician has just read about generalized cardinals! ;-) –  Asaf Karagila Jul 11 '13 at 11:09
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See here.

More specifically, see here.

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Isn't this essentially Igor's answer? –  Benjamin Steinberg Jul 11 '13 at 15:07
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