I am unable to understand the motivation behind the set theoretic definition of a natural number. The definition given in the book by Goldrei is as follows:
First he defines an inductive set:
A set $y$ is inductive if $\emptyset \in y$ and $x^{+} \in y$ whenever $x \in y$, where $x^{+} = x \cup \{x\}$.
So far fine.
But then the set of natural number is defined as follows:
The set of natural numbers $\mathbb{N}$ is the intersection of all inductive subsets of any inductive set $y$, i.e.
$\mathbb{N} = \cap \{z:z \text{ is an inductive subset of }y\} = \{x: x \in z, \forall \text{ inductive }z \subseteq y\}$.
I am actually lost with this definition. I feel any inductive set is isomorphic to $\mathbb{N}$. What is the reason/motivation for defining $\mathbb{N}$ as the intersection of all the inductive subsets of an inductive set $y$? What are we missing if we were to define $\mathbb{N}$ as just an inductive set?
If my statements/questions don't make sense, it is because I am confused. I would appreciate if someone could throw light on this.