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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers

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Von Neumann defined ordinals as the set of all smaller ordinals [Neumann, by Poundstone]

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity

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Transfinite ordinals are needed in proof theory, and for recursive functions and computability [Hossack]