I once posted this a long time ago. But I starting thinking about this again.
Definition: A set is "finite" if and only if and a bijective map .
Warning: Everything from here might be considered Naive Set Theory. This is why I will use the term "proper class" instead. (And I will not be so careful, because I do not want to spend time being careful).
Defintion: Let be the proper class of all finite sets. So that if then .
Definition: Define an relation on as for iff there exists a bijection .
Theorem: The above relation is an equivalence relation.
Definition: Define to be the proper class of modulo .
Definition: Define the cardinality of each element in the proper class of to be an "natural".
Definition: For and naturals we define to be the cardinality of the union of the two equivalence classes.
(And this is well-defined).
Now with this with continue to define some properties ....