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 ....