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.
Best Answer
Usually in set theory we write $\omega$ instead of $\mathbb{N}$. A concrete example of an inductive set other than $\omega$ is the ordinal $\omega + \omega$, which is the closure of $\omega \cup \{\omega\}$ under ${}^+$. This set has the structure of two copies of $\omega$, one after the other.
Given any set $x$, if you take the closure of $x \cup \{\emptyset\}$ under ${}^+$ then you get an inductive set containing every member of $x$. So it is not true that any inductive set is even isomorphic to $\omega$. For example there are uncountable inductive sets.
What you really want is a $\subseteq$-*minimal* inductive set. There is only one of these, which is $\omega$ (which, as before, you are identifying with $\mathbb{N}$).
The most likely reason that Goldrei says "the intersection of all inductive subsets of a fixed inductive set" rather than "the intersection of all inductive sets" is that the latter of those is an intersection of a proper class worth of sets, which he may not have defined formally. But the result is the same: if you take the intersection of all inductive sets, or the intersection of all inductive subsets of an inductive set, either way you get the unique $\subseteq$-minimal inductive set.