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 $\mathbb{N}$ with an element $1\in \mathbb{N}$ and a function $s:\mathbb{N}\to \mathbb{N}$ that satisfies the following three properties.

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

b. The function s is injective.

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

Definition 1.2.2. The set of natural numbers, denoted $\mathbb{N}$, 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 $\{1,3,5,7,\dots \}$, or the powers of 5 $\{1,5,25,625\dots \}$, 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?

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

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

b. The function s is injective.

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

Definition 1.2.2. The set of natural numbers, denoted $\mathbb{N}$, 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 $\{1,3,5,7,\dots \}$, or the powers of 5 $\{1,5,25,625\dots \}$, 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?