In many theorems, certain results hold only if a given set is finite, or infinite. This got me thinking about the nature of finite and infinite quantities - how do we define them? Obviously the simplest way to define what it means for a set S to be finite is if there exists a natural number n such that a bijective function S -> (0,n) exists. Or in other words, you can use the natural numbers to count their elements.

However, this has always bugged me. Natural numbers are normally defined in convoluted and abstract ways using certain set-theoretic models, and have always seemed divorced from the concepts they describe. In our daily lives, concepts like "three" and "seventeen" are properties first, and objects second, whereas in mathematics we construct objects that we later demonstrate can usefully map to the property we desire. But what if we didn't?

To me, the first thing to establish is what it means for a set to be finite (or infinite), without reference to the naturals, ideally as some property that can be said about sets in reference to themselves.

Do you have any good definitions? Ones that are particularly elegant, or concise, or which really get to the heart of what makes a finite quantity finite, or just really f*%&ing clever? In fact, what definition do you think is the "best" one (which is totally an objective and uncontroversial qualifier)?