Is there a natural number between $0$ and $1$? > Is there a natural number between $0$ and $1$? A proof, s'il vous plaît, not your personal opinion. (Assume the Peano Postulates.)

Every natural number $m$ is either $0$ or $s(n)$, where $n$ is a natural number.

Proof: It can't be both, because $s(n)$ can't be $0$. Set of all natural numbers which are either $0$ or $s(n)$ for some $n$ satisfies induction principle, so it contains all natural numbers.

Direct consequence: Every natural number is either $0$, or $s(0)$ or $s(s(n))$ for some natural number $n$.

Suppose there is $m$ such that $0 < m < s(0)$. Either $m$ is $0$, $s(0)$ or $s(s(n))$. First two cannot hold, so you have $s(s(n)) < s(0)$, i.e., $s(n) < 0$.