Fractional cardinalities of sets Is there any extension of the usual notion of cardinalities of sets such that there is some sets with fractional cardinalities such as 5/2, ie a set with 2.5 elements, what would be an example of such a set?
Basically is there any consistent set theory where there is a set whose cardinality is less than that of {cat,dog,fish} but greater than that of {47,83} ?
 A: There are no notion of a fraction of an element in ZF. However without the axiom of choice one may have a model in which there is a set of cardinals isomorphic to the real numbers. 
What does that mean? It means that there is a family of sets of cardinality continuum and the cardinalities of the sets in this family ordered as the real numbers. 
The reason this happens is that cardinalities in the usual notion of set theory count the number of elements. Every element can either be in the set or not in the set. It cannot have a partial value. If one wants the $\in$ relation to have fractional values one would have to go to fuzzy logic, continuous logic, or Boolean valued logic. 
That been said I have to admit that at the end of my freshman year I tried to develop such notion of fractional cardinality, to some degree I believe that I did that. However it was completely useless later on and was nothing more than a game, which now I know how to develop using other and better tools.
A: The category theorists have a notion of fractional cardinality, see Baez week 147. In particular, the category of finite sets turns out to have cardinality $e$. 
A: Yes, there does.  If you do this in the context of fuzzy set theory (one possible way to extend things), there exist two different notions of cardinality.  There's an entire book on cardinality in fuzzy set theory actually.  Scalar cardinality comes as the relevant concept here.
For a possible example of such a set consider the set of your eight closest friends in the set of "tall people" (or your "Top Eight" in your MySpace page if you had one a few years ago).  I'd at least guess that, you'll have fractional "degrees of membership" for each person in the set of "tall people" as say a 6 foot person, I would expect, you would consider only somewhat tall and thus perhaps classify as .8 tall or something like that.  E. G. say I have friends Bob, Sally, Sue, Steve, Chris, John, Elmer, and Guido, with degrees of membership in the set of tall people of .3, .5, .6, .2, .86, .35, .6, and .1 respectively.  Then, then scalar cardinality of the set of my friends in the set of tall people equals 3.51 out a possible 8.0.
As another example, go to the room where you have the most books in your house.  Find the most common book cover color where each book gets assigned one of {red, blue, green, cyan, magenta, yellow}.  Now consider each book according to how red/blue/green/cyan/magenta/yellow that book is.  So, one book cover might have a degree of membership of .6 in the set of red books, and another .1 degree of membership.  Finally, sum all these together, and at least in plenty of cases the set of say red book covers in that room will have a non-natural number for its cardinality.
In my opinion, if you think about things in terms of "how well does x fit with y", it doesn't come as all that hard to find examples of real-world sets with fractional scalar cardinalities (or more generally it's NOT hard to find fuzzy sets).  In my opinion, it actually comes as harder to find real-world sets with natural number cardinalities, but I've almost surely started to digress.
A: One can extend the notion of cardinality to include negative and non-integer values by using the Euler characteristic and homotopy cardinality.  For example, the space of finite sets has homotopy cardinality $e=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\dotsi$.  The idea is to sum over each finite set, inversely weighted by the size of their symmetry group.  John Baez discusses this in detail on his blog.  He has plenty of references, as well as lecture notes, course notes, and blog posts about the topic here.  The first sentence on the linked page:
"We all know what it means for a set to have 6 elements, but what sort of thing has -1 elements, or 5/2? Believe it or not, these questions have nice answers." -Baez
