Axiom 1.2.1 (Peano Postulates). There exists a set with an element and a function that satisfies the following three properties.
a. There is no such that .
b. The function s is injective.
c. Let be a set. Suppose that , and that . Then .
Definition 1.2.2. The set of natural numbers, denoted , 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 , or the powers of 5 , could be constructed (with a different , of course, since is not defined in the postulates anyway). How do these properties uniquely identify the set of the natural numbers?