The Natural Numbers Are Not Finite

Let $\mathbb{N}$ be the set of natural numbers, with successor map $S$ as in that definition. The notions finite and has $n$ elements are those of the indicated definitions. A map $u:X\to Y$ between sets is called injective if $u(x)=u(x')$ implies $x=x'$ for all $x,x'\in X$.

1. (The natural numbers)¶ The set $\mathbb{N}$ is not finite.

2. (Sets receiving an injection)¶ Let $X$ be a set and let $h:\mathbb{N}\to X$ be injective. Then $X$ is not finite.