I am beginning real analysis and got stuck on the first page (Peano Postulates). It reads as follows, at least in my textbook.

**Axiom 1.2.1 (Peano Postulates)**. There exists a set $\Bbb N$ with an element $1 \in \Bbb N$ and a function $s:\Bbb N \to \Bbb N$ that satisfies the following three properties.

a. There is no $n \in \Bbb N$ such that $s(n) = 1$.

b. The function $s$ is injective.

c. Let $G \subseteq \Bbb N$ be a set. Suppose that $1 \in G$, and that $g \in G \Rightarrow s(g) \in G$. Then $G = \Bbb N$.

**Definition 1.2.2.** The set of natural numbers, denoted $\Bbb N$, is the set the existence of which is given in the Peano Postulates.

My question is: From my understanding of the postulates, we could construct an infinite set which satisfies the three properties. For example, the odd numbers $\{1,3,5,7, \ldots \}$, or the powers of 5 $\{1,5,25,625 \ldots \}$, could be constructed (with a different $s(n)$, of course, since $s(n)$ is not defined in the postulates anyway). How do these properties uniquely identify the set of the natural numbers?