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} ?

Solution 1:

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

Solution 2:

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.