How are natural numbers constructed from the empty set in Zermelo-Fraenkel set theory?

In Zermelo-Fraenkel set theory, natural numbers are constructed by defining them as sets, starting with the empty set ($\varnothing$) as the representation for 0. This construction relies on a successor function, denoted as $S(x)=x\cup \{x\}$, which generates each subsequent natural number from the previous one.123

The sequence is built as follows:

• 0 = $\varnothing$23
• 1 = $S(0)=0\cup \{0\}=\{0\}=\{\varnothing \}$2
• 2 = $S(1)=1\cup \{1\}=\{0,1\}=\{\varnothing ,\{\varnothing \}\}$2
• 3 = $S(2)=2\cup \{2\}=\{0,1,2\}=\{\varnothing ,\{\varnothing \},\{\varnothing ,\{\varnothing \}\}\}$2

In general, any natural number $n$ is defined as the set of all preceding natural numbers: $n=\{0,1,2,\dots ,n-1\}$. The existence of the complete set of these numbers, denoted as $\omega$, is guaranteed by the axiom of infinity. These constructed entities satisfy the Peano postulates, providing a rigorous set-theoretic foundation for arithmetic.213