Natural Numbers are Infinite

Theorem

The set $\N$ of natural numbers is infinite.

Proof

Let the mapping $s: \N \to \N$ be defined as:

$\forall n \in \N: \map s n = n + 1$

$s$ is clearly an injection.

Aiming for a contradiction, suppose $\N$ were finite.

By Equivalence of Mappings between Sets of Same Cardinality it follows that $s$ is a surjection.

But:

$\forall n \in \N: \map s n \ge 0 + 1 > 0$

So:

$0 \notin s \sqbrk \N$

and $s$ is not a surjection.

From this contradiction it is seen that $\N$ cannot be finite.

So, by definition, $\N$ is infinite.

$\blacksquare$