Theorem 4.11: Any finite collection of natural numbers is a set.

Proof: We proceed by induction on n, the number of things in the collection. If n = 0, then the collection is empty, and by Axiom 4, the empty set is a set. We assume that it is true for a collection of n numbers and seek to show that it is true for a collection of n + 1 numbers. If the numbers are x1, x2, . . . , xn, xn+1, then by the inductive hypothesis

{x1, x2, . . . , xn }

is a set. Since xn+1 is a natural number, it is a set by Theorem 4.4, so

is a set by Axiom 3. Then

is a set by Axiom 2.