How does the axiom of infinity support the construction of natural numbers in set theory?

In set theory, the axiom of infinity is essential because it guarantees the existence of an infinite set, which allows for the formal construction of the set of all natural numbers, denoted as $\omega$.12

While the other axioms of Zermelo-Fraenkel set theory can describe finite sets, they do not inherently necessitate the existence of an infinite collection. The axiom of infinity provides the foundation to establish the existence of a unique minimal set $\omega$ that contains the empty set ($\varnothing$) and is closed under the successor operation.1

Using this framework, natural numbers are constructed as sets:
* 0 is defined as the empty set $\varnothing$.13
* Successor function: For any set $n$, its successor is defined as $S(n)=n\cup \{n\}$.23
* Construction: Following this, $1=S(0)=\{0\}$, $2=S(1)=\{0,1\}$, and so on.13

The axiom of infinity ensures that the collection of these sets forms a set itself, providing the basis for proving the Peano postulates as theorems within set theory. This construction paves the way for developing arithmetic and the theory of real numbers.1