When studying natural numbers we could do as we did for sets, and start with some primitive notion of a number and use axioms to define how it behaves. However, with our already developed ideas in set theory we can construct the numbers from sets.
Usually, we define numbers from sets as follows
0 = \varnothing\\
1 = \{0\} = \{\varnothing\}\\
2 = \{0,1\} = \{\varnothing,\{\varnothing\}\}\\
Let us define this truly.
Given a set a, its successor a^+ is given by
a^+ = a \cup \{a\}.
This is all well and good, and we can construct such successors from the axioms of union and pairing. We now give another definition
A set A is inductive if \varnothing \in A and the following statement holds
(\forall a \in A) a^+ \in A
It is clear that any inductive set will be infinite, but we have not yet encountered infinite sets. This is where the axiom of infinity comes in.
There exists an inductive set:
(\exists A)[\varnothing \in A \land (\forall a \in A)a^+ \in A].
A natural number is a set that belongs to every inductive set.
There exists a set whose members are exactly the natural numbers.
Let A be an inductive set, which exists thanks to the axiom of infinity. Then using the separation axiom, we create a set w such that
x \in w \implies x \text{ in every inductive set}
\omega is the set of all natural numbers.
\omega is inductive, and is a subset of every other inductive set.
A Peano system is an ordered triple (N,S,e) consisting of a set N, a function S \colon N \to N, and some member e \in N such that
(i) e \not \in ran(S)
(ii) S is injective
(iii) Any subset A of N that contains e and is closed under S equals N.
A set A is transitive if every member of a member of A is also a member of A:
x \in a \in A \implies x \in A.
[TO-DO] Non transitive, transitive sets.