1. ## Integer Construction

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

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

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

Theorem: The above relation is an equivalence relation.

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

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

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

2. Originally Posted by ThePerfectHacker

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

Isn't this the definition of a "non-finite" set?

RonL

3. Originally Posted by CaptainBlank
????
Isn't this the definition of a "non-finite" set?
Yes, sorry. (When I was writing this I had a lot to drink). A infinite set is exactly as described above. So a finite set is a non-infinite set. The negation of that definition.