Finite von Neumann ordinal ($n=\{0,\dots ,n-1\}$)

A finite von Neumann ordinal is obtained from $0=\varnothing$ by finitely many applications of the successor operation $S(x)=x\cup \{x\}$. Equivalently, it is an ordinal every nonempty subset of which has a greatest element. This characterization defines the individual natural numbers without quantifying over all inductive sets.