Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 120 / 4 / b

Define a natural number to be a Finite von Neumann ordinal: an ordinal $n$ such that every nonempty subset of $n$ has a greatest member. This avoids the usual impredicative description of $\mathbb{N}$ as the intersection of all inductive sets.

Every usual von Neumann natural numberhas this property. The proof is by induction: a nonempty subset of $n+1$ either contains $n$, which is then greatest, or is a nonempty subset of $n$.

Conversely, let $\alpha$ be an ordinal with the stated property. If $\alpha$ were not one of the finite von Neumann ordinals, it would contain every finite ordinal. Indeed, if $n$ were the least finite ordinal not in $\alpha$, ordinal comparability and the presence of all $m<n$ would force $\alpha =n$ or $\alpha =m$ for some $m<n$. The subsetwould then be nonempty and have no greatest element, a contradiction. Thus this definition picks out exactly the natural numbers given by the usual least-inductive-set definition.