Naturalness of reducing mathematics to set theory (via ZFC and von Neumann numerals)

Determine whether representing natural numbers by the von Neumann set-theoretic construction and building the rationals and reals from them within Zermelo–Fraenkel set theory with the Axiom of Choice constitutes a natural foundational reduction of mathematics.

Background

The essay reviews how Russell’s paradox motivated axiomatizations of set theory, culminating in ZFC. It then describes the common modern reduction of mathematics to set theory, in which numbers are represented as sets (e.g., 0 = ∅, 1 = {∅}, 2 = {∅, {∅}}, etc.) and higher number systems are constructed from these.

The author questions whether this reduction aligns with our intuitive understanding of numbers, noting that while collections of three objects are naturally seen as sets, the number 3 itself may be less naturally identified with a specific set. This motivates asking if the set-theoretic reduction is an appropriate or ‘natural’ foundation.