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Von Neumann ordinals. In the area of mathematics called **set** theory, a specific construction due to John von Neumann defines the **natural** **numbers** as follows: . **Set** 0 = { }, the empty **set**,; Define S(a) = a ∪ {a} for every **set** a. S(a) is the successor of a, and S is called the successor function.; By the axiom of infinity, there exists a **set** which contains 0 and is closed under the successor.

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