Natural Numbers are Infinite

Theorem
The set $$\N$$ is infinite.

Proof
Let the mapping $$s: \N \to \N$$ be defined as:
 * $$\forall n \in \N: s \left({n}\right) = n + 1$$

$$s$$ is clearly an injection. But:
 * $$\forall n \in \N: s \left({n}\right) \ge 0 + 1 > 0$$

So $$0 \notin s \left({\N}\right)$$, and $$s$$ is not a surjection.

Therefore $$\N$$ is not finite and so by Same Cardinality Bijective Injective Surjective is therefore infinite.