Is there a natural number between $0$ and $1$?

Question

Is there a natural number between $0$ and $1$?

A proof, s'il vous plaît, not your personal opinion. (Assume the Peano Postulates.)

Answer 1

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., $s(n) < 0$.

Answer 2

HINT $\rm\ \ S\:n\ =\ S\:0\ \Rightarrow\ n\: =\: 0\: \ne\: S\: m$

Answer 3

One is defined as $\{\emptyset\}$. That a number n is between 0 and 1 means that $0\in n$ and $n\in 1$. Since $n\in 1$, it follows that $n=0$, a contradiction.