I once posted this a long time ago. But I starting thinking about this again.

Definition:A set $\displaystyle S$ is "finite" if and only if $\displaystyle \exists \ T\subset S$ and a bijective map $\displaystyle \phi: T\mapsto S$.

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 $\displaystyle \Omega$ be the proper class of all finite sets. So that if $\displaystyle S_1,S_2 \in \Omega$ then $\displaystyle S_1\cap S_2 = \{ \}$.

Definition:Define an relation $\displaystyle R$ on $\displaystyle \Omega$ as $\displaystyle (X,Y)\in R$ for $\displaystyle X,Y\in \Omega$ iff there exists a bijection $\displaystyle \phi: X\mapsto Y$.

Theorem:The above relation is an equivalence relation.

Definition:Define $\displaystyle \Phi$ to be the proper class of $\displaystyle \Omega$ modulo $\displaystyle R$.

Definition:Define the cardinality of each element in the proper class of $\displaystyle \Phi$ to be an "natural".

Definition:For $\displaystyle n_1$ and $\displaystyle n_2$ naturals we define $\displaystyle n_1+n_2$ 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 ....