How can natural numbers be represented by sets in axiomatic set theory?

In axiomatic set theory, natural numbers are represented as sets by defining them as a sequence of finite ordinal numbers. This construction, often attributed to John von Neumann, begins with the empty set ($\varnothing$) as the representation for 0.12

Each subsequent natural number is defined using a successor function, $S(x)=x\cup \{x\}$, which adjoins a set to its own elements. Under this framework, the natural numbers are constructed as follows:12

• 0 = $\varnothing$
• 1 = $S(0)=0\cup \{0\}=\{0\}=\{\varnothing \}$
• 2 = $S(1)=1\cup \{1\}=\{0,1\}=\{\varnothing ,\{\varnothing \}\}$
• 3 = $S(2)=2\cup \{2\}=\{0,1,2\}=\{\varnothing ,\{\varnothing \},\{\varnothing ,\{\varnothing \}\}\}$

In this system, each natural number $n$ is defined as the set of all preceding natural numbers $\{0,1,\dots ,n-1\}$. The existence of the complete set of these numbers, denoted as $\omega$, is guaranteed by the axiom of infinity. These sets are well-ordered by the membership relation ($\in$), and they serve as the foundation for finite ordinal arithmetic.123