The Contemporary Standard
In standard, Zermelo–Fraenkel (ZF) set theory the natural numbers are defined recursively by 0 = {} (the empty set) and n + 1 = n ∪ {n}. Then n = {0, 1, ..., n − 1} for each natural number n. The first few numbers defined this way are 0 = {}, 1 = {0} = {{}}, 2 = {0,1} = {{},{{}}}, 3 = {0,1,2} = {{},{{}},{{},{{}}}}.
The set N of natural numbers is defined as the smallest set containing 0 and closed under the successor function S defined by S(n) = n ∪ {n}. (For the existence of such a set we need an Axiom of Infinity.) The structure ⟨N,0,S⟩ is a model of Peano arithmetic.
The set N and its elements, when constructed this way, are examples of von Neumann ordinals.
Read more about this topic: Set-theoretic Definition Of Natural Numbers
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