Axiomatizing Natural Numbers

One can not prove theorems by assuming nothing. But what is more fascinating is that a large number of concepts can be explained by a fixed number of elementary notions, known as axioms.
Axioms are assumed to be self-evident truths. One such set of axioms about Natural Numbers was proposed by Peano:

Peano’s Axioms

Let $\mathbb{N}$ denote a set. Suppose we are not permitted to see its contents. Instead, we are given the following information about $\mathbb{N}$:

1. $0 \in \mathbb{N}$
2. A mapping $s: \mathbb{N} -> \mathbb{N}$, different from indentity map, such that $\mathbb{N}$ is invariant under $s$, i.e., $a \in \mathbb{N} \implies s(a) \in \mathbb{N}$
3. Pre-image of $0$ does not exist under $s$,
   i.e., there does not exist an $a \in \mathbb{N}$ such that $s(a) = 0$
4. The mapping $s$ is injective, i.e., $s(a)=s(b)\implies a=b$
5. If $S \subseteq \mathbb{N}$ with $0 \in S$ and $a \in S \implies s(a) \in S$, then $S=\mathbb{N}$

With these axioms alone, we can describe/generate the entire set $N$, without actually seeing its elements. Let us see how.
Let us start by analyzing what are we provided with. We are sure that there exists an element denoted by $0$ in $\mathbb{N}$. So $\mathbb{N}$ is non-empty. Additionally there also exists a map $s$ from $\mathbb{N}$ to itself, which is different from identity map.
This map is defined in such a way that when fed with an element of $\mathbb{N}$, it gives us another element from $\mathbb{N}$. Notice here, that had the map been identity map, it would be useless to us, since it would return the same element. So $s$ must be a non-identity map, so that we can get new information about $\mathbb{N}$.

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