TITLE: How does Peano Postulates construct Natural numbers only?
QUESTION [25 upvotes]: 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?

REPLY [21 votes]: This axioms define the natural numbers up to isomorphism; That is, given another Peano system $S$, there always is a bijective function $f: \mathbb{N}\to S$ such that $f(s(n))=s_1(f(n)); f(1)=1_S$. This substantially means that from the Peano's structure point of view, the two are equal. Given $1, 1_S$, we can just identify as equal the two of them and the other numbers will follow, as if we called the natural numbers with different symbols $1_S$ instead that $1$, but the structure behind is the same (this theorem can actually be formally proved by using the principle of recursion)
A little note, though: this isomorphism is sometimes not very evident end extremely helpful. For example, $\mathbb{N}^2$ can be given a Peano structure, and this can be identified with $\mathbb{N}$ by means of an isomorhpism (namely Cantor pairing function), which is highly non trivial.