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Natural numbers are constructed set-theoretically: 0 = ∅, n+1 = n ∪ {n}, yielding ℕ = {0, 1, 2, 3, ...} = {∅, {∅}, {∅,{∅}}, ...}. This von Neumann construction embeds ℕ into the set-theoretic universe and allows ordinal numbers to generalize the concept of 'counting' to infinite cases.
The von Neumann construction of natural numbers is a remarkable application of minimalist set theory: it builds the counting numbers out of nothing but the empty set and the operations you have already studied. Recall that you know what finite sets are and how recursion on finite structures works. The construction exploits both. Start with 0 = ∅ (the empty set, which exists by the Empty Set Axiom), then define each successor by n+1 = n ∪ {n}. So 1 = ∅ ∪ {∅} = {∅}, 2 = {∅} ∪ {{∅}} = {∅, {∅}}, 3 = {∅, {∅}, {∅,{∅}}}, and so on. Every natural number is the set of all natural numbers that preceded it.
This construction does elegant double duty. Not only does it produce the natural numbers — it encodes their ordering for free: n < m if and only if n ∈ m. Membership doubles as "less than." This is not a coincidence; it is the design. Each von Neumann natural number is simultaneously an ordinal and the set of all smaller ordinals. When you later study von Neumann ordinals and then ω (the first infinite ordinal), you will find that ω = {0, 1, 2, 3, ...} = {∅, {∅}, {∅,{∅}}, ...} — precisely the set of all von Neumann naturals. The Axiom of Infinity guarantees that this set exists, lifting the construction from finite recursion into the infinite.
The recursion principle you know from finite structures guarantees uniqueness: given a well-defined base case and step, there is exactly one function satisfying both. This is why the von Neumann construction is not merely one way to model ℕ but the canonical one in ZFC — it is uniquely determined by the axioms. The philosophical payoff is that the natural numbers need not be treated as abstract Platonic objects requiring a separate foundation: they are specific sets, constructed step by step from ∅ using union and pairing. Any arithmetic theorem — commutativity, induction, the division algorithm — can in principle be traced back to membership and union operations on these sets. This is what "reducing arithmetic to set theory" concretely means: not that counting becomes complicated, but that successor, order, and finiteness are shown to be derivable from the axioms alone, with no extra primitive concepts added.