The Set-Theoretic Perspective: von Neumann Ordinals

We examine the set-theoretic level, in which natural numbers are reduced to particular elements of the cumulative hierarchy of pure sets. Working primarily in ZFC, we focus on von Neumann's ordinal encoding where 58 is realized as the set {0,1,...,57}. We contrast this with Zermelo's earlier encoding and, following Benacerraf's 1965 paper, argue that the discrepancy is not a defect but a structural fact: numbers are not sets, and any attempt to identify them with sets generates encoding-relative artifacts.